# Koszul duality for Coxeter groups

Simon Riche  Université Clermont Auvergne, CNRS, LMBP, F-63000 Clermont-
Ferrand, France<EMAIL_ADDRESS>and  Cristian Vay  Universidad Nacional
de Córdoba, Facultad de Matemática, Astronomía, Física y Computación, CIEM –
CONICET, Córdoba, Argentina<EMAIL_ADDRESS>
###### Abstract.

We construct a “Koszul duality” equivalence relating the (diagrammatic) Hecke
category attached to a Coxeter system and a given realization to the Hecke
category attached to the same Coxeter system and the dual realization. This
extends a construction of Beĭlinson–Ginzburg–Soergel [BGS] and
Bezrukavnikov–Yun [BY] in a geometric context, and of the first author with
Achar, Makisumi and Williamson [AMRW2]. As an application, we show that the
combinatorics of the “tilting perverse sheaves” considered in [ARV] is encoded
in the combinatorics of the canonical basis of the Hecke algebra of $(W,S)$
attached to the dual realization.

This project has received funding from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation programme
(S.R., grant agreement No. 101002592). C. V. is partially supported by CONICET
PIP 11220200102916CO, Foncyt PICT 2020-SERIEA-02847 and Secyt (UNC)

##  1\. Introduction

###  1.1. Koszul duality for general Coxeter groups

The utility of Koszul duality in Representation Theory has been first
emphasized by Beĭlinson–Ginzburg–Soergel [BGS] in the setting of the
Kazhdan–Lusztig conjecture on characters of simple highest weight modules for
a complex semisimple Lie algebra. In this paper the authors explained in
particular the relation between this construction and some “mixed” properties
of $\ell$-adic perverse sheaves on flag varieties of reductive algebraic
groups. A modified form of this Koszul duality for constructible sheaves on
flag varieties was later generalized to Kac–Moody groups by Bezrukavnikov–Yun
[BY], which allowed to explain the relations between several equivalences of
categories constructed by Bezrukavnikov with various collaborators, and
related to local geometric Langlands duality and representations of quantum
groups at roots of unity.

More recently, as part of a program aiming at generalizing some of
Bezrukavnikov’s equivalences in the setting of reductive algebraic groups over
fields of positive characteristic, a version of the Koszul duality of [BY] was
obtained by the first author with Achar, Makisumi and Williamson in [AMRW2].
This construction is more formal and less geometric than that of [BY]; in
particular it involves the “mixed derived categories” constructed with Achar
(see [AR]), which are useful but rather ad-hoc. A more natural setting for
this construction seems to be that of the Hecke category attached to a Coxeter
system and a given realization by Elias–Williamson [EW2] and, although the
final construction of Koszul duality was restricted to the case of Cartan
realizations of crystallographic Coxeter systems, i.e. the case when the Hecke
category can be described geometrically in terms of parity complexes on flag
varieties of Kac–Moody groups, part of the constructions involved were already
treated in full generality in [AMRW1].

This raised the question of the existence of a version of Koszul duality in
the general setting of [EW2], involving in particular a general Coxeter
system.111A first suggestion of the existence of such a construction can be
found in [EW2, Remark 3.5]. The main result of this paper is a realization of
this idea based on some prior work with Achar [ARV], under some technical
conditions that we discuss in §1.4 below.

###  1.2. Statement

Let us consider a Coxeter system $(W,S)$ and a realization $\mathfrak{h}$ of
$(W,S)$ over a field $\Bbbk$ satisfying appropriate assumptions (see §1.4). To
these data Elias–Williamson attach a $\Bbbk$-linear monoidal category
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ endowed with a “shift”
autoequivalence $(1)$, defined by generators and relations, and whose split
Grothendieck group identifies with the Hecke algebra of $(W,S)$. For any
objects $X$ and $Y$ in $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$, the
$\Bbbk$-vector space

$\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}(X,Y(n))$

admits a canonical structure of graded bimodule over the symmetric algebra $R$
of $V^{*}$, where $V$ is the representation underlying $\mathfrak{h}$, and by
“killing” the left, resp. right, action of this algebra one obtains a category
$\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$, resp.
$\underline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$. We then introduce the
“biequivariant”, “right equivariant” and “left equivariant” categories
attached to $(\mathfrak{h},W)$ as

$\displaystyle\mathsf{BE}(\mathfrak{h},W)$
$\displaystyle=K^{\mathrm{b}}\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W),$
$\displaystyle\mathsf{RE}(\mathfrak{h},W)$
$\displaystyle=K^{\mathrm{b}}\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W),$
$\displaystyle\mathsf{LE}(\mathfrak{h},W)$
$\displaystyle=K^{\mathrm{b}}\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$

where the subscript “$\oplus$” indicates the additive hull. (The terminology,
taken from [AMRW1], is motivated by the special case when the Hecke category
can be described in terms of constructible sheaves: the biequivariant category
involves sheaves on the group which are equivariant for a Borel subgroup on
both sides, while the right, resp. left, equivariant category involves sheaves
which are equivariant for the action on the right, resp. left.)

With this notation, in the special case considered there, one form of the
Koszul duality of [AMRW2] is an equivalence of triangulated categories

$\kappa:\mathsf{RE}(\mathfrak{h},W)\xrightarrow{\sim}\mathsf{LE}(\mathfrak{h}^{*},W)$

which satisfies $\kappa\circ(1)=(-1)[1]\circ\kappa$, where $\mathfrak{h}^{*}$
is the dual realization (obtained by switching roots and coroots; in the case
related to geometry this amounts to Langlands duality). To state further
properties of this equivalence one needs to recall that the categories
$\mathsf{RE}(\mathfrak{h},W)$ and $\mathsf{LE}(\mathfrak{h}^{*},W)$ admit
canonical “perverse” t-structures (again, the terminology comes from geometry)
whose hearts admit canonical highest weight structures. More precisely, at the
time when [AMRW1, AMRW2] were written this was known only in the case of
Cartan realizations of crystallographic Coxeter systems, but in the meantime
this construction was extended to the general setting in [ARV]. As in any
highest weight category one can consider the indecomposable tilting objects in
these categories, and the main property of Koszul duality can be roughly
stated as the fact that it exchanges the indecomposable objects in the
karoubian envelope of
$\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$, resp.
$\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h}^{*},W)$, with the
indecomposable tilting objects in the heart of the perverse t-structure on
$\mathsf{LE}(\mathfrak{h}^{*},W)$, resp. $\mathsf{RE}(\mathfrak{h},W)$. One of
the main results of the present paper is a version of this statement for
general realizations of general Coxeter groups, see Theorem 7.4.

###  1.3. Strategy

The main strategy of our proof is similar to that used in [AMRW2] (which,
itself, is an adaptation of constructions considered in [S1, BBM, BGS, BY]).
Namely, instead of constructing the equivalence $\kappa$ directly, we first
consider a “free monodromic” variant, from which (the inverse of) $\kappa$
will be obtained by essentially “killing” a left $R$-action on morphisms. The
main point is that in this setting one works with _monoidal_ categories, where
the definition of $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ by generators
and relations can be used with great effect.

We therefore construct a category of “free monodromic tilting objects”
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ attached to $(\mathfrak{h},W)$,
and then an equivalence of monoidal categories

$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h}^{*},W)\xrightarrow{\sim}\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W),$

see Theorem 7.1. The main difficulty lies in the _definition_ of such a
functor; once this is known the same arguments as in [AMRW2] can be developed
to prove that it is an equivalence. To define this functor, in view of the
definition of $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$ we need to
describe the images of the generating morphisms, and then check that these
images satisfy the relations imposed in
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$. The construction of all the
morphisms involving only one color can be copied from [AMRW2], but the
definition of the morphism corresponding the $2$-colors generators given there
relies partially on geometry. Here we give a different (and general)
construction of this morphism in Section 6.

This construction relies on the prior construction of a “functor
${\mathbb{V}}$” in case $W$ is finite, explained in Section 5. This
construction is similar to that in [AMRW2], with one notable exception: in
[AMRW2] this functor takes values in “usual” Soergel bimodules, which leads to
imposing technical assumptions on the characteristic of $\Bbbk$; these
assumptions can be removed later in the paper using some change-of-scalars
arguments which make sense only for Cartan realizations. Here we use a variant
of the construction of Soergel bimodules developed in the meantime by Abe
[A1], which allows to avoid these technical assumptions completely.

The proof that these morphisms satisfy the required relations is again similar
to the corresponding part of [AMRW2], but using Abe’s category of bimodules
rather than plain bimodules.

###  1.4. Assumptions

The assumptions that we have to impose on our relization $\mathfrak{h}$ are
explained in detail in §2.1, §2.3 and §4.9. The assumptions of §§2.1–2.3 are
“standard” assumptions that are required for the theories in [EW2] and [A1] to
behave appropriately. They are known to hold in the main examples of
realizations that arise “in nature,” i.e. the Cartan realizations of
crystallographic Coxeter systems and the geometric realization (and its
variants considered by Soergel) of any Coxeter system not involving type
$\mathsf{H}_{3}$.

The assumption of §4.9 is of a different kind: in [AMRW1] an ad-hoc version of
the “free monodromic completed category” of [BY] was constructed in the
diagrammatic setting. This category (or a subcategory) should be monoidal, and
all the structures involved can indeed be constructed, but the question of
whether these structures satisfy the appropriate “interchange law” was left
open. Here we assume that this property is satisfied for appropriate objects.
A proof that it is indeed the case in full generality has been announced by
Hogancamp and Makisumi, but no written account of their work is available as
of now. In [AMRW1] it was proved that this property holds for Cartan
realizations of crystallographic Coxeter systems, and here we remark that the
same approach, combined with the results of [ARV], also applies under some
technical assumptions that are satisfied for geometric and Soergel
realizations.

In particular, the presently available literature is enough to show that all
of our statements hold at least for the 2 main families of examples of
realizations of Coxeter systems that are known.

###  1.5. Application

As an application, we show in §7.7 that the combinatorics of the
indecomposable tilting objects in $\mathsf{RE}(\mathfrak{h},W)$ (constructed
in [ARV]) is governed by the combinatorics of the “canonical basis” attached
to the dual realization. In particular, in the case of Soergel realizations,
this combinatorics is governed by Kazhdan–Lusztig polynomials (see Example
7.6), and the heart of the perverse t-structure on
$\mathsf{RE}(\mathfrak{h},W)$ is equivalent to the category of finite-
dimensional graded modules over a finite-dimensional Koszul ring (see Remark
7.7).

###  1.6. Acknowledgements

The results in this paper are the realization of a project initiated with
Pramod Achar, Shotaro Makisumi and Geordie Williamson. We thank them for very
helpful discussions at various stages of its completion, and for their
contributions in the formulation of several key ideas. This work also owes
much to the work of Noriyuki Abe [A1, A2], which provided the necessary
ingredients to generalize the approach used in [AMRW2].

A large part of our work was accomplished during a visit of the second author
in Clermont-Ferrand funded by CIMPA-ICTP fellowships program “Research in
Pairs” and ERC.

##  2\. Preliminaries

###  2.1. Conventions and assumptions

Throughout the work $(W,S)$ denotes a Coxeter system with $S$ finite. (As
usual, we will usually only indicate $W$ in our notations, although all the
structures we consider also depend on the choice of Coxeter generators $S$.)
We consider on $W$ the Bruhat order $\leq$ and the length function $\ell$. We
will use the standard terminology regarding Coxeter systems, as recalled e.g.
in [ARV, §3.1]. In particular, an expression is a word
${\underline{w}}=(s_{1},\dots,s_{n})$ in $S$ and
$\pi({\underline{w}})=s_{1}\cdots s_{n}$ denotes the element in $W$ expressed
by ${\underline{w}}$. The set of expressions will be denoted
$\operatorname{Exp}(W)$. The _length_ of an expression ${\underline{w}}$ is
its length as a word; it will be denoted $\ell({\underline{w}})$. We will
identify simple reflections with the corresponding $1$-letter expression
whenever convenient. Given a pair $(s,t)$ of simple reflections, we will
denote by $m_{s,t}\in{\mathbb{Z}}_{\geq 1}\cup\\{\infty\\}$ the order of $st$
in $W$, and by $\langle s,t\rangle$ the subgroup generated by $s$ and $t$.

We fix a field $\Bbbk$ and a realization

$\mathfrak{h}=\bigl{(}V,(\alpha_{s}^{\vee}:s\in S),(\alpha_{s}:s\in
S)\bigr{)}$

of $(W,S)$ over $\Bbbk$ in the sense of Elias–Williamson [EW2, Definition
3.1]. In particular, $V$ is a finite-dimensional $\Bbbk$-vector space,
$(\alpha_{s}^{\vee}:s\in S)$ is a collection of vectors in $V$,
$(\alpha_{s}:s\in S)$ is a collection of vectors in
$V^{*}:=\operatorname{Hom}_{\Bbbk}(V,\Bbbk)$, and there exists an action of
$W$ on $V$ such that for $s\in S$ and $v\in V$ we have

$s\cdot v=v-\langle\alpha_{s},v\rangle\alpha_{s}^{\vee}.$

Realizations can be restricted to parabolic subsystems of $(W,S)$, by simply
forgetting part of the elements $\alpha_{s}$ and $\alpha_{s}^{\vee}$: if
$S^{\prime}\subset S$ is a subset, and $W^{\prime}\subset W$ is the subgroup
generated by $S^{\prime}$, we will denote by $\mathfrak{h}_{|W^{\prime}}$ the
realization $(V,(\alpha_{s}^{\vee}:s\in S^{\prime}),(\alpha_{s}:s\in
S^{\prime}))$ of $(W^{\prime},S^{\prime})$.

In addition to the conditions appearing in this definition, it has been
recently explained in [EW3] that some further restrictions have to be imposed
in order for the theory developed in [EW2] to behave as expected, most of
which were made more explicit in [Ha]. Here we will assume that the following
conditions are satisfied.

  1. (1)

The realization is balanced (see [EW2, Definition 3.7]) and satisfies Demazure
surjectivity (see [EW2, Assumption 3.9]).

  2. (2)

In case $W$ admits a parabolic subgroup of type $\mathsf{H}_{3}$, we assume
that there exists a linear combination of diagrams as in [EW2, Equation
(5.12)] which is sent to $0$ under the operation described in [EW3, §2].

  3. (3)

For any pair $(s,t)$ of distinct simple reflections such that $m_{s,t}<\infty$
we have

$\genfrac{[}{]}{0.0pt}{}{m_{s,t}}{k}_{s}(\langle\alpha_{s}^{\vee},\alpha_{t}\rangle,\langle\alpha_{t}^{\vee},\alpha_{s}\rangle)=\genfrac{[}{]}{0.0pt}{}{m_{s,t}}{k}_{t}(\langle\alpha_{s}^{\vee},\alpha_{t}\rangle,\langle\alpha_{t}^{\vee},\alpha_{s}\rangle)=0$

for all integers $1\leq k\leq m_{st}-1$, where we use the notation of [Ha].

The assumptions in (1) are standard in this theory. The second one is really
necessary; the first one can usually be relaxed at the cost of a more
complicated combinatorics (see e.g. [EW3, §7]), but we will not consider this
question here. Assumptions (2) and (3) are also necessary for the theory of
[EW2], hence also for all of its applications, although this was not made
explicit before [EW3, Ha]. (In particular, they should be imposed in [AMRW1]
and in [ARV].) Here, by the main result of [Ha], (3) implies the existence and
rotatability of Jones–Wenzl projectors, which as explained in [EW3] plays a
crucial role in this story. Note that (3) is also the technical condition
imposed in [A2] to ensure that the theory of [A1] applies.

It is important to note that if the assumptions (1)–(3) are satisfied by a
realization, then they are satisfied by its restriction to any parabolic
subsystem of $(W,S)$. Further assumptions will be imposed and discussed in
§2.3 and §4.9; they are also stable under restriction to a parabolic
subsystem.

######  Example 2.1.

The main examples of data as above the reader can keep in mind are the
following.

  1. (1)

Let $A=(a_{i,j})_{i,j\in I}$ be a generalized Cartan matrix. A _Kac–Moody root
datum_ associated with $A$ is a triple

$(\mathbf{X},(\alpha_{i}:i\in I),(\alpha_{i}^{\vee}:i\in I))$

where $\mathbf{X}$ is a finite free ${\mathbb{Z}}$-module, $(\alpha_{i}:i\in
I)$ is a family of elements of $\mathbf{X}$, and $(\alpha_{i}^{\vee}:i\in I)$
is a family of elements of
$\operatorname{Hom}_{{\mathbb{Z}}}(\mathbf{X},{\mathbb{Z}})$, such that
$\langle\alpha_{i}^{\vee},\alpha_{j}\rangle=a_{i,j}$ for any $i,j\in I$. To
$A$ one can associate a Coxeter system $(W,S)$ where $S$ is in bijection with
$I$ (through a map $s\mapsto i_{s}$), and the order $m_{s,t}$ of $st$ is
determined as follows:

$m_{s,t}=\begin{cases}2&\text{if $a_{i_{s}i_{t}}a_{i_{t}i_{s}}=0$;}\\\
3&\text{if $a_{i_{s}i_{t}}a_{i_{t}i_{s}}=1$;}\\\ 4&\text{if
$a_{i_{s}i_{t}}a_{i_{t}i_{s}}=2$;}\\\ 6&\text{if
$a_{i_{s}i_{t}}a_{i_{t}i_{s}}=3$;}\\\ \infty&\text{if
$a_{i_{s}i_{t}}a_{i_{t}i_{s}}\geq 4$.}\\\ \end{cases}$

To $(\mathbf{X},(\alpha_{i}:i\in I),(\alpha_{i}^{\vee}:i\in I))$ one can
associate a realization of $(W,S)$ over any field $\Bbbk$ by setting
$V:=\Bbbk\otimes_{{\mathbb{Z}}}\operatorname{Hom}_{{\mathbb{Z}}}(\mathbf{X},{\mathbb{Z}})$,
and choosing for $(\alpha_{s}^{\vee}:s\in S)$ and $(\alpha_{s}:s\in S)$ the
images of $(\alpha_{i}^{\vee}:i\in I)$ and $(\alpha_{i}:i\in I)$ in $V$ and
$V^{*}$ respectively.

Such realizations are called _Cartan realizations of crystallographic Coxeter
groups_. The status of our assumptions above for such realizations is
discussed at length in [R4, Chap. II, §2.4 and §3.2]. Regarding (1), such
realizations are always balanced; they satisfy Demazure surjectivity at least
when $\mathrm{char}(\Bbbk)\neq 2$. Assumption (2) is irrelevant since
crystallographic Coxeter groups do not admit parabolic subgroups of type
$\mathsf{H}_{3}$. Assumption (3) is automatically satisfied.

Cartan realizations of crystallographic Coxeter groups are the realizations
considered in [AMRW1, Chap. 10–11] and [AMRW2] (where more general coefficient
rings are allowed.) For such a realization, the associated Hecke category (see
§3.1) can be realized geometrically as a category of parity complexes on the
flag variety of the Kac–Moody group associated with
$(\mathbf{X},(\alpha_{i}:i\in I),(\alpha_{i}^{\vee}:i\in I))$; see [RW, Part
III] for details.

  2. (2)

In the special case when $A$ is a Cartan matrix, the datum of a Kac–Moody root
datum of $A$ is equivalent to the datum of a based root datum with associated
Cartan matrix $A$. In this case, $W$ is the Weyl group of the associated
connected reductive algebraic group (over any algebraically closed field).

  3. (3)

Let now $(W,S)$ be an arbitrary Coxeter system with $S$ finite. Let $V$ be the
associated _geometric representation_ of $W$; it is a representation over
$\mathbb{R}$, and comes with a basis $(e_{s}:s\in S)$ indexed by $S$ and a
bilinear form $\langle-,-\rangle$. One can “upgrade” this representation to a
realization of $(W,S)$ over $\mathbb{R}$, called the geometric realization, by
setting $\alpha_{s}^{\vee}:=e_{s}$ and $\alpha_{s}:=2\langle e_{s},-\rangle$.
As explained in [R4, Chap. II, §2.4 and §3.2], for this realization our
assumptions (1) and (3) are satisfied. The status of assumption (2) (in case
$(W,S)$ has a parabolic subsystem of type $\mathsf{H}_{3}$) is unclear to us.

  4. (4)

For an arbitrary Coxeter system $(W,S)$ with $S$ finite, one can also consider
variants of the geometric realization considered by Soergel in [S3], see e.g.
[R4, Chap. II, §1.2.2]. Namely, consider a vector space $V$ endowed with
linearly independent families $(e_{s}:s\in S)$ of vectors of $V$ and
$(e_{s}^{*}:s\in S)$ of vectors of $V^{*}$ such that

$\langle e_{t},e_{s}^{*}\rangle=-2\cos\left(\frac{\pi}{m_{s,t}}\right)$

where $m_{s,t}$ is the order of $st$ in $W$. (We use the convention that
$\frac{\pi}{\infty}=0$. Note also that such data always exist.) Then
$(V,(e_{s}:s\in S),(e_{s}^{*}:s\in S))$ is a realization of $(W,S)$, see [R4,
Chap. II, Remark 2.7]. These realizations will be called Soergel realizations.
(Note that in case $W$ is finite, the geometric realization is an example of a
Soergel realization.) In this case again, our assumptions (1) and (3) are
satisfied (see [R4, Chap. II, §2.4 and §3.2]), but the status of assumption
(2) (in case $(W,S)$ has a parabolic subsystem of type $\mathsf{H}_{3}$) is
unclear to us. For such a realization, the Hecke category is equivalent to the
corresponding category of Soergel bimodules by [EW2, §6.7].

###  2.2. Gradings

By a “graded” (resp. “bigraded”) vector space, we mean a ${\mathbb{Z}}$-graded
(resp. ${\mathbb{Z}}^{2}$-graded) vector space. Whenever convenient, we will
identify graded vector spaces with bigraded vector spaces which are zero in
all degrees belonging to
$({\mathbb{Z}}\smallsetminus\\{0\\})\times{\mathbb{Z}}$. If $M$ is graded,
resp. bigraded, its component in degree $n$, resp. in bidegree $(n,m)$, will
be denoted $M_{n}$, resp. $M^{n}_{m}$. The shift-of-grading functor $(1)$ on a
bigraded vector space $M=\oplus_{i,j\in{\mathbb{Z}}}M_{j}^{i}$ is defined by

$M(1)^{i}_{j}=M_{j+1}^{i+1}.$

We also have shift functors $[1]$ and $\langle 1\rangle:=(-1)[1]$, which
satisfy $(M[1])^{i}_{j}=M_{j}^{i+1}$ and $(M\langle
1\rangle)^{i}_{j}=M_{j-1}^{i}$. Note that $\langle 1\rangle$ stabilizes graded
vectors spaces.

We will work in particular with the symmetric algebras

$R:=\operatorname{Sym}(V^{*}),\quad R^{\vee}:=\operatorname{Sym}(V),\quad
R^{\wedge}:=\operatorname{Sym}(V),$

considered as graded rings where $V^{*}\subset R$ is in degree $2$, $V\subset
R^{\vee}$ is in degree $-2$ and $V\subset R^{\wedge}$ is in degree $2$. We
will also consider the localization $Q$ of the ring $R$ with respect to the
multiplicative subset generated by $\\{w(\alpha_{s}):s\in S,\,w\in W\\}$; this
ring has a natural grading where the elements $\frac{1}{w(\alpha_{s})}$ are in
degree $-2$. Analogously, we denote by $Q^{\vee}$ and $Q^{\wedge}$ the
corresponding localizations of $R^{\vee}$ and $R^{\wedge}$.

Let $R^{\vee}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\vee}$ denote the category of graded $R^{\vee}$-bimodules, and define
analogously $R^{\wedge}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar
45\relax R^{\wedge}$. Then $\langle 1\rangle$ induces autoequivalences of
these categories, which will be denoted similarly. If $M$ belongs to
$R^{\vee}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\vee}$, we let $M^{\wedge}$ be the graded $R^{\wedge}$-bimodule which is
$M$ as ungraded bimodule and whose homogeneous components are
$(M^{\wedge})_{i}=M_{-i}$, $i\in{\mathbb{Z}}$. This induces a functor from
$R^{\vee}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\vee}$ to $R^{\wedge}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar
45\relax R^{\wedge}$ which satisfies

(2.1) $(M\langle 1\rangle)^{\wedge}=M^{\wedge}\langle-1\rangle.$

Given a graded free right $R^{\vee}$-module, resp. vector space, of finite
rank $M\simeq\oplus_{i}R^{\vee}(n_{i})$, resp.
$M\simeq\oplus_{i}\Bbbk(n_{i})$, we set

$\operatorname{grk}_{R^{\vee}}M:=\sum_{i}v^{n_{i}},\quad\text{resp.}\quad\operatorname{grk}_{\Bbbk}M:=\sum_{i}v^{n_{i}},$

considered as elements in ${\mathbb{Z}}[v,v^{-1}]$ where $v$ is an
indeterminate. Of course, if $N$ is another such module, then $M\simeq N$ if
and only if $\operatorname{grk}_{R^{\vee}}M=\operatorname{grk}_{R^{\vee}}N$.
We define analogously the function $\operatorname{grk}_{R^{\wedge}}$. Note
that if $M$ is a graded free right $R^{\vee}$-module, then $M^{\wedge}$ is a
graded free right $R^{\wedge}$-module and we have

(2.2)
$\operatorname{grk}_{R^{\wedge}}M^{\wedge}=\overline{\operatorname{grk}_{R^{\vee}}M}$

where $\overline{\cdot}$ is the unique ring automorphism of
${\mathbb{Z}}[v,v^{-1}]$ such that $\overline{v}=v^{-1}$.

Below we will need the following application of the graded Nakayama lemma,
where we consider $\Bbbk$ (concentrated in degree $0$) as a graded
$R^{\vee}$-module by letting $V$ act by zero.

######  Lemma 2.2.

Let $M$ and $N$ be graded free right $R^{\vee}$-modules of finite rank and
$f:M\to N$ be a morphism of graded $R^{\vee}$-modules. If

$f\otimes_{R^{\vee}}\Bbbk:M\otimes_{R^{\vee}}\Bbbk\to
N\otimes_{R^{\vee}}\Bbbk$

is injective (resp. surjective), then $f$ is injective (resp. surjective).

###### Proof.

Let $C$ denote the complex

$\cdots 0\to M\xrightarrow{f}N\to 0\cdots$

with $M$, resp. $N$, placed in degree $-1$, resp. $0$. If
$f\otimes_{R^{\vee}}\Bbbk$ is injective, this complex satisfies the
assumptions of [AMRW1, Lemma 3.4.1]. In particular, it follows that
$H^{-1}(C)=0$, that is $f$ is injective. If $f\otimes_{R^{\vee}}\Bbbk$ is
surjective, we use [AMRW1, Lemma 3.4.1] with the same complex with $M$, resp.
$N$, placed in degree $0$, resp. $1$, to deduce that $f$ is also surjective. ∎

###  2.3. Dual realization

We will also consider the dual realization

$\mathfrak{h}^{*}=(V^{*},(\alpha_{s}:s\in S),(\alpha_{s}^{\vee}:s\in S))$

of $(W,S)$ over $\Bbbk$. It is clear that this realization satisfies our
assumptions (1) and (3). It is not clear (to us) wether assumption (2) is
automatically satisfied; in doubt, we will assume that it is also satisfied by
$\mathfrak{h}^{*}$. Note that the graded ring playing, with respect to
$\mathfrak{h}^{*}$, the role that $R$ plays for $\mathfrak{h}$ is
$R^{\wedge}$.

######  Example 2.3.

  1. (1)

In the setting of Example 2.1(1), if $(\mathbf{X},(\alpha_{i}:i\in
I),(\alpha_{i}^{\vee}:i\in I))$ is a Kac–Moody root datum associated with a
generalized Cartan matrix $A$, then

$(\operatorname{Hom}_{{\mathbb{Z}}}(\mathbf{X},{\mathbb{Z}}),(\alpha_{i}^{\vee}:i\in
I),(\alpha_{i}:i\in I))$

is a Kac–Moody root datum associated with the generalized Cartan matrix
${}^{\mathrm{t}}A$. These generalized Cartan matrices share the same
associated Coxeter system $(W,S)$, and for any field $\Bbbk$ the dual of the
realization over $\Bbbk$ associated with $(\mathbf{X},(\alpha_{i}:i\in
I),(\alpha_{i}^{\vee}:i\in I))$ is the realization over $\Bbbk$ associated
with
$(\operatorname{Hom}_{{\mathbb{Z}}}(\mathbf{X},{\mathbb{Z}}),(\alpha_{i}^{\vee}:i\in
I),(\alpha_{i}:i\in I))$. Note that this “duality” of Kac–Moody root data
restricts to Langlands’ duality in the setting of Example 2.1(2).

  2. (2)

Consider the setting of Example 2.1(4). By definition the dual of a Soergel
realization is also a Soergel realization. In particular, in case $W$ is
finite, the geometric realization of Example 2.1(3) is self dual.

##  3\. Two incarnations of the Hecke category

We continue with our data $(W,S)$ and $\mathfrak{h}$ as in Section 2, which
satisfy the conditions of §2.1 and §2.3. We recall in this section the
definitions of the Elias–Williamson diagrammatic category and of Abe’s
category attached to $\mathfrak{h}$ and $(W,S)$.

###  3.1. The Elias–Williamson diagrammatic category

We will denote by

$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$

the category attached to $(W,S)$ and $\mathfrak{h}$ in [EW2, §5.2] (see also
[R4, Chap. II, §2.5]). This category is a $\Bbbk$-linear monoidal category,
which can be considered equivalently as enriched over graded vector spaces or
endowed with a shift-of-grading autoequivalence $(1)$, see [ARV, §2.1]. From
the first of these points of view, the objects in
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ are the symbols
$B_{{\underline{w}}}$ for ${\underline{w}}\in\operatorname{Exp}(W)$. The
monoidal product is defined by $B_{{\underline{v}}}\star
B_{{\underline{w}}}=B_{{\underline{v}}{\underline{w}}}$ where
${\underline{v}}{\underline{w}}$ is the concatenation of ${\underline{v}}$ and
${\underline{w}}$. The morphisms are generated (under horizontal and vertical
concatenation, and $\Bbbk$-linear combinations) by morphisms depicted by some
diagrams recalled below, and are subject to a number of relations for which we
refer to [EW2], [AMRW1, §2.3] or [R4]. The generating morphisms are (to be
read from bottom to top):

  1. (1)

for any homogeneous $f\in R$, a morphism

$f$

from $B_{\varnothing}$ to itself, of degree $\deg(f)$;

  2. (2)

for any $s\in S$, “dot” morphisms

$\bullet$$s$ and $\bullet$$s$

from $B_{s}$ to $B_{\varnothing}$ and from $B_{\varnothing}$ to $B_{s}$
respectively, of degree $1$;

  3. (3)

for any $s\in S$, trivalent morphisms

$s$$s$$s$ and $s$$s$$s$

from $B_{s}$ to $B_{(s,s)}$ and from $B_{(s,s)}$ to $B_{s}$ respectively, of
degree $-1$;

  4. (4)

for any pair $(s,t)$ of distinct simple reflections such that $st$ has finite
order $m_{st}$ in $W$, a morphism

$s$$t$$s$$t$$s$$t$$t$$t$$t$$\cdots$$\cdots$$s$$t$$t$ if $m_{st}$ is odd
or$s$$t$$s$$t$$s$$t$$t$$t$$t$$\cdots$$\cdots$$t$$s$$t$ if $m_{st}$ is even

from $B_{(s,t,\cdots)}$ to $B_{(t,s,\cdots)}$ (where each expression has
length $m_{st}$, and colors alternate), of degree $0$.

The graded vector space of morphisms from $B_{{\underline{w}}}$ to
$B_{{\underline{v}}}$ will be denoted

$\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}^{\bullet}(B_{{\underline{w}}},B_{{\underline{v}}}).$

Considering $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ as a category with
shift-of-grading autoequivalence $(1)$, its objects are the symbols
$B_{{\underline{w}}}(n)$ where ${\underline{w}}\in\operatorname{Exp}(W)$ and
$n\in{\mathbb{Z}}$, and the vector space of morphisms from
$B_{{\underline{w}}}(n)$ to $B_{{\underline{v}}}(m)$ is
$\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}^{m-n}(B_{{\underline{w}}},B_{{\underline{v}}})$.
This is the point of view we will mostly use below.

More generally, given $X,Y$ in $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ we
will set

$\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}^{\bullet}(X,Y):=\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}(X,Y(n)).$

This graded vector space has a natural structure of graded $R$-bimodule, where
the left (resp. right) action of $f\in R_{n}$ is given by adding a box
labelled by $f$ to the left (resp. right) of a diagram. With this structure,
$\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}^{\bullet}(X,Y)$
is graded free of finite rank as a left $R$-module and as a right $R$-module,
see [EW2, Corollary 6.14].

We will denote by $\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ the
additive hull of $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$, and by
$\mathscr{D}(\mathfrak{h},W)$ the karoubian envelope of
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$. The latter category is
Krull–Schmidt, and there exists a family $(B_{w}:w\in W)$ of objects in
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ characterized in [EW2, Theorem
6.26] and such that the assignment $(w,n)\mapsto B_{w}(n)$ induces a bijection
between $W\times{\mathbb{Z}}$ and the set of isomorphism classes of
indecomposable objects in $\mathscr{D}(\mathfrak{h},W)$.

We will also consider the category
$\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$ which has the same
objects as $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$, and such that the
morphism space from $X$ to $Y$ is the subspace of degree-$0$ elements in the
graded vector space

$\Bbbk\otimes_{R}\operatorname{Hom}_{\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)}^{\bullet}(X,Y)$

(where $\Bbbk$ is in degree $0$ and $R$ acts via the quotient $R/V\cdot
R=\Bbbk$). The functor $(1)$ induces an autoequivalence of
$\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$ which will be denoted
similarly, and $\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$ is
naturally a right module category for the monoidal category
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$. We have a natural functor

(3.1)
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)\to\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W);$

the image of $B_{{\underline{w}}}$ under this functor will be denoted
$\overline{B}_{{\underline{w}}}$. As for
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$, we will denote by
$\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ the additive
hull of $\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$, and by
$\overline{\mathscr{D}}(\mathfrak{h},W)$ the karoubian envelope of
$\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$.

The functor (3.1) induces a functor
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)\to\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$,
and then a functor
$\mathscr{D}(\mathfrak{h},W)\to\overline{\mathscr{D}}(\mathfrak{h},W)$. The
category $\overline{\mathscr{D}}(\mathfrak{h},W)$ is Krull–Schmidt, being
karoubian and with finite-dimensional morphism spaces, see [CYZ, Corollary
A.2]. For $w\in W$, we will denote by $\overline{B}_{w}$ the image of $B_{w}$
in $\overline{\mathscr{D}}(\mathfrak{h},W)$. Then
$\operatorname{End}_{\overline{\mathscr{D}}(\mathfrak{h},W)}(\overline{B}_{w})$
is a quotient of $\operatorname{End}_{\mathscr{D}(\mathfrak{h},W)}(B_{w})$,
hence is a local ring, which implies that $\overline{B}_{w}$ is an
indecomposable object. Using this, it is easily seen that the assignment
$(w,n)\mapsto\overline{B}_{w}(n)$ induces a bijection between
$W\times{\mathbb{Z}}$ and the set of isomorphism classes of indecomposable
objects in $\overline{\mathscr{D}}(\mathfrak{h},W)$.

We will also denote by
$\underline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$,
$\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ and
$\underline{\mathscr{D}}(\mathfrak{h},W)$ the categories obtained in the same
way using the tensor product on the _right_ with $\Bbbk$ over $R$. Here
$\underline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$ is naturally a _left_
module category over $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$, and the same
considerations as above for
$\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$ apply. We will use
obvious variants of the notations introduced above; in particular, for $w\in
W$, resp. for ${\underline{w}}\in\operatorname{Exp}(W)$, we define the object
$\underline{B}_{w}\in\underline{\mathscr{D}}(\mathfrak{h},W)$, resp.
$\underline{B}_{{\underline{w}}}\in\underline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$,
in the same way as for $\overline{B}_{w}$, resp.
$\overline{B}_{{\underline{w}}}$.

######  Remark 3.1.

  1. (1)

Of course, all the constructions above can also be considered for the
realization $\mathfrak{h}^{*}$ of §2.3, giving rise to the category
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$ and all its cousins. To
distinguish the two cases, the object of
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$ attached to an expression
${\underline{w}}$ will be denoted $B^{\wedge}_{\underline{w}}$. Similar
conventions will be used for the objects $B_{w}$, $\overline{B}_{w}$,
$\underline{B}_{w}$.

  2. (2)

It is a standard fact that the category
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ admits a canonical autoequivalence
induced by reflecting diagrams along a horizontal axis. This autoequivalence
exchanges left and right multiplication by polynomials, hence induces an
equivalence between $\underline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$
and $\overline{\mathscr{D}}_{\mathrm{BS}}(\mathfrak{h},W)$.

###  3.2. Abe’s category

Below we will also use a different incarnation of the Hecke category attached
to $(W,S)$ and $\mathfrak{h}$, which we will denote by
$\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$, and which was introduced by Abe
in [A1].

######  Remark 3.2.

Although this is not written explicitly, the conventions on realizations in
[A1, A2] are different from those of [EW2, EW3] (which we follow here).
Namely, in [A1, A2] a realization is a triple $(V,(\alpha_{s}:s\in
S),(\alpha_{s}^{\vee}:s\in S))$ where $\alpha_{s}\in V$ and
$\alpha_{s}^{\vee}\in V^{*}$, and the algebra $R$ is defined as the symmetric
algebra of $V$. In other words, the module “$V$” of [A1, A2] is the module
“$V^{*}$” of [EW2, EW3]. Here we have decided to follow the conventions of
[EW2, EW3]; we will therefore translate all the results and constructions from
[A1, A2] into these conventions.

In order to construct the category
$\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$, Abe first introduces the category
$\mathscr{C}(\mathfrak{h},W)$ (denoted $\mathcal{C}^{\prime}$ in [A1, A2]222In
fact the definition of $\mathcal{C}^{\prime}$ in [A1] is slightly different,
but this creates difficulties. The definition we use below solves these
problems, as explained in [A2].) whose objects are the triples

$(M,(M_{Q}^{w})_{w\in W},\xi_{M})$

where $M$ is a graded $R$-bimodule, each $M_{Q}^{w}$ is a graded
$(R,Q)$-bimodule such that $m\cdot f=w(f)\cdot m$ for any $m\in M_{Q}^{w}$ and
$f\in R$, these bimodules being $0$ except for finitely many $w$’s, and

(3.2) $\xi_{M}:M\otimes_{R}Q\to\bigoplus_{w\in W}M_{Q}^{w}$

is an isomorphism of graded $(R,Q)$-bimodules. A morphism in
$\mathscr{C}(\mathfrak{h},W)$ from the object $(M,(M_{Q}^{w})_{w\in
W},\xi_{M})$ to $(N,(N_{Q}^{w})_{w\in W},\xi_{N})$ is a morphism of graded
$R$-bimodules $\varphi:M\to N$ such that

$\left(\xi_{N}\circ(\varphi\otimes_{R}Q)\circ\xi_{M}^{-1}\right)(M_{Q}^{w})\subset
N_{Q}^{w}$

for any $w\in W$. This category has a natural monoidal structure induced by
$\otimes_{R}$, with neutral object the $R$-bimodule $R$ (upgraded in the
obvious way to an object of $\mathscr{C}(\mathfrak{h},W)$). The shift-of-
grading functor $\langle 1\rangle$ (see §2.2) induces in the natural way an
autoequivalence of $\mathscr{C}(\mathfrak{h},W)$, which will be denoted
similarly. For simplicity, we often write $M$ for $(M,(M_{Q}^{w})_{w\in
W},\xi_{M})$.

For $s\in S$, let

$R^{s}=\\{f\in R\mid s\cdot f=f\\}$

be the subring of $s$-invariant elements in $R$, and choose an element
$\delta_{s}\in V^{*}$ such that
$\langle\delta_{s},\alpha_{s}^{\vee}\rangle=1$. (Such a vector exists because
our realization is assumed to satisfy Demazure surjectivity.) The $R$-bimodule
$B_{s}=R\otimes_{R^{s}}R\langle-1\rangle$ can be upgraded to an object in
$\mathscr{C}(\mathfrak{h},W)$ by setting

(3.3) $(B_{s})^{e}_{Q}=Q(\delta_{s}\otimes 1-1\otimes
s(\delta_{s})),\quad(B_{s})^{s}_{Q}=Q(\delta_{s}\otimes 1-1\otimes\delta_{s})$

and $(B_{s})^{w}_{Q}=0$ for all $w\notin\\{e,s\\}$; see [A1, §2.4] or [R4,
Chap. II, §3.1.4].

The category $\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$ is defined as the
smallest full subcategory of $\mathscr{C}(\mathfrak{h},W)$ which contains the
neutral object $R$ and the objects $(B_{s}:s\in S)$ and is stable under the
monoidal product $\otimes_{R}$ and the shift functor $\langle 1\rangle$. In
other words, the objects in $\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$ are
the objects of the form

$B_{s_{1}}\otimes_{R}\cdots\otimes_{R}B_{s_{r}}\langle n\rangle$

with $r\in{\mathbb{Z}}_{\geq 0}$, $s_{1},\cdots,s_{r}\in S$ and
$n\in{\mathbb{Z}}$. As in the diagrammatic category we will set

$\operatorname{Hom}^{\bullet}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)}(X,Y)=\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)}(X,Y\langle-n\rangle),$

considered as a graded vector space with
$\operatorname{Hom}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)}(X,Y\langle-n\rangle)$
in degree $n$. We will denote by
$\mathscr{A}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ the additive hull of
$\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$, and by
$\mathscr{A}(\mathfrak{h},W)$ the karoubian envelope of
$\mathscr{A}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$.

For an expression ${\underline{w}}=(s_{1},\cdots,s_{n})$, we define the object
$B_{{\underline{w}}}$ in $\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$ as

$B_{{\underline{w}}}:=B_{s_{1}}\otimes_{R}\cdots\otimes_{R}B_{s_{n}}=R\otimes_{R^{s_{1}}}\cdots\otimes_{R^{s_{n}}}R\langle-n\rangle$

if $n\geq 1$, and $B_{\varnothing}=R$. The so-called $1$-tensor element
$u_{{\underline{w}}}=(1\otimes 1)\otimes_{R}\cdots\otimes_{R}(1\otimes 1)\in
B_{{\underline{w}}}$ plays a singular role in the theory. (In case
${\underline{w}}=\varnothing$ is the empty expression, the element
$u_{\varnothing}$ is interpreted as $1\in R$.)

As the reader might have noticed, we have used the same notation as for some
objects in $\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$. This should not lead
to any confusion, because of the following result due to Abe (see [A2, Theorem
3.15]).

######  Theorem 3.3.

There exists an equivalence of monoidal categories

$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)\xrightarrow{\sim}\mathscr{A}_{\mathrm{BS}}(\mathfrak{h},W)$

which intertwines the autoequivalences $(1)$ and $\langle-1\rangle$ and sends
$B_{\underline{w}}$ to $B_{\underline{w}}$ for any
${\underline{w}}\in\operatorname{Exp}(W)$.

######  Remark 3.4.

The proof of Theorem 3.3 has two parts: (a) the construction of the functor,
and (b) the proof that it is an equivalence. Once (a) has been solved in the
appropriate way, (b) is guaranteed by the results of [A1]. The current proof
of (a) relies on the computations in [A2]. We believe that considerations
similar (or, in a sense, “Koszul dual”) to those in Sections 5–6 (using, for
$W$ finite, the indecomposable object $B_{w_{0}}$ rather than
$\mathcal{T}_{w_{0}}$) can be used to provide an alternative construction of
this functor. Details will appear elsewhere if this finds any application.

##  4\. Perverse and tilting sheaves

In this section we briefly recall the definitions of a series of homotopy-type
categories constructed from the Hecke category. These categories are the main
objects of study of [AMRW1] and [ARV]. We also extend some results of [AMRW1,
Chap. 10–11] to our present setting.

###  4.1. The biequivariant category

The biequivariant category $\mathsf{BE}(\mathfrak{h},W)$ is defined in [AMRW1,
§4.2] as

$\mathsf{BE}(\mathfrak{h},W):=K^{\mathrm{b}}\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W);$

the natural functor from $\mathsf{BE}(\mathfrak{h},W)$ to
$K^{\mathrm{b}}\mathscr{D}(\mathfrak{h},W)$ is an equivalence, see [AMRW1,
Lemma 4.9.1], and we will therefore identify these categories whenever
convient. The biequivariant category is monoidal for a certain product
$\,\underline{\star}\,$ which restricted to
$\mathscr{D}_{\mathrm{BS}}(\mathfrak{h},W)$ coincides with $\star$ and is
triangulated on both sides; see [AMRW1, §4.2] for details.

The cohomological shift functors on the triangulated category
$\mathsf{BE}(\mathfrak{h},W)$ is denoted $[1]$. The shift-of-grading functor
$(1)$ on $\mathsf{BE}(\mathfrak{h},W)$ is the functor sending a complex
$(\mathcal{F}^{n},d^{n})_{n\in{\mathbb{Z}}}$ to the complex
$(\mathcal{F}^{n}(1),-d^{n})_{n\in{\mathbb{Z}}}$. We also have the shift
functor $\langle 1\rangle=(-1)[1]$.

As in [AMRW1, §4.2], given $\mathcal{F},\mathcal{G}$ in
$\mathsf{BE}(\mathfrak{h},W)$, we will denote by

$\mathbb{H}\operatorname{om}_{\mathsf{BE}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$

the bigraded $\Bbbk$-vector space whose homogeneous components are

$\mathbb{H}\operatorname{om}_{\mathsf{BE}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})_{j}^{i}:=\operatorname{Hom}_{\mathsf{BE}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G}[i]\langle-j\rangle)$

for all $i,j\in{\mathbb{Z}}$. We also set
$\mathbb{E}\operatorname{nd}_{\mathsf{BE}(\mathfrak{h},W)}(\mathcal{F})=\mathbb{H}\operatorname{om}_{\mathsf{BE}(\mathfrak{h},W)}(\mathcal{F},\mathcal{F})$.

Following [AMRW1, Example 4.2.2], we define the standard object
$\Delta_{\underline{w}}$ and the costandard object $\nabla_{\underline{w}}$
for any expression ${\underline{w}}=(s_{1},\dots,s_{n})$ as

$\Delta_{\underline{w}}=\Delta_{s_{1}}\,\underline{\star}\,\cdots\,\underline{\star}\,\Delta_{s_{n}}\quad\mbox{and}\quad\nabla_{\underline{w}}=\nabla_{s_{1}}\,\underline{\star}\,\cdots\,\underline{\star}\,\nabla_{s_{n}},$

where $\Delta_{s}$ and $\nabla_{s}$ denote the complexes

(4.1) $\cdots 0\to B_{s}\xrightarrow{\leavevmode\hbox to11.67pt{\vbox
to15.91pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-10.3595pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-9.9595pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}B_{\varnothing}(1)\to
0\cdots\quad\mbox{and}\quad\cdots 0\to
B_{\varnothing}(-1)\xrightarrow{\leavevmode\hbox to11.67pt{\vbox
to15.91pt{\pgfpicture\makeatletter\hbox{\hskip 5.83301pt\lower-5.55522pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{9.9595pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}B_{s}\to 0\cdots$

concentrated in degrees $0$ and $1$, and $-1$ and $0$, respectively. (By
convention, $\Delta_{\varnothing}=\nabla_{\varnothing}=B_{\varnothing}$.)

On the other hand, standard and costandard objects $\Delta_{w}$ and
$\nabla_{w}$ in $\mathsf{BE}(\mathfrak{h},W)$ were defined for every $w\in W$
in [ARV, §6.3].333In [ARV, §6.3] such objects are defined for certain subsets
$I\subset W$ containing $w$. Here we take $I=W$, and omit it from the
notation, following the conventions in [ARV]. The same comment applies to
various constructions from [ARV] considered below. We have

(4.2)
$\Delta_{w}\simeq\Delta_{\underline{w}}\quad\mbox{and}\quad\nabla_{w}\simeq\nabla_{\underline{w}}$

if ${\underline{w}}$ is a reduced expression for $w$, see [ARV, Proposition
6.11].

A t-structure on $\mathsf{BE}(\mathfrak{h},W)$ is constructed in [ARV, §7.2];
its heart will be denoted $\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)$. It turns
out that the (co)standard objects $\Delta_{w}$ and $\nabla_{w}$ ($w\in W$)
belong to $\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)$, see [ARV, Proposition
7.8], and that the shift functor $\langle 1\rangle$ is t-exact, see [ARV,
Lemma 7.3]. For every $w\in W$, there is (up to scalar) a unique nonzero
morphism $f_{w}:\Delta_{w}\to\nabla_{w}$ [ARV, Lemma 6.6]; we let

$\mathscr{L}_{w}:=\operatorname{Im}(f_{w})$

be the image of this morphism in $\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)$.
Then the assignment $(w,n)\mapsto\mathscr{L}_{w}\langle n\rangle$ induces a
bijection between $W\times{\mathbb{Z}}$ and the set of isomorphism classes of
simple objects in $\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)$, see [ARV, §8.1].

The following statement, which follows from the construction of standard and
costandard objects via the recollement formalism of [ARV, §5], will be
required below.

######  Lemma 4.1.

For any $w\in W$, the cone of any nonzero morphism $\Delta_{w}\to\nabla_{w}$
in $\mathsf{BE}(\mathfrak{h},W)$ belongs to the triangulated subcategory
generated by the objects of the form $\Delta_{v}(n)$ with $v\in W$ satisfying
$v<w$ and $n\in{\mathbb{Z}}$.

###  4.2. The right-equivariant category

The right-equivariant category is defined in [AMRW1, §4.3] as

$\mathsf{RE}(\mathfrak{h},W):=K^{\mathrm{b}}\overline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W);$

the natural functor from $\mathsf{RE}(\mathfrak{h},W)$ to
$K^{\mathrm{b}}\overline{\mathscr{D}}(\mathfrak{h},W)$ is an equivalence, see
[AMRW1, Lemma 4.9.1], and we will therefore identify these two categories
whenever convenient. The category $\mathsf{RE}(\mathfrak{h},W)$ is in a
natural way a right module category for the monoidal category
$(\mathsf{BE}(\mathfrak{h},W),\,\underline{\star}\,)$; the action bifunctor
will also be denoted $\,\underline{\star}\,$, it is triangulated on both
sides. There exists a natural “forgetful” functor

$\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}:\mathsf{BE}(\mathfrak{h},W)\to\mathsf{RE}(\mathfrak{h},W)$

induced by (3.1); this functor satisfies

(4.3)
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\mathcal{F}\,\underline{\star}\,\mathcal{G})=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\mathcal{F})\,\underline{\star}\,\mathcal{G}$

for $\mathcal{F},\mathcal{G}$ in $\mathsf{BE}(\mathfrak{h},W)$. The shifts
functors $[1]$, $(1)$ and $\langle 1\rangle$, and the bigraded vector spaces
$\mathbb{H}\operatorname{om}_{\mathsf{RE}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
are defined as for the biequivariant category. Then the functor
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}$ commutes with the functors $[1]$,
$(1)$ and $\langle 1\rangle$ in the obvious way.

A t-structure on $\mathsf{RE}(\mathfrak{h},W)$ is also constructed in [ARV,
§9]; its heart will be denoted $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$. For
this structure, the functors $\langle 1\rangle$ and
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}$ are t-exact. The category
$\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ shares many properties with the
categories of Bruhat-constructible perverse sheaves on flag varieties of
Kac–Moody groups, but “with an extra grading;” in particular it has a natural
structure of graded highest weight category (in the sense of [AR,
Appendix]444Compared to this reference, we make two modifications. First we
use the term “highest weight” instead of “quasihereditary.” Second, we allow
our weight poset $\mathcal{S}$ to be infinite, but with the condition that for
any $s\in\mathcal{S}$ the set $\\{t\in\mathcal{S}\mid t\leq s\\}$ is finite;
this does not affect the theory, except for the existence of enough projective
objects, which is not used here. See e.g. [R4, §A] for the ungraded setting.)
with weight poset $(W,\leq)$ and normalized standard and costandard objects

$\overline{\Delta}_{w}:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\Delta_{w})\quad\mbox{and}\quad\overline{\nabla}_{w}=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\nabla_{w})$

for $w\in W$, see [ARV, Theorem 9.6]. (In case $W$ is a the Weyl group of a
reductive group, and for an appropriate choice of $\mathfrak{h}$, one can in
fact relate explicitly $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ with the
corresponding category of perverse sheaves; see [AR] for details.) The
restriction of the functor $\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}$ to the
hearts of the perverse t-structures defines a fully faithful functor
$\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)\to\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$,
see [ARV, Proposition 9.4]. Up to isomorphism, the simple objects in
$\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ are the objects

$\overline{\mathscr{L}}_{w}\langle
n\rangle:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\mathscr{L}_{w}\langle
n\rangle)$

for $(w,n)\in W\times{\mathbb{Z}}$.

###  4.3. Tilting objects

Since $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ has a natural structure of
graded highest weight category, it makes sense to consider its tilting
objects, i.e. the objects admitting both a filtration with subquotients of the
form $\overline{\Delta}_{w}\langle n\rangle$ ($w\in W$, $n\in{\mathbb{Z}}$)
and a filtration with subquotients of the form $\overline{\nabla}_{w}\langle
n\rangle$ ($w\in W$, $n\in{\mathbb{Z}}$). The full subcategory of
$\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ whose objects are tilting will be
denoted $\mathsf{Tilt}_{\mathsf{RE}}(\mathfrak{h},W)$. The category
$\mathsf{Tilt}_{\mathsf{RE}}(\mathfrak{h},W)$ is Krull–Schmidt, and its
isomorphism classes of indecomposable objects is in a natural bijection with
$W\times{\mathbb{Z}}$; for $w\in W$ we will denote by
$\overline{\mathcal{T}}_{w}$ the indecomposable object corresponding to
$(w,0)$. Then, for any $n\in{\mathbb{Z}}$, the indecomposable object
corresponding to $(w,n)$ is $\overline{\mathcal{T}}_{w}\langle n\rangle$. (For
all of this, see [AR, Appendix].) It follows from [AR, Lemma A.5 and Lemma
A.6] that the natural functors

(4.4) $K^{\mathrm{b}}\mathsf{Tilt}_{\mathsf{RE}}(\mathfrak{h},W)\to
D^{\mathrm{b}}\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)\to\mathsf{RE}(\mathfrak{h},W)$

are equivalences of triangulated categories.

######  Example 4.2.

The simple object in $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ corresponding
to the neutral element of $W$ is $\overline{B}_{\varnothing}$, viewed as a
complex concentrated in degree $0$, and it coincides with the object
$\overline{\Delta}_{1}=\overline{\nabla}_{1}=\overline{\mathcal{T}}_{1}$.

######  Example 4.3.

Let $s\in S$. The indecomposable tilting object $\overline{\mathcal{T}}_{s}$
in $\mathsf{Tilt}_{\mathsf{RE}}(\mathfrak{h},W)$ is the complex

$\cdots 0\to\overline{B}_{\varnothing}(-1)\xrightarrow{\leavevmode\hbox
to11.67pt{\vbox to15.91pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-5.55522pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{9.9595pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\overline{B}_{s}\xrightarrow{\leavevmode\hbox
to11.67pt{\vbox to15.91pt{\pgfpicture\makeatletter\hbox{\hskip
5.83301pt\lower-10.3595pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{{}}\pgfsys@setlinewidth{0.8pt}\pgfsys@invoke{ } {}{{}}{}
{}{}{}\pgfsys@moveto{0.0pt}{-9.9595pt}\pgfsys@lineto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
{{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-2.5pt}{-2.22221pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$\bullet$}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\overline{B}_{\varnothing}(1)\to
0\cdots$

concentrated in degrees $-1$, $0$ and $1$. It fits in exact sequences

$0\to\overline{\Delta}_{s}\to\overline{\mathcal{T}}_{s}\to\overline{\Delta}_{1}\langle
1\rangle\to 0\quad\text{and}\quad
0\to\overline{\nabla}_{1}\langle-1\rangle\to\overline{\mathcal{T}}_{s}\to\overline{\nabla}_{s}\to
0.$

For this, see [AMRW1, Example 4.3.4].

###  4.4. The left-equivariant category

We now define the left-equivariant category $\mathsf{LE}(\mathfrak{h},W)$ by

$\mathsf{LE}(\mathfrak{h},W):=K^{\mathrm{b}}\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W).$

Of course, all the constructions and properties of
$\mathsf{RE}(\mathfrak{h},W)$ have analogues for
$\mathsf{LE}(\mathfrak{h},W)$. (In fact, these two categories are equivalent
by Remark 3.1(2).) In particular, we will denote by
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{LE}}:\mathsf{BE}(\mathfrak{h},W)\to\mathsf{LE}(\mathfrak{h},W)$
the natural “forgetful" functor (induced by the natural functor
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)\to\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$),
and by

$\underline{\Delta}_{w}:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LE}}(\Delta_{w}),\quad\text{resp.}\quad\underline{\nabla}_{w}=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LE}}(\nabla_{w})$

the standard, resp. costandard, object associated with $w\in W$. The category
$\mathsf{LE}(\mathfrak{h},W)$ admits a natural “perverse” t-structure whose
heart, denoted $\mathsf{P}_{\mathsf{LE}}(\mathfrak{h},W)$, admits a canonical
structure of graded highest weight category with weight poset $(W,\leq)$ and
normalized standard, resp. costandard, objects the objects
$(\underline{\Delta}_{w}:w\in W)$, resp. $(\underline{\nabla}_{w}:w\in W)$.
Its full subcategory of tilting objects will be denoted
$\mathsf{Tilt}_{\mathsf{LE}}(\mathfrak{h},W)$. The indecomposable objects in
$\mathsf{Tilt}_{\mathsf{LE}}(\mathfrak{h},W)$ are in a canonical bijection
with $W\times{\mathbb{Z}}$, and the indecomposable object associated with
$(w,0)$ will be denoted $\underline{\mathcal{T}}_{w}$ (for $w\in W$).

Recall that $\underline{\mathscr{D}}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$
is a left module category for the monoidal category
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$. This structure “extends”
to a bifunctor

$\,\underline{\star}\,:\mathsf{BE}(\mathfrak{h},W)\times\mathsf{LE}(\mathfrak{h},W)\to\mathsf{LE}(\mathfrak{h},W)$

which defines a left action of the monoidal category
$(\mathsf{BE}(\mathfrak{h},W),\,\underline{\star}\,)$ on
$\mathsf{LE}(\mathfrak{h},W)$. This bifunctor is triangulated on both sides,
and for $\mathcal{F},\mathcal{G}\in\mathsf{BE}(\mathfrak{h},W)$ we have a
canonical (in particular, bifunctorial) isomorphism

$\mathsf{For}^{\mathsf{BE}}_{\mathsf{LE}}(\mathcal{F}\,\underline{\star}\,\mathcal{G})\cong\mathcal{F}\,\underline{\star}\,\mathsf{For}^{\mathsf{BE}}_{\mathsf{LE}}(\mathcal{G}).$

######  Remark 4.4.

As in Remark 3.1(1), below we will also consider all the constructions above
for the realization $\mathfrak{h}^{*}$ instead of $\mathfrak{h}$. We will add
a superscript “$\wedge$” to all our notations in this case.

###  4.5. Free-monodromic category

We will denote by

$\mathsf{FM}(\mathfrak{h},W)$

the “free-monodromic" category defined in [AMRW1, §5.1]. It is a
$\Bbbk$-linear additive category, whose objects are sequences of objects in
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ endowed with a kind of
differential; the precise construction is rather technical, and will not be
recalled here. The category $\mathsf{FM}(\mathfrak{h},W)$ has shift functors
$[1]$, $(1)$ and $\langle 1\rangle$ commuting with each other and such that
$\langle 1\rangle=(-1)[1]$. (Note that $[1]$ is a priori not the suspension
functor for a triangulated structure on $\mathsf{FM}(\mathfrak{h},W)$; in fact
it is not known at this point whether this category admits a triangulated
structure.) As in the biequivariant category (see §4.1), for
$\mathcal{F},\mathcal{G}\in\mathsf{FM}(\mathfrak{h},W)$ we define the bigraded
vector space
$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
by setting

$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})_{j}^{i}:=\operatorname{Hom}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G}[i]\langle-j\rangle)$

and
$\mathbb{E}\operatorname{nd}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F})=\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{F})$.
We point out that we have

(4.5)
$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G}\langle
1\rangle)=\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})\langle
1\rangle$

for all $\mathcal{F},\mathcal{G}\in\mathsf{FM}(\mathfrak{h},W)$.

As in §2.2 we consider the graded ring $R^{\vee}$ as a bigraded ring with
nonzero components concentrated in $\\{0\\}\times{\mathbb{Z}}$. Then as
explained in [AMRW1, §5.2], for
$\mathcal{F},\mathcal{G}\in\mathsf{FM}(\mathfrak{h},W)$ the bigraded vector
space
$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
has a natural structure of bigraded $R^{\vee}$-bimodule. Both actions are
denoted $\,\widehat{\star}\,$ and are compatible with composition in the sense
that $x\,\widehat{\star}\,(f\circ g)=(x\,\widehat{\star}\,f)\circ
g=f\circ(x\,\widehat{\star}\,g)$ for $x\in R^{\vee}$, and similarly for the
right action. In particular, the left action is induced by certain bigraded
algebra homomorphisms

$\mu_{\mathcal{F}}:R^{\vee}\to\mathbb{E}\operatorname{nd}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F})$

for any object $\mathcal{F}$; if
$f\in\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
and $x\in R^{\vee}$, then $x\,\widehat{\star}\,f=\mu_{\mathcal{G}}(x)\circ
f=f\circ\mu_{\mathcal{F}}(x)$.

It should be the case that $\mathsf{FM}(\mathfrak{h},W)$ (or an appropriate
subcategory) admits a structure of monoidal category. Unfortunately this
construction turns out to be delicate, and no general answer is known at
present. But at least part of this structure has been constructed in [AMRW1,
Chap. 6–7]. Explicitly, there is a notion of “convolutive complexes” in
$\mathsf{FM}(\mathfrak{h},W)$, see [AMRW1, Definition 6.1.1], and the full
subcategory of $\mathsf{FM}(\mathfrak{h},W)$ whose objects are the convolutive
complexes is equipped with an operation $\,\widehat{\star}\,$ such that
$\mathcal{F}\,\widehat{\star}\,(-)$ and $(-)\,\widehat{\star}\,\mathcal{F}$
are functors for any fixed convolutive object $\mathcal{F}$. Moreover, for a
fixed convolutive complex $\mathcal{F}$ the functor
$\mathcal{F}\,\widehat{\star}\,(-)$ has an extension to the whole category
$\mathsf{FM}(\mathfrak{h},W)$, see [AMRW1, Proposition 7.6.3]. The operation
$\,\widehat{\star}\,$ satisfies all the axioms of a monoidal category except
for the “interchange law” stating that for $f:\mathcal{F}\to\mathcal{G}$,
$g:\mathcal{G}\to\mathcal{H}$, $h:\mathcal{F}^{\prime}\to\mathcal{G}^{\prime}$
and $k:\mathcal{G}^{\prime}\to\mathcal{H}^{\prime}$ morphisms between
convolutive complexes we have

$(g\circ f)\,\widehat{\star}\,(k\circ
h)=(g\,\widehat{\star}\,k)\circ(f\,\widehat{\star}\,h);$

see [AMRW1, Chap. 7] for details. In particular:

  * •

the operation $\,\widehat{\star}\,$ is associative, so that we can omit
parentheses when considering multiple instances of $\,\widehat{\star}\,$, see
[AMRW1, §7.2];

  * •

we have a convolutive object $\widetilde{\mathcal{T}}_{\varnothing}$
constructed in [AMRW1, §5.3.1] and which behaves like a unit object; see
[AMRW1, §7.1].

We will return to this question in §4.9 below.

###  4.6. Left-monodromic category

We will denote by

$\mathsf{LM}(\mathfrak{h},W)$

the “left monodromic" category defined in [AMRW1, §4.4]. As for the free-
monodromic category, the objects in this category are sequences of objects in
$\mathscr{D}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ endowed with (another
kind of) “differential”. There are shift functors $[1]$, $(1)$ and $\langle
1\rangle$ with analogous properties to those in $\mathsf{FM}(\mathfrak{h},W)$,
which allows to defined bigraded vector spaces
$\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
for $\mathcal{F},\mathcal{G}\in\mathsf{LM}(\mathfrak{h},W)$ by the same recipe
as in $\mathsf{FM}(\mathfrak{h},W)$. This time, the bigraded space
$\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
has a canonical structure of bigraded left $R^{\vee}$-module. There exists a
functor

$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}:\mathsf{FM}(\mathfrak{h},W)\to\mathsf{LM}(\mathfrak{h},W)$

which satisfies

$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}\circ(1)=(1)\circ\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}},\quad\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}\circ[1]=[1]\circ\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}},\quad\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}\circ\langle
1\rangle=\langle 1\rangle\circ\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}.$

Moreover, $\mathsf{LM}(\mathfrak{h},W)$ has a natural structure of
triangulated category with suspension functor $[1]$. There is also a natural
right action of the monoidal category
$(\mathsf{BE}(\mathfrak{h},W),\,\underline{\star}\,)$ on
$\mathsf{LM}(\mathfrak{h},W)$; the corresponding bifunctor will again be
denoted $\,\underline{\star}\,$. (See [AMRW1, (4.23)] for an explicit
construction.)

There is a notion of convolutive objects in $\mathsf{LM}(\mathfrak{h},W)$ and,
for $\mathcal{F}\in\mathsf{FM}(\mathfrak{h},W)$ and
$\mathcal{G}\in\mathsf{LM}(\mathfrak{h},W)$ both convolutive, a convolutive
object
$\mathcal{F}\,\widehat{\star}\,\mathcal{G}\in\mathsf{LM}(\mathfrak{h},W)$.
There is also a “convolution” operation on morphisms, such that for
$\mathcal{F}\in\mathsf{FM}(\mathfrak{h},W)$ and
$\mathcal{G}\in\mathsf{LM}(\mathfrak{h},W)$ convolutive the operations
$\mathcal{F}\,\widehat{\star}\,(-)$ and $(-)\,\widehat{\star}\,\mathcal{G}$
are functorial; see [AMRW1, §6.6]. Moreover, for a fixed convolutive object
$\mathcal{F}\in\mathsf{FM}(\mathfrak{h},W)$ the functor
$\mathcal{F}\,\widehat{\star}\,(-)$ extends to a triangulated functor from
$\mathsf{LM}(\mathfrak{h},W)$ to itself, see [AMRW1, Proposition 7.6.4]. The
functor $\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}$ sends convolutive complexes
to convolutive complexes and satisfies

(4.6)
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\mathcal{F}\,\widehat{\star}\,\mathcal{G})\simeq\mathcal{F}\,\widehat{\star}\,\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\mathcal{G})$

for all $\mathcal{F},\mathcal{G}$ in $\mathsf{FM}(\mathfrak{h},W)$
convolutive, see [AMRW1, $(6.18)$].

There is also an equivalence of triangulated categories

(4.7)
$\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}:\mathsf{LM}(\mathfrak{h},W)\xrightarrow{\sim}\mathsf{RE}(\mathfrak{h},W)$

given in [AMRW1, Theorem 4.6.2] and we have a functor

$\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}:\mathsf{BE}(\mathfrak{h},W)\to\mathsf{LM}(\mathfrak{h},W)$

such that
$\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}\circ\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}$,
cf. [AMRW1, §4.6]. By [AMRW1, Lemma 6.6.1], for any
$\mathcal{G}\in\mathsf{BE}(\mathfrak{h},W)$ the object
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\mathcal{G})$ is convolutive, and
for all $\mathcal{F}\in\mathsf{FM}(\mathfrak{h},W)$ we have a canonical
isomorphism

$\mathcal{F}\,\widehat{\star}\,\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\mathcal{G})\simeq\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\mathcal{F})\,\underline{\star}\,\mathcal{G}.$

The functors $\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}$ and
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}$ commute with the shift functors in
the obvious way.

We can use that $\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}$ is an equivalence
to translate all of the structures and properties of
$\mathsf{RE}(\mathfrak{h},W)$ to $\mathsf{LM}(\mathfrak{h},W)$. Explicitly, we
endow $\mathsf{LM}(\mathfrak{h},W)$ with the t-structure obtained from that of
$\mathsf{RE}(\mathfrak{h},W)$ (see §4.2) and denote by
$\mathsf{P}_{\mathsf{LM}}(\mathfrak{h},W)$ its heart, i.e. the inverse image
of $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$ under the equivalence
$\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}$. Of course, this category is stable
under $\langle 1\rangle$ and inherites the graded highest weight structure
from $\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$. The normalized standard and
costandard objects are

(4.8) $\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\Delta_{w})\quad\mbox{and}\quad\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\nabla_{w})$

for all $w\in W$, since
$\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}(\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\Delta_{w}))=\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}(\Delta_{w})=\overline{\Delta}_{w}$
and similarly for the costandard object. The objects

$\mathcal{L}_{w}\langle
n\rangle=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(\mathscr{L}_{w}\langle
n\rangle),$

$(w,n)\in W\times{\mathbb{Z}}$, form a family of representatives of the
isomorphism classes of simple objects in
$\mathsf{P}_{\mathsf{LM}}(\mathfrak{h},W)$. By Example 4.2, we have
$\mathcal{L}_{1}=\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{1}=\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{1}$.

The following property is useful to extend results from [AMRW2] to our more
general context.

######  Proposition 4.5.

Let $w\in W$.

  1. (1)

The socle of $\hskip 1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}$
is isomorphic to $\mathcal{L}_{1}\langle-\ell(w)\rangle$, and the cokernel of
the inclusion $\mathcal{L}_{1}\langle-\ell(w)\rangle\hookrightarrow\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}$ has no composition
factor of the form $\mathcal{L}_{1}\langle n\rangle$ with $n\in{\mathbb{Z}}$.

  2. (2)

The head of
$\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}$
is isomorphic to $\mathcal{L}_{1}\langle\ell(w)\rangle$, and the kernel of the
surjection
$\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\twoheadrightarrow\mathcal{L}_{1}\langle\ell(w)\rangle$
has no composition factor of the form $\mathcal{L}_{1}\langle n\rangle$ with
$n\in{\mathbb{Z}}$.

###### Proof.

The objects $\mathscr{L}_{1}$, $\Delta_{w}$ and $\nabla_{w}$ satisfy an
analogous statement in $\mathsf{BE}(\mathfrak{h},W)$, see [ARV, Proposition
8.3]. Since the functor
$\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}:\mathsf{P}_{\mathsf{BE}}(\mathfrak{h},W)\to\mathsf{P}_{\mathsf{RE}}(\mathfrak{h},W)$
is fully faithful, and since its essential image contains all simple objects,
we deduce that the socle of $\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}$ is isomorphic to
$\mathcal{L}_{1}\langle-\ell(w)\rangle$ and the head of
$\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}$
is isomorphic to $\mathcal{L}_{1}\langle\ell(w)\rangle$. The other claims also
follow, since $\mathsf{For}^{\mathsf{BE}}_{\mathsf{RE}}$ is exact and sends
simple objects to simple objects. ∎

###  4.7. Left-monodromic tilting category

The following results were established for Cartan realizations of
crystallographic Coxeter groups in [AMRW1, Chap. 10]. Using the theory
developed in [ARV], we can extend them to our present setting, with identical
proofs.

For $s\in S$, recall the object
$\widetilde{\mathcal{T}}_{s}\in\mathsf{FM}(\mathfrak{h},W)$ defined in [AMRW1,
§5.3.2]; this object is convolutive by definition. We therefore have a
triangulated functor
$\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,(-):\mathsf{LM}(\mathfrak{h},W)\to\mathsf{LM}(\mathfrak{h},W)$,
see §4.6.

######  Lemma 4.6.

Let $w\in W$ and $s\in S$.

  1. (1)

If $sw>w$, we have distinguished triangles

$\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{sw}\to\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\to\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle
1\rangle\xrightarrow{[1]}\quad\mbox{and}\quad\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\langle-1\rangle\to\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\to\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{sw}\xrightarrow{[1]}$

in $\mathsf{LM}(\mathfrak{h},W)$, where in each case the second morphism is
nonzero.

  2. (2)

If $sw<w$, we have distinguished triangles

$\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle-1\rangle\to\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\to\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{sw}\xrightarrow{[1]}\quad\mbox{and}\quad\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{sw}\to\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\to\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\langle
1\rangle\xrightarrow{[1]}$

in $\mathsf{LM}(\mathfrak{h},W)$, where in each case the second morphism is
nonzero.

###### Proof.

The existence of the distinguished triangles can be obtained as in [AMRW1,
Lemma 10.5.3], citing [ARV, Proposition 6.11] instead of [AR, Proposition 4.4]
when tensoring with (co)standard objects. We prove that the second morphism in
the first triangle in (1) is nonzero; the other cases are similar. Assume for
a contradiction that this is not the case; then we have $\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{sw}\cong\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\oplus\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle 1\rangle[-1]$.
This contradicts the fact that $\operatorname{Hom}(\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{sw},\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle
1\rangle[-1])=0$, which follows e.g. from the fact that both $\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{sw}$ and $\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle 1\rangle$ belong
to $\mathsf{P}_{\mathsf{LM}}(\mathfrak{h},W)$. ∎

######  Lemma 4.7.

The triangulated functor
$\widetilde{\mathcal{T}}_{s}\,\widehat{\star}\,(-):\mathsf{LM}(\mathfrak{h},W)\to\mathsf{LM}(\mathfrak{h},W)$
is t-exact with respect to the perverse t-structure.

###### Proof.

By [ARV, Lemma 7.5 and Remark 7.6], the nonpositive and nonnegative parts of
the t-structure are generated under extensions by appropriate shifts of
standard and costandard objects, respectively. Thus, the claim follows from
Lemma 4.6. ∎

We set
$\mathcal{T}_{1}:=\mathsf{For}^{\mathsf{BE}}_{\mathsf{LM}}(B_{\varnothing})$
and, given an expression ${\underline{w}}=(s_{1},\dots,s_{r})$, we set

$\mathcal{T}_{\underline{w}}:=\widetilde{\mathcal{T}}_{s_{1}}\,\widehat{\star}\,\cdots\,\widehat{\star}\,\widetilde{\mathcal{T}}_{s_{r}}\,\widehat{\star}\,\mathcal{T}_{1}\in\mathsf{LM}(\mathfrak{h},W).$

Let $\mathsf{Tilt}_{\mathsf{LM}}^{\oplus}(\mathfrak{h},W)$ be the full
subcategory of $\mathsf{LM}(\mathfrak{h},W)$ whose objects are the direct sums
of objects of the form $\mathcal{T}_{{\underline{w}}}\langle n\rangle$ with
${\underline{w}}\in\operatorname{Exp}(W)$ and $n\in{\mathbb{Z}}$. We define
the left-monodromic tilting category
$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ as its karoubian envelope. (Note
that $\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ is a full subcategory of
$\mathsf{LM}(\mathfrak{h},W)$ as the latter has a bounded t-structure and
hence is karoubian by the main result of [LC].) This definition is justified
by the following result.

######  Proposition 4.8.

The functor $\mathsf{For}^{\mathsf{LM}}_{\mathsf{RE}}$ induces an equivalence
of additive categories

$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\xrightarrow{\sim}\mathsf{Tilt}_{\mathsf{RE}}(\mathfrak{h},W).$

As a consequence, the natural functors

(4.9) $K^{\mathrm{b}}\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\to
D^{\mathrm{b}}\mathsf{P}_{\mathsf{LM}}(\mathfrak{h},W)\to\mathsf{LM}(\mathfrak{h},W)$

are equivalences of triangulated categories.

###### Proof.

For the first claim, the proof of [AMRW1, Proposition 10.5.1] applies in the
present setting, using Lemma 4.6 instead of [AMRW1, Lemma 10.5.3]. Then, the
fact that the functors in (4.9) are equivalences follows from the similar
property for the functors in (4.4). ∎

######  Remark 4.9.

Standard arguments show that the obvious functor

$K^{\mathrm{b}}\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\to
K^{\mathrm{b}}\mathsf{Tilt}_{\mathsf{LM}}^{\oplus}(\mathfrak{h},W)$

is an equivalence of triangulated categories. Hence we also obtain an
equivalence of categories

(4.10)
$K^{\mathrm{b}}\mathsf{Tilt}_{\mathsf{LM}}^{\oplus}(\mathfrak{h},W)\xrightarrow{\sim}\mathsf{LM}(\mathfrak{h},W),$

as in [AMRW2, Lemma 2.4].

Using Proposition 4.8 we can transfer to
$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ the usual properties satisfied
by the tilting objects of a graded highest weight category, cf. [ARV, §9.5],
and obtain the following statement.

######  Corollary 4.10.

The category $\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ is Krull–Schmidt.
For any $w\in W$, there exists a unique (up to isomorphism) indecomposable
object $\mathcal{T}_{w}$ characterized by the following properties:

  1. (1)

for any reduced expression ${\underline{w}}$ expressing $w$, $\mathcal{T}_{w}$
occurs as a direct summand of $\mathcal{T}_{{\underline{w}}}$ with
multiplicity $1$;

  2. (2)

$\mathcal{T}_{w}$ does not occur as a direct summand of any object
$\mathcal{T}_{\underline{v}}\langle n\rangle$ with ${\underline{v}}$ an
expression such that $\ell({\underline{v}})<\ell(w)$ and $n\in{\mathbb{Z}}$.

Moreover, the assignment $(w,n)\mapsto\mathcal{T}_{w}\langle n\rangle$ induces
a bijection bewteen $W\times{\mathbb{Z}}$ and the set of isomorphism classes
of indecomposable objects in $\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$. ∎

###  4.8. Free-monodromic tilting category

Given an expression ${\underline{w}}=(s_{1},\dots,s_{r})$, we set

$\widetilde{\mathcal{T}}_{\underline{w}}=\widetilde{\mathcal{T}}_{s_{1}}\,\widehat{\star}\,\cdots\,\widehat{\star}\,\widetilde{\mathcal{T}}_{s_{r}}\in\mathsf{FM}(\mathfrak{h},W).$

Note that by (4.6) we have
$\mathcal{T}_{\underline{w}}=\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widetilde{\mathcal{T}}_{\underline{w}})$.

Proposition 4.8 implies that for any expressions
${\underline{v}},{\underline{w}}$ and $i,j\in{\mathbb{Z}}$ we have

$\operatorname{Hom}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{T}_{\underline{v}},\mathcal{T}_{\underline{w}}[i]\langle
j\rangle)=0\quad\text{unless $i=0$}$

(see [ARV, $(9.2)$]). As in the proof of [AMRW1, Corollary 10.6.2], the above
equality and [AMRW1, Lemma 5.2.3] imply the following statement.

######  Proposition 4.11.

For any expressions ${\underline{v}}$, ${\underline{w}}$ and any
$i,j\in{\mathbb{Z}}$, we have

$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})^{i}_{j}=0\quad\mbox{unless}\quad
i=0.$

Moreover
$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})^{0}_{\bullet}$
is graded free as a right $R^{\vee}$-module, and the morphism

$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})^{0}_{\bullet}\otimes_{R^{\vee}}\Bbbk\to\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{T}_{\underline{v}},\mathcal{T}_{\underline{w}})^{0}_{\bullet}$

induced by the functor $\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}$ is an
isomorphism. ∎

We define the free-monodromic tilting category
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ as the full subcategory of
$\mathsf{FM}(\mathfrak{h},W)$ whose objects are those of the form
$\widetilde{\mathcal{T}}_{\underline{w}}\langle n\rangle$ with
${\underline{w}}\in\operatorname{Exp}(W)$ and $n\in{\mathbb{Z}}$. We denote by
$\mathscr{T}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ the additive hull of
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$, and by
$\mathscr{T}(\mathfrak{h},W)$ the karoubian envelope of
$\mathscr{T}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$. By construction, the
convolution operation $\,\widehat{\star}\,$ restricts to an operation

(4.11)
$\,\widehat{\star}\,:\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\times\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\to\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W).$

The functor
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}:\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\to\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$
extends to $\mathscr{T}(\mathfrak{h},W)$ by the universal property of the
karoubian envelope. By a minor abuse of notation we also denote this extension
by $\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}$. With this definition,
Proposition 4.11 extends to all objects in $\mathscr{T}(\mathfrak{h},W)$, as
follows.

######  Proposition 4.12.

Let $X,Y$ be objects in $\mathscr{T}(\mathfrak{h},W)$. Then the graded vector
space
$\bigoplus_{n}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(X,Y\langle
n\rangle)$ is graded free as a right $R^{\vee}$-module, and the morphism

$\left(\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(X,Y\langle
n\rangle)\right)\otimes_{R^{\vee}}\Bbbk\to\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(X),\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(Y))^{0}_{\bullet}$

induced by the functor $\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}$ is an
isomorphism.

###### Proof.

By construction of the Karoubian envelope,
$\bigoplus_{n}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(X,Y\langle
n\rangle)$ identifies with a direct summand of a similar space of morphisms in
$\mathscr{T}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$. Hence this space is a
projective graded $R^{\vee}$-module by Proposition 4.11 and thus is graded-
free. The fact that $\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}$ induces an
isomorphism also follows from Proposition 4.11. ∎

The classification of the indecomposable tilting objects in
$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ can be “upgraded” to
$\mathscr{T}(\mathfrak{h},W)$ using Proposition 4.12, as in [AMRW1, Theorem
10.7.1].

######  Theorem 4.13.

The category $\mathscr{T}(\mathfrak{h},W)$ is Krull–Schmidt. For any $w\in W$,
there exists a unique (up to isomorphism) indecomposable object
$\widetilde{\mathcal{T}}_{w}$ such that

$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widetilde{\mathcal{T}}_{w})=\mathcal{T}_{w}.$

In addition, $\widetilde{\mathcal{T}}_{w}$ is characterized by the following
properties:

  1. (1)

for any reduced expression ${\underline{w}}$ expressing $w$,
$\widetilde{\mathcal{T}}_{w}$ occurs as a direct summand of
$\widetilde{\mathcal{T}}_{{\underline{w}}}$ with multiplicity $1$;

  2. (2)

$\widetilde{\mathcal{T}}_{w}$ does not occur as a direct summand of any object
$\widetilde{\mathcal{T}}_{\underline{v}}$ with ${\underline{v}}$ an expression
such that $\ell({\underline{v}})<\ell(w)$ and $n\in{\mathbb{Z}}$.

Moreover, the assignment $(w,n)\mapsto\widetilde{\mathcal{T}}_{w}\langle
n\rangle$ induces a bijection bewteen $W\times{\mathbb{Z}}$ and the set of
isomorphism classes of indecomposable objects in
$\mathscr{T}(\mathfrak{h},W)$. ∎

In case $w=1$ we have
$\widetilde{\mathcal{T}}_{1}=\widetilde{\mathcal{T}}_{\varnothing}$, and if
${\underline{w}}=s\in S$ then $\widetilde{\mathcal{T}}_{s}$ is the object
denoted in this way above. In these cases the object
$\widetilde{\mathcal{T}}_{w}$ is canonical; in general, it is only defined up
to isomorphism.

###  4.9. The bifunctoriality assumption

As explained in §4.5, it is not known whether the operation
$\,\widehat{\star}\,$ is a bifunctor on the subcategory of convolutive objects
in $\mathsf{FM}(\mathfrak{h},W)$. As explained in [AMRW1, §7.7], it is
expected that this condition holds at least on the subcategory
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$, which boils down to the fact that
for $f:\mathcal{F}\to\mathcal{G}$, $g:\mathcal{G}\to\mathcal{H}$,
$h:\mathcal{F}^{\prime}\to\mathcal{G}^{\prime}$ and
$k:\mathcal{G}^{\prime}\to\mathcal{H}^{\prime}$ morphisms between objects in
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ we have

(4.12) $(g\circ f)\,\widehat{\star}\,(k\circ
h)=(g\,\widehat{\star}\,k)\circ(f\,\widehat{\star}\,h).$

Below we will assume that this property holds for our given realization, which
implies that $\,\widehat{\star}\,$ induces a monoidal structure on the
category $\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ (and then on
$\mathscr{T}_{\mathrm{BS}}^{\oplus}(\mathfrak{h},W)$ and on
$\mathscr{T}(\mathfrak{h},W)$).

######  Remark 4.14.

  1. (1)

In case our realization is a Cartan realization of a crystallographic Coxeter
group, the main result of [AMRW1, Chap. 11] states that the condition above is
satisfied.

  2. (2)

Using the results of §4.8, the methods of [AMRW1, Chap. 11] also apply for a
general realization of a general Coxeter group, provided for any pair $s,t\in
S$ of distinct simple reflections generating a finite subgroup of $W$, the
conditions in [AMRW1, Chap. 8] are satisfied by $\mathfrak{h}_{|\langle
s,t\rangle}$. This case covers at least, for any Coxeter system, the geometric
realization and the Soergel realizations (see §2.1), provided the condition
(2) of §2.1 is satisfied in case $(W,S)$ admits a parabolic subsystem of type
$\mathsf{H}_{3}$.

  3. (3)

Matthew Hogancamp and Shotaro Makisumi have announced a proof of this
condition in full generality (provided the assumptions (1)–(2)–(3) of §2.1 are
satisfied). Unfortunately, no written account of their proof is available at
this point.

  4. (4)

Using Proposition 4.11 one obtains that, if the condition above is satisfied,
the operation (4.11) is a bifunctor, and defines on
$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$ a structure of module category
for the monoidal category $(\mathscr{T}(\mathfrak{h},W),\,\widehat{\star}\,)$.

##  5\. A functor ${\mathbb{V}}$

In this section we assume that the conditions of §2.1, §2.3 and §4.9 are
satisfied. In addition, we assume that $W$ is finite.

###  5.1. Statement

Recall the dual realization $\mathfrak{h}^{*}$ of $(W,S)$ considered in §2.3.
Our aim is to prove the following statement.

######  Theorem 5.1.

There exists a canonical equivalence of monoidal additive categories

${\mathbb{V}}:\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\xrightarrow{\sim}\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$

which satisfies ${\mathbb{V}}\circ\langle 1\rangle=\langle
1\rangle\circ{\mathbb{V}}$ and
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})\simeq
B_{\underline{w}}^{\wedge}$ for any expression ${\underline{w}}$.

We construct the functor ${\mathbb{V}}$ and prove the theorem in §5.4 and
§5.5. Before that we need some preliminaries.

###  5.2. The big tilting object

Let $w_{0}$ be the longest element in $W$, and consider the corresponding
indecomposable tilting object $\mathcal{T}_{w_{0}}$ in
$\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)$, see Corollary 4.10. We set

$\mathcal{P}:=\mathcal{T}_{w_{0}}\langle-\ell(w_{0})\rangle.$

By [ARV, Theorem 10.3], $\mathcal{P}$ is the projective cover of the simple
object $\mathcal{T}_{1}=\mathcal{L}_{1}$, and [ARV, $(10.5)$] implies that

(5.1)
$\dim_{\Bbbk}\operatorname{Hom}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{P},\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\langle
m\rangle)=\begin{cases}1&\mbox{if $m=-\ell(w)$;}\\\
0&\mbox{otherwise.}\end{cases}$

In particular, we have

(5.2)
$\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{P},\mathcal{T}_{1})^{i}_{j}=\begin{cases}\Bbbk&\text{if
$i=j=0$;}\\\ 0&\text{otherwise.}\end{cases}$

We fix an indecomposable tilting object $\widetilde{\mathcal{T}}_{w_{0}}$ in
$\mathscr{T}(\mathfrak{h},W)$ such that
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widetilde{\mathcal{T}}_{w_{0}})=\mathcal{T}_{w_{0}}$,
see Theorem 4.13, and set

(5.3)
$\widetilde{\mathcal{P}}:=\widetilde{\mathcal{T}}_{w_{0}}\langle-\ell(w_{0})\rangle.$

Proposition 4.12 and (5.2) imply that we have an isomorphism of graded
$R^{\vee}$-modules

$\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{1}\langle
n\rangle)\simeq R^{\vee},$

hence in particular that

$\dim_{\Bbbk}(\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{1}))=1.$

We fix from now on a nonzero morphism

(5.4) $\xi:\widetilde{\mathcal{P}}\to\widetilde{\mathcal{T}}_{1},$

which is automatically a generator of
$\bigoplus_{n}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{1}\langle
n\rangle)$ as a right $R^{\vee}$-module. We also set
$\xi^{\prime}=\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\xi)$, a generator of
$\operatorname{Hom}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{P},\mathcal{T}_{1})$.

The objects $\widetilde{\mathcal{P}}$ and $\mathcal{P}$ are studied in [AMRW2,
§§3.1–3.4] for Cartan realizations of (finite) crystallographic Coxeter
groups. As in §4.8, these results hold in our present setting, and their
proofs can be copied from [AMRW2], replacing the references to [AR] by
references to [ARV]. Below we state the results we will use, and give sketches
of proofs.

######  Lemma 5.2.

In the abelian category $\mathsf{P}_{\mathsf{LM}}(\mathfrak{h},W)$ we have

$[\mathcal{P}:\mathcal{L}_{1}\langle m\rangle]=0$

unless $m\leq 0$, and moreover $[\mathcal{P}:\mathcal{L}_{1}]=1$. In
particular,
$\operatorname{End}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{P})=\Bbbk\cdot\operatorname{id}$.

###### Proof.

The proof is similar to that of [AMRW2, Lemma 3.1]. Namely, as $\mathcal{P}$
is tilting it admits a standard filtration. Using the reciprocity formula (see
e.g. [ARV, $(10.1)$]), for $w\in W$ and $n\in{\mathbb{Z}}$ we have

$(\mathcal{P}:\hskip
1.0pt{\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta_{w}\langle
n\rangle)=[\rotatebox[origin={c}]{180.0}{${\mbox{\scriptsize$\Delta$}}\kern-8.0pt\Delta$}_{w}\langle
n\rangle:\mathcal{L}_{1}],$

which is equal to $1$ if $n=-\ell(w)$ and to $0$ otherwise by Proposition 4.5.
Using again Proposition 4.5 we deduce that

$[\mathcal{P}:\mathcal{L}_{1}\langle n\rangle]=\\#\\{w\in W\mid
n=-2\ell(w)\\},$

which implies the statement. ∎

Let $s\in S$ and
$\widehat{\epsilon}_{s}:\widetilde{\mathcal{T}}_{s}\to\widetilde{\mathcal{T}}_{1}\langle
1\rangle$ be the morphism defined in [AMRW1, §5.3.4]. As in [AMRW2, §3.3],
there exists a unique morphism

$\zeta_{s}^{\prime}:\mathcal{P}\to\mathcal{T}_{s}\langle-1\rangle$

such that
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widehat{\epsilon}_{s}\langle-1\rangle)\circ\zeta_{s}^{\prime}=\xi^{\prime}$.
In turn, as in [AMRW2, Lemma 3.5], there exist a unique morphism

(5.5)
$\zeta_{s}:\widetilde{\mathcal{P}}\to\widetilde{\mathcal{T}}_{s}\langle-1\rangle$

such that
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\zeta_{s})=\zeta_{s}^{\prime}$, and
this morphism satisfies

(5.6) $\widehat{\epsilon}_{s}\langle-1\rangle\circ\zeta_{s}=\xi.$

And there is a unique isomorphism of graded $R^{\vee}$-bimodules

(5.7) $\gamma_{s}^{\prime}:R^{\vee}\otimes_{(R^{\vee})^{s}}R^{\vee}\langle
1\rangle\xrightarrow{\sim}\bigoplus_{m}\operatorname{Hom}^{\bullet}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{s}\langle-m\rangle)$

sending $u_{s}=1\otimes 1$ to $\zeta_{s}$.

The following statement is the analogue in our setting of [AMRW2, Proposition
3.6], for which the same proof applies.

######  Proposition 5.3.

The object $\widetilde{\mathcal{P}}$ admits a canonical coalgebra structure in
the monoidal category $\mathscr{T}(\mathfrak{h},W)$, with counit
$\xi:\widetilde{\mathcal{P}}\to\widetilde{\mathcal{T}}_{1}$ and the
comultiplication morphism
$\delta:\widetilde{\mathcal{P}}\to\widetilde{\mathcal{P}}\,\widehat{\star}\,\widetilde{\mathcal{P}}$
characterized as the unique morphism such that
$(\xi\,\widehat{\star}\,\xi)\circ\delta=\xi$. ∎

###  5.3. Localization

We now recall the localization procedure from [AMRW1, §11.1]. (Once again, in
[AMRW1], this construction is considered only for Cartan realizations of
crystallographic Coxeter groups, but it now makes sense in our present
setting.)

Recall the graded ring $Q^{\vee}$ from §2.2. If $\mathcal{F},\mathcal{G}$ are
objects in $\mathsf{FM}(\mathfrak{h},W)$ we set

$\mathbb{H}\operatorname{om}_{\mathrm{loc}}(\mathcal{F},\mathcal{G}):=\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})\otimes_{R^{\vee}}Q^{\vee},$

where we consider the _right_ action of $R^{\vee}$ on
$\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G})$
from §4.5. Let

$\mathsf{Loc}(\mathfrak{h},W)$

be the category whose objects are the same as those of
$\mathsf{FM}(\mathfrak{h},W)$, and such that the space of morphisms from
$\mathcal{F}$ to $\mathcal{G}$ consists of the elements of bi-degree $(0,0)$
in $\mathbb{H}\operatorname{om}_{\mathrm{loc}}(\mathcal{F},\mathcal{G})$. By
construction there exists a canonical functor

$\mathsf{Loc}:\mathsf{FM}(\mathfrak{h},W)\to\mathsf{Loc}(\mathfrak{h},W).$

Below we will need to consider also the karoubian envelope
$\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)$ of the category
$\mathsf{FM}(\mathfrak{h},W)$. Using the fact that the right action of
$R^{\vee}$ on morphism spaces is compatible with composition, one sees that
for any $\mathcal{F},\mathcal{G}$ in
$\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)$ we have a canonical (right)
action of $R^{\vee}$ on
$\bigoplus_{n,m}\operatorname{Hom}_{\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)}(\mathcal{F},\mathcal{G}\langle-m\rangle[n])$,
so that one can consider the category $\mathsf{Loc}^{\prime}(\mathfrak{h},W)$
obtained from $\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)$ by the same
procedure as $\mathsf{Loc}(\mathfrak{h},W)$ is obtained from
$\mathsf{FM}(\mathfrak{h},W)$. Then there exists a canonical fully faithful
functor
$\mathsf{Loc}(\mathfrak{h},W)\to\mathsf{Loc}^{\prime}(\mathfrak{h},W)$.

Recall from [AMRW2, §7.4] that for any $s\in S$ we have a certain convolutive
object $\widetilde{\nabla}_{s}$ in $\mathsf{FM}(\mathfrak{h},W)$. Then, for an
expression ${\underline{w}}=(s_{1},\cdots,s_{n})$ we can consider the object

$\widetilde{\nabla}_{{\underline{w}}}=\widetilde{\nabla}_{s_{1}}\,\widehat{\star}\,\cdots\,\widehat{\star}\,\widetilde{\nabla}_{s_{n}}.$

When ${\underline{w}}=\varnothing$ we set
$\widetilde{\nabla}_{\varnothing}:=\widetilde{\mathcal{T}}_{\varnothing}$.
Using [AMRW2, Lemma 7.4.2 and Corollary 11.3.2], one sees that we have
isomorphisms of graded $Q^{\vee}$-modules

(5.8)
$\mathbb{H}\operatorname{om}_{\mathrm{loc}}(\widetilde{\nabla}_{{\underline{x}}},\widetilde{\nabla}_{{\underline{y}}})\cong\begin{cases}Q^{\vee}&\text{if
$\pi({\underline{x}})=\pi({\underline{y}})$;}\\\
0&\mbox{otherwise}\end{cases}$

where $\pi$ is as in §2.1.

Consider the morphism

(5.9)
$\phi_{s}:\widetilde{\mathcal{T}}_{s}\to\widetilde{\nabla}_{\varnothing}\langle
1\rangle\oplus\widetilde{\nabla}_{s}$

in $\mathsf{FM}(\mathfrak{h},W)$ denoted $\varkappa^{2}_{s}$ in [AMRW1,
Proposition 11.2.1]. Given an expression
${\underline{w}}=(s_{1},\cdots,s_{n})$, we set

(5.10)
$\phi_{\underline{w}}:=\phi_{s_{1}}\,\widehat{\star}\,\cdots\,\widehat{\star}\,\phi_{s_{n}}:\widetilde{\mathcal{T}}_{\underline{w}}\to(\widetilde{\nabla}_{\varnothing}\langle
1\rangle\oplus\widetilde{\nabla}_{s_{1}})\,\widehat{\star}\,\cdots\,\widehat{\star}\,(\widetilde{\nabla}_{\varnothing}\langle
1\rangle\oplus\widetilde{\nabla}_{s_{n}}).$

With this definition, it is clear that for any expressions
${\underline{v}},{\underline{w}}$ we have

(5.11)
$\phi_{\underline{v}}\,\widehat{\star}\,\phi_{\underline{w}}=\phi_{{\underline{v}}{\underline{w}}}.$

The following statement is our version of [AMRW1, Corollary 11.2.2], which
follows from the same arguments.

######  Lemma 5.4.

For any expression ${\underline{w}}$, the morphism
$\mathsf{Loc}(\phi_{\underline{w}})$ is an isomorphism.∎

###  5.4. Construction of the functor ${\mathbb{V}}$

Recall the functor $(-)^{\wedge}$ considered in §2.2. For
$\mathcal{F}\in\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ we set

${\mathbb{V}}(\mathcal{F}):=\left(\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\mathcal{F}\langle
n\rangle)\right)^{\wedge}.$

We observe that ${\mathbb{V}}\circ\langle 1\rangle=\langle
1\rangle\circ{\mathbb{V}}$ by definition and (2.1). This defines a functor
from $\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ to $R^{\wedge}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax R^{\wedge}$. Our goal in
this subsection is to show that this functor factors through a monoidal
functor from $\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ to
$\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$.

If ${\underline{w}}=(s_{1},\cdots,s_{n})$ is an expression, we denote by
$S_{\underline{w}}$ the multiset of subexpressions of ${\underline{w}}$, i.e.
expressions which can be obtained from ${\underline{w}}$ by removing some
letters. (The multiplicities come from the fact that the same expression can
sometimes be obtained by removing letters from ${\underline{w}}$ in several
ways.) Then we have

$(\widetilde{\nabla}_{\varnothing}\langle
1\rangle\oplus\widetilde{\nabla}_{s_{1}})\,\widehat{\star}\,\cdots\,\widehat{\star}\,(\widetilde{\nabla}_{\varnothing}\langle
1\rangle\oplus\widetilde{\nabla}_{s_{n}})\cong\bigoplus_{{\underline{v}}\in
S_{{\underline{w}}}}\widetilde{\nabla}_{{\underline{v}}}\langle\ell({\underline{w}})-\ell({\underline{v}})\rangle.$

For $u\in W$, we set

$\widetilde{\mathcal{T}}_{\underline{w}}^{u}:=\bigoplus_{\begin{subarray}{c}{\underline{v}}\in
S_{{\underline{w}}}\\\
\pi({\underline{v}})=u\end{subarray}}\widetilde{\nabla}_{{\underline{v}}}\langle\ell({\underline{w}})-\ell({\underline{v}})\rangle.$

Then $\phi_{\underline{w}}$ defines a morphism

$\widetilde{\mathcal{T}}_{{\underline{w}}}\to\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{{\underline{w}}}^{u},$

and if ${\underline{v}}$ and ${\underline{w}}$ are expressions, for $x,y\in W$
we have

(5.12)
$\mathbb{H}\operatorname{om}_{\mathrm{loc}}(\widetilde{\mathcal{T}}_{\underline{v}}^{x},\widetilde{\mathcal{T}}_{\underline{w}}^{y})=0\quad\text{unless
$x=y$}$

by (5.8).

If ${\underline{w}}$ is an expression and $u\in W$, we consider the
$(R^{\wedge},Q^{\wedge})$-bimodule

${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})_{Q^{\wedge}}^{u}:=\left(\bigoplus_{n\in{\mathbb{Z}}}\operatorname{Hom}_{\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{\underline{w}}^{u}\langle
n\rangle)\right)^{\wedge}\otimes_{R^{\wedge}}Q^{\wedge},$

and the morphism of graded $(R^{\wedge},Q^{\wedge})$-bimodules

$\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}:{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})\otimes_{R^{\wedge}}Q^{\wedge}\to\bigoplus_{u\in
W}{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})_{Q^{\wedge}}^{u}$

induced by the assignment $g\mapsto\phi_{\underline{w}}\circ g$ for
$g\in\operatorname{Hom}_{\mathscr{T}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\mathcal{T}}_{{\underline{w}}}\langle
n\rangle)$. It follows from Lemma 5.4 that
$\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}$ is an
isomorphism.

Recall the definition of the category $\mathscr{C}(\mathfrak{h}^{*},W)$ in
§3.2.

######  Lemma 5.5.

  1. (1)

For any expression ${\underline{w}}$, the triple

$\left({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}}),({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})_{Q^{\wedge}}^{u})_{u\in
W},\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}\right)$

is an object of $\mathscr{C}(\mathfrak{h}^{*},W)$.

  2. (2)

For any expressions ${\underline{v}},{\underline{w}}$ and any
$\varphi\in\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})$,
${\mathbb{V}}(\varphi)$ is a morphism in $\mathscr{C}(\mathfrak{h}^{*},W)$
from the triple
$({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}}),({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})_{Q^{\wedge}}^{u})_{u\in
W},\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})})$ to the
triple
$({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}}),({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})_{Q^{\wedge}}^{u})_{u\in
W},\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})})$.

###### Proof.

(1) Let ${\underline{w}}$ be an expression, and let $x\in R^{\vee}$. For any
expression ${\underline{y}}$ and any
$g\in\mathbb{H}\operatorname{om}_{\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)}(\widetilde{\mathcal{P}},\widetilde{\nabla}_{{\underline{y}}})$
we have

$x\,\widehat{\star}\,g=g\circ\mu_{\widetilde{\nabla}_{{\underline{y}}}}(x)=g\,\widehat{\star}\,\pi({\underline{y}})^{-1}(x)$

by [AMRW1, Lemma 7.4.1]. Therefore the left and right $R^{\wedge}$-action on
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})_{Q^{\wedge}}^{u}$
satisfy the compatibility property in the definition of
$\mathscr{C}(\mathfrak{h}^{*},W)$ for all $u\in W$. Hence our triple is an
object of $\mathscr{C}(\mathfrak{h}^{*},W)$, as desired.

(2) Fix ${\underline{v}},{\underline{w}}$ and $\varphi$ as in the statement.
What we have to prove is that for any $u\in W$ and any
$\widetilde{f}\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})^{u}_{Q^{\wedge}}$
the element

$h:=\left(\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}\circ({\mathbb{V}}(\varphi)\otimes_{R^{\wedge}}1)\circ(\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})})^{-1}\right)(\widetilde{f})$

belongs to
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{u}_{Q^{\wedge}}$.

First, let us assume that
$\widetilde{f}=\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})}(f)$
for some $f\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})$. In this
case, $h=\phi_{\underline{w}}\circ\varphi\circ f$ in
$\mathsf{Loc}^{\prime}(\mathfrak{h},W)$. Since $\phi_{\underline{v}}$ is an
isomorphism in this category, we have

$h=(\phi_{\underline{w}}\circ\varphi\circ\phi_{\underline{v}}^{-1})\circ(\phi_{\underline{v}}\circ
f)=(\phi_{\underline{w}}\circ\varphi\circ\phi_{\underline{v}}^{-1})\circ\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})}(f)$

and hence
$h\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{u}_{Q^{\wedge}}$
by our assumption on $f$ and (5.12), see Figure 1.

$\widetilde{\mathcal{P}}$$\widetilde{\mathcal{T}}_{\underline{v}}$$\widetilde{\mathcal{T}}_{\underline{w}}$$\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}$$\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{w}}^{u}$$f$$\varphi$$\phi_{\underline{v}}$$\phi_{\underline{w}}$$\widetilde{\mathcal{T}}_{\underline{w}}$$\widetilde{\mathcal{P}}$$\circlearrowright$$\widetilde{\mathcal{T}}_{\underline{v}}$$\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}$$\widetilde{\mathcal{T}}_{\underline{v}}^{u}$$\widetilde{\mathcal{T}}_{\underline{w}}$$\circlearrowright$$\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}$$\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{w}}^{u}$$f$$\xi_{{\mathbb{V}}_{\widetilde{\mathcal{T}}_{\underline{v}}}}(f)$$\varphi$$\phi_{\underline{v}}$$\phi_{\underline{w}}$$\phi_{\underline{w}}\circ\varphi\circ\phi_{\underline{v}}^{-1}$$\mathsf{Loc}^{\prime}$
Figure 1. The diagram on the left-hand side is in
$\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)$ and the other one is in
$\mathsf{Loc}^{\prime}(\mathfrak{h},W)$ where the dashed morphisms exist and
their composition equals $h$.

Now we consider the general case. There exist $a\in R^{\wedge}$ and
$f\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})$ such that
$\widetilde{f}=\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})}(f\frac{1}{a})$.
Then
$\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})}(f)\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})^{u}_{Q^{\wedge}}$
since $\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})}$ is
morphism of right $Q^{\wedge}$-modules, hence
$ha\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{u}_{Q^{\wedge}}$,
which implies that $h$ also belongs to
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{u}_{Q^{\wedge}}$. ∎

Lemma 5.5 shows that the functor ${\mathbb{V}}$ factors through a functor

$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\to\mathscr{C}(\mathfrak{h}^{*},W),$

which will be denoted similarly. We will next show that this functor can be
endowed with a monoidal structure. For that, we introduce the morphism

(5.13)
$\gamma_{\varnothing}:B^{\wedge}_{\varnothing}=R^{\wedge}\to{\mathbb{V}}(\widetilde{\mathcal{T}}_{\varnothing})={\mathbb{V}}(\widetilde{\mathcal{T}}_{1})$

given by the assignment $x\mapsto\xi\,\widehat{\star}\,x$ (where $\xi$ is as
in (5.4)), and the bifunctorial morphism

$\beta_{\mathcal{F},\mathcal{G}}:{\mathbb{V}}(\mathcal{F})\otimes_{R^{\wedge}}{\mathbb{V}}(\mathcal{G})\to{\mathbb{V}}(\mathcal{F}\,\widehat{\star}\,\mathcal{G})$

defined by $\beta_{\mathcal{F},\mathcal{G}}(f\otimes
g)=(f\,\widehat{\star}\,g)\circ\delta\in{\mathbb{V}}(\mathcal{F}\,\widehat{\star}\,\mathcal{G})_{m+n}$
for all $f\in{\mathbb{V}}(\mathcal{F})_{m}$ and
$g\in{\mathbb{V}}(\mathcal{G})_{n}$ (where $\delta$ is as in Proposition 5.3).

######  Proposition 5.6.

The triple $({\mathbb{V}},\beta,\gamma)$ is a monoidal functor from
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ to
$\mathscr{C}(\mathfrak{h}^{*},W)$.

###### Proof.

We first observe that the proof of [AMRW2, Proposition 3.7] shows that
$({\mathbb{V}},\beta,\gamma)$ is a monoidal functor from
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ to $R^{\wedge}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax R^{\wedge}$. In fact, all
the ingredients of this proof have been repeated above, except for [AMRW2,
Lemma 2.6], which can be proved by the same considerations using Proposition
4.5 instead of [AR, Lemma 4.9].

It is clear also that $\gamma$ is a morphism in
$\mathscr{C}(\mathfrak{h}^{*},W)$, so all that remains to be justified is that
$\beta_{\mathcal{F},\mathcal{G}}$ is a morphism in
$\mathscr{C}(\mathfrak{h}^{*},W)$ for any
$\mathcal{F},\mathcal{G}\in\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$, i.e.
that for any expressions ${\underline{v}},{\underline{w}}$ the morphism

$\beta_{{\underline{v}},{\underline{w}}}:=\beta_{\widetilde{\mathcal{T}}_{{\underline{v}}},\widetilde{\mathcal{T}}_{{\underline{w}}}}:{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})\otimes_{R^{\wedge}}{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})\to{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}}\,\widehat{\star}\,\widetilde{\mathcal{T}}_{{\underline{w}}})$

is a morphism in $\mathscr{C}(\mathfrak{h}^{*},W)$. Let $x,y\in W$,
$\widetilde{f}\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})^{x}_{Q^{\wedge}}$
and
$\widetilde{g}\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{y}_{Q^{\wedge}}$.
What we have to prove that the element

$h:=\left(\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}})}\circ(\beta_{{\underline{v}},{\underline{w}}}\otimes_{R^{\wedge}}1)\circ(\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})}^{-1}\otimes_{Q^{\wedge}}\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}^{-1})\right)(\widetilde{f}\otimes_{Q^{\wedge}}\widetilde{g})$

belongs to
${\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}})^{xy}_{Q^{\wedge}}$.
As in the proof of Lemma 5.5(2), we can assume that there exist
$f\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})$ and
$g\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})$ such that
$\widetilde{f}=\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}})}(f)\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})^{x}_{Q^{\wedge}}$
and
$\widetilde{g}=\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{w}}})}(g)\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{y}_{Q^{\wedge}}$.
Thus, in $\mathsf{Loc}^{\prime}(\mathfrak{h},W)$ we have

$\displaystyle
h=\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}})}\left((\beta_{{\underline{v}},{\underline{w}}}\otimes_{R^{\wedge}}1)((f\otimes_{R^{\wedge}}1)\otimes_{Q^{\wedge}}(g\otimes_{R^{\wedge}}1))\right)$
$\displaystyle\overset{(a)}{=}\phi_{{\underline{v}}{\underline{w}}}\circ(f\,\widehat{\star}\,g)\circ\delta$
$\displaystyle\overset{(b)}{=}(\phi_{{\underline{v}}}\,\widehat{\star}\,\phi_{{\underline{w}}})\circ(f\,\widehat{\star}\,g)\circ\delta$
$\displaystyle\overset{(c)}{=}(\phi_{{\underline{v}}}\circ
f)\,\widehat{\star}\,(\phi_{{\underline{w}}}\circ g)\circ\delta$
$\displaystyle\overset{(d)}{=}(\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})}(f)\,\widehat{\star}\,\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})}(g))\circ\delta.$

Here:

  * •

$(a)$ and $(d)$ follow from the definitions of
$\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}})}$
and $\beta$;

  * •

$(b)$ follows from (5.11);

  * •

$(c)$ follows from the interchange law, see §4.9.

Therefore
$h\in{\mathbb{V}}(\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}})^{xy}_{Q^{\wedge}}$
by our assumptions and (5.12), see Figure 2. ∎

$\widetilde{\mathcal{P}}$$\widetilde{\mathcal{P}}\,\widehat{\star}\,\widetilde{\mathcal{P}}$$\widetilde{\mathcal{T}}_{\underline{v}}\,\widehat{\star}\,\widetilde{\mathcal{T}}_{\underline{w}}$$\vphantom{\displaystyle\left(\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}\right)\,\widehat{\star}\,\left(\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}\right)}\widetilde{\mathcal{T}}_{\underline{v}}^{x}\,\widehat{\star}\,\widetilde{\mathcal{T}}_{\underline{w}}^{y}$$\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}}$$\displaystyle\left(\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}\right)\,\widehat{\star}\,\left(\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{w}}^{u}\right)$$\vphantom{\displaystyle\left(\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{\underline{v}}^{u}\right)\,\widehat{\star}\,\left(\bigoplus_{u\in
W}\right)}\displaystyle\bigoplus_{u\in
W}\widetilde{\mathcal{T}}_{{\underline{v}}{\underline{w}}}^{u}$$\delta$$f\,\widehat{\star}\,g$$\phi_{\underline{v}}\,\widehat{\star}\,\phi_{\underline{w}}$$\phi_{{\underline{v}}{\underline{w}}}$$\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})}(f)\,\widehat{\star}\,\xi_{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})}(g)$$\circlearrowright$$\circlearrowright$
Figure 2. The dashed morphisms are in $\mathsf{Loc}^{\prime}(\mathfrak{h},W)$
and the others in $\mathsf{FM}^{\operatorname{Kar}}(\mathfrak{h},W)$.

Recall the morphism $\gamma_{s}^{\prime}$ defined in (5.7).

######  Lemma 5.7.

For any $s\in S$, the morphism $\gamma_{s}:=(\gamma_{s}^{\prime})^{\wedge}$
defines an isomorphism
$B_{s}^{\wedge}\xrightarrow{\sim}{\mathbb{V}}(\widetilde{\mathcal{T}}_{s})$ in
$\mathscr{C}(\mathfrak{h}^{*},W)$.

###### Proof.

We already know that $\gamma_{s}$ is an isomorphism of graded
$R^{\wedge}$-bimodules, so we only have to show that it defines a morphism
from $B_{s}^{\wedge}$ to ${\mathbb{V}}(\widetilde{\mathcal{T}}_{s})$ in
$\mathscr{C}(\mathfrak{h}^{*},W)$. However, the bimodules
$(B_{s}^{\wedge})_{Q^{\wedge}}^{u}$ and
$({\mathbb{V}}(\widetilde{\mathcal{T}}_{s}))^{u}_{Q^{\wedge}}$ vanish unless
$u\in\\{1,s\\}$, and identify with the subset of elements $m$ which satisfy
$m\cdot x=u(x)\cdot m$ for any $x\in R^{\wedge}$ if $u\in\\{1,s\\}$ (see the
comments at the beginning of [A1, §2.4]), so that the forgetful functor
induces an isomorphism

$\operatorname{End}_{\mathscr{C}(\mathfrak{h}^{*},W)}(B^{\wedge}_{s},{\mathbb{V}}(\widetilde{\mathcal{T}}_{s}))\xrightarrow{\sim}\operatorname{End}_{R^{\wedge}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\wedge}}(B^{\wedge}_{s},{\mathbb{V}}(\widetilde{\mathcal{T}}_{s})).$

Hence the desired property is automatically satisfied. ∎

Using Proposition 5.6 and Lemma 5.7 we obtain, for any nonempty expression
${\underline{w}}$, a canonical isomorphism

(5.14)
$\gamma_{{\underline{w}}}:B_{\underline{w}}^{\wedge}\xrightarrow{\sim}{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})$

in $\mathscr{C}(\mathfrak{h}^{*},W)$. (For the empty expression, such an
isomorphism was already constructed in (5.13).)

We have therefore proved that the functor ${\mathbb{V}}$ defines a monoidal
functor

${\mathbb{V}}:\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\to\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$

such that ${\mathbb{V}}\circ\langle 1\rangle=\langle
1\rangle\circ{\mathbb{V}}$, and canonical isomorphisms
$\gamma_{\underline{w}}:B_{\underline{w}}^{\wedge}\xrightarrow{\sim}{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})$
for all expressions ${\underline{w}}$. To finish the proof of Theorem 5.1, it
only remains to prove that this functor is fully faithful, which will be done
in the next subsection.

###  5.5. Invertibility of ${\mathbb{V}}$

We start by comparing the graded ranks of morphisms in the source and target
categories of ${\mathbb{V}}$.

######  Lemma 5.8.

For any expressions ${\underline{v}},{\underline{w}}$ we have

$\operatorname{grk}_{R^{\wedge}}\mathbb{H}\operatorname{om}_{\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})^{\wedge}=\operatorname{grk}_{R^{\wedge}}\operatorname{Hom}^{\bullet}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)}(B^{\wedge}_{\underline{v}},B^{\wedge}_{\underline{w}}).$

###### Proof.

As in [AMRW2, §2.7], we consider the Hecke algebra $\mathcal{H}_{(W,S)}$ of
$(W,S)$, its “standard” basis $(H_{w}:w\in W)$ (as a
${\mathbb{Z}}[v,v^{-1}]$-module) and the ${\mathbb{Z}}[v,v^{-1}]$-bilinear
pairing $\langle-,-\rangle$ on $\mathcal{H}_{(W,S)}$ which satisfies $\langle
H_{x},H_{y}\rangle=\delta_{x,y}$, for $x,y\in W$. For $s\in S$ we set
$\underline{H}_{s}=H_{s}+v$, and if ${\underline{w}}=(s_{1},\cdots,s_{n})$ is
an expression we set

$\underline{H}_{{\underline{w}}}=\underline{H}_{s_{1}}\cdots\underline{H}_{s_{n}}.$

The expansion of this element in the basis $(H_{w}:w\in W)$ will be denoted

$\underline{H}_{\underline{v}}=\sum_{x\in W}p_{{\underline{v}}}^{x}(v)H_{x}.$

By [AMRW2, Lemma 2.8] (which holds in our setting, with the same proof), for
any expressions ${\underline{v}},{\underline{w}}$ we have

(5.15)
$\langle\underline{H}_{\underline{v}},\underline{H}_{\underline{w}}\rangle=\operatorname{grk}_{\Bbbk}\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{T}_{\underline{v}},\mathcal{T}_{\underline{w}})=\operatorname{grk}_{R^{\vee}}\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})$

where the second equality is due to Proposition 4.11. On the other hand, by
[A1, Corollary 4.7] we have

$\overline{\langle\underline{H}_{\underline{v}},\underline{H}_{\underline{w}}\rangle}=\overline{\sum_{x\in
W}p_{{\underline{v}}}^{x}(q)p_{{\underline{w}}}^{x}(q)}=\operatorname{grk}_{R^{\wedge}}\operatorname{Hom}^{\bullet}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)}(B_{\underline{v}}^{\wedge},B_{\underline{w}}^{\wedge})$

This proves the lemma, in view of (2.2). ∎

Let us now denote by $R^{\vee}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}$,
resp. $R^{\wedge}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}$, the category of
graded left $R^{\vee}$-modules, resp. $R^{\wedge}$-modules. Then as for
bimodules we have a natural equivalence of categories

$(-)^{\wedge}:R^{\vee}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}\to
R^{\wedge}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}$

which replaces the grading by the opposite grading. We consider the functor

(5.16) ${\mathbb{V}}^{\prime}:\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\to
R^{\wedge}\mathchar 45\relax\mathrm{Mod}^{\mathbb{Z}}$

defined by

${\mathbb{V}}^{\prime}(\mathcal{F})=\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{P},\mathcal{F})^{\wedge}.$

Then by Proposition 4.12 we have a commutative diagram

$\textstyle{\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathbb{V}}}$$\scriptstyle{\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}}$$\textstyle{R^{\wedge}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\wedge}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)\otimes_{R^{\wedge}}\Bbbk}$$\textstyle{\mathsf{Tilt}_{\mathsf{LM}}(\mathfrak{h},W)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\mathbb{V}}^{\prime}}$$\textstyle{R^{\wedge}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}.}$

The following lemma is an analogue of [AMRW2, Lemma 3.9], which follows from
the same arguments using Proposition 4.5 instead of [AR, Lemma 4.9].

######  Lemma 5.9.

The functor ${\mathbb{V}}^{\prime}$ is faithful. ∎

We are now ready to complete the proof of Theorem 5.1.

###### Proof of Theorem 5.1.

As explained at the end of §5.4, it only remains to prove that ${\mathbb{V}}$
is fully faithful. First we show that ${\mathbb{V}}$ is faithful. Let
${\underline{w}}$ and ${\underline{v}}$ be expressions. Fix a finite family of
$n$ homogeneous generators of the $R^{\vee}$-bimodule
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})$; by Proposition 4.12
this also provides a family of homogeneous generators of the left
$R^{\vee}$-module ${\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{v}})$.
Consider the commutative diagram

$\textstyle{\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{m}\operatorname{Hom}_{R^{\vee}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}\mathchar 45\relax
R^{\vee}}({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}}),{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})\langle
m\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})^{\oplus
n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{H}\operatorname{om}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{T}_{\underline{v}},\mathcal{T}_{\underline{w}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus_{m}\operatorname{Hom}_{R^{\vee}\mathchar
45\relax\mathrm{Mod}^{\mathbb{Z}}}({\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{v}}),{\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{w}})\langle
m\rangle)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{w}})^{\oplus
n}}$

where:

  * •

the left horizontal arrows are induced by the functors ${\mathbb{V}}$ and
${\mathbb{V}}^{\prime}$;

  * •

the right horizontal arrows are induced by our choice of generators of
${\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}})$ and
${\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{v}})$;

  * •

the vertical arrows are induced by the functors
$\mathsf{For}_{\mathsf{LM}}^{\mathsf{FM}}$ and $(-)\otimes_{R^{\wedge}}\Bbbk$.

Proposition 4.12 shows that in the leftmost column and in the rightmost
column, the $\Bbbk$-vector space on the bottom line is obtained from the free
right $R^{\vee}$-module on the upper line by applying the functor
$(-)\otimes_{R^{\wedge}}\Bbbk$. The lower composition is injective by Lemma
5.9 and the fact that our family generates
${\mathbb{V}}^{\prime}(\mathcal{T}_{\underline{v}})$. By Lemma 2.2, it follows
that the upper composition is also injective, hence so is the upper left
morphism, showing that ${\mathbb{V}}$ is faithful.

Now, for any $m\in{\mathbb{Z}}$, the functor ${\mathbb{V}}$ induces a morphism
between the homogeneous components of degree $-m$

$\left(\mathbb{H}\operatorname{om}_{\mathsf{FM}(\mathfrak{h},W)}(\widetilde{\mathcal{T}}_{\underline{v}},\widetilde{\mathcal{T}}_{\underline{w}})^{\wedge}\right)_{m}\quad\mbox{and}\quad\operatorname{Hom}_{\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)}({\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{v}}),{\mathbb{V}}(\widetilde{\mathcal{T}}_{\underline{w}})\langle
m\rangle)$

which is injective as ${\mathbb{V}}$ is faithful. By Lemma 5.8, these vector
spaces have the same finite dimension. Therefore this morphism is an
isomorphism, proving that ${\mathbb{V}}$ is fully faithful. ∎

######  Remark 5.10.

Our proof of full faithfulness above is less direct than that of the
corresponding claim in [AMRW2]. This is due to the fact that there is no
theory of “Soergel modules” in the setting of [A1].

##  6\. Construction of the $2m_{st}$-valent morphisms

From now on we drop the assumption that $W$ is finite. We therefore consider
an arbitrary Coxeter system $(W,S)$ and a realization $\mathfrak{h}$ of
$(W,S)$ which satisfies the conditions of §2.1, §2.3 and §4.9.

###  6.1. Overview

Given a pair $s,t$ of distinct simple reflections such that the product $st$
have finite order $m_{s,t}$, we will denote by $w(s,t)$ the word
$(s,t,\cdots)$ of length $m_{s,t}$ where the letters alternate between $s$ and
$t$. Note that the relation $\pi(w(s,t))=\pi(w(t,s))$ in $W$ is precisely the
braid relation associated with $s$ and $t$. Our goal in this section is to
construct, for any such pair $s,t$, a canonical morphism

$f_{s,t}:\widetilde{\mathcal{T}}_{w(s,t)}\to\widetilde{\mathcal{T}}_{w(t,s)}$

which will eventually be the image under Koszul duality of the
$2m_{s,t}$-valent morphism associated with $(s,t)$ in the Hecke category.

We will proceed as follows. By Abe’s theory there exists a unique morphism

$\varphi_{s,t}:B^{\wedge}_{w(s,t)}\to B^{\wedge}_{w(t,s)}$

in $\mathscr{A}_{\mathrm{BS}}(\mathfrak{h}^{*},W)$ such that

(6.1) $\varphi_{s,t}(u_{w(s,t)})=u_{w(t,s)},$

see [A2, Theorem 3.9]. (More specifically, this reference states the existence
of such a morphism. Unicity follows from [A1, Theorem 4.6] and the
computations in [Li, §4.1].) Now, choose a subset $S^{\prime}\subset S$ which
contains $s$ and $t$ and generates a finite subgroup $W^{\prime}$ of $W$.
Choose also the data that allow to define a functor
${\mathbb{V}}_{W^{\prime}}$ as in §5.4 and its monoidal structure, for the
Coxeter system $(W^{\prime},S^{\prime})$ and its realization
$\mathfrak{h}_{|W^{\prime}}$ (see (5.3)–(5.4)). Then the category
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h}_{|W^{\prime}},W^{\prime})$ identifies
in the natural way with a full subcategory of
$\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$, in such a way that the objects
$\widetilde{\mathcal{T}}_{w(s,t)}$ and $\widetilde{\mathcal{T}}_{w(t,s)}$ in
these two categories coincide. By full faithfulness of
${\mathbb{V}}_{W^{\prime}}$ (see Theorem 5.1) there exists a unique morphism
$f_{s,t}$ as above such that

(6.2)
${\mathbb{V}}_{W^{\prime}}(f_{s,t})=\gamma_{w(t,s)}\circ\varphi_{s,t}\circ(\gamma_{w(s,t)})^{-1},$

which provides the desired morphism.

Obviously there is a problem with this definition, since it might depend on
the extra data we have introduced: the subset $S^{\prime}$, and the data
involved in the definition of ${\mathbb{V}}_{W^{\prime}}$. Our goal is exactly
to prove that, in fact, it is not the case. Note that we could have solved the
problem of the dependence on the choice of $S^{\prime}$ by setting
$S^{\prime}=\\{s,t\\}$; however, later we will need to use that (6.2) holds
also for some other choices of $S^{\prime}$.

###  6.2. Proof of independence

Our goal is therefore to prove the following claim.

######  Proposition 6.1.

For any pair $(s,t)$ of simple reflections generating a finite subgroup of
$W$, there exists a morphism

$f_{s,t}:\widetilde{\mathcal{T}}_{w(s,t)}\to\widetilde{\mathcal{T}}_{w(t,s)}$

which satisfies the following property. For any subset $S^{\prime}\subset S$
containing $s$ and $t$ and generating a finite subgroup $W^{\prime}$ of $W$,
and for any choices (5.3)–(5.4) allowing to define a monoidal functor
${\mathbb{V}}_{W^{\prime}}$ as in §5.4 for the Coxeter system
$(W^{\prime},S^{\prime})$ and its realization $\mathfrak{h}_{|W^{\prime}}$,
the equality (6.2) holds.

The proof of Proposition 6.1 will use the following preliminary result.
Recall, for any $s\in S$, the morphism

(6.3)
$\widehat{\epsilon}_{s}:\widetilde{\mathcal{T}}_{s}\to\widetilde{\mathcal{T}}_{1}\langle
1\rangle$

in $\mathscr{T}_{\mathrm{BS}}(\mathfrak{h},W)$ constructed in [AMRW1, §5.3.4].
For an expression ${\underline{w}}=(s_{1},\cdots,s_{r})$ we set

(6.4)
$\widehat{\epsilon}_{{\underline{w}}}:=\widehat{\epsilon}_{s_{1}}\,\widehat{\star}\,\cdots\,\widehat{\star}\,\widehat{\epsilon}_{s_{r}}:\widetilde{\mathcal{T}}_{{\underline{w}}}\to\widetilde{\mathcal{T}}_{1}\langle\ell({\underline{w}})\rangle.$

######  Lemma 6.2.

Let $(s,t)$ be a pair of simple reflections generating a finite subgroup of
$W$. The $\Bbbk$-vector space

$\operatorname{Hom}_{\mathsf{LM}(\mathfrak{h},W)}(\mathcal{T}_{w(s,t)},\mathcal{T}_{1}\langle\ell(w(s,t))\rangle)$

is $1$-dimensional, and is spanned by
$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widehat{\epsilon}_{w(s,t)})$.
Moreover, for any subset $S^{\prime}\subset S$ generating a finite subgroup
$W^{\prime}$ of $W$, and for any data (5.3)–(5.4) allowing to define a
monoidal functor ${\mathbb{V}}_{W^{\prime}}$ as in §5.4, if $f_{s,t}$ is the
unique morphism which satisfies (6.2) we have

$\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widehat{\epsilon}_{w(s,t)})=\mathsf{For}^{\mathsf{FM}}_{\mathsf{LM}}(\widehat{\epsilon}_{w(t,s)}\circ
f_{s,t}).$

###### Proof.