# Affine symmetries in quantum cohomology: corrections and new results

P.–E. Chaput  Domaine Scientifique Victor Grignard, 239, Boulevard des
Aiguillettes, Université Henri Poincaré Nancy 1, B.P. 70239, F-54506
Vandoeuvre-lès-Nancy Cedex, France<EMAIL_ADDRESS>and  N. Perrin  Université Paris-Saclay, UVSQ, CNRS, Laboratoire de
Mathématiques de Versailles, 78000, Versailles, France.
<EMAIL_ADDRESS>
(Date: June 18, 2020)

###### Abstract.

In [CMP09] a general formula was given for the multiplication by some special
Schubert classes in the quantum cohomology of any homogeneous space. Although
this formula is true in the non equivariant setting, the stated equivariant
version was wrong. We provide correction for the equivariant formula, thus
giving a correct argument for the non equivariant formula. We also give new
formulas in the equivariant homology of the affine grassmannian that could
lead to non-equivariant Pieri formulas.

###### 2000 Mathematics Subject Classification:

14M15, 14N35

##  1\. Introduction

In [CMP09] a general formula was given for the multiplication in the quantum
cohomology of any homogeneous space by some special Schubert classes coming
from cominuscule weights. Although this formula is true in the non equivariant
setting, the stated equivariant version is wrong. We provide correction for
the equivariant formula, thus giving a correct argument for the non
equivariant formula. We also provide new product formulas in the equivariant
homology of the affine grassmannian.

Let $G$ be a semisimple simply connected algebraic group and fix $T\subset B$
a maximal torus and a Borel subgroup containing it. Denote by $P^{\vee}$ and
$Q^{\vee}$ be the coweight and corrot lattices. A dominant coweight
$\lambda^{\vee}\in P^{\vee}$ is minuscule if
$\langle\lambda^{\vee},\alpha\rangle\in\\{0,1\\}$ for any positive root
$\alpha$. A minuscule dominant coweight is a fundamental coweight. Denote by
$I_{m}$ the subset of the set $I$ of vertices of the Dynkin diagram of $G$
parametrising minuscule coweights.

We consider a finite group $Z$ which has several interpretation. Define $Z$
has

$Z:=P^{\vee}/Q^{\vee}.$

Representatives for this quotient are for example the opposites of the
minuscule fundamental coweights $(-\varpi^{\vee}_{i})_{i\in I_{m}}$. The group
$Z$ is also the center of $G$ and if $G^{\rm ad}\ \\!$ the the adjoint group
associated to $G$, then $Z=\pi_{1}(G^{\rm ad}\ \\!)$.

The group $Z$ can be realised as a subgroup of the Weyl group $W$ of $G$ as
follows. Let $w_{0}$ be the longest element in $W$. For $i\in I_{m}$ define
$v_{i}\in W$ to be the smallest element in $W$ such that
$v_{i}\varpi_{i}=w_{0}\varpi_{i}$. Then the family $(v_{i})_{i\in I_{m}}$
forms a finite subgroup of $W$ isomorphic to $Z$. Finally $Z$ can be realised
as a subgroup of the extended affine Weyl group ${W_{\rm aff}}=W\ltimes
P^{\vee}$ (see Section 2.2 below) by $i\mapsto\tau_{i}:=v_{i}t_{-\varpi_{i}}$.

For $P\subset G$ a parabolic subgroup, let $I_{P}$ be the set of vertices in
the Dynkin diagram such that, for $i\in I$, the simple root $\alpha_{i}$ is a
root of $P$ if and only if $i\in I_{P}$. For $w\in W$, denote by
$\sigma^{P}(w)$ the Schubert class in $H^{2\ell(w)}(G/P,{\mathbb{Z}})$ defined
by $w$. Denote by $Q_{P}^{\vee}$ the coroot lattice of $P$ and consider
$\eta_{P}:Q^{\vee}\to Q^{\vee}/Q_{P}^{\vee}$ the quotient map. We define an
action of the Weyl group $W$ of $G$ on the equivariant cohomology
$H^{*}_{T}(G/P)$ using, for $w\in W$, the pull-back in cohomology of the left
multiplication by $w$ (see Subsection 6.1). We denote this action by $w^{*}$.
This action is trivial in non-equivariant cohomology and extends to an action
on equivariant quantum cohomology $\operatorname{QH}_{T}^{*}(G/P)$. In this
paper we obtain the following formula in the quantum equivariant cohomology
$\operatorname{QH}^{*}_{T}(G/P)$ for any parabolic subgroup $P\subset G$ (see
Theorem 6.9).

######  Theorem 1.1.

Let $i$ be a cominuscule node. In $QH^{*}_{T}(G/P)$ we have

$\sigma^{P}(v_{i})\times
v_{i}^{*}(\sigma^{P}(w))=q_{\eta_{P}(\varpi_{i}^{\vee}-w^{-1}(\varpi_{i}^{\vee}))}\sigma^{P}(v_{i}w).$

This result corrects our formula in [CMP09, Theorem 1] which was wrong in the
equivariant setting (the action $v_{i}^{*}$ on the second factor on the LHS
was missing). The error in [CMP09] comes from an incorrect description of the
ring structure of $H_{*}^{T}({\Omega K^{\rm ad}})$ the equivariant homology of
the adjoint affine grassmannian (see Section 5). If ${\Omega K}$ is the affine
grassmannian for $G$, the incorrect claim ([CMP09, Page 12]) was that
$H_{*}^{T}({\Omega K})$ should be isomorphic to $Z\otimes H_{*}^{T}({\Omega
K})$. This is not true as explained in Section 5 (see Remark 5.6). This is
corrected in the present paper. Especially, in Proposition 5.2, we prove the
$S$-algebra isomorphism (here $S=H^{*}_{T}({\rm pt})$):

$H_{*}^{T}({\Omega K^{\rm ad}})\simeq
S[P^{\vee}]\otimes_{S[{Q^{\vee}}]}H_{*}^{T}({\Omega K}).$

The incorrect product formula was then used only once in [CMP09, Proposition
3.16]. We give a correct version of Proposition 3.16 in [CMP09] in Proposition
5.14.

We tried to write this paper as independently from [CMP09] as possible and
included many preliminary results on the algebra and the module structure of
the extended affine Hecke algebra ${\widetilde{\mathbb{A}}_{\rm aff}}$ (see
Section 3) and on its module structure ${\widetilde{M}}$ which is isomorphic
to $H_{*}^{T}({\Omega K^{\rm ad}})$ the homology of the adjoint affine
grassmannian. We also added new results. Especially we provide a
generalization of a formula in [Lam08, Proposition 5.4] to coweights for the
map $j^{\rm ad}\ \\!:H_{*}^{T}({\Omega K^{\rm ad}})\to
Z_{{\widetilde{\mathbb{A}}_{\rm aff}}}(S)$ (see Proposition 7.1).

######  Proposition 1.2.

Let ${\mu^{\vee}}\in P^{\vee}$ be antidominant and set $W_{\mu^{\vee}}=\langle
s_{\alpha_{i}}\ |\ i\in[1,r]\textrm{ and
}\langle\alpha_{i},{\mu^{\vee}}\rangle=0\rangle=\\{w\in W\ |\
w({\mu^{\vee}})={\mu^{\vee}}\\}$. Then

${j^{\rm ad}}(\xi_{t_{\mu^{\vee}}})=\sum_{w\in
W/W_{\mu^{\vee}}}{\widetilde{A}}_{t_{w({\mu^{\vee}})}}\,.$

Finally, we use this formula to give an explicit formula for the image of the
map $j:H_{*}^{T}({\Omega K})\to Z_{{\widetilde{\mathbb{A}}_{\rm aff}}}(S)$ for
the special elements $\tau_{i}(v_{i})=\tau_{i}v_{i}\tau_{i}^{-1}$ (see
Proposition 7.4). Here ${{{\widetilde{W}_{\rm aff}}^{-}}}$ denotes the set of
minimal representatives of the quotient ${\widetilde{W}_{\rm aff}}/W$.

######  Proposition 1.3.

We have $\tau_{i}(v_{i})\in{{{\widetilde{W}_{\rm aff}}^{-}}}$ and

$j(\xi_{\tau_{i}(v_{i})})=\sum_{w\leq_{L}v_{i}}\sum_{v\leq
v_{I}^{-1}}\tau_{i}(\xi^{v}(v_{i}^{-1}))A_{\tau_{i}(w)v_{i}w^{-1}}A_{v},$

where $\leq$ is the Bruhat order and $\leq_{L}$ the weak left Bruhat order.

We hope to use the above formula to prove Pieri type formulas in
$H_{*}^{T}({\Omega K})$ in the spirit of what Lam, Lapointe, Morse and
Shimozono [LLMS10] did in type $A$.

Acknowledgement: we thank Elizabeth Milićević for showing us a counterexample
to [CMP09, Theorem 1] which lead to the present correction and development.

###### Contents

  1. 1 Introduction
  2. 2 Notations
    1. 2.1 Affine Lie algebras
    2. 2.2 Affine Weyl groups
    3. 2.3 Translations
  3. 3 extended nil-Hecke ring
    1. 3.1 Definition
    2. 3.2 Definition and properties of $A_{\alpha}$
  4. 4 Module and ring structures of ${\widetilde{\mathbb{A}}_{\rm aff}}$
    1. 4.1 $S$-module structure of ${\widetilde{\mathbb{A}}_{\rm aff}}$
    2. 4.2 Ring structure of ${\widetilde{\mathbb{A}}_{\rm aff}}$
    3. 4.3 Module over ${\widetilde{\mathbb{A}}_{\rm aff}}$
  5. 5 Homology of the adjoint affine Grassmannian ${\Omega K^{\rm ad}}$
    1. 5.1 Cohomology of the finite-dimensional flag manifold $G/B$
    2. 5.2 Affine Grassmannian and the Pontryagin ring structure
    3. 5.3 Geometry of fixed points in ${\Omega K^{\rm ad}}$
    4. 5.4 Reminder on $H_{*}^{T}({\Omega K})$
    5. 5.5 $S$-algebra structure on $H_{*}^{T}({\Omega K^{\rm ad}})$
    6. 5.6 Compatibility between the ring and the ${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure
    7. 5.7 Translations modulo $P$
  6. 6 Affine symmetries
    1. 6.1 Peterson’s isomorphism
    2. 6.2 A Weyl group action on $\operatorname{QH}^{*}_{T}(G/P)$.
    3. 6.3 Compatibility of Peterson’s isomorphism
    4. 6.4 The result
  7. 7 Pieri formulas

##  2\. Notations

In this section, we fix notation for affine Kac-Moody Lie algebras, we
introduce the finite group $Z$ and define the extended affine Weyl group
${\widetilde{W}_{\rm aff}}$.

###  2.1. Affine Lie algebras

We denote by $\mathfrak{g}$ a simple finite-dimensional Lie algebra of rank
$r$, and by $\mathfrak{h}$ a Cartan subalgebra. We denote by $G$ the simply-
connected group corresponding to $\mathfrak{g}$ and by ${G^{\rm ad}}$ the
adjoint group. The affine Kac-Moody group corresponding to $G$ will be denoted
by $\mathcal{G}$ and $\mathcal{P}\subset\mathcal{G}$ is the parabolic subgroup
such that $\mathcal{G}/\mathcal{P}$ is the affine Grassmannian.

The corresponding affine Lie algebra will be denoted by $\mathfrak{g}_{\rm
aff}$, with Cartan subalgebra $\mathfrak{h}_{\rm aff}$. The simple roots are
denoted $(\alpha_{i})_{i\in[1,r]}$ and the null-root, orthogonal to all the
simple roots $(\alpha_{i})_{i\in[1,r]}$, will be denoted by $\epsilon$. Recall
that we have the equality $\epsilon=\Theta+\alpha_{0}$, where $\Theta$ is the
highest root of $\mathfrak{g}$. As in [Kac90, p.82] we will use the
decompositions $\mathfrak{h}_{\rm
aff}^{\vee}=\mathfrak{h}^{\vee}\oplus{\mathbb{C}}\Lambda_{0}\oplus{\mathbb{C}}\epsilon$
and $\mathfrak{h}_{\rm
aff}=\mathfrak{h}\oplus{\mathbb{C}}K\oplus{\mathbb{C}}d$. We denote by
${R_{\rm aff}}$ the set of roots of $\mathfrak{g}_{\rm aff}$ and by $R$ those
of $\mathfrak{g}$.

We denote by $Q,P,{Q^{\vee}},{P^{\vee}}$ the root, weight, coroot, coweight
lattices of $\mathfrak{g}$. We also denote by $S$ the symmetric algebra on
$P$.

###  2.2. Affine Weyl groups

Let $W$ be the Weyl group of $\mathfrak{g}$ and let ${W_{\rm
aff}}=Q^{\vee}\rtimes W$ be the affine Weyl group. For $\lambda^{\vee}\in
Q^{\vee}$, the corresponding element in ${W_{\rm aff}}$ will be denoted by
$t_{\lambda^{\vee}}$. The reflection associated to a root $\alpha$ will be
denoted by $s_{\alpha}$. The group ${W_{\rm aff}}$ is a Coxeter group with
Coxeter generators $s_{i}$ for $1\leq i\leq r$ and
$s_{0}=t_{\Theta^{\vee}}s_{\Theta}$ ([Kum02, Prop 13.1.7], see also Lemma
2.7).

Define the extended affine Weyl group ${\widetilde{W}_{\rm
aff}}:=P^{\vee}\rtimes W\supset{W_{\rm aff}}$. The group ${W_{\rm aff}}$ acts
on $P\oplus{\mathbb{Z}}\epsilon$ while the group by ${\widetilde{W}_{\rm
aff}}$ acts only on $Q\oplus{\mathbb{Z}}\epsilon$ via

(1)
$\begin{array}[]{c}wt_{\lambda^{\vee}}\cdot(\mu+n\epsilon)=w(\mu)+(n-\langle\mu,{\lambda^{\vee}}\rangle)\epsilon\\\
wt_{\mu^{\vee}}\cdot(\lambda+n\epsilon)=w(\lambda)+(n-\langle\lambda,{\mu^{\vee}}\rangle)\epsilon\end{array}$

where we have $\lambda\in Q,{\lambda^{\vee}}\in{Q^{\vee}},\mu\in
P,{\mu^{\vee}}\in{P^{\vee}}$. Note that in general ${\widetilde{W}_{\rm aff}}$
does not act on $P\oplus{\mathbb{Z}}\epsilon$ since $\langle
P,P^{\vee}\rangle\not\subset{\mathbb{Z}}$ in general.

We may however define an actions of ${\widetilde{W}_{\rm aff}}$ on $P^{\vee}$
(and therefore on $Q^{\vee}$) by prescribing that translations do not act: we
simply set $wt_{\lambda^{\vee}}({\mu^{\vee}})=w({\mu^{\vee}})$ for $w\in W$
and ${\lambda^{\vee}},{\mu^{\vee}}\in P^{\vee}$.

######  Notation 2.1.

Since an element in ${Q^{\vee}}$ is also an element in
${Q^{\vee}}\oplus{\mathbb{Z}}\delta$, we will denote by
$w\cdot{\lambda^{\vee}}$ the result of the action of $w\in{\widetilde{W}_{\rm
aff}}$ on ${\lambda^{\vee}}$ as an element in
${Q^{\vee}}\oplus{\mathbb{Z}}\delta$ and by $w({\lambda^{\vee}})$ the element
in ${Q^{\vee}}$.

Recall the definition of the fundamental alcove

$A_{\circ}=\\{\lambda\in\mathfrak{h}_{\mathbb{R}}^{\vee}\ |\
\langle\lambda,\alpha_{i}^{\vee}\rangle\geq 0\textrm{ for all
}i\in[1,r]\textrm{ and }\langle\lambda,\Theta^{\vee}\rangle\leq 1\\}.$

The stabiliser of $A_{\circ}$ in ${\widetilde{W}_{\rm aff}}$ will be denoted
by $Z$; it is a subgroup of ${\widetilde{W}_{\rm aff}}$ isomorphic to
${P^{\vee}}/{Q^{\vee}}$ [LS10, §10.1]. In _loc. cit._ , the authors also prove
the following result.

######  Lemma 2.2.

Let $\tau\in Z$. The conjugation by $\tau$ is an automorphism of the Coxeter
group ${W_{\rm aff}}$. In fact, there exists an automorphism $f_{\tau}$ of the
affine Dynkin diagram such that

$\forall i\in I\cup\\{0\\}\ ,\ \tau
s_{\alpha_{i}}\tau^{-1}=s_{\tau\cdot\alpha_{i}}=s_{\alpha_{f_{\tau}(i)}}\ .$

In particular, we have $\tau\cdot\epsilon=\epsilon$.

######  Notation 2.3.

For ${\widehat{x}}\in{W_{\rm aff}}$, set
$\tau({\widehat{x}}):=\tau{\widehat{x}}\tau^{-1}\in{W_{\rm aff}}$. We have
$\ell(\tau({\widehat{x}}))=\ell({\widehat{x}})$.

######  Lemma 2.4.

An element $\tau$ in $Z$ permutes the positive real roots.

###### Proof.

According to Lemma 2.2, we have $\tau\cdot(\sum n_{j}\alpha_{j}+n\delta)=\sum
n_{j}\alpha_{f_{\tau}(j)}+n\delta$. Since a real root $\alpha+n\delta$ is
positive if and only if $n>0$ or $n=0$ and $\alpha>0$, $\tau$ indeed permutes
positive roots. $\square$

As explained in [CMP09], ${\widetilde{W}_{\rm aff}}$ is not a Coxeter group,
but we have a well defined length function.

######  Definition 2.5.

Every element $x\in{\widetilde{W}_{\rm aff}}$ can be uniquely written as
$\tau{\widehat{x}}$ with $\tau\in Z$ and ${\widehat{x}}\in{W_{\rm aff}}$

  1. (1)

Define the length function by $\ell(x):=\ell({\widehat{x}})$.

  2. (2)

Define a partial order on ${\widetilde{W}_{\rm aff}}$ by
$\displaystyle\tau{\widehat{x}}\leq\sigma{\widehat{y}}\Longleftrightarrow\tau=\sigma\mbox{
and }{\widehat{x}}\leq{\widehat{y}}.$

Covering relations in ${\widetilde{W}_{\rm aff}}$ for the above partial order
are defined by $x\lessdot y$ if $x\leq y$ and $\ell(x)=\ell(y)-1$.

######  Remark 2.6.

The length of $x\in{\widetilde{W}_{\rm aff}}$ is also the number of
inversions, namely the cardinal of the set $I(x)=\\{\alpha\in{R_{\rm aff}}\ |\
\alpha>0,\alpha\textrm{ is real and }x(\alpha)<0\\}$. Indeed, for
$x=\tau{\widehat{x}}$, by Lemma 2.4, we have $I(x)=I({\widehat{x}})$.

###  2.3. Translations

We will need the following lemma.

######  Lemma 2.7.

Let $\alpha\in R$. We have
$t_{{\alpha^{\vee}}}=s_{\epsilon-\alpha}s_{\alpha}$.

###### Proof.

Set $K^{\bot}=\\{\mu\in\mathfrak{h}_{\rm aff}^{\vee}\ |\
\langle\mu,K\rangle=0\\}$. By [Kac90, p.87], it is enough to compute
$s_{\epsilon-\alpha}s_{\alpha}(\mu)$ for $\mu\in K^{\bot}$. We have

$s_{\epsilon-\alpha}s_{\alpha}(\mu)=\mu-\langle\mu,(\epsilon-\alpha)^{\vee}\rangle(\epsilon-\alpha)-\langle\mu,{\alpha^{\vee}}\rangle\alpha+\langle\mu,{\alpha^{\vee}}\rangle\langle\alpha,(\epsilon-\alpha)^{\vee}\rangle(\epsilon-\alpha).$

Now, for any $\beta\in{R_{\rm aff}}$, we have by [Kac90, §2.3.5 and §6.2.3]:

$\langle\mu,(\epsilon+\beta)^{\vee}\rangle=\frac{2(\epsilon+\beta,\mu)}{(\epsilon+\beta,\epsilon+\beta)}=\frac{2(\beta,\mu)}{(\beta,\beta)}=\langle\mu,{\beta^{\vee}}\rangle.$

Therefore,

$\begin{array}[]{rcl}s_{\epsilon-\alpha}s_{\alpha}(\mu)&=&\mu+\langle\mu,\alpha^{\vee}\rangle(\epsilon-\alpha)-\langle\mu,{\alpha^{\vee}}\rangle\alpha-2\langle\mu,{\alpha^{\vee}}\rangle(\epsilon-\alpha)\\\
&=&\mu-\langle\mu,{\alpha^{\vee}}\rangle\epsilon=t_{\alpha^{\vee}}(\mu)\
,\end{array}$

where the last equality follows from the definition of $t_{\alpha^{\vee}}$ in
[Kac90, §6.5.5]. $\square$

######  Corollary 2.8.

For $\alpha\in R,k\in{\mathbb{Z}}$ and ${\mu^{\vee}}\in P^{\vee}$, we have
$s_{\alpha+k\epsilon}({\mu^{\vee}})=s_{\alpha}({\mu^{\vee}})$.

###### Proof.

We have
$s_{\alpha+\epsilon}({\mu^{\vee}})=s_{\alpha+\epsilon}t_{-\alpha}({\mu^{\vee}})=s_{\alpha+\epsilon}s_{\alpha+\epsilon}s_{-\alpha}({\mu^{\vee}})=s_{-\alpha}({\mu^{\vee}})=s_{\alpha}({\mu^{\vee}})$.
The result follows by induction. $\square$

##  3\. extended nil-Hecke ring

The goal of this section is to extend the notion of the nil-Hecke ring defined
by Kostant and Kumar [KK86]. This ring was used in [LS10] to compare the
quantum cohomology of $G/P$ and the homology of affine Grassmannians ${\Omega
K}$. We need a refined version of this nil-Hecke ring that enables dealing
with ${\Omega K^{\rm ad}}$ the adjoint affine grassmannian (see Section 5).

###  3.1. Definition

We extend several classical object in particular the affine nil-Hecke algebra.
Our reference for these classical objects is Kumar’s book [Kum02].

######  Definition 3.1.

Recall that the ring ${Q_{\rm aff}}$ is

(2) ${Q_{\rm aff}}=\bigoplus_{w\in{W_{\rm aff}}}{\rm{Frac}(S)}\,\delta_{w}$

We define the following extended version:

(3) ${\widetilde{Q}_{\rm aff}}=\bigoplus_{w\in{\widetilde{W}_{\rm
aff}}}{\rm{Frac}(S)}\,\delta_{w}$

In both cases, the ring structure is defined by the equations
$\delta_{u}\delta_{v}=\delta_{uv}$ and $\delta_{u}s=u(s)\delta_{u}$, for
$u,v\in{W_{\rm aff}}$ resp. $u,v\in{\widetilde{W}_{\rm aff}}$, and $s\in S$.

######  Definition 3.2.

As in [Kum02], we consider particular elements in ${Q_{\rm aff}}$.

  1. (1)

For $i\in I$, set $A_{i}=\frac{1}{\alpha_{i}}(\delta_{e}-\delta_{s_{i}})$. For
$i=0$, set $A_{0}=\frac{-1}{\Theta}(\delta_{e}-\delta_{s_{0}})$ (note that
this is coherent with the forthcoming Definition 3.6).

  2. (2)

For $w\in{W_{\rm aff}}$ and for $w=s_{i_{1}}\cdots s_{i_{\ell}}$ a reduced
expression. We set:

$A_{w}=A_{i_{1}}\cdots A_{i_{\ell}}\,.$

By [Kum02, Theorem 11.1.2], the right hand side does not depend on the chosen
reduced expression.

Recall that for $x\in{\widetilde{W}_{\rm aff}}$, there is a unique
decomposition $x=\tau{\widehat{x}}$ with $\tau\in Z$ and
${\widehat{x}}\in{W_{\rm aff}}$.

######  Definition 3.3.

Let $x=\tau{\widehat{x}}\in{\widetilde{W}_{\rm aff}}$, we set
${\widetilde{A}}_{w}=\delta_{\tau}A_{\widehat{x}}.$

######  Definition 3.4.

As in [KK86], the (extended) nil-Hecke ring is generated over $S$ by the
elements $A_{w}$.

  1. (1)

The nil-Hecke ring is

${\mathbb{A}_{\rm aff}}=\bigoplus_{w\in{W_{\rm aff}}}S\cdot
A_{w}\subset{Q_{\rm aff}}\,.$

  2. (2)

The extended nil-Hecke ring is

${\widetilde{\mathbb{A}}_{\rm aff}}=\bigoplus_{w\in{\widetilde{W}_{\rm
aff}}}S\cdot{\widetilde{A}}_{w}\subset{\widetilde{Q}_{\rm aff}}\,.$

######  Remark 3.5.

We will see below that both are indeed subrings of ${\widetilde{Q}_{\rm
aff}}$.

###  3.2. Definition and properties of $A_{\alpha}$

It will be helpful to generalize the definition of $A_{i}$ in the following
way.

######  Definition 3.6.

For a real root $\alpha=\gamma+k\epsilon$ with $\gamma\in R$, set
$A_{\alpha}=\frac{1}{\gamma}(\delta_{e}-\delta_{s_{\alpha}}).$

These elements satisfy the following properties.

######  Proposition 3.7.

Let $w\in{\widetilde{W}_{\rm aff}},\alpha\in{R_{\rm aff}}$ a real root and
$\lambda\in Q$. Then we have:

  1. (1)

$\delta_{w}A_{\alpha}\delta_{w^{-1}}=A_{w(\alpha)}$.

  2. (2)

$A_{\alpha}\lambda=s_{\alpha}(\lambda)A_{\alpha}+\langle\lambda,\alpha^{\vee}\rangle
1$.

###### Proof.

Let $w=ut_{\mu^{\vee}},\alpha=\gamma+k\epsilon,\lambda\in P$ be as in the
proposition. Then,

$\begin{array}[]{rcccl}\delta_{w}A_{\alpha}\delta_{w^{-1}}&=&\delta_{w}\frac{1}{\gamma}(\delta_{e}-\delta_{s_{\alpha}})\delta_{w^{-1}}&=&\frac{1}{u(\gamma)}\delta_{w}(\delta_{e}-\delta_{s_{\alpha}})\delta_{w^{-1}}\\\
&=&\frac{1}{u(\gamma)}(\delta_{e}-\delta_{ws_{\alpha}w^{-1}})&=&\frac{1}{u(\gamma)}(\delta_{e}-\delta_{s_{w(\alpha)}})=A_{w(\alpha)}\,.\end{array}$

For the second point, we use [LS10, §6.1] and the above conjugation relation.
Let $w\in{W_{\rm aff}}$ and $i\in I\cup\\{0\\}$ be such that
$\alpha=w(\alpha_{i})$. We have

$\begin{array}[]{rcl}A_{\alpha}\lambda&=&\delta_{w}A_{i}\delta_{w^{-1}}\lambda\\\
&=&\delta_{w}A_{i}w^{-1}(\lambda)\delta_{w^{-1}}\\\
&=&\delta_{w}(s_{i}w^{-1}(\lambda)A_{i}+\langle
w^{-1}(\lambda),\alpha_{i}^{\vee}\rangle)\delta_{w^{-1}}\\\
&=&ws_{i}w^{-1}(\lambda)\delta_{w}A_{i}\delta_{w^{-1}}+\langle\lambda,w(\alpha_{i})^{\vee}\rangle\\\
&=&s_{\alpha}(\lambda)A_{\alpha}+\langle\lambda,\alpha^{\vee}\rangle\,.\end{array}$

$\square$

######  Remark 3.8.

The second formula in the above proposition generalizes the usual relation
satisfied by the elements $A_{i}$ (see for example [LS10, §6.1]).

######  Corollary 3.9.

For any real root $\alpha$, we have $A_{\alpha}\in{\mathbb{A}_{\rm aff}}$.

######  Corollary 3.10.

The (extended) nil Hecke rings ${\mathbb{A}_{\rm aff}}$ and
${\widetilde{\mathbb{A}}_{\rm aff}}$ are subrings of ${\widetilde{Q}_{\rm
aff}}$.

###### Proof.

The second formula above shows that for $s,s^{\prime}\in S$ and $u,v\in{W_{\rm
aff}}$, the product $sA_{u}s^{\prime}A_{v}$ lies in ${\mathbb{A}_{\rm aff}}$
therefore ${\mathbb{A}_{\rm aff}}$ is a ring. The first formula proves that
for $\tau,\sigma\in Z$ and for $u,v\in{W_{\rm aff}}$, we have
$\delta_{\tau}A_{u}\delta_{\sigma}A_{v}=\delta_{\tau}\delta_{\sigma}A_{\sigma^{-1}(u)}A_{v}\in{\widetilde{\mathbb{A}}_{\rm
aff}}$ proving that ${\widetilde{\mathbb{A}}_{\rm aff}}$ is a ring. $\square$

##  4\. Module and ring structures of ${\widetilde{\mathbb{A}}_{\rm aff}}$

In this section we present three different descriptions of
${\widetilde{\mathbb{A}}_{\rm aff}}$ and describe its $S$-module structure and
its ring structure in each case.

###  4.1. $S$-module structure of ${\widetilde{\mathbb{A}}_{\rm aff}}$

Recall that we have an injection of ${W_{\rm aff}}$ in the group of
invertibles of ${\mathbb{A}_{\rm aff}}$, given by $w\mapsto\delta_{w}$: in
fact $\delta_{s_{i}}=1-\alpha_{i}A_{i}\in{\mathbb{A}_{\rm aff}}$ thus
$\delta_{w}\in{\mathbb{A}_{\rm aff}}$ for all $w\in{W_{\rm aff}}$. Therefore
the subgroup $Q^{\vee}\subset{W_{\rm aff}}$ also injects in ${\mathbb{A}_{\rm
aff}}$, and since ${\mathbb{A}_{\rm aff}}$ is a ring we have an injection of
the Laurent polynomial algebra ${\mathbb{Z}}[{Q^{\vee}}]$ inside
${\mathbb{A}_{\rm aff}}$. Thus ${\mathbb{A}_{\rm aff}}$ is
${\mathbb{Z}}[{Q^{\vee}}]$-module via left mutliplication. The natural
${\mathbb{Z}}$-module basis of ${\mathbb{Z}}[{Q^{\vee}}]$ will be denoted by
$(h_{\lambda^{\vee}})_{{\lambda^{\vee}}\in{Q^{\vee}}}$.

We now introduce two new algebraic models of ${\widetilde{\mathbb{A}}_{\rm
aff}}$.

######  Definition 4.1.

Let $\varphi_{1},\varphi_{2}$ be the following morphisms of
${\mathbb{Z}}$-modules:

$\begin{array}[t]{rrcll}\varphi_{1}&:&{\mathbb{Z}}[{P^{\vee}}]\otimes_{{\mathbb{Z}}[{Q^{\vee}}]}{\mathbb{A}_{\rm
aff}}&\rightarrow&{\widetilde{Q}_{\rm aff}}\\\ &&h_{{\lambda^{\vee}}}\otimes
A_{w}&\mapsto&\delta_{t_{\lambda^{\vee}}}A_{w}\,,\end{array}$
$\begin{array}[t]{rrcll}\varphi_{2}&:&{\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{\mathbb{A}_{\rm
aff}}&\rightarrow&{\widetilde{Q}_{\rm aff}}\\\ &&\tau\otimes
A_{w}&\mapsto&\delta_{\tau}A_{w}\,.\end{array}$

Note that ${\mathbb{A}_{\rm aff}}$ has a structure of $S$-bimodule, thus also
the two tensor products in this definition. Both maps $\varphi_{1}$ and
$\varphi_{2}$ are $S$-linear on the right, moreover $\varphi_{1}$ is also
$S$-linear on the left whereas $\varphi_{2}$ is not.

######  Proposition 4.2.

With the above notations, ${\rm Im}(\varphi_{1})={\rm
Im}(\varphi_{2})={\widetilde{\mathbb{A}}_{\rm aff}}$. Moreover, if
$J\subset{\mathbb{A}_{\rm aff}}$ is a left ideal, then
$\varphi_{1}({\mathbb{Z}}[{P^{\vee}}]\otimes_{{\mathbb{Z}}[{Q^{\vee}}]}J)=\varphi_{2}({\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}J)$.

###### Proof.

Observe that $\varphi_{1}$ is well-defined:
$\varphi_{1}(h_{\lambda^{\vee}}\otimes
1)=\varphi_{1}(1\otimes\delta_{t_{\lambda^{\vee}}})=\delta_{t_{\lambda^{\vee}}}$
for ${\lambda^{\vee}}\in{Q^{\vee}}$. We now prove that
$\varphi_{1}({\mathbb{Z}}[{P^{\vee}}]\otimes
J)\subset\varphi_{2}({\mathbb{Z}}[Z]\otimes J)$. Let
${\lambda^{\vee}}\in{P^{\vee}}$: there exists $\tau\in Z$ and
${\widehat{w}}\in{W_{\rm aff}}$ such that
$t_{\lambda^{\vee}}=\tau{\widehat{w}}$. Then for $a\in J$, we have
$\varphi_{1}(h_{\lambda^{\vee}}\otimes
a)=\delta_{t_{\lambda^{\vee}}}a=\delta_{\tau}\delta_{\widehat{w}}a\in\varphi_{2}({\mathbb{Z}}[Z]\otimes
J)$ since $\delta_{\widehat{w}}a\in J$.

The reverse inclusion $\varphi_{2}({\mathbb{Z}}[Z]\otimes
J)\subset\varphi_{1}({\mathbb{Z}}[{P^{\vee}}]\otimes J)$ follows similarly
from the fact that any element in $Z$ can be written as a product
$t_{\lambda^{\vee}}u$ for some ${\lambda^{\vee}}\in{P^{\vee}}$ and $u\in W$.
Finally, the equality ${\rm Im}(\varphi_{2})={\widetilde{\mathbb{A}}_{\rm
aff}}$ follows from the definition of ${\widetilde{\mathbb{A}}_{\rm aff}}$
(Definition 3.4). $\square$

###  4.2. Ring structure of ${\widetilde{\mathbb{A}}_{\rm aff}}$

We give the description of the ring structure of ${\widetilde{\mathbb{A}}_{\rm
aff}}$ according to the given three equivalent definitions of this module.

######  Proposition 4.3.

Let $x,y\in{\widetilde{W}_{\rm aff}}$, then we have

${\widetilde{A}}_{x}{\widetilde{A}}_{y}=\left\\{\begin{array}[]{l}{\widetilde{A}}_{xy}\mbox{
if }\ell(xy)=\ell(x)+\ell(y)\\\ 0\mbox{ otherwise}\,.\end{array}\right.$

###### Proof.

Write $x=\sigma{\widehat{x}}$ and $y=\tau{\widehat{y}}$ with $\sigma,\tau\in
Z$ and ${\widehat{x}},{\widehat{y}}\in{W_{\rm aff}}$. Recall that for
$u,v\in{W_{\rm aff}}$, we have:

$A_{u}A_{v}=\left\\{\begin{array}[]{l}A_{uv}\mbox{ if
}\ell(uv)=\ell(u)+\ell(v)\\\ 0\mbox{ otherwise}\,.\end{array}\right.$

By Lemma 2.2, we have

$\begin{array}[]{rcl}{\widetilde{A}}_{x}{\widetilde{A}}_{y}&=&\delta_{\sigma}A_{\widehat{x}}\delta_{\tau}A_{\widehat{y}}\\\
&=&\delta_{\sigma}\delta_{\tau}A_{\tau^{-1}({\widehat{x}})}A_{\widehat{y}}\\\
&=&\left\\{\begin{array}[]{l}\delta_{\sigma\tau}A_{\tau^{-1}({\widehat{x}}){\widehat{y}}}\mbox{
if
}\ell(\tau^{-1}({\widehat{x}}){\widehat{y}})=\ell(\tau^{-1}({\widehat{x}}))+\ell({\widehat{y}})\\\
0\mbox{ otherwise}\end{array}\right.\end{array}$

But $\ell(\tau^{-1}({\widehat{x}}){\widehat{y}})=\ell(xy)$ since
$xy=\sigma{\widehat{x}}\tau{\widehat{y}}=\sigma\tau\tau^{-1}({\widehat{x}}){\widehat{y}}$,
and
$\ell(\tau^{-1}({\widehat{x}}))+\ell({\widehat{y}})=\ell({\widehat{x}})+\ell({\widehat{y}})=\ell(x)+\ell(y)$.
The result follows. $\square$

We now express the product in ${\widetilde{\mathbb{A}}_{\rm
aff}}=\varphi_{1}({\mathbb{Z}}[{P^{\vee}}]\otimes_{{\mathbb{Z}}[{Q^{\vee}}]}{\mathbb{A}_{\rm
aff}})$. Note that we need to compute the product
$(\delta_{t_{\lambda^{\vee}}}A_{u})(\delta_{t_{\mu^{\vee}}}A_{v})$. We
therefore need to “move” $\delta_{t_{\mu^{\vee}}}$ to the left of $A_{u}$. The
following proposition gives formulas for this.

######  Proposition 4.4.

Let ${\lambda^{\vee}}\in{P^{\vee}}$ and let $\alpha=\gamma+k\epsilon\in{R_{\rm
aff}}$. Then:

  1. (1)

$A_{\alpha}\delta_{t_{{\lambda^{\vee}}}}=\delta_{s_{\alpha}(t_{\lambda^{\vee}})}A_{\alpha}+\frac{1}{\gamma}(\delta_{t_{\lambda^{\vee}}}-\delta_{t_{s_{\alpha}({\lambda^{\vee}})}})$.

  2. (2)

$\delta_{t_{\lambda^{\vee}}}-\delta_{t_{s_{\alpha}({\lambda^{\vee}})}}=\delta_{t_{\lambda^{\vee}}}(1-\delta_{t_{-\langle\alpha,{\lambda^{\vee}}\rangle{\alpha^{\vee}}}}),$

  3. (3)

$1-\delta_{t_{n{\alpha^{\vee}}}}=(1+\delta_{t_{\alpha^{\vee}}}+\cdots+\delta_{t_{(n-1){\alpha^{\vee}}}})(1-\delta_{t_{\alpha^{\vee}}})\,\textrm{
for }n\in{\mathbb{N}},$

  4. (4)

$1-\delta_{t_{-{\alpha^{\vee}}}}=\gamma(A_{\alpha}-\gamma
A_{\alpha}A_{\epsilon-\alpha}+A_{\epsilon-\alpha})$.

###### Proof.

(1) From the equality
$s_{\alpha}t_{\lambda^{\vee}}=t_{s_{\alpha}({\lambda^{\vee}})}s_{\alpha}$, we
get
$\delta_{s_{\alpha}}\delta_{t_{\lambda^{\vee}}}=\delta_{t_{s_{\alpha}({\lambda^{\vee}})}}\delta_{s_{\alpha}}$
in ${\widetilde{\mathbb{A}}_{\rm aff}}$. By definition of $A_{\alpha}$
(Definition 3.6), this relation implies

$(1-\gamma
A_{\alpha})\delta_{t_{\lambda^{\vee}}}=\delta_{t_{s_{\alpha}({\lambda^{\vee}})}}(1-\gamma
A_{\alpha})\,.$

Thus we get $\gamma
A_{\alpha}\delta_{t_{\lambda^{\vee}}}=\gamma\delta_{t_{s_{\alpha}({\lambda^{\vee}})}}A_{\alpha}+\delta_{t_{\lambda^{\vee}}}-\delta_{t_{s_{\alpha}({\lambda^{\vee}})}}$.

(2) and (3) are easy consequences of the product formulas in
${\widetilde{Q}_{\rm aff}}$.

(4) By Lemma 2.7, we have

$\begin{array}[]{rcl}1-\delta_{t_{-{\alpha^{\vee}}}}&=&1-\delta_{s_{\alpha}s_{\epsilon-\alpha}}\\\
&=&1-(1-\gamma A_{\alpha})(1+\gamma A_{\epsilon-\alpha})\\\ &=&\gamma
A_{\alpha}-\gamma A_{\epsilon-\alpha}+\gamma A_{\alpha}\gamma
A_{\epsilon-\alpha}\\\ &=&\gamma A_{\alpha}-\gamma
A_{\epsilon-\alpha}+\gamma\left(s_{\alpha}(\gamma)A_{\alpha}+\langle{\alpha^{\vee}},\gamma\rangle\right)A_{\epsilon-\alpha}\\\
&=&\gamma A_{\alpha}+\gamma
A_{\epsilon-\alpha}-\gamma^{2}A_{\alpha}A_{\epsilon-\alpha}\,,\end{array}$

where we used Proposition 3.7 on the fourth line. $\square$

The ring structure in ${\widetilde{\mathbb{A}}_{\rm
aff}}=\varphi_{2}({\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{\mathbb{A}_{\rm aff}})$
is easy to describe:

######  Proposition 4.5.

Let $\sigma,\tau\in Z$ and let $a,b\in{\mathbb{A}_{\rm aff}}$. Then:

$\varphi_{2}(\sigma\otimes a)\cdot\varphi_{2}(\tau\otimes
b)=\varphi_{2}(\sigma\tau\otimes\tau^{-1}(a)b)\,.$

###### Proof.

This follows from the fact that in ${\widetilde{\mathbb{A}}_{\rm aff}}$, we
have
$\delta_{\sigma}a\delta_{\tau}b=\delta_{\sigma}\delta_{\tau}\tau^{-1}(a)b\,.$
$\square$

In the next proposition, we give an explicit formula for the commuting
relation of the elements ${\widetilde{A}}_{x}$ and $\lambda\in P$,
generalizing [KK86, Proposition 4.3.b]:

######  Proposition 4.6.

Let $x\in{\widetilde{W}_{\rm aff}}$ and let $\lambda\in P$. We have:

${\widetilde{A}}_{x}\lambda=x(\lambda){\widetilde{A}}_{x}+\sum_{{\alpha}:\
xs_{\alpha}\lessdot
x}\langle\lambda,{\alpha}^{\vee}\rangle{\widetilde{A}}_{xs_{\alpha}}\,,$

where the sum runs over positive real roots ${\alpha}$ such that
$xs_{\alpha}\lessdot x$.

###### Proof.

Let $x=\tau{\widehat{x}}\in{\widetilde{W}_{\rm aff}}$ with $\tau\in Z$ and
${\widehat{x}}\in{W_{\rm aff}}$. Let $\lambda\in P$. According to Definition
3.3, we have ${\widetilde{A}}_{x}=\delta_{\tau}A_{\widehat{x}}$. Using [KK86,
Proposition 4.3.b], we get (sums always run over positive real roots):

$\displaystyle{\widetilde{A}}_{x}\lambda=\delta_{\tau}A_{\widehat{x}}\lambda$
$\displaystyle=\delta_{\tau}{\widehat{x}}(\lambda)A_{\widehat{x}}+\delta_{\tau}\sum_{{\alpha}:\
{\widehat{x}}s_{\alpha}\lessdot{\widehat{x}}}\langle\lambda,{\alpha}^{\vee}\rangle
A_{{\widehat{x}}s_{\alpha}}$
$\displaystyle=\tau{\widehat{x}}(\lambda)\delta_{\tau}A_{\widehat{x}}+\sum_{{\alpha}:\
{\widehat{x}}s_{\alpha}\lessdot{\widehat{x}}}\langle\lambda,{\alpha}^{\vee}\rangle\delta_{\tau}A_{{\widehat{x}}s_{\alpha}}$
$\displaystyle=x(\lambda){\widetilde{A}}_{x}+\sum_{{\alpha}:\
{\widehat{x}}s_{\alpha}\lessdot{\widehat{x}}}\langle\lambda,{\alpha}^{\vee}\rangle{\widetilde{A}}_{xs_{\alpha}}\,.$

Since, by Definition 3.3, the relation
${\widehat{x}}s_{\alpha}\lessdot{\widehat{x}}$ holds if and only if the
relation $xs_{\alpha}\lessdot x$ holds, we get the result. $\square$

###  4.3. Module over ${\widetilde{\mathbb{A}}_{\rm aff}}$

We now define a natural module over ${\widetilde{\mathbb{A}}_{\rm aff}}$ which
will be identified in the next section with the homology of the adjoint affine
Grassmmannian ${\Omega K^{\rm ad}}$.

######  Definition 4.7.

Let ${W_{{\rm aff}}^{-}}$ resp. ${{{\widetilde{W}_{\rm aff}}^{-}}}$ be the set
of minimal length representatives of the quotient ${W_{\rm aff}}/W$ resp.
${\widetilde{W}_{\rm aff}}/W$. By [LS10, Lemma 3.3], ${W_{{\rm aff}}^{-}}$ is
the set of elements $w=ut_{\lambda^{\vee}}$ such that $\lambda^{\vee}\leq 0$
and $\forall i\in I,\langle{\lambda^{\vee}},\alpha_{i}\rangle=0\implies
u(\alpha_{i})>0\,.$

We generalize the characterization of ${W_{{\rm aff}}^{-}}$ as follows:

######  Lemma 4.8.

We have $ut_{\lambda^{\vee}}\in{{{\widetilde{W}_{\rm aff}}^{-}}}$ if and only
if ${\lambda^{\vee}}\leq 0$ and for all $i$ in $I$ it holds

$\langle{\lambda^{\vee}},\alpha_{i}\rangle=0\implies u(\alpha_{i})>0\,.$

###### Proof.

Recall that we have a length formula in ${\widetilde{W}_{\rm aff}}$ similar to
the one in ${W_{\rm aff}}$:

$\ell(ut_{\lambda^{\vee}})=\sum_{{\alpha}\in
R^{+}}\left|\langle{\lambda^{\vee}},{\alpha}\rangle+\chi(u({\alpha})<0)\right|\,,$

where $\chi(\mathcal{P})=1$ if $\mathcal{P}$ is true and $\chi(\mathcal{P})=0$
if $\mathcal{P}$ is false. This is proved in [CMP09, Corollary 3.13]. It
follows that

$\ell(ut_{\lambda^{\vee}}s_{i})-\ell(ut_{\lambda^{\vee}})=|\langle{\lambda^{\vee}},-{\alpha}_{i}\rangle+\chi(u({\alpha}_{i})>0)|-|\langle{\lambda^{\vee}},{\alpha}_{i}\rangle+\chi(u({\alpha}_{i})<0)|\,.$

This is non-negative for all $i$ in $I$ if and only if for all $i$,
$\langle{\lambda^{\vee}},{\alpha}_{i}\rangle\leq 0$, and
$\langle{\lambda^{\vee}},\alpha_{i}\rangle=0$ implies $u(\alpha_{i})>0$.
$\square$

######  Definition 4.9.

For each $w\in{W_{{\rm aff}}^{-}}$, we define a variable $\xi_{w}$ and we set

$\displaystyle M=\bigoplus_{w\in{W_{{\rm aff}}^{-}}}S\cdot\xi_{w}.$

Recall [LS10, §6.2] that we may define a left ${\mathbb{A}_{\rm aff}}$-module
structure on $M$ via:

$A_{w}\cdot\xi_{u}=\left\\{\begin{array}[]{l}\xi_{wu}\mbox{ if
}\ell(wu)=\ell(w)+\ell(u)\mbox{ and }wu\in{W_{{\rm aff}}^{-}}\,,\\\ 0\mbox{
otherwise}\,.\end{array}\right.$

As left ${\mathbb{A}_{\rm aff}}$-module, we have an isomorphism

$M\simeq{\mathbb{A}_{\rm aff}}/J,\textrm{ where }J=\bigoplus_{w\not\in{W_{{\rm
aff}}^{-}}}S\cdot A_{w}.$

Using Proposition 4.2, we define similarly a left ideal in
${\widetilde{\mathbb{A}}_{\rm aff}}$.

######  Definition 4.10.

Let
$\displaystyle{\widetilde{J}}=\varphi_{1}({\mathbb{Z}}[{P^{\vee}}]\otimes_{{\mathbb{Z}}[{Q^{\vee}}]}J)=\varphi_{2}({\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}J)=\bigoplus_{w\not\in{{{\widetilde{W}_{\rm
aff}}^{-}}}}{\widetilde{A}}_{w}\,.$

######  Definition 4.11.

We introduce the following three modules.

  * •

Let ${\widetilde{M}}_{1}$ be the $S$-module
${\mathbb{Z}}[{P^{\vee}}]\otimes_{{\mathbb{Z}}[{Q^{\vee}}]}M$.

  * •

Let ${\widetilde{M}}_{2}={\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}M$. This is an
${\widetilde{\mathbb{A}}_{\rm aff}}$-module with the action given by
$(\sigma\otimes
a)\cdot(\tau\otimes\xi)=\sigma\tau\otimes\tau^{-1}(a)\cdot\xi$, for
$\sigma\otimes a\in{\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{\mathbb{A}_{\rm
aff}}={\widetilde{\mathbb{A}}_{\rm aff}}$.

  * •

Let $\displaystyle{\widetilde{M}}_{3}=\bigoplus_{w\in{{{\widetilde{W}_{\rm
aff}}^{-}}}}S\cdot{\widetilde{\xi}}_{w}$. This is an
${\widetilde{\mathbb{A}}_{\rm aff}}$-module with the action given by

${\widetilde{A}}_{w}\cdot{\widetilde{\xi}}_{u}=\left\\{\begin{array}[]{l}{\widetilde{\xi}}_{wu}\mbox{
if }\ell(wu)=\ell(w)+\ell(u)\mbox{ and }wu\in{{{\widetilde{W}_{\rm
aff}}^{-}}}\,,\\\ 0\mbox{ otherwise}\,,\end{array}\right.$

for $\displaystyle{\widetilde{A}}_{w}\in{\widetilde{\mathbb{A}}_{\rm
aff}}=\bigoplus_{w\in{\widetilde{W}_{\rm aff}}}S\cdot{\widetilde{A}}_{w}$.

######  Proposition 4.12.

With the above definitions,

  1. (1)

Moding out by ${\widetilde{J}}$, the morphism $\varphi_{1}$ induces an
$S$-module isomorphism ${\widetilde{M}}_{1}\to{\widetilde{\mathbb{A}}_{\rm
aff}}/{\widetilde{J}}$.

  2. (2)

Moding out by ${\widetilde{J}}$, the morphism $\varphi_{2}$ induces an
${\widetilde{\mathbb{A}}_{\rm aff}}$-module isomorphism
${\widetilde{M}}_{2}\to{\widetilde{\mathbb{A}}_{\rm aff}}/{\widetilde{J}}$
(which is _not_ $S$-linear if we give ${\widetilde{M}}_{2}$ the tensor product
$S$-module structure).

  3. (3)

The left ${\widetilde{\mathbb{A}}_{\rm aff}}$-modules
${\widetilde{\mathbb{A}}_{\rm aff}}/{\widetilde{J}}$ and ${\widetilde{M}}_{3}$
are isomorphic.

###### Proof.

This proposition follows easily from Propositions 4.3 and 4.5. $\square$

######  Remark 4.13.

The ${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure induced on
${\widetilde{M}}_{1}$ by the isomorphism in Proposition 4.12.(1) can also be
described via Proposition 4.4.

######  Definition 4.14.

The ${\widetilde{\mathbb{A}}_{\rm aff}}$-module defined by one of the above
equivalent definitions will be denoted by ${\widetilde{M}}$.

##  5\. Homology of the adjoint affine Grassmannian ${\Omega K^{\rm ad}}$

In this section, we recall the adjoint affine Grassmannian ${\Omega K^{\rm
ad}}$, we prove that the ${\widetilde{\mathbb{A}}_{\rm aff}}$-module
${\widetilde{M}}$ is isomorphic to the homology of ${\Omega K^{\rm ad}}$, we
define a ring structure on this module and study the compatibility of these
two structures.

###  5.1. Cohomology of the finite-dimensional flag manifold $G/B$

Recall, see for example [Kum02, Chapter 11], that $H^{*}_{T}(G/B)$ has an
$S$-basis $(\xi^{w})_{w\in W}=(\sigma^{B}(w))_{w\in W}$ indexed by the Weyl
group. The pull-back along the map $(G/B)^{T}\to G/B$ induces an inclusion

$H^{*}_{T}(G/B)\to H_{T}^{*}((G/B)^{T})=S^{W}.$

Viewing $\xi^{w}$ as a function on $W$, Kumar [Kum02, 11.1.6.(3)] sets
$d_{u,v}=\xi^{u}(v)=\langle\xi^{u},v\rangle$ and $D=(d_{u,v})_{u,v\in W}$. If
$(f^{w})_{w\in W}$ is the basis of $S^{W}$ given by $\langle
f^{u},v\rangle=f^{u}(v)=\delta_{u,v}$, then we have
$(\xi^{u})_{u}=D(f^{u})_{u}$. Given the identification [Kum02, 11.1.4(2)], we
also have $\langle f^{u},\delta_{v}\rangle=\delta_{u,v}$.

The dual of $H^{*}_{T}(G/B)$ is $H_{*}^{T}(G/B)$ and identifies as an
$S$-module with the $S$-subalgebra ${\mathbb{A}}$ of ${\mathbb{A}_{\rm aff}}$
generated by $(A_{w})_{w\in W}$:

(4) $H_{*}^{T}(G/B)\simeq\bigoplus_{w\in W}S\cdot A_{w}$

Note that $(A_{w})_{w\in W}$ is the dual basis to $(\xi^{w})_{w\in W}$ _i.e._
$\langle\xi^{u},A_{v}\rangle=\xi^{u}(A_{v})=\delta_{u,v}$ (see [Kum02,
11.1.5], were $A_{u}$ is denoted by $x_{u}$). Over $F={\rm Frac}(S)$ we also
have the basis $(\delta_{w})_{w\in W}$ for $H_{*}^{T}(G/B)$. Kumar, in [Kum02,
11.1.2.(e)], describes the base change:

$A_{u}=\sum_{v}c_{u,v}\delta_{v}$

with $C=(c_{u,v})_{u,v\in W}$ a matrix with coefficients in $S$, in
particular, we have $(A_{v})_{v}=C(\delta_{v})_{v}$. We have the following
relation between the matrices $C$ and $D$.

######  Fact 5.1 (See [Kum02, 11.1.7.(a)]).

We have $D^{-1}=C^{T}$. Thus,

$\delta_{v}=\sum_{w\leq v}\xi^{w}(v)A_{w}\,.$

###### Proof.

In fact, from the identity $(A_{v})_{v}=C(\delta_{v})_{v}$, we deduce that
$(\delta_{v})_{v}=C^{-1}(A_{v})_{v}=D^{T}(A_{v})$. Since
$D^{T}_{v,w}=\xi^{w}(v)$ and the matrix $D$ is triangular, we get the result.
$\square$

Note also that an explicit formula for the coefficients $\xi^{w}(v)$ is known:
see [Kum02, Proposition 11.1.11].

###  5.2. Affine Grassmannian and the Pontryagin ring structure

Let $G$ be the simply-connected almost simple group associated to
$\mathfrak{g}$, and let ${G^{\rm ad}}$ be the adjoint quotient of this group.
Let $K$ resp. ${K^{\rm ad}}$ be a maximal compact subgroup in $G$ resp.
${G^{\rm ad}}$. Let ${\Omega K}$ resp. ${\Omega K^{\rm ad}}$ be the group of
loops $l$ with values in $K$ resp. ${K^{\rm ad}}$ such that $l(0)$ is the unit
element in $K$ resp. ${K^{\rm ad}}$. By a loop we mean a map
$l:\mathbb{S}^{1}\to K^{(\rm ad)}$ that extends to a meromorphic map
$\mathbb{D}^{\circ}\to G^{(\rm ad)}$, where $\mathbb{D}^{\circ}$ denotes the
pointed disk. Moding out a loop by the center of $K$ yields an inclusion
${\Omega K}\subset{\Omega K^{\rm ad}}$. The action of $T\cap K$ on ${\Omega
K}$ resp. ${\Omega K^{\rm ad}}$ is given by conjugation.

This implies that the equivariant homology of ${\Omega K}$ and ${\Omega K^{\rm
ad}}$ have a natural structure of an algebra, given by the Pontryagin product
which is also $(T\cap K)$-equivariantly homotopy equivalent to the point-wise
product of loops. In this section, we will recall an algebraic model for
$H_{*}^{T\cap K}({\Omega K})$ and give one for $H_{*}^{T\cap K}({\Omega K^{\rm
ad}})$. In particular we will describe the ring structure as well as an
${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure on $H_{*}^{T\cap
K}({\Omega K^{\rm ad}})$ extending the ring structure and the
${\mathbb{A}_{\rm aff}}$-module structure on $H_{*}^{T\cap K}({\Omega K})$.

###  5.3. Geometry of fixed points in ${\Omega K^{\rm ad}}$

Since $K\to K^{\rm ad}\ \\!$ is the universal cover of $K^{\rm ad}\ \\!$, the
connected components of ${\Omega K^{\rm ad}}$ are isomorphic to ${\Omega K}$
and are indexed by $\pi_{1}(G^{\rm ad}\ \\!)=\pi_{1}(K^{\rm ad}\ \\!)=Z$. We
now describe the $T$-fixed points in ${\Omega K^{\rm ad}}$. We have, in the
loop space picture

$({\Omega K^{\rm
ad}})^{T}=\\{{\widetilde{\psi}}_{t_{\lambda^{\vee}}}:\mathbb{S}^{1}\to K^{\rm
ad}\ \\!\ |\ {\lambda^{\vee}}\in P^{\vee}\\},$

where ${\widetilde{\psi}}_{t_{\lambda^{\vee}}}(t)=\exp(2i\pi
t{\lambda^{\vee}})$ is the loop induced by the one-parameter subgroup
${\lambda^{\vee}}$ of $T^{\rm ad}\ \\!$ (the maximal torus of $K^{\rm ad}\
\\!$). For ${\lambda^{\vee}}\in P^{\vee}$, let $[{\lambda^{\vee}}]$ be its
class in $P^{\vee}/Q^{\vee}=\pi_{1}(K^{\rm ad}\ \\!)$ and denote by ${\Omega
K}_{[{\lambda^{\vee}}]}^{\rm ad}\ \\!$ be the connected component of ${\Omega
K^{\rm ad}}$ containing ${\widetilde{\psi}}_{t_{\lambda^{\vee}}}$. We have

${\Omega K^{\rm ad}}=\coprod_{[{\lambda^{\vee}}]\in P^{\vee}/Q^{\vee}}{\Omega
K}_{[{\lambda^{\vee}}]}^{\rm ad}\ \\!.$

Let $m_{\lambda^{\vee}}:{\Omega K}\to{\Omega K}_{[{\lambda^{\vee}}]}^{\rm ad}\
\\!$ be the left multiplication by ${\widetilde{\psi}}_{t_{\lambda^{\vee}}}$.
Since $T$ and ${\widetilde{\psi}}_{t_{\lambda^{\vee}}}$ commute, this is a
$T$-equivariant isomorphism. Thus, $H_{*}^{T}({\Omega
K}_{[{\lambda^{\vee}}]}^{\rm ad}\ \\!)\simeq H_{*}^{T}({\Omega K})$.

###  5.4. Reminder on $H_{*}^{T}({\Omega K})$

Recall from [KK86] that ${\Omega K}$ has a cellular decomposition whose cells
are indexed by ${W_{{\rm aff}}^{-}}$. This implies that, as $S$-module, we
have

$H_{*}^{T}({\Omega K})=\bigoplus_{w\in{W_{{\rm aff}}^{-}}}S\cdot\xi_{w}\simeq
M.$

Furthermore, according to [Lam08, (3.1) and (3.2)], ${\mathbb{A}_{\rm aff}}$
acts on $H_{*}^{T}(\mathcal{G}/\mathcal{P})$ by

$A_{v}\cdot\xi_{w}=\left\\{\begin{array}[]{l}\xi_{vw}\mbox{ if
}\ell(vw)=\ell(v)+\ell(w)\mbox{ and }vw\in{W_{{\rm aff}}^{-}}\\\ 0\mbox{
otherwise}\end{array}\right.$

and ${\mathbb{A}_{\rm aff}}$ acts on $H^{*}_{T}(\mathcal{G}/\mathcal{P})$ by

$A_{v}\cdot\xi^{w}=\left\\{\begin{array}[]{l}\xi^{vw}\mbox{ if
}\ell(vw)=\ell(w)-\ell(w)\mbox{ and }vw\in{W_{{\rm aff}}^{-}}\\\ 0\mbox{
otherwise}\end{array}\right.$

###  5.5. $S$-algebra structure on $H_{*}^{T}({\Omega K^{\rm ad}})$

We use the $T\cap K$-equivariant homology of the $T\cap K$-space ${\Omega
K^{\rm ad}}$, where $T\cap K$ acts on ${\Omega K^{\rm ad}}$ via $T\cap
K\to(T\cap K)^{\rm ad}\to{G^{\rm ad}}$. The inclusion $T\cap K\to T$ induces
an isomorphism in equivariant cohomology $H_{T}^{*}(pt)\to H_{T\cap
K}^{*}(pt)$. Note that we have $H_{T\cap K}({\Omega K})\simeq H_{T\cap
K}(\mathcal{G}/\mathcal{P})\simeq H_{T}(\mathcal{G}/\mathcal{P})$, where
$\mathcal{G}/\mathcal{P}$ is the affine Grassmannian. Abusing notations
slightly, we will denote in the following $H_{T\cap K}({\Omega K})$ simply by
$H_{T}({\Omega K})$, and similarly for $H_{T}({\Omega K^{\rm ad}})$. The
$T$-equivariant cohomology of the point is the symmetric algebra on $P$,
namely $S$, see [Br98, p.5], so that the homology $H_{*}^{T}({\Omega K^{\rm
ad}})$ will be an $S$-module and even an $S$-algebra. We are not considering
$T^{\rm ad}$-equivariant homology.

######  Proposition 5.2.

As $S$-algebras, we have: $H_{*}^{T}({\Omega K^{\rm ad}})\simeq
S[P^{\vee}]\otimes_{S[{Q^{\vee}}]}H_{*}^{T}({\Omega K}).$

###### Proof.

We have the following inclusions that are compatible with pointwise
multiplication and $T$-equivariant inducing $S$-algebra morphisms

$\textstyle{{\Omega
K}^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega
K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\Omega
K^{\rm
ad}})^{T}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\Omega
K^{\rm ad}},}$ $\textstyle{H_{*}^{T}({\Omega
K}^{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}^{T}({\Omega
K})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}^{T}(({\Omega
K^{\rm
ad}})^{T})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{H_{*}^{T}({\Omega
K^{\rm ad}}).}$

Recall that we have bijections $({\Omega K^{\rm ad}})^{T}\simeq P^{\vee}$ and
${\Omega K}^{T}\simeq Q^{\vee}$ that are group homomorphisms since
${\widetilde{\psi}}_{t_{\lambda^{\vee}}}{\widetilde{\psi}}_{t_{\mu^{\vee}}}={\widetilde{\psi}}_{t_{{\lambda^{\vee}}+{\mu^{\vee}}}}$.
We thus have $H_{*}^{T}(({\Omega K^{\rm ad}})^{T})\simeq S[P^{\vee}]$ and
$H_{*}^{T}({\Omega K}^{T})\simeq S[Q^{\vee}]$. In particular, the above
diagram induces an $S$-algebra morphism
$S[P^{\vee}]\otimes_{S[Q^{\vee}]}H_{*}^{T}({\Omega K})\to H_{*}^{T}({\Omega
K^{\rm ad}})$. The restriction of this map to
${\widetilde{\psi}}_{t_{\lambda^{\vee}}}\otimes H_{*}^{T}({\Omega K}^{T})\to
H_{*}^{T}(({\Omega K^{\rm ad}})^{T})$ is the multiplication
$m_{\lambda^{\vee}}$. The above decomposition of ${\Omega K^{\rm ad}}$ in
connected components gives an isomorphism of $S$-modules

$H_{*}^{T}({\Omega K^{\rm ad}})=\bigoplus_{[{\lambda^{\vee}}]\in
P^{\vee}/Q^{\vee}}H_{*}^{T}({\Omega K}_{[{\lambda^{\vee}}]}^{\rm ad}\ \\!)$

proving that the map $S[P^{\vee}]\otimes_{S[Q^{\vee}]}H_{*}^{T}({\Omega K})\to
H_{*}^{T}({\Omega K^{\rm ad}})$ is surjective.

To prove injectivity, first note that, since $H_{*}^{T}({\Omega K})$ is a free
$S$-module and $S[P^{\vee}]$ is a free $S[Q^{\vee}]$-module, the $S$-module
$S[P^{\vee}]\otimes_{S[Q^{\vee}]}H_{*}^{T}({\Omega K})$ is free. We therefore
only need to prove the injectivity of the map after base extension to $F={{\rm
Frac}}(S)$ the field of fractions of $S$. Now recall the following general
result (see [Kum02, C.8 Theorem]: on the level of $T$-equivariant cohomology
we have isomorphisms $H^{*}_{T}({\Omega K})\otimes_{S}F\simeq
H^{*}_{T}({\Omega K}^{T})\otimes_{S}F$ and $H^{*}_{T}({\Omega K^{\rm
ad}})\otimes_{S}F\simeq H^{*}_{T}(({\Omega K^{\rm ad}})^{T})\otimes_{S}F$.
This induces isomorphisms in $T$-equivariant homology:

$H_{*}^{T}({\Omega K}^{T})\otimes_{S}F\simeq H_{*}^{T}({\Omega
K})\otimes_{S}F\textrm{ and }H_{*}^{T}(({\Omega K^{\rm
ad}})^{T})\otimes_{S}F\simeq H_{*}^{T}({\Omega K^{\rm ad}})\otimes_{S}F.$

After base change to $F$, since $H_{*}^{T}({\Omega K}^{T})\simeq F[Q^{\vee}]$
and $H_{*}^{T}(({\Omega K^{\rm ad}})^{T})\otimes_{S}F\simeq F[P^{\vee}]$, our
map is given by

$F[P^{\vee}]\otimes_{F[Q^{\vee}]}F[Q^{\vee}]\to F[P^{\vee}]$

and is therefore injective. $\square$

Recall that, as $S$-module, we have an isomorphism $H_{*}^{T}({\Omega K})=M$.
In particular the above results identifies $H_{*}^{T}({\Omega K^{\rm ad}})$
with the $S$-module ${\widetilde{M}}$ of Definition 4.14:

######  Corollary 5.3.

As $S$-modules, we have: $H_{*}^{T}({\Omega K^{\rm
ad}})\simeq{\widetilde{M}}$.

######  Corollary 5.4.

The exists an ${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure on
$H_{*}^{T}({\Omega K^{\rm ad}})$ compatible with the ${\mathbb{A}_{\rm
aff}}$-module structure on $H_{*}^{T}({\Omega K})$. Furthermore, for this
structure, we have an isomorphism of ${\widetilde{\mathbb{A}}_{\rm
aff}}$-modules $H_{*}^{T}({\Omega K^{\rm ad}})\simeq{\widetilde{M}}.$

###### Proof.

We define the ${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure on
$H_{*}^{T}({\Omega K^{\rm ad}})$. Since we have the isomorphism of $S$-modules
$H_{*}^{T}({\Omega K^{\rm
ad}})\simeq{\widetilde{M}}_{1}=S[P^{\vee}]\otimes_{S[Q^{\vee}]}M$, we may
extend the ${\mathbb{A}_{\rm aff}}$-module structure on $M$ to the
${\widetilde{\mathbb{A}}_{\rm aff}}-$module structure ${\widetilde{M}}$.
$\square$

######  Remark 5.5.

The above ${\widetilde{\mathbb{A}}_{\rm aff}}$-module structure on
$H_{*}^{T}({\Omega K^{\rm ad}})$ also has a geometric description, see [CMP09,
Proposition 3.3].

######  Remark 5.6.

The above result shows that our claim on [CMP09, Page 12] that
${H_{*}^{T}}({\Omega K^{\rm ad}})$ is the tensor product _ring_
${\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{H_{*}^{T}}({\Omega K})$ is wrong: by
localization ${H_{*}^{T}}({\Omega K^{\rm ad}})$ is a subring of
$F[{P^{\vee}}]$ and this Laurent polynomial algebra contains no roots of
unity, whereas ${\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{H_{*}^{T}}({\Omega K})$
does.

Recall that ${\widetilde{W}_{\rm aff}}$ can be embedded in
${\widetilde{\mathbb{A}}_{\rm aff}}$ via $w\mapsto\delta_{w}$. The induced
action is denoted by
$x\cdot{\widetilde{\xi}}:=\delta_{x}\cdot{\widetilde{\xi}}$ for
$x\in{\widetilde{W}_{\rm aff}}$ and ${\widetilde{\xi}}\in H_{*}^{T}({\Omega
K^{\rm ad}})$.

######  Corollary 5.7.

Let $w\in W$ and ${\lambda^{\vee}},{\mu^{\vee}}\in P^{\vee}$, we have

$wt_{\lambda^{\vee}}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}:=\delta_{wt_{\lambda^{\vee}}}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}={\widetilde{\psi}}_{t_{w({\lambda^{\vee}}+{\mu^{\vee}})}}.$

###### Proof.

As already explained in the proof of Proposition 5.2, we have
$\delta_{t_{\lambda^{\vee}}}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}={\widetilde{\psi}}_{t_{{\lambda^{\vee}}+{\mu^{\vee}}}}$.
We need to check that
$\delta_{w}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}={\widetilde{\psi}}_{t_{w({\mu^{\vee}})}}$.
But our identification of $H_{*}^{T}({\Omega K^{\rm ad}})$ with
${\widetilde{M}}$ identifies ${\widetilde{\psi}}_{t_{\mu^{\vee}}}$ with
$h_{{\mu^{\vee}}}\otimes 1$. Recall that
$1=[\delta]\in{\widetilde{\mathbb{A}}_{\rm aff}}/{\widetilde{J}}$, so that
$h_{\mu^{\vee}}\otimes 1=[\delta_{t_{\mu^{\vee}}}]$ and
$\delta_{w}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}=\delta_{w}\cdot[\delta_{t_{\mu^{\vee}}}]=[\delta_{w}\delta_{t_{\mu^{\vee}}}]=[\delta_{w}\delta_{t_{\mu^{\vee}}}\delta_{w^{-}1}]$
since $\delta_{w^{-1}}\in{\widetilde{J}}$. We get

$\delta_{w}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}}=[\delta_{w}\delta_{t_{\mu^{\vee}}}\delta_{w^{-}1}]=[\delta_{t_{w}({\mu^{\vee}})}]={\widetilde{\psi}}_{t_{w}({\mu^{\vee}})}$

proving the result. $\square$

###  5.6. Compatibility between the ring and the ${\widetilde{\mathbb{A}}_{\rm
aff}}$-module structure

The above description of ${H_{*}^{T}}({\Omega K^{\rm ad}})$ as ring and as
${\widetilde{\mathbb{A}}_{\rm aff}}$-module is not enough for our purposes: we
need to be able to multiply two classes of the form $\sigma\otimes\xi_{x}$ and
$\tau\otimes\xi_{y}$, see also Remark 5.6. To this end, we recall the
definition and properties of ${j^{\rm ad}}$ given in [CMP09, §3.3].

######  Proposition 5.8.

There is an $S$-algebra isomorphism ${j^{\rm ad}}:{H_{*}^{T}}({\Omega K^{\rm
ad}})\to Z_{{\widetilde{\mathbb{A}}_{\rm aff}}}(S)$. It satisfies:

  1. (1)

${j^{\rm ad}}(\xi)\cdot\xi^{\prime}=\xi\xi^{\prime}$ for
$\xi,\xi^{\prime}\in{H_{*}^{T}}({\Omega K^{\rm ad}})$ ;

  2. (2)

${j^{\rm
ad}}({\widetilde{\psi}}_{t_{\lambda^{\vee}}})=\delta_{t_{\lambda^{\vee}}}$ for
${\lambda^{\vee}}\in{P^{\vee}}$.

For $w\in{{{\widetilde{W}_{\rm aff}}^{-}}}$, ${j^{\rm
ad}}({\widetilde{\xi}}_{w})$ is characterized by the two following properties:

  1. (a)

${j^{\rm ad}}({\widetilde{\xi}}_{w})$ is congruent to $A_{w}$ modulo
$\sum_{x\in W\setminus\\{e\\}}{\widetilde{\mathbb{A}}_{\rm aff}}\cdot A_{x}$ ;

  2. (b)

${j^{\rm ad}}({\widetilde{\xi}}_{w})$ belongs to
$Z_{{\widetilde{\mathbb{A}}_{\rm aff}}}(S)$.

The map ${j^{\rm ad}}$ has the following equivariance property:

######  Proposition 5.9.

Let $u\in
W,{\lambda^{\vee}}\in{P^{\vee}},{\widetilde{\xi}}\in{H_{*}^{T}}({\Omega K^{\rm
ad}})$. Then

  1. (1)

${j^{\rm
ad}}(ut_{\lambda^{\vee}}\cdot{\widetilde{\xi}})=\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}{j^{\rm
ad}}({\widetilde{\xi}})\delta_{u^{-1}}=\delta_{ut_{\lambda^{\vee}}}{j^{\rm
ad}}({\widetilde{\xi}})\delta_{u^{-1}}$;

  2. (2)

$\delta_{t_{\lambda^{\vee}}}{j^{\rm ad}}({\widetilde{\xi}})={j^{\rm
ad}}({\widetilde{\xi}})\delta_{t_{{\lambda^{\vee}}}}.$

###### Proof.

(1) Let $s\in S$ be a scalar, we have:

  * •

${j^{\rm ad}}(ut_{\lambda^{\vee}}\cdot s{\widetilde{\xi}})={j^{\rm
ad}}(u(s)ut_{\lambda^{\vee}}\cdot{\widetilde{\xi}})=u(s){j^{\rm
ad}}(ut_{\lambda^{\vee}}\cdot{\widetilde{\xi}})$ ;

  * •

$\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}{j^{\rm
ad}}(s{\widetilde{\xi}})\delta_{u^{-1}}=\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}s{j^{\rm
ad}}({\widetilde{\xi}})\delta_{u^{-1}}=u(s)\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}{j^{\rm
ad}}({\widetilde{\xi}})\delta_{u^{-1}}$.

Thus, by semi-linearity, it is enough to prove the result for
${\widetilde{\xi}}={\widetilde{\psi}}_{t_{\mu^{\vee}}}$. For
${\widetilde{\xi}}={\widetilde{\psi}}_{t_{\mu^{\vee}}}$, we have ${j^{\rm
ad}}(ut_{\lambda^{\vee}}\cdot{\widetilde{\psi}}_{t_{\mu^{\vee}}})={j^{\rm
ad}}(u\cdot{\widetilde{\psi}}_{t_{{\lambda^{\vee}}+{\mu^{\vee}}}})={j^{\rm
ad}}({\widetilde{\psi}}_{t_{u({\lambda^{\vee}})+u({\mu^{\vee}})}})=\delta_{t_{u({\lambda^{\vee}})+u({\mu^{\vee}})}}.$
We also have $\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}{j^{\rm
ad}}({\widetilde{\psi}}_{t_{\mu^{\vee}}})\delta_{u^{-1}}=\delta_{t_{u({\lambda^{\vee}})}}\delta_{u}\delta_{t_{\mu^{\vee}}}\delta_{u^{-1}}=\delta_{t_{u({\lambda^{\vee}})+u({\mu^{\vee}})}}.$
Thus the result is proved.

(2) Both terms are $S$-linear so we only need to check this for
${\widetilde{\xi}}={\widetilde{\psi}}_{t_{\mu^{\vee}}}$ but we have
$\delta_{t_{\lambda^{\vee}}}{j^{\rm
ad}}({\widetilde{\psi}}_{t_{\mu^{\vee}}})=\delta_{t_{\lambda^{\vee}}}\delta_{t_{\mu^{\vee}}}=\delta_{t_{\mu^{\vee}}}\delta_{t_{\lambda^{\vee}}}={j^{\rm
ad}}({\widetilde{\psi}}_{t_{\mu^{\vee}}})\delta_{t_{{\lambda^{\vee}}}}$.
$\square$

In particular, the previous Proposition allows computing ${j^{\rm ad}}$ in
terms of $j$:

######  Example 5.10.

Let $\tau_{i}=v_{i}t_{-\varpi_{i}}\in Z$ and let $\xi\in{H_{*}^{T}}({\Omega
K})$. Then

${j^{\rm ad}}(\tau_{i}\cdot\xi)=\delta_{\tau_{i}}\ j(\xi)\
\delta_{v_{i}^{-1}}\,.$

We deduce a formula allowing reducing products in the homology of ${\Omega
K^{\rm ad}}$ to products in the homology of ${\Omega K}$:

######  Corollary 5.11.

Let $\sigma=ut_{\lambda^{\vee}},\tau=vt_{\mu^{\vee}}$ be elements in $Z$. Let
${\widetilde{\xi}},{\widetilde{\xi}}^{\prime}\in{H_{*}^{T}}({\Omega K^{\rm
ad}})$. Then

$(\sigma\cdot{\widetilde{\xi}})\times(\tau\cdot{\widetilde{\xi}}^{\prime})=\sigma\tau\cdot(\psi_{\sigma,\tau}\times(v^{-1}_{*}{\widetilde{\xi}})\times(u^{-1}_{*}{\widetilde{\xi}}^{\prime}))\,,$

where
$\psi_{\sigma,\tau}=\psi_{u^{-1}({\mu^{\vee}})-{\mu^{\vee}}}=\psi_{v^{-1}({\lambda^{\vee}})-{\lambda^{\vee}}}$.

###### Proof.

Since $Z$ is abelian, $\sigma\tau=\tau\sigma$. We have
$\sigma\tau=ut_{\lambda^{\vee}}vt_{\mu^{\vee}}=uvt_{v^{-1}({\lambda^{\vee}})+{\mu^{\vee}}}$
and
$\tau\sigma=vt_{\mu^{\vee}}ut_{\lambda^{\vee}}=vut_{u^{-1}({\mu^{\vee}})+{\lambda^{\vee}}}$.
We get
$v^{-1}({\lambda^{\vee}})+{\lambda^{\vee}}=u^{-1}({\mu^{\vee}})+{\mu^{\vee}}$
so
$u^{-1}({\mu^{\vee}})-{\mu^{\vee}}=v^{-1}({\lambda^{\vee}})-{\lambda^{\vee}}$,
so that $\psi_{\sigma,\tau}$ is well defined. We also get $uv=vu$.

Using Proposition 5.9, we compute:

$\begin{array}[]{rcl}(\sigma\cdot{\widetilde{\xi}})\times(\tau\cdot{\widetilde{\xi}}^{\prime})&=&{j^{\rm
ad}}(\sigma\cdot{\widetilde{\xi}})\cdot(\tau\cdot{\widetilde{\xi}}^{\prime})\\\
&=&\delta_{u}\delta_{t_{\lambda^{\vee}}}{j^{\rm
ad}}({\widetilde{\xi}})\delta_{u^{-1}}\delta_{v}\delta_{t_{\mu^{\vee}}}\cdot{\widetilde{\xi}}^{\prime}\\\
&=&\delta_{u}\delta_{t_{\lambda^{\vee}}}{j^{\rm
ad}}({\widetilde{\xi}})\delta_{v}\delta_{u^{-1}}\delta_{t_{\mu^{\vee}}}\cdot{\widetilde{\xi}}^{\prime}\\\
&=&\delta_{u}\delta_{t_{\lambda^{\vee}}}\delta_{v}{j^{\rm
ad}}(v^{-1}\cdot{\widetilde{\xi}})\delta_{t_{u^{-1}({\mu^{\vee}})}}\delta_{u^{-1}}\cdot{\widetilde{\xi}}^{\prime}\\\
&=&\delta_{u}\delta_{t_{\lambda^{\vee}}}\delta_{v}\delta_{t_{u^{-1}({\mu^{\vee}})}}{j^{\rm
ad}}(v^{-1}\cdot{\widetilde{\xi}})\delta_{u^{-1}}\cdot{\widetilde{\xi}}^{\prime}\\\
&=&\delta_{u}\delta_{t_{\lambda^{\vee}}}\delta_{v}\delta_{t_{\mu^{\vee}}}\delta_{t_{u^{-1}({\mu^{\vee}})-{\mu^{\vee}}}}{j^{\rm
ad}}(v^{-1}\cdot{\widetilde{\xi}})\delta_{u^{-1}}\cdot{\widetilde{\xi}}^{\prime}\\\
&=&\sigma\tau\cdot(\psi_{\sigma,\tau}\times(v^{-1}_{*}{\widetilde{\xi}})\times(u^{-1}_{*}{\widetilde{\xi}}^{\prime}))\,.\end{array}$

$\square$

######  Remark 5.12.

In [CMP09, p.12], it is claimed that ${H_{*}^{T}}({\Omega K^{\rm ad}})$ is the
tensor product ring ${\mathbb{Z}}[Z]\otimes_{\mathbb{Z}}{H_{*}^{T}}({\Omega
K})$. As explained in Remark 5.6, this is not true. However as the next
corollary shows, this is true in the non equivariant homology.

######  Corollary 5.13.

In non equivariant homology, let
${\widetilde{\xi}},{\widetilde{\xi}}^{\prime}\in H_{*}({\Omega K^{\rm ad}})$
and $\sigma,\tau\in Z$, then

$(\sigma\cdot{\widetilde{\xi}})\times(\tau\cdot{\widetilde{\xi}}^{\prime})=\sigma\tau\cdot({\widetilde{\xi}}\times{\widetilde{\xi}}^{\prime})\,.$

###### Proof.

Indeed, push-forwards $u^{-1}_{*}$ and $v^{-1}_{*}$ are trivial in non
equivariant homology. Moreover, the equivariant classes
${\widetilde{\psi}}_{\lambda^{\vee}}$ restrict to the class of a point in
$H_{*}({\Omega K^{\rm ad}})$, which is the unit in $H_{*}({\Omega K^{\rm
ad}})$. $\square$

###  5.7. Translations modulo $P$

We use [LS10, Lemma 10.1] and [CMP09, Corollary 3.15] as a definition:

(5) ${(W^{P})_{\rm aff}}=\left\\{ut_{\nu^{\vee}}|\,\forall\gamma\in
R_{P}^{+},\langle{\nu^{\vee}},\gamma\rangle=\left\\{\begin{array}[]{cl}0&\mbox{if
}u(\gamma)>0\\\ -1&\mbox{if }u(\gamma)<0\end{array}\right.\right\\}\,$ (6)
${(\widetilde{W}^{P})_{\rm aff}}=\left\\{ut_{\nu^{\vee}}\in{\widetilde{W}_{\rm
aff}}\ \Big{|}\ \forall\gamma\in
R_{P}^{+},\langle{\nu^{\vee}},\gamma\rangle=\left\\{\begin{array}[]{cl}0&\mbox{if
}u(\gamma)>0\\\ -1&\mbox{if }u(\gamma)<0\end{array}\right.\right\\}\,.$

Following [LS10, §10.2 and 10.3], we also define $(W_{P})_{\rm
aff}=\\{wt_{\lambda^{\vee}}\,|\,w\in
W_{P},{\lambda^{\vee}}\in{Q^{\vee}_{P}}\\}$. Recall, from [CMP09, Section 3.4]
that any element $w\in{\widetilde{W}_{\rm aff}}$ can be uniquely factorized as
$w_{1}w_{2}$ with $w_{1}\in{(\widetilde{W}^{P})_{\rm aff}}$ and
$w_{2}\in(W_{P})_{\rm aff}$ and $\ell(w)=\ell(w_{1})+\ell(w_{2})$. We denote
$w_{1}=\pi_{P}(w)$. Thus ${(\widetilde{W}^{P})_{\rm aff}}$ is a set of
representatives for the quotient ${\widetilde{W}_{\rm aff}}/(W_{P})_{\rm aff}$
which will be relevant for Peterson’s isomorphism (7).

Following [LS10, Section 10.4] and [CMP09, Section 3.4], define the ideals
$J_{P}\subset M$ and ${\widetilde{J}}_{P}\subset{\widetilde{M}}$ as follows:

$J_{P}=\sum_{x\in{W_{{\rm aff}}^{-}}\setminus{(W^{P})_{\rm
aff}}}S\xi_{x}\textrm{ and
}{\widetilde{J}}_{P}=\sum_{x\in{{{\widetilde{W}_{\rm
aff}}^{-}}}\setminus{(\widetilde{W}^{P})_{\rm aff}}}S{\widetilde{\xi}}_{x}.$

The following result corrects [CMP09, Proposition 3.16] which used the wrong
product structure, see Remark 5.6.

######  Proposition 5.14.

Let $x\in{{{\widetilde{W}_{\rm aff}}^{-}}}\cap{(\widetilde{W}^{P})_{\rm aff}}$
and let ${\nu^{\vee}}\in P^{\vee}_{-}$. Then
$x\pi_{P}(t_{\nu^{\vee}})\in{{{\widetilde{W}_{\rm
aff}}^{-}}}\cap{(\widetilde{W}^{P})_{\rm aff}}$. Let us write as usual
$x=\sigma{\widehat{x}}$ and
$\pi_{P}(t_{\nu^{\vee}})=\tau\widehat{\pi_{P}(t_{\nu^{\vee}})}$ with
$\sigma=ut_{\lambda^{\vee}},\tau=vt_{\mu^{\vee}}$. Then

$(v^{-1}_{*}\xi_{\widehat{x}})\times(u^{-1}_{*}\xi_{\widehat{\pi_{P}(t_{\nu^{\vee}})}})=\psi_{\sigma,\tau}^{-1}\xi_{\widehat{x\pi_{P}(t_{\nu^{\vee}})}}\textrm{
modulo }J_{P}.$

###### Proof.

The proof follows the arguments in [CMP09]. In particular, we get

$(\sigma\cdot\xi_{\widehat{x}})\times(\tau\cdot\xi_{\widehat{\pi_{P}(t_{\nu^{\vee}})}})=\sigma\tau\cdot\xi_{\widehat{x\pi_{P}(t_{\nu^{\vee}})}}\textrm{
modulo }{\widetilde{J}}_{P}.$

Using the correct product formula given in Corollary 5.11, the left hand side
is

$\sigma\tau\cdot(\psi_{\sigma,\tau}(v^{-1}_{*}\xi_{\widehat{x}})\times(u^{-1}_{*}\xi_{\widehat{\pi_{P}(t_{\nu^{\vee}})}}))\,.$

This proves the result since ${\widetilde{J}}_{P}\cap M=J_{P}$ (as
$\sigma{\widehat{x}}\in{{{\widetilde{W}_{\rm
aff}}^{-}}}\Leftrightarrow{\widehat{x}}\in{W_{{\rm aff}}^{-}}$). $\square$

In particular, the case $P=B$ yields:

######  Corollary 5.15.

Let $x\in{{{\widetilde{W}_{\rm aff}}^{-}}}$ and let ${\nu^{\vee}}\in
P^{\vee}_{-}$. Then $xt_{\nu^{\vee}}\in{{{\widetilde{W}_{\rm aff}}^{-}}}$. Let
us write as usual $x=\sigma{\widehat{x}}$ and
$t_{\nu^{\vee}}=\tau\widehat{t_{\nu^{\vee}}}$ with
$\sigma=ut_{\lambda^{\vee}},\tau=vt_{\mu^{\vee}}$. Then

$(v^{-1}_{*}\xi_{\widehat{x}})\times(u^{-1}_{*}\xi_{\widehat{t_{\nu^{\vee}}}})=\psi_{\sigma,\tau}^{-1}\xi_{\widehat{xt_{\nu^{\vee}}}}\,.$

##  6\. Affine symmetries

In this section, we correct [CMP09, Section 3.5], see Remark 5.6 using the
correct product formula given in Corollary 5.11 and Proposition 5.14. In
particular we prove that the formulas given in [CMP09] are correct in the non
equivariant setting.

###  6.1. Peterson’s isomorphism

Proposition 5.14 is our needed result in the equivariant homology of the
affine Grassmannian. Translating this formula in the quantum cohomology of
$G/P$, we prove our main theorem. We use Peterson’s isomorphism proved in
[LS10] to relate ${H_{*}^{T}}({\Omega K})$ and ${QH_{T}^{*}}(G/P)$.

Let $\eta_{P}:{Q^{\vee}}\to{Q^{\vee}_{P}}$ be the projection on the coroot
subspace generated by simple roots $\alpha_{i}$ with $\alpha_{i}\not\in
R_{P}$. Peterson’s isomorphism is the map

(7) $\begin{array}[t]{rrcll}\psi_{P}&:&{H_{*}^{T}}({\Omega
K})_{P}&\rightarrow&{\mathbb{Z}}[{Q^{\vee}_{P}}]\otimes_{\mathbb{Z}}H_{T}^{*}(G/P)\\\
&&\xi_{w\pi_{P}(t_{\lambda^{\vee}})}\xi_{\pi_{P}(t_{\mu^{\vee}})}^{-1}&\mapsto&q_{\eta_{P}({\lambda^{\vee}}-{\mu^{\vee}})}\sigma^{P}(w)\end{array}$

where $w\in W^{P}$ and ${\lambda^{\vee}},{\mu^{\vee}}\in{Q^{\vee}_{-}}$ with
${Q^{\vee}_{-}}$ the set of antidominant elements in $Q^{\vee}$.

######  Remark 6.1.

  1. In the above statement we have:

  2. (1)

The space ${H_{*}^{T}}({\Omega K})_{P}$ is a quotient and a localization of
${H_{*}^{T}}({\Omega K})$ defined in [CMP09, §2.2]. The family
$\\{\sigma^{P}(w),w\in W/W_{P}\\}$ is the Schubert base of $H_{T}^{*}(G/P)$,
and the element in ${\mathbb{Z}}[{Q^{\vee}_{P}}]$ corresponding to
${\nu^{\vee}}\in{Q^{\vee}_{P}}$ is denoted by $q_{{\nu^{\vee}}}$. We have for
${\nu^{\vee}}\in{Q^{\vee}_{-}}$ the formula

$\deg(q_{\nu^{\vee}})=\sum_{\alpha\in R^{+}\setminus
R_{P}^{+}}\langle{\nu^{\vee}},\alpha\rangle=-\ell(t_{\nu^{\vee}}).$

  3. (2)

This isomorphism is graded. In fact, for very negative coweights
${\lambda^{\vee}},{\mu^{\vee}}$, the element
$\xi_{w\pi_{P}(t_{\lambda^{\vee}})}\xi_{\pi_{P}(t_{\mu^{\vee}})}^{-1}$ has
homological degree
$\ell(\pi_{P}(t_{\lambda^{\vee}}))-\ell(w)-\ell(\pi_{P}(t_{\mu^{\vee}}))$, by
[LS10, Lemma 3.3]. On the other hand, in quantum cohomology, the element
$q_{\eta_{P}({\lambda^{\vee}}-{\mu^{\vee}})}\sigma^{P}(w)$ has degree
$-\ell(\pi_{P}(t_{\lambda^{\vee}}))+\ell(\pi_{P}(t_{\mu^{\vee}}))+\ell(w)$.

###  6.2. A Weyl group action on $\operatorname{QH}^{*}_{T}(G/P)$.

In this subsection, we define an action of the Weyl group on $H^{*}_{T}(G/P)$
and on $\operatorname{QH}^{*}_{T}(G/P)$ by left translation. We will prove the
compatibility of this action with Peterson’s isomorphism in the next
subsection. Since this action is different from the action defined in Kumar
[Kum02, 11.3.4] we define it carefully. We start with the action on $G/B$ and
then deal with the general situation for $G/P$.

We define an algebraic and a geometric action of the Weyl group $W$ on
$H^{*}_{T}(G/B)$. We then prove that these actions coincide.

Let $n\in G$ be in the normalizer of $T$ and let $w$ be the corresponding
element of the Weyl group. Define the left action $L_{n}:G/B\to G/B$ by left
multiplication: $L_{n}\cdot[x]=[n^{-1}x]$. This action is $T$-equivariant if
we consider the $w$-twisted action of $T$ on $G/B$ given by
$t\cdot[x]=[w(t)x]$. It therefore induces a $w$-semilinear map
$H^{*}_{T}(G/B)\to H^{*}_{T}(G/B)$, denoted $L_{n}^{*}$:
$L_{n}^{*}(s\xi)=w(s)L_{n}^{*}(\xi)$ for $s\in S$ and $\xi\in H^{*}_{T}(G/B)$.

######  Fact 6.2.

The above action $L_{n}^{*}$ satisfies the following properties:

  1. (1)

$L_{n}^{*}$ depends on $w$ and not on $n$ itself; it will be denoted by
$w^{*}$ in the sequel.

  2. (2)

Via the inclusion $H^{*}_{T}(G/P)\subset H^{*}_{T}(G/B)$ given by pulling back
the projection $G/B\to G/P$, we have $w^{*}H^{*}_{T}(G/P)\subset
H^{*}_{T}(G/P)$.

  3. (3)

The induced action of $w^{*}$ on the non equivariant cohomology $H^{*}(G/B)$
is trivial.

###### Proof.

(1) Let $N$ denote the normalizer of $T$. The map $N\times G/B\to
G/B,(n,[x])\mapsto L_{n}\cdot[x]$ is continuous and therefore for $\xi\in
H^{*}_{T}(G/B)$, the map $N\mapsto L_{n}^{*}\xi$ is locally constant.

(2) For $n$ in the normalizer of $T$, we have a commutative diagram:

${G/B}$${G/B}$${G/P}$${G/P}$$\scriptstyle{L_{n}^{B}}$$\scriptstyle{L_{n}^{P}}$

Here we made a difference between the action of $n$ on $G/B$ and $G/P$ using
superscripts. It follows that for $\xi\in H^{*}_{T}(G/P)$, we have
$w^{*}\xi=(L_{n}^{B})^{*}\xi=(L_{n}^{P})^{*}\xi\in H^{*}_{T}(G/P)$.

(3) For $g\in G$, we can consider the action of left translation $L_{g}^{*}$
on non equivariant cohomology $H^{*}(G/B)$. By the same argument as in (1),
this action is trivial. In particular, for $g=n$ in $N$, we obtain that the
action $L_{n}^{*}$ on non equivariant cohomology is trivial. $\square$

Recall that $W$ can be embedded in ${\mathbb{A}_{\rm aff}}$ via
$v\mapsto\delta_{v}$.

######  Definition 6.3.

Let $w\in W$. Consider $H^{*}_{T}(G/B)$ as the dual of
$H_{*}^{T}(G/B)\subset{\mathbb{A}_{\rm aff}}$ and set

$(w\bullet f)(x)=f(\delta_{w^{-1}}x)\textrm{ for $x\in
H_{*}^{T}(G/B)={\mathbb{A}}$}.$

######  Proposition 6.4.

For $f\in H^{*}_{T}(G/B)$ and $w\in W$, we have $w\bullet f=w^{*}f$.

###### Proof.

Using ${\rm Frac}(S)$-linearity, we only need to compare these actions on the
elements $\xi^{v}$. We have
$(w\bullet\xi^{v})(\delta_{u})=\xi^{v}(\delta_{w^{-1}}\delta_{u})=\xi^{v}(\delta_{w^{-1}u})=\delta_{v,w^{-1}u}=\xi^{v}(w^{-1}u)=(w^{*}\xi^{v})(u)$,
proving the result. $\square$

######  Corollary 6.5.

Let $\alpha$ be a simple root and $w\in W^{P}$. We have

$(s_{\alpha})^{*}\sigma^{P}(w)=\left\\{\begin{array}[]{ll}\sigma^{P}(w)&\textrm{if
}s_{\alpha}w>w;\\\ \sigma^{P}(w)-\alpha\sigma^{P}(s_{\alpha}w)&\textrm{if
}s_{\alpha}w<w.\\\ \end{array}\right.$

###### Proof.

We compute
$((s_{\alpha})^{*}\sigma^{P}(w))(A_{u})=\sigma^{P}(w)(\delta_{s_{\alpha}}A_{u})=\sigma^{P}(w)((1-\alpha
A_{\alpha})A_{u})=\sigma^{P}(w)(A_{u})-\alpha\sigma^{P}(w)(A_{\alpha}A_{u})$.
Now we have

$A_{\alpha}A_{u}=\left\\{\begin{array}[]{ll}0&\textrm{if }s_{\alpha}u<u\\\
A_{s_{\alpha}u}&\textrm{if }s_{\alpha}u>u.\\\ \end{array}\right.$

Since $\sigma^{P}(w)(A_{v})=\delta_{v,w}$, we get

$((s_{\alpha})^{*}\sigma^{P}(w))(A_{u})=\left\\{\begin{array}[]{ll}\delta_{u,w}&\textrm{if
}s_{\alpha}u<u\\\ \delta_{u,w}-\alpha\delta_{s_{\alpha}u,w}&\textrm{if
}s_{\alpha}u>u.\\\ \end{array}\right.$

This in turn gives the result. $\square$

######  Remark 6.6.

  1. (1)

Note that, for $\alpha$ simple, the two conditions $w\in W^{P}$ and
$s_{\alpha}w<w$ imply the inclusion $s_{\alpha}w\in W^{P}$ since the inversion
set of $s_{\alpha}w$ is contained in the inversion set of $w$. In particular,
in the second case of the above formula, the class $\sigma^{P}(s_{\alpha}w)$
is well defined.

  2. (2)

This formula also shows that the action $w^{*}$ is trivial in the non
equivariant setting (indeed, in that case wet set $\alpha=0$).

The action $w^{*}$ is extended to $\operatorname{QH}^{*}_{T}(G/P)$ by
linearity on quantum parameters.

###  6.3. Compatibility of Peterson’s isomorphism

In this subsection we prove that Peterson’s isomorphism is compatible with the
actions $u_{*}$ in homology and $u^{*}$ in cohomology. We start with a useful
lemma.

######  Lemma 6.7.

Let $w\in W^{P}$ and let ${\lambda^{\vee}}\in{Q^{\vee}_{-}}$ be such that
$x=w\pi_{P}(t_{\lambda^{\vee}})\in{(W^{P})_{\rm aff}}$. Write
$x=w\pi_{P}(t_{\lambda^{\vee}})=vt_{\mu^{\vee}}$ with $v\in W$ and
${\mu^{\vee}}\in Q^{\vee}$.

Let $\alpha$ be a simple root and let $\beta=w^{-1}(\alpha)$,
$\beta^{\prime}=v^{-1}(\alpha)$.

  1. (1)

We have $w^{-1}v\in W_{P}$ and ${\mu^{\vee}}\in Q^{\vee}_{-}$.

  2. (2)

We have $\beta\in R_{P}\Longleftrightarrow\beta^{\prime}\in R_{P}$.

  3. (3)

We have $s_{\alpha}x\in{W_{{\rm
aff}}^{-}}\Longleftrightarrow\langle{\mu^{\vee}},\beta\rangle\neq 0$.

  4. (4)

We have $s_{\alpha}x\in{(W^{P})_{\rm aff}}\Longleftrightarrow\beta\not\in
R_{P}\Longleftrightarrow s_{\alpha}w\in W^{P}$.

  5. (5)

We have the equivalence:

$\left(s_{\alpha}x\in{W_{{\rm aff}}^{-}}\cap{(W^{P})_{\rm aff}}\textrm{ and
}\ell(s_{\alpha}x)>\ell(x)\right)\Longleftrightarrow\left(s_{\alpha}w\in
W^{P}\textrm{ and }\ell(s_{\alpha}w)<\ell(w)\right).$

###### Proof.

(1) By [LS10, Lemma 10.7], we have
$\pi_{P}(t_{\lambda^{\vee}})=ut_{\mu^{\vee}}$ with $u\in W_{P}$. This give
$w^{-1}v=u\in W_{P}$. Since $vt_{\mu^{\vee}}\in{W_{{\rm aff}}^{-}}$ we have
${\mu^{\vee}}\in Q^{\vee}_{-}$.

(2) Since $u=w^{-1}v\in W_{P}$ and $\beta=u(\beta^{\prime})$, we have
$\beta\in R_{P}\Leftrightarrow\beta^{\prime}\in R_{P}$.

(3) We have $vt_{\mu^{\vee}}\in{W_{{\rm aff}}^{-}}$ therefore ${\mu^{\vee}}\in
Q^{\vee}_{-}$ and for $\gamma>0$, we have the implication
$(\langle{\mu^{\vee}},\gamma\rangle=0\Rightarrow v(\gamma)>0)$. The condition
$s_{\alpha}vt_{\mu}\in{W_{{\rm aff}}^{-}}$ is thus equivalent to
$(\langle{\mu^{\vee}},\gamma\rangle=0\Rightarrow s_{\alpha}v(\gamma)>0)$ for
$\gamma>0$. But since for $\gamma=\beta^{\prime}$, the roots $v(\gamma)$ and
$s_{\alpha}v(\gamma)$ have opposite signs, the condition
$s_{\alpha}vt_{\mu^{\vee}}\in{W_{{\rm aff}}^{-}}$ is equivalent to
$\langle{\mu^{\vee}},\beta^{\prime}\rangle=0$.

(4) We have $vt_{\mu^{\vee}}\in{(W^{P})_{\rm aff}}$ therefore, for $\gamma\in
R_{P}^{+}$, we have the equivalences
$(\langle{\mu^{\vee}},\gamma\rangle=0\Leftrightarrow v(\gamma)>0)$ and
$(\langle{\mu^{\vee}},\gamma\rangle=-1\Leftrightarrow v(\gamma)<0)$. The
condition $s_{\alpha}vt_{\mu}\in{(W^{P})_{\rm aff}}$ is equivalent to having
the equivalences $(\langle{\mu^{\vee}},\gamma\rangle=0\Leftrightarrow
s_{\alpha}v(\gamma)>0)$ and
$(\langle{\mu^{\vee}},\gamma\rangle=-1\Leftrightarrow s_{\alpha}v(\gamma)<0)$.
Since for $\gamma=\beta^{\prime}$, the roots $v(\gamma)$ and
$s_{\alpha}v(\gamma)$ have opposite signs, the last equivalences occur if and
only if $\beta^{\prime}\not\in R_{P}$. This in turn is equivalent to
$\beta\not\in R_{P}$ by (2).

For the last equivalence, note that by definition, we have $(s_{\alpha}w\in
W^{P}\Leftrightarrow s_{\alpha}w(R_{P}^{+})\subset R^{+})$. Since $w\in
W^{P}$, we have $w(R_{P}^{+})\subset R^{+}$. Since the inversion sets of $w$
and $s_{\alpha}w$ only differ by $\beta$ (or its opposite, depending on the
sign of $\beta$) we get the last equivalence.

(5) Note that we have the equivalence
$(\ell(s_{\alpha}w)<\ell(w)\Leftrightarrow\beta<0)$. We therefore need to
prove that the left hand side of the equivalence is equivalent to
$\beta\not\in R_{P}$ and $\beta<0$. Note that since $w\in W^{P}$, this is
equivalent to $\beta<0$.

First assume that $s_{\alpha}x\in{W_{{\rm aff}}^{-}}\cap{(W^{P})_{\rm aff}}$
and $\ell(s_{\alpha}x)>\ell(x)$. By [LS10, Lemma 3.3], since
$vt_{\mu^{\vee}},s_{\alpha}vt_{\mu^{\vee}}\in Waffm$, we have
$\ell(s_{\alpha}x)=\ell(t_{\mu^{\vee}})-\ell(s_{\alpha}v)$ and
$\ell(x)=\ell(t_{\mu^{\vee}})-\ell(v)$. In particular, we have
$\ell(s_{\alpha}v)<\ell(v)$, thus $\beta^{\prime}<0$. Since
$s_{\alpha}x\in{(W^{P})_{\rm aff}}$, we also have $\beta\not\in R_{P}$ thus
$\beta^{\prime}\not\in R_{P}$. Now, since $u\in W_{P}$, this implies
$\beta<0$.

Conversely, assume $\beta<0$. By the above arguments, this implies
$\beta\not\in R_{P}$ and thus $s_{\alpha}x\in{(W^{P})_{\rm aff}}$. This also
implies $\beta^{\prime}\not\in R_{P}$ and since $u\in W_{P}$ and
$\beta^{\prime}=u^{-1}(\beta)$, we get $\beta^{\prime}<0$. Since
$vt_{\mu}\in{W_{{\rm aff}}^{-}}$, $-\beta^{\prime}>0$ and
$v(-\beta^{\prime})=-\alpha<0$, we must have
$\langle{\mu^{\vee}},\beta^{\prime}\rangle\neq 0$ and by (3), this implies
$s_{\alpha}x\in{W_{{\rm aff}}^{-}}$. $\square$

We have the following equivariance property of $\psi_{P}$.

######  Proposition 6.8.

For $\xi\in{H_{*}^{T}}({\Omega K})_{P}$, we have
$\psi_{P}(u_{*}\xi)=u^{*}\xi$.

###### Proof.

We may assume that $u=s_{i}$, with $\alpha_{i}$ a simple root. Then
$u_{*}\xi_{w\pi_{P}(t_{\lambda^{\vee}})}=\delta_{u}\cdot\xi_{w\pi_{P}(t_{\lambda^{\vee}})}=(1-\alpha_{i}A_{i})\cdot\xi_{w\pi_{P}(t_{\lambda^{\vee}})}$.
If
$\ell(s_{i}w\pi_{P}(t_{\lambda^{\vee}}))>\ell(w\pi_{P}(t_{\lambda^{\vee}}))$
and $s_{i}w\pi_{P}(t_{\lambda^{\vee}})\in{(W^{P})_{\rm aff}}\cap{W_{{\rm
aff}}^{-}}$, then this is equal to
$\xi_{w\pi_{P}(t_{\lambda^{\vee}})}-\alpha_{i}\xi_{s_{i}w\pi_{P}(t_{\lambda^{\vee}})}$.
Otherwise, this is equal to $\xi_{w\pi_{P}(t_{\lambda^{\vee}})}$.

The action ${s_{i}}^{*}\sigma^{P}(w)$ is computed in Corollary 6.5. If
$\ell(s_{i}w)<\ell(w)$ and $s_{i}w\in W^{P}$, then this is equal to
$\sigma^{P}(w)-\alpha_{i}\sigma^{P}(s_{i}w)$. Otherwise, this is equal to
$\sigma^{P}(w)$.

Let $\beta=w^{-1}(\alpha_{i})$. The condition $s_{i}w\in W^{P}$ and
$\ell(s_{i}w)<\ell(w)$ is equivalent to the condition
$\ell(s_{i}w\pi_{P}(t_{\lambda^{\vee}}))>\ell(w\pi_{P}(t_{\lambda^{\vee}}))$
and $s_{i}w\pi_{P}(t_{\lambda^{\vee}})\in{(W^{P})_{\rm aff}}\cap{W_{{\rm
aff}}^{-}}$ by Lemma 6.7.(5). This proves the result. $\square$

###  6.4. The result

We now prove our main result. For $i$ a cominuscule node, _i.e._ such that
$\varpi_{i}^{\vee}$ is a minuscule coweight, we let $v_{i}$ be the smallest
element in $W$ such that $v_{i}(\varpi_{i}^{\vee})=w_{0}(\varpi_{i}^{\vee})$
($w_{0}$ is the longest element in $W$). The coweight
$v_{i}(\varpi_{i}^{\vee})=w_{0}(\varpi_{i}^{\vee})$ is the opposite of a
fundamental coweight: there exists $f(i)\in I$ such that
$v_{i}(\varpi_{i}^{\vee})=-\varpi_{f(i)}^{\vee}$. Actually we have
$\alpha_{f(i)}=-w_{0}(\alpha_{i})$ and $v_{f(i)}=v_{i}^{-1}$.

######  Theorem 6.9.

Let $i$ be a cominuscule node. In $QH^{*}_{T}(G/P)$ we have

$\sigma^{P}(v_{i})\times
v_{i}^{*}(\sigma^{P}(w))=q_{\eta_{P}(\varpi_{i}^{\vee}-w^{-1}(\varpi_{i}^{\vee}))}\sigma^{P}(v_{i}w)\,.$

###### Proof.

Let $w\in W^{P}$, we have $\pi_{P}(w)=w$. Let $\varpi_{i}^{\vee}$ be the
minuscule coweight associated to $i$ and let ${\mu^{\vee}}$ and ${\nu^{\vee}}$
be in $Q^{\vee}$ and dominant enough. As in [CMP09, §3.5], we get

$\begin{array}[]{l}\pi_{P}(t_{-\varpi^{\vee}_{i}-\mu})=\tau_{i}\pi_{P}(v_{f(i)})\pi_{P}(t_{-(\varpi_{i}^{\vee}+\varpi_{f(i)}^{\vee}+\mu)}\textrm{
and }\\\
\pi_{P}(wt_{-\nu}t_{-\varpi^{\vee}_{i}-\mu})=\tau_{i}\pi_{P}(v_{f(i)}w)\pi_{P}(t_{-(\varpi_{i}^{\vee}+w^{-1}(\varpi_{f(i)}^{\vee})+\mu+\nu)}.\\\
\end{array}$

For $\mu$ and $\nu$ dominant enough, the elements $wt_{-\nu}$,
$t_{-\varpi_{i}^{\vee}-\mu}$ and $wt_{-\nu}t_{-\varpi_{i}^{\vee}-\mu}$ are in
${{{\widetilde{W}_{\rm aff}}^{-}}}$ and their image by $\pi_{P}$ are in
${(\widetilde{W}^{P})_{\rm aff}}\cap{{{\widetilde{W}_{\rm aff}}^{-}}}$. We may
therefore apply Proposition 5.14 to the elements $wt_{-\nu}$ and
$t_{-\varpi_{i}^{\vee}-\mu}$ to get:

$\begin{array}[]{cl}&(v_{f(i)})_{*}\xi_{w\pi_{P}(t_{-{\nu^{\vee}}})}\times\xi_{\pi_{P}(v_{f(i)})\pi_{P}(t_{-(\varpi_{i}^{\vee}+\varpi_{f(i)}^{\vee}+{\mu^{\vee}})})}\\\
\equiv&\xi_{\pi_{P}(v_{f(i)}w)\pi_{P}(t_{-(\varpi_{i}^{\vee}+w^{-1}(\varpi_{f(i)}^{\vee}+{\mu^{\vee}}+{\nu^{\vee}}))})}\,,\end{array}$

where $\equiv$ means equality in ${H_{*}^{T}}({\Omega K})_{P}$ or equivalently
equality modulo $J_{P}$. Applying Peterson’s map (7), we get thanks to
Proposition 6.8 the corresponding formula in the quantum cohomology ring:

$\begin{array}[]{cl}&v_{f(i)}^{*}\sigma^{P}(w)q_{-\eta_{P}({\nu^{\vee}})}*\times\sigma^{P}(v_{f(i)})q_{-\eta_{P}((\varpi_{i}^{\vee}+\varpi_{f(i)}^{\vee}+{\mu^{\vee}}))}\\\
=&\sigma^{P}(v_{f(i)}w)q_{-\eta_{P}((\varpi_{i}^{\vee}+w^{-1}(\varpi_{f(i)}^{\vee})+{\mu^{\vee}}+{\nu^{\vee}}))}\,,\end{array}$

hence finally:

$v_{f(i)}^{*}\sigma^{P}(w)*\sigma^{P}(v_{f(i)})=q_{\eta_{P}(\varpi_{f(i)}^{\vee}-w^{-1}(\varpi_{f(i)}^{\vee}))}\sigma^{P}(v_{f(i)}w).$

This concludes the proof of the theorem. $\square$

######  Corollary 6.10.

Let $i$ be such that $\varpi_{i}^{\vee}$ is a minuscule coweight. In
$QH^{*}(G/P)$, we have

$\sigma^{P}(v_{i})\times\sigma^{P}(w)=q_{\eta_{P}(\varpi_{i}^{\vee}-w^{-1}(\varpi_{i}^{\vee}))}\sigma^{P}(v_{i}w)\,.$

######  Example 6.11.

Let $G$ be of type $A_{1}$, so that $G/B={\mathbb{P}}^{1}$. Let $s$ be the non
trivial element of $W$ and $\alpha$ the simple root. We have

$\sigma^{B}(s)*(\sigma^{B}(s)-\alpha)=q\,.$

###### Proof.

Let $i$ be the unique node of the Dynkin diagram of $G$. Then $v_{i}=s$. To
apply Theorem 6.9, we also set $w=s$. Let $x$ resp. $y$ be the $B$-stable
resp. $B^{-}$-stable point in ${\mathbb{P}}^{1}$. The class $\sigma^{B}(s)$ is
the $T$-equivariant class of $x$, and $v_{i}^{*}\sigma^{B}(s)$ is the
$T$-equivariant class of $y$. Since $[x]-[y]=\alpha$, we have
$v_{i}^{*}\sigma^{B}(s)=\sigma^{B}(s)-\alpha$. Denoting $h=\sigma^{B}(s)$, the
theorem yields $h\times(h-\alpha)=q$, as claimed. Note that $h^{2}=q+\alpha h$
is also predicted eg by [Mi07, Theorem 1]. $\square$

##  7\. Pieri formulas

We now give another application of Proposition 5.8 to prove a formula for
$j(\xi_{\tau_{i}(v_{i})})$, see Proposition 7.4. This gives the multiplication
in $H^{T}_{*}({\Omega K})$ by the class $\xi_{\tau_{i}(v_{i})}$. We hope in
subsequent work to deduce Pieri formulas for the non-equivariant
multiplication by classes generating $H_{*}({\Omega K})$ in all classical
types.

We first provide a generalization of [Lam08, Proposition 5.4] to coweights.
For ${\mu^{\vee}}\in P^{\vee}$, set $W_{\mu^{\vee}}=\langle s_{\alpha_{i}}\ |\
i\in[1,r]\textrm{ and }\langle\alpha_{i},{\mu^{\vee}}\rangle=0\rangle=\\{w\in
W\ |\ w({\mu^{\vee}})={\mu^{\vee}}\\}$.

######  Proposition 7.1.

Let ${\mu^{\vee}}\in{P^{\vee}}$ be antidominant. Then

${j^{\rm ad}}(\xi_{t_{\mu^{\vee}}})=\sum_{w\in
W/W_{\mu^{\vee}}}{\widetilde{A}}_{t_{w({\mu^{\vee}})}}\,.$

###### Proof.

We follow the idea of proof given in [Lam08, Proposition 5.4]. Using Lemma
4.8, we see that for $w\in W/W_{{\mu^{\vee}}}$ non trivial,
$t_{w({\mu^{\vee}})}\not\in{{{\widetilde{W}_{\rm aff}}^{-}}}$, so that
$A_{t_{w({\mu^{\vee}})}}$ belongs to the ideal $\sum_{x\in
W\setminus\\{e\\}}{\widetilde{\mathbb{A}}_{\rm aff}}\cdot A_{x}$ of
Proposition 5.8. Thus, using Proposition 5.8, we only need to prove that
$\sum_{w\in W/W_{\mu^{\vee}}}A_{t_{w({\mu^{\vee}})}}\in
Z_{{\widetilde{\mathbb{A}}_{\rm aff}}}(S)$.

To prove that $c:=\sum_{w\in
W/W_{\mu^{\vee}}}{\widetilde{A}}_{t_{w({\mu^{\vee}})}}$ centralizes $S$, or
equivalently commutes with any $\lambda$ in $P$, we use Proposition 4.6 to
compute ${\widetilde{A}}_{t_{\nu^{\vee}}}\lambda$. In this formula, the term
$t_{\nu^{\vee}}(\lambda)$ is equal to $\lambda$ by (1) in §2.2. Let
$\mathcal{P}$ be the set of pairs $({\nu^{\vee}},\beta)$ where
${\nu^{\vee}}\in W\cdot{\mu^{\vee}}$, $\beta$ is a positive real root, and
$t_{\nu^{\vee}}s_{\beta}\lessdot t_{\nu^{\vee}}$. We have:

(8) $c\lambda-\lambda
c=\sum_{({\nu^{\vee}},\beta)\in\mathcal{P}}\langle\lambda,\beta^{\vee}\rangle{\widetilde{A}}_{t_{{\nu^{\vee}}}s_{\beta}}\,,$

so our concern now is to prove that this sum vanishes.

We consider the map $\iota:\mathcal{P}\to\mathcal{P}$ defined by
$\iota({\nu^{\vee}},\beta)=(s_{\beta}({\nu^{\vee}}),-t_{\nu^{\vee}}(\beta))$.
Let $({\nu^{\vee}},\beta)\in\mathcal{P}$. We have

$\begin{array}[]{rclcl}t_{\nu^{\vee}}s_{\beta}=t_{\nu^{\vee}}s_{\beta}t_{-{\nu^{\vee}}}t_{\nu^{\vee}}&=&s_{t_{\nu^{\vee}}(\beta)}t_{\nu^{\vee}}\\\
&=&t_{s_{t_{\nu^{\vee}}(\beta)}({\nu^{\vee}})}s_{t_{\nu^{\vee}}(\beta)}&=&t_{s_{\beta}({\nu^{\vee}})}s_{-t_{\nu^{\vee}}(\beta)}\,,\end{array}$

where the last equality follows from (1) and Lemma 2.8. By the length formula
in [CMP09, Corollary 3.13],
$\ell(t_{s_{\beta}({\nu^{\vee}})})=\ell(t_{\nu^{\vee}})$ and by definition of
$\mathcal{P}$, $\ell(t_{\nu^{\vee}}s_{\beta})=\ell(t_{\beta^{\vee}})-1$. Thus,
$\ell(t_{s_{\beta}({\nu^{\vee}})}s_{-t_{\nu^{\vee}}(\beta)})=\ell(t_{s_{\beta}({\nu^{\vee}})})-1$.
Moreover, by [BB05, Proposition 4.4.6], $t_{\nu^{\vee}}(\beta)<0$, which
implies $t_{s_{\beta}({\nu^{\vee}})}s_{-t_{\nu^{\vee}}(\beta)}\lessdot
t_{s_{\beta}({\nu^{\vee}})}$ and $-t_{\nu^{\vee}}(\beta)>0$, so
$(s_{\beta}({\nu^{\vee}}),-t_{\nu^{\vee}}(\beta))\in\mathcal{P}$ as claimed.

We also observe that
$\langle\lambda,-t_{\nu^{\vee}}(\beta)^{\vee}\rangle=\langle\lambda,-\beta^{\vee}\rangle=-\langle\lambda,\beta^{\vee}\rangle$.
Finally,

$-t_{s_{\beta}({\nu^{\vee}})}(-t_{\nu^{\vee}}(\beta))=t_{s_{\beta}({\nu^{\vee}})}t_{\nu^{\vee}}(\beta)=s_{\beta}t_{\nu^{\vee}}s_{\beta}t_{\nu^{\vee}}(\beta)\,.$

One can check that this root is equal to $\beta$, so that $\iota$ is an
involution and the terms in (8) cancel pairwise. $\square$

We now prove some preliminary lemmas.

######  Lemma 7.2.

Let $i\in{I_{\rm aff}}$. We have $\displaystyle{j^{\rm
ad}}({\widetilde{\xi}}_{w_{0}(\varpi_{i}^{\vee})})=\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\,.$

###### Proof.

Since $w_{0}(\varpi_{i}^{\vee})\leq 0$ we may apply Proposition 7.1 and get

${j^{\rm ad}}(\xi_{w_{0}(\varpi_{i}^{\vee})})=\sum_{{\mu^{\vee}}\in W\cdot
w_{0}(\varpi_{i}^{\vee})}A_{t_{\mu^{\vee}}}\,.$

Thus,

$\begin{array}[]{rcccl}{j^{\rm
ad}}(\xi_{w_{0}(\varpi_{i}^{\vee})})&=&\sum_{{\mu^{\vee}}\in
W\cdot\varpi_{i}^{\vee}}A_{t_{\mu^{\vee}}}&=&\sum_{w\leq_{L}v_{i}}A_{wt_{\varpi_{i}^{\vee}}w^{-1}}\\\
&=&\sum_{w\leq_{L}v_{i}}A_{w\tau_{i}^{-1}v_{i}w^{-1}}&=&\sum_{w\leq_{L}v_{i}}A_{\tau_{i}^{-1}\tau_{i}(w)v_{i}w^{-1}}\\\
&=&\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\,.\end{array}$

$\square$

######  Lemma 7.3.

Let $s\in S$ and $i\in{I_{\rm aff}}$. We have

$\left(\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\right)s=\tau_{i}(s)\left(\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\right)\,.$

###### Proof.

Let $i\in{I_{\rm aff}}$. Since ${j^{\rm
ad}}(\xi_{w_{0}(\varpi_{i}^{\vee})})=\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}$,
we deduce that

$\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\in
Z_{\mathbb{A}_{\rm aff}}(S)\,.$

Let $s\in S$, we have:

$\delta_{\tau_{i}^{-1}}(\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}})\,s=s\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}=\delta_{\tau_{i}^{-1}}\tau_{i}(s)\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\,,$

which proves the lemma. $\square$

######  Proposition 7.4.

Let $i\in{I_{\rm aff}}$, let as above $v_{i}$ the maximal element in
$W^{P_{i}}$ and $\tau_{i}$ the automorphism of the affine Dynkin diagram
defined by $i$. Then $\tau_{i}(v_{i})\in{W_{{\rm aff}}^{-}}$ and we have:

$j(\xi_{\tau_{i}(v_{i})})=\sum_{w\leq_{L}v_{i}}\sum_{v\leq
v_{i}^{-1}}\tau_{i}(\xi^{v}(v_{i}^{-1}))A_{\tau_{i}(w)v_{i}w^{-1}}A_{v}\,.$

###### Proof.

We first prove that $\tau_{i}(v_{i})\in{W_{{\rm aff}}^{-}}$. We know that
$\tau_{i}(\alpha_{i})=\alpha_{0}$. Since $v_{i}\in W^{P_{i}}$, we have for
$1\leq j\leq n$ with $j\neq i$, $\ell(v_{i}s_{j})>\ell(v_{i})$. Since
$v_{i}\in W$, $\ell(v_{i}s_{0})>\ell(v_{i})$. Applying $\tau_{i}$, we deduce
that for all $k>0$, $\ell(\tau_{i}(v_{i})s_{k})>\ell(\tau_{i}(v_{i}))$. Thus,
$\tau_{i}(v_{i})\in{W_{{\rm aff}}^{-}}$.

Moreover, we know that $\tau_{i}=v_{i}t_{-\varpi_{i}^{\vee}}$. Therefore,
$v_{i}=\tau_{i}t_{\varpi_{i}^{\vee}}=t_{w_{0}(\varpi_{i}^{\vee})}\tau_{i}$, so
that
$t_{w_{0}(\varpi_{i}^{\vee})}=v_{i}\tau_{i}^{-1}=\tau_{i}^{-1}\tau_{i}(v_{i})$.
By Proposition 5.9, we deduce that ${j^{\rm
ad}}(\xi_{t_{w_{0}(\varpi_{i}^{\vee})}})=\delta_{\tau_{i}^{-1}}j(\xi_{\tau_{i}(v_{i})})\delta_{v_{i}}$.

By Lemma 7.2, we deduce that

$\delta_{\tau_{i}^{-1}}j(\xi_{\tau_{i}(v_{i})})\delta_{v_{i}}=\delta_{\tau_{i}^{-1}}\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\,.$

Therefore, using Fact 5.1 and then Lemma 7.3, we find

$\begin{array}[]{rcl}j(\xi_{\tau_{i}(v_{i})})&=&\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\delta_{v_{i}^{-1}}\\\
&=&\sum_{w\leq_{L}v_{i}}A_{\tau_{i}(w)v_{i}w^{-1}}\sum_{v\leq
v_{i}^{-1}}\xi^{v}(v_{i}^{-1})A_{v}\\\ &=&\sum_{w\leq_{L}v_{i}}\sum_{v\leq
v_{i}^{-1}}\tau_{i}(\xi^{v}(v_{i}^{-1}))A_{\tau_{i}(w)v_{i}w^{-1}}A_{v}\,.\end{array}$

$\square$

######  Remark 7.5.

Let $x\in{W_{{\rm aff}}^{-}}$. In the non equivariant homology, we thus have

$\xi_{\tau_{i}(v_{i})}\cdot\xi_{x}=\sum\xi_{\tau_{i}(w)v_{i}w^{-1}x}\,,$

where the sum is over $w\leq_{L}v_{i}$ such that
$\ell(\tau_{i}(w)v_{i}w^{-1}x)=\ell(v_{i})+\ell(x)$ and
$\tau_{i}(w)v_{i}w^{-1}x\in{W_{{\rm aff}}^{-}}$.

By Corollary 5.15, we know that there is only one Schubert class in the
product $\xi_{\tau_{i}(v_{i})}\cdot\xi_{x}$, from which we deduce that there
is exactly one $w\leq_{L}v_{i}$ such that
$\ell(\tau_{i}(w)v_{i}w^{-1}x)=\ell(v_{i})+\ell(x)$ and
$\tau_{i}(w)v_{i}w^{-1}x\in{W_{{\rm aff}}^{-}}$.

######  Example 7.6.

Let us assume we are in type $\widetilde{A}_{3}$ and let us write for short
$A_{210}$ instead of $A_{s_{2}s_{1}s_{0}}$ and similarly for $\xi_{210}$ and
$\delta_{210}$. Let $i=1$ so that $v_{i}=s_{3}s_{2}s_{1}$ and
$\tau_{i}(v_{i})=s_{2}s_{1}s_{0}$. First we observe that

$\begin{array}[]{rcl}\delta_{\tau_{i}^{-1}}=\delta_{123}&=&(1-\alpha_{1}A_{1})(1-\alpha_{2}A_{2})(1-\alpha_{3}A_{3})\\\
&=&1-{\alpha}_{1}A_{1}-({\alpha}_{1}+{\alpha}_{2})A_{2}-({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})A_{3}\\\
&+&{\alpha}_{1}({\alpha}_{1}+{\alpha}_{2})A_{12}+{\alpha}_{1}({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})A_{13}+({\alpha}_{1}+{\alpha}_{2})({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})A_{23}\\\
&-&{\alpha}_{1}({\alpha}_{1}+{\alpha}_{2})({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})A_{123}\,.\end{array}$

Since
$\tau_{i}({\alpha}_{1})=-\theta=-({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})$,
$\tau_{i}({\alpha}_{2})={\alpha}_{1}$ and
$\tau_{i}({\alpha}_{3})={\alpha}_{2}$, we get:

$\begin{array}[]{rcl}j(\xi_{210})&=&A_{210}+A_{321}+A_{032}+A_{103}\\\
&+&{\alpha}_{3}(A_{2103}+A_{3213}+A_{0323})\\\
&+&({\alpha}_{2}+{\alpha}_{3})(A_{2102}+A_{3212}A_{1032})\\\
&+&({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})(A_{2101}+A_{0321}+A_{1031})\\\
&+&{\alpha}_{3}({\alpha}_{2}+{\alpha}_{3})(A_{21023}+A_{32123}+A_{10323})\\\
&+&{\alpha}_{3}({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})(A_{21031}+A_{03231})\\\
&+&({\alpha}_{2}+{\alpha}_{3})({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})(A_{21012}+A_{03212}+A_{10312})\\\
&+&{\alpha}_{3}({\alpha}_{2}+{\alpha}_{3})({\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3})(A_{210123}+A_{032123}+A_{103123})\,.\end{array}$

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