# Novikov algebras and their primitive ideals

Aditya Dwarkesh  Indian Institute of Science Education and Research - Kolkata,
Mohanpur, Nadia - 741246, West Bengal, India<EMAIL_ADDRESS>Amartya Goswami  [1] Department of Mathematics and Applied Mathematics,
University of Johannesburg, P.O. Box 524, Auckland Park 2006, South Africa.
[2] National Institute for Theoretical and Computational Sciences (NITheCS),
South Africa<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to study the primitive ideals of Novikov algebras. In
terms of modular maximal right ideals, a characterization of the primitive
ideals of a Novikov algebra has been obtained. We prove a Chevalley-Jacobson
density-type theorem for primitive Novikov algebras. We obtain some
equivalences between prime, simple, and primitive Novikov algebras. We
describe a subalgebra of a Novikov algebra as a Novikov algebra of
endomorphisms.

###### Key words and phrases:

Novikov algebra; primitive ideal; simple module; division ring; density
theorem.

###### 1991 Mathematics Subject Classification:

17A30; 17D25

##  1\. Introduction

Although the terminology ‘Novikov algebra’ has been introduced by Osborn [16],
the structure was first introduced by Gel’fand & Dorfman [8], corresponding to
a certain type of Hamiltonian operators. The same algebra has also been used
by Balinskii & Novikov [1] in connection with Poisson brackets of the
hydrodynamic type. Novikov algebras were first studied as abstract algebraic
structures by Zelmanov [25], who demonstrated that simple finite-dimensional
Novikov algebras of characteristic zero are one-dimensional. Further studies
of finite dimensional Novikov algebras with characteristic $p$ can be found in
Cherkasin [4], Fillippov [6], and in Osborn [16, 17, 19], whereas for the case
of infinite-dimensional Novikov algebras, see Osborn [18]. A complete
classification of finite-dimensional simple Novikov algebras and their
irreducible modules over an algebraically closed field with a prime
characteristic have been done in Xu [23].

It has been shown in Shestakov & Zhang [22] that solvability and right-
nilpotency are equivalent in the variety of Novikov algebras. Panasenko, in
[20], has shown that an ideal of a semiprime Novikov algebra is semiprime
Novikov algebra, and furthermore, a minimal ideal of a Novikov algebra is
either trivial, or a simple algebra.

Jacobson [10] (see also [11]) introduced the primitive ideals of
noncommutative rings, and this notion plays a crucial role in determining the
structure theory of rings. Primitive ideals have been demonstrated to be
important in understanding structural aspects of modules (see Brešar [2] and
Rowen [21]), Lie algebras (see A. A. Kucherov, O. A. Pikhtilkova, & S. A.
Pikhtilkov [15]), enveloping algebras (see Dixmier [5] and Joseph [13]), PI-
algebras (see Jacobson [12]), quantum groups (see Joseph [14]), skew
polynomial rings (see Irving [9]), and others. Goswami [7] introduced the
concept of a primitive ideal of a Novikov algebra with the goal of studying
Jacobson topology on the spectrum of primitive ideals of a Novikov algebra.
The purpose of the present work is to study in detail those primitive ideals
of a Novikov algebra.

The main results of this paper are briefly summarized as follows: In Section
2, using the notion of a module over a Novikov algebra, we define a primitive
ideal of a Novikov algebra. In a Novikov algebra, we show that every maximal
ideal is primitive and every primitive ideal is prime (Corollary 2.8 and Lemma
2.2). We characterize primitive ideals in terms of modular maximal right
ideals (Corollary 2.13). A Chevalley-Jacobson density-type theorem (Theorem
3.2) has been formulated and proved in Section 3. We obtain a consequence
(Corollary 3.3) of this density theorem, which also shows equivalences between
prime, simple, and primitive Novikov algebras. Finally, we describe a
subalgebra of a Novikov algebra as a Novikov algebra of endomorphisms (Theorem
3.5).

##  2\. Primitive ideals

Recall that a (_left_) _Novikov algebra_ is a vector space $\mathcal{A}$ over
a field $\mathds{k}$ endowed with a bilinear map such that

  1. (i)

$(x\circ y)\circ z=(x\circ z)\circ y;$

  2. (ii)

$(x,y,z)=(y,x,z),$

where $(x,y,z)=(x\circ y)\circ z-x\circ(y\circ z)$ for all $x,$ $y,$
$z\in\mathcal{A}$.

Like in rings, the notion of a primitive ideal of a Novikov algebra depends on
the the notion of a module over a Novikov algebra. Osborn [19] (see also [16])
introduced a module over a Novikov algebra as a vector space endowed with two
bilinear maps satisfying the conditions (i) and (ii). We explicitly list below
the set of axioms of a module over a Novikov algebra.

If $\mathcal{A}$ is a Novikov algebra over a field $\mathds{k}$, then a
_module over_ $\mathcal{A}$ is a vector space $M$ (over $\mathds{k}$) equipped
with two maps $\mathcal{A}\times M\to M$ and $M\times\mathcal{A}\to M$
satisfying the following axioms:

  1. (I)

$(x\circ y)m=(xm)y$;

  2. (II)

$(mx)y=(my)x;$

  3. (III)

$x(\alpha m+\beta m^{\prime})=\alpha(xm)+\beta(xm^{\prime});$

  4. (IV)

$(\alpha x+\beta y)m=\alpha(xm)+\beta(ym);$

  5. (V)

$m(x\circ y)=(mx)y+x(my)-(xm)y;$

  6. (VI)

$(x\circ y)m=x(ym)+(y\circ x)m-y(xm);$

  7. (VII)

$(\alpha m+\beta m^{\prime})x=\alpha(mx)+\beta(mx^{\prime});$

  8. (VIII)

$m(\alpha x+\beta y)=\alpha(mx)+\beta(my),$

for all $\alpha,$ $\beta\in\mathds{k},$ $x,$ $y\in\mathcal{A},$ and $m,$
$m^{\prime}\in M$. A nonzero $\mathcal{A}$-module $M$ is said to be simple if
$M$ has no non-trivial proper submodules.

######  Remark 2.1.

Note that to define a left or a right module over a ring requires respectively
a left or a right action maps, whereas to obtain a module over a Novikov
algebra requires both. Since every Novikov algebra must be a module over
itself, it must satisfy condition (i), and hence only one of the action map is
not sufficient. For an alternative (but equivalent) formulation of an
$\mathcal{A}$-module, see Xu [23, 24].

A vector subspace $I$ of a Novikov algebra $\mathcal{A}$ is called a left
(right) ideal if $xI\subseteq I$ ($Ix\subseteq I$) for all $x\in\mathcal{A}$.
A Novikov algebra $\mathcal{A}$ is called simple if $\mathcal{A}$ has no non-
zero proper ideals. In this paper the word “ideal” without modifiers will
always mean two-sided ideal.

######  Lemma 2.2.

Let $M$ be an $\mathcal{A}$-module and let
$\mathrm{Ann}_{\mathcal{A}}M=\\{n\in\mathcal{A}\mid nm=mn=0,\;\forall m\in
M\\}.$ Then $\mathrm{Ann}_{\mathcal{A}}M$ is a two-sided ideal of $N$.

###### Proof.

Note that the lack of associativity in a Novikov algebra makes the proof
different from rings. Let $x,y\in\mathrm{Ann}_{\mathcal{A}}M,$
$n\in\mathcal{A}$, and $m\in M$. Axiom (I) implies

$(xn)m=(xm)n=0n=0,\quad\text{and}\quad(nx)m=(nm)x=0.$

Using Axiom (V), we obtain

$m(xn)=(mx)n+x(mn)-(xm)n=0,\quad\text{and}\quad m(nx)=(mx)n+n(mx)-(nm)x=0.$

By Axiom (IV) and Axiom (VIII), we respectively get $(x-y)m=0$ and $m(x-y)=0.$
∎

An ideal $P$ of a Novikov algebra $\mathcal{A}$ is called a _primitive ideal_
if there exists a simple $\mathcal{A}$-module $M$ such that $P$ is the
annihilator $\mathrm{Ann}_{\mathcal{A}}M$. We call an ideal $Q$ of a Novikov
algebra $\mathcal{A}$ _prime_ if $IJ\subseteq Q$ implies $I\subseteq Q$ or
$J\subseteq Q$ for any two ideals $I$ and $J$ of $\mathcal{A}$. Like rings, we
also have the following implication for a Novikov algebra.

######  Lemma 2.3.

Every primitive ideal of a Novikov algebra is prime.

###### Proof.

Although the argument is similar to the case of rings, the first equality in
(2.1) follows from Axiom (I). To prove the claim, let $P$ be the annihilator
of some simple $\mathcal{A}$-module $M$ and $J$ is an ideal of $\mathcal{A}$
such that $JM\neq 0$ (i.e., $J\not\subseteq P$). Since $M$ is simple, we must
have $JM=M.$ Now, if $I$ is another nonzero ideal of $\mathcal{A}$ such that
$I\not\subseteq P$, then

$(JI)M=(JM)I=MI=M.$ (2.1)

Therefore, from (2.1) it follows that $JI\not\subseteq P.$ ∎

We now give an example of a primitive ideal of a Novikov algebra. In the
following example, the construction of the Novikov algebra is taken from Burde
& De Graaf [3].

######  Example 2.4.

Consider the Novikov algebra $\mathds{R}^{3}$ over the field $\mathds{R}$ with
basis vectors $\\{e_{1},e_{2},e_{3}\\}$ and the product defined as follows:

$e_{1}\circ e_{1}=e_{1},\;e_{1}\circ e_{2}=2e_{2},\;e_{1}\circ
e_{3}=2e_{3},\;e_{2}\circ e_{1}=e_{2},\;e_{3}\circ e_{1}=e_{3},$

and all other products are zero. Consider the subspace $P$ generated by
$\\{e_{2},e_{3}\\}$. Then $P$ is the annihilator of the simple
$\mathds{R}^{3}$-module $M$ generated by $\\{e_{2}\\}.$ Since $M$ is cyclic,
$M$ is indeed a simple $\mathds{R}^{3}$-module, and $P$ annihilates $M$
follows from the definition of the product operation of the Novikov algebra.

An $\mathcal{A}$-module $M$ is called faithful if
$\mathrm{Ann}_{\mathcal{A}}M=0$ and a Novikov algebra is called primitive if
it has a faithful simple module. A Novikov algebra $N$ is called _prime_ if
$IJ=0$ implies either $I=0$ or $J=0$, for any two ideals $I,$ $J$ of
$\mathcal{A}$. The following lemma is lifted from ring theory.

######  Lemma 2.5.

Let $\mathcal{A}$ be a Novikov algebra and $I$ be an ideal of $\mathcal{A}$.
Let $M$ be an $\mathcal{A}/I$-module. Then, $M$ can also be considered as an
$\mathcal{A}$-module. Conversely, if $M$ is an $\mathcal{A}$-module and
$I\subseteq\mathrm{Ann}_{\mathcal{A}}M$, then $M$ can be considered as an
$\mathcal{A}/I$-module.

Similar to noncommutative primitive and prime rings, the following is a
characterization of primitive and prime Novikov algebras. The proof is
analogues to rings and is a consequence of Lemma 2.5.

######  Proposition 2.6.

If $\mathcal{A}$ is a Novikov algebra, then $P$ is a primitive (prime) ideal
of $\mathcal{A}$ if and only if $\mathcal{A}/P$ is a primitive (prime) Novikov
algebra.

In Lemma 2.2 we have seen that every primitive ideal of a Novikov algebra is
prime. The following result gives a partial converse of it.

######  Theorem 2.7.

Let $P$ be a prime ideal of a Novikov algebra $\mathcal{A}$ and there exist an
ideal $I$ of $\mathcal{A}$ satisfying the following property:

$P\subsetneq I\;\text{such that}\;P\subseteq J\subseteq I\implies
J=I\;\text{or}\;J=P,\;\;\text{for all}\;J\in\mathcal{A}.$ (2.2)

Then $P$ is a primitive ideal of $\mathcal{A}$.

###### Proof.

Let $I$ be an ideal of $\mathcal{A}$ satifying the property (2.2). Then it
follows from the correspondence theorem that $\mathcal{A}/P$ has a minimal
ideal. To have our claim, it thus suffices to show that any prime Novikov
algebra $\mathcal{A}$ with a minimal ideal $I$ is primitive. It is immediate
that $I$ is a simple $\mathcal{A}$-module. What remains is to show $I$ is
faithful. Note that by Axiom (I), for $j,$ $n\in\mathcal{A}$

$jI=0\implies(jn)I=(jI)n=0.$

This implies we have a right ideal $J$ generated by $j$ such that $JI=0$. But,
$JI=0$ implies $J=0$, since $\mathcal{A}$ is prime and $I\neq 0$. Therefore,
$j\in\mathrm{Ann}_{\mathcal{A}}I\implies
j\in\mathrm{Ann}_{\mathcal{A}}\mathcal{A}$. But
$\mathrm{Ann}_{\mathcal{A}}\mathcal{A}=0$. Therefore, $I$ is faithful. ∎

######  Corollary 2.8.

Every maximal ideal of a Novikov algebra is primitive.

###### Proof.

The proof follows by setting $I=N$ in Theorem 2.7. ∎

Based on Lemma 2.2 and Corollary 2.8, we have now the following implications
in a Novikov algebra:

$\text{Maximal ideal }\Rightarrow\text{Primitive ideal}\Rightarrow\text{Prime
ideal}.$

It is therefore natural to ask for an example of a primitive ideal that is not
maximal, and we provide that below. Once again, the underlying Novikov algebra
is taken from Burde & De Graaf [3].

######  Example 2.9.

Consider $\mathds{R}^{4}$ as a vector space over $\mathds{R}$ with basis
vectors $\\{e_{1},e_{2},e_{3},e_{4}\\}$ and the bilinear map is defined as
follows:

$e_{1}e_{1}=e_{1},\;e_{1}e_{2}=e_{2}+e_{3},\;e_{1}e_{3}=e_{3}e_{1}=e_{3},\;e_{2}e_{1}=e_{2},$

and all other products are zero. Consider the subspace $I$ generated by
$\\{e_{4}\\}$. Since $I$ is properly contained in the subspace generated by
$\\{e_{1},e_{4}\\}\subsetneq\mathds{R}^{4}$, the ideal $I$ is not maximal. On
the other hand, $I=\mathrm{Ann}_{\mathcal{A}}M$, where $M$ is the simple
$\mathcal{A}$-module generated by $\\{e_{1}\\}$.

The following lemma is lifted from ring theory.

######  Lemma 2.10.

Every simple Novikov algebra is primitive.

######  Remark 2.11.

The proof of Lemma 2.10 follows the usual procedure of showing that the
algebra forms a simple faithful module over itself. However, note that for
associative rings, the assumption of being unital is also required.

This is because, in general, simple rings do not form a simple module over
themselves. Since modules are typically one-sided in the usual setting, and a
ring being simple means they have no non-trivial two-sided ideals, the one-
sided module they form over themselves could fail to be simple because the
ring has, say, a non-trivial left ideal.

However, as per Remark 2.1, since a Novikov algebra module must necessarily be
two-sided, this complication is avoided. Because of the same reasoning, in
Corollary 2.8, we also do not require a unit element in a Novikov algebra.

Our next step is to characterize primitive ideals of a Novikov algebra in
terms of maximal modular (one-sided) ideals (see Corollary 2.13). Similar to
rings, we say that a right ideal $I$ of a Novikov algebra $\mathcal{A}$ is
called _modular_ if there exists an $e\in\mathcal{A}$ such that for all
$a\in\mathcal{A}$, $a-ae\in I.$ Using modular maximal left ideals, a
characterization of simple $\mathcal{A}$-modules is given below.

######  Theorem 2.12.

Let $\mathcal{A}$ be a Novikov algebra. An $\mathcal{A}$-module $M$ is a
simple if and only if $M$ is isomorphic to $\mathcal{A}/J$, where $J$ is a
modular maximal left ideal of $\mathcal{A}$.

###### Proof.

Suppose $M$ is a simple $\mathcal{A}$-module. If $0\neq u\in M$, then we claim
that $\mathcal{A}u=M.$ To show this, it suffices to show that $\mathcal{A}u$
is an additive subgroup of $M$; equality will then follow from the fact that
$M$ is non-zero and simple. Let $x=au$ and $x^{\prime}=a^{\prime}u$ for some
$a,$ $a^{\prime}\in\mathcal{A}$. Then, by Axiom (VIII),
$x-x^{\prime}=(a-a^{\prime})u\in\mathcal{A}u$.

Next, consider the map $\phi\colon\mathcal{A}\to\mathcal{A}u,$ defined by
$\phi(a)=au$. That $\phi$ is a $\mathcal{A}$-module homomorphism follows from
the bilinearity of $\circ$. Furthermore,
$\ker\phi=\mathrm{Ann}_{\mathcal{A}}u$. By the first isomorphism theorem, it
follows that $M=\mathcal{A}u\cong\mathcal{A}/\mathrm{Ann}_{\mathcal{A}}u.$

It remains to show that $J:=\mathrm{Ann}_{\mathcal{A}}u$ is a modular maximal
left ideal. From Lemma 2.2, it follows that $J$ is a left ideal of
$\mathcal{A}$. Since $M=u\mathcal{A}$, there exists an $e\in\mathcal{A}$ such
that $u=ue.$ Then for $a\in\mathcal{A},$ $au=(au)e=(ae)u$ (by Axiom (I)) or,
$(a-ae)u=0,$ which implies $a-ae\in J$, proving that $J$ is a modular right
ideal. The maximality of $J$ now follows from Lemma 2.5, namely the
correspondence between submodules of $\mathcal{A}/J$ and ideals of
$\mathcal{A}$ containing $J$.

To show the converse, let $M=\mathcal{A}/J$, where $J$ is a modular maximal
right ideal of $\mathcal{A}$. Then it is immediate that $M$ has no proper
submodule. For if $M^{\prime}$ were a non-zero proper submodule of
$\mathcal{A}/J,$ its pre-image under the quotient map would be a proper right
ideal containing $J$, contradicting its maximality. Thus, $M$ is simple. ∎

######  Corollary 2.13.

An ideal $I$ is primitive in a Novikov algebra $\mathcal{A}$ if and only if

$I=\\{x\in\mathcal{A}\mid xa,\,ax\in J\;\text{for all}\;a\in\mathcal{A}\\}$

for some modular maximal right ideal $J$ of $\mathcal{A}$.

An ideal $I$ of a Novikov algebra $\mathcal{A}$ is called a nil ideal if for
each $x\in I$, there is some positive integer $n$ such that $x^{n}=0$. The
Jacobson radical of a Novikov algebra, denoted by $\mathrm{Jac}\,\mathcal{A}$,
is defined to be the intersection of all primitive ideals of $\mathcal{A}$. If
$\mathcal{A}$ has no primitive ideals, we define
$\mathrm{Jac}\,\mathcal{A}=\mathcal{A}$. Observe that
$\mathrm{Jac}\,\mathcal{A}$ is also a two-sided ideal of $\mathcal{A}$. The
relation between a nil ideal and the Jacobson radical of a Novikov algebra is
as follows.

######  Proposition 2.14.

If $I$ is a nil ideal of $\mathcal{A},$ then
$I\subseteq\mathrm{Jac}\,\mathcal{A}$.

###### Proof.

We prove the claim by contradiction. Let $I$ be a nil ideal of $\mathcal{A}$
such that $I\subsetneq\mathrm{Jac}\,\mathcal{A}$. This implies there exists an
$u\in I$, a simple $\mathcal{A}$-module $M$, and an $m\in M$ such that
$n=mu\neq 0$. Since $\mathcal{A}n=M$, there is an $x\in\mathcal{A}$ such that
$xn=m$. Set $v:=xu\in I$. Then

$vn=(xu)n=(xn)u=mu=n,$

where the second equality follows by Axiom (I). Therefore, $v^{k}n=n,$ for all
positive integers $k$, contradicting the fact that $v$ is nilpotent. ∎

##  3\. Density Theorems

Our aim in this section is to obtain a Chevalley-Jacobson density-type theorem
for a primitive Novikov algebra. Once we have that, similar to rings, we
obtain a series of consequences. To this end, we first construct a Novikov
algebra of endomorphisms. Let $\mathcal{A}$ be a Novikov algebra. We define a
bilinear map
$\circ^{\prime\prime}\colon\mathrm{End}_{\mathds{k}}\mathcal{A}\times\mathrm{End}_{\mathds{k}}\mathcal{A}\to\mathrm{End}_{\mathds{k}}\mathcal{A}$
as

$(f\circ^{\prime\prime}g)(a)=f(a)\circ g(a),$

where $f,$ $g\in\mathrm{End}_{\mathds{k}}\mathcal{A}$ and $a\in\mathcal{A}$.
We claim the following.

######  Lemma 3.1.

$(\mathrm{End}_{\mathds{k}}\mathcal{A},\circ^{\prime\prime})$ is a Novikov
algebra.

###### Proof.

For $f,$ $g,$ $h\in\mathrm{End}_{\mathds{k}}\mathcal{A},$ we have

$\displaystyle((f\circ^{\prime\prime}g)\circ^{\prime\prime}h)(x)$
$\displaystyle=((f\circ^{\prime\prime}g)(x))\circ h(x)$
$\displaystyle=(f(x)\circ g(x))\circ h(x)$ $\displaystyle=(f(x)\circ
h(x))\circ g(x)=((f\circ^{\prime\prime}h)\circ^{\prime\prime}g)(x),$

hence satisfies Axiom (i). For Axiom (ii),

$\displaystyle(f,g,h)(a)$
$\displaystyle=((f\circ^{\prime\prime}g)\circ^{\prime\prime}h)(a)-(f\circ^{\prime\prime}(g\circ
h))(a)$ $\displaystyle=(f\circ^{\prime\prime}g)(a)\circ
h(a)-f(a)\circ(g\circ^{\prime\prime}h)(a)$ $\displaystyle=(f(a)\circ
g(a))\circ h(a)-f(a)\circ(g(a)\circ h(a))$ $\displaystyle=(g(a)\circ
f(a))\circ h(a)-g(a)\circ(f(a)\circ h(a))$
$\displaystyle=((g\circ^{\prime\prime}f)\circ^{\prime\prime}h)(a)-(g\circ(f\circ^{\prime\prime}h))(a)=(g,f,h)(a).\qed$

Let $\mathcal{A}$ be a Novikov algebra over a division ring $\Delta$, and let
$\mathrm{End}_{\Delta}(\mathcal{A})$ be the endomorphism algebra. A subalgebra
$\mathcal{E}$ of $\mathrm{End}_{\Delta}(\mathcal{A})$ is called a dense
algebra of linear operators of $\mathcal{A}$ if between any two finite subsets
$U$ and $V$ of $\mathcal{A}$ of equal cardinality wherein $U$ is linearly
independent, there is an $f\in\mathcal{E}$ such that $f(u)=v$, for each $u\in
U$ and $v\in V$.

######  Theorem 3.2 (Density Theorem).

Let $\mathcal{A}$ be a Novikov algebra over a division ring. Then, if
$\mathcal{A}$ is primitive, $\mathcal{A}$ is isomorphic to a dense algebra of
operators of a Novikov algebra over a division ring.

###### Proof.

Suppose $\mathcal{A}$ is a primitive Novikov algebra and $M$ is a simple
faithful $\mathcal{A}$-module. Observe that $\mathrm{End}_{\mathcal{A}}M$ is a
division ring. For addition being pointwise and composition as multiplication,
$\mathrm{End}_{\mathcal{A}}M$ is a ring and since $M$ is simple, for every
nonzero $f\in\mathrm{End}_{\mathcal{A}}M$, we have $\mathrm{im}\,f=M$ and
$\mathrm{ker}\,f=0$, so that $f$ is invertible.

We impose a Novikov algebra (over the division ring
$\mathrm{End}_{\mathcal{A}}M$) structure on $M$. The vector space (over
$\mathrm{End}_{\mathcal{A}}M$) structure on $M$ comes naturally from the
scalar multiplication $\mathrm{End}_{\mathcal{A}}M\times M\to M$ defined by
$(f,m)\mapsto f(m)$. Next we endow a bilinear map $\circ^{\prime}:M\times
M\rightarrow M$ defined as

$m_{1}\circ^{\prime}m_{2}=(a\circ a^{\prime})m,$

where $m_{1}=am,m_{2}=a^{\prime}m$, $a,$ $a^{\prime}\in\mathcal{A}$, for a
fixed $m\in M.$ Thanks to the faithfulness of $M$, note that such $a$ and
$a^{\prime}$ in $\mathcal{A}$ always exist. The $\mathcal{A}$-module $M$
endowed with the above-mentioned $\mathrm{End}_{\mathcal{A}}M$-Novikov algebra
structure is now going to be denoted by $\mathcal{M}.$

By Lemma 3.1, $\mathrm{End}_{\mathrm{End}_{\mathcal{A}}M}\mathcal{M}$ is also
a Novikov algebra. We claim that $\mathcal{A}$ is isomorphic to a dense
algebra $\mathcal{E}$ of operators of $\mathcal{M}$. We first obtain such a
subalgebra $\mathcal{E}$ of $\mathcal{M}$. For each $a\in\mathcal{A}$, we
define $f_{a}\colon\mathcal{M}\rightarrow\mathcal{M}$ by $f_{a}(m)=am$. Since

$f_{a}(m)\circ^{\prime}f_{a}(m^{\prime})=(am)\circ^{\prime}(am^{\prime})=a(m\circ^{\prime}m^{\prime})=f_{a}(m\circ^{\prime}m^{\prime}),$

we have $f_{a}\in\mathrm{End}_{\mathrm{End}_{\mathcal{A}}M}\mathcal{M}$. Take
$\mathcal{E}=\\{f_{a}\mid a\in\mathcal{A}\\}$. To prove the isomorphism,
consider the map $\xi\colon\mathcal{A}\to\mathcal{E}$ defined by
$\xi(a)=f_{a}$. For $m\in\mathcal{M},$

$f_{a\circ a^{\prime}}(m)=(a\circ
a^{\prime})m=(am)\circ^{\prime}(a^{\prime}m)=(f_{a}\circ^{\prime\prime}f_{a^{\prime}})(m),$

that is $f_{a\circ a^{\prime}}=f_{a}\circ^{\prime\prime}f_{a^{\prime}}$. Hence
$\xi$ is a homomorphism. Faithfulness of $M$ implies $\xi$ is injective,
whereas surjectivity of $\xi$ is obvious.

Finally, the proof is complete if we show $\mathcal{E}$ is dense. Since the
argument for the proof of denseness is identical to that of rings, we do not
repeat it here. ∎

######  Corollary 3.3.

If $\mathcal{A}$ is a nonzero finite dimensional Novikov algebra, then the
following statements are equivalent:

  1. (i)

$\mathcal{A}$ is prime.

  2. (ii)

$\mathcal{A}$ is simple.

  3. (iii)

There exists a division ring $\Delta$ and a finite dimensional Novikov algebra
$\mathcal{M}$ over $\Delta$ such that $\mathcal{A}$ is isomorphic to
$\mathrm{End}_{\Delta}\mathcal{M}$.

  4. (iv)

$\mathcal{A}$ is primitive.

We conclude this paper with a result (see Theorem 3.5) that brings a finite
dimensional Novikov algebra over an algebraically closed Novikov algebra. To
obtain that we first need the following lemma.

######  Lemma 3.4.

Let $\mathcal{F}$ be a unital division Novikov algebra over a field
$\mathds{k}$ and $\mathcal{A}$ be a Novikov algebra over $\mathcal{F}$. Let
$\mathcal{M}=\mathrm{End}_{\mathcal{F}}\mathcal{A}$, and $\mathcal{B}$ be a
$\mathds{k}$-subalgebra of $\mathcal{M}$. If
$a\mathcal{B}=\mathcal{B}a=\mathcal{A}$ for every nonzero $a\in\mathcal{A}$
and $\mathrm{End}_{\mathcal{B}}\mathcal{A}=\mathcal{F}$, then $\mathcal{B}$ is
a dense algebra of linear operators of $\mathcal{A}$.

###### Proof.

In the proof of Theorem 3.2, we have seen that $\mathcal{A}$ has a natural
$\mathcal{B}$-module structure; and, furthermore, that it is a faithful. The
assumption $a\mathcal{B}=\mathcal{B}a=\mathcal{A}$ tells us that it is also
simple. Thus, $\mathcal{B}$ is a primitive Novikov algebra. Therefore, it
follows from Theorem 3.2 that $\mathcal{B}$ is isomorphic to some dense
algebra of linear operators of a Novikov algebra $\mathcal{A}^{\prime}$ over a
unital division Novikov algebra $\mathcal{F}^{\prime}$. Since by hypothesis,
$\mathrm{End}_{\mathcal{B}}\mathcal{A}=\mathcal{F}$, we must have
$\mathcal{A}^{\prime}=\mathcal{A}$ and $\mathcal{F}^{\prime}=\mathcal{F}$, so
that $\mathcal{B}$ itself is dense as required. ∎

######  Theorem 3.5.

Let $\mathcal{A}$ be a finite dimensional Novikov algebra over an
algebraically closed Novikov algebra $\mathcal{F}$. If $\mathcal{B}$ is a
subalgebra of $\mathrm{End}_{\mathcal{F}}\mathcal{A}$ such that $\mathcal{A}$
is simple as a module over $\mathcal{B}$, then
$\mathcal{B}=\mathrm{End}_{\mathcal{F}}\mathcal{A}$.

###### Proof.

First, we show that a finite dimensional division Novikov algebra
$\mathcal{D}$ over an algebraically closed Novikov algebra $\mathcal{F}$
implies $\mathcal{D}=\mathcal{F}$. It will follow that
$\mathrm{End}_{\mathcal{B}}\mathcal{A}=\mathcal{F}$ (since the left-hand side
is a finite dimensional division algebra over $\mathcal{F}$), so that
$\mathcal{B}$ is dense (setting $\mathds{k}=\mathcal{F}$ in Lemma 3.4).

Let $x\in\mathcal{D}$. The set of all powers of any given element is linearly
dependent. Thus, there exists a polynomial $f\in\mathcal{F}[x]$ such that
$f(x)=0$, assuming without loss of generality that its leading coefficient is
$1$. Since $\mathcal{F}$ is algebraically closed, $f$ factors linearly. Thus,
$(x-\alpha_{1})\cdots(x-\alpha_{r})=0$. Since $\mathcal{D}$ is a division
Novikov algebra, $x-\alpha_{i}=0$ for some $\alpha_{i}$ and that implies
$x=\alpha_{i}\in\mathcal{F}$, and we are done.

Next, we show that the only dense algebra of linear operators on a finite
dimensional Novikov algebra will be the entire homomorphism algebra, and this
will complete the proof. Let $\mathcal{A}$ be a finite dimensional Novikov
algebra over $\mathcal{F}$ with basis $\\{u_{i}\\}_{i=1}^{n}$, and pick
$g\in\mathrm{End}_{\mathcal{F}}\mathcal{A}$, of which $\mathcal{E}$ is a dense
algebra. Then, there exists $f\in\mathcal{E}$ such that $f(u_{i})=g(u_{i})$,
which implies $f=g$, and hence $f\in\mathrm{End}_{\mathcal{F}}\mathcal{A}$. ∎

## References

  * [1] A. A. Balinskii and S. P. Novikov, Poisson brackets of hydrodynamic type, Frobenius algebras and Lie algebras, Sov. Math. Dokl., 32 (1985), 228–231.
  * [2] M. Brešar, Introduction to noncommutative algebra, Springer, 2014.
  * [3] D. Burde and W. de Graaf, Classification of Novikov algebras, Appl. Algebra Engrg. Comm. Comput., 24 (2013), 1–15.
  * [4] V. P. Cherkashin, Left-symmetric algebras with commuting right multiplications, Moscow Univ. Math. Bull., 43(5) (1988), 49–52.
  * [5] J. Dixmier, Enveloping algebras, Amer. Math. Soc., 1996.
  * [6] V. T. Filipov, A class of simple nonassociative algebras, Mat. Zametki, 45 (1989), 101–105.
  * [7] A. Goswami, Jacobson’s structure spaces of Novikov algebras (submitted).
  * [8] I. M. Gel’fand and I. Ya. Dorfman, Hamiltonian operators and algebraic structures related to them, Funkts. Anal. Prilozhen, 13 (1979), 13–30.
  * [9] R. S. Irving, Prime Ideals of Ore extensions over commutative rings, J. Algebra, 56 (1979), 315–342.
  * [10] N. Jacobson, A topology for the set of primitive ideals in an arbitrary Novikov algebra, Proc. Nat. Acad. Sei. U.S.A., 31 (1945), 333–338.
  * [11] ———, Structure of Novikov algebras, Amer. Math. Soc. Colloquium Publications, vol. 37, Providence, (1956)
  * [12] ———, PI-algebras. An introduction, Springer-Verlag, 1975.
  * [13] A. Joseph, Primitive ideals in enveloping algebras, Proc. ICM (Warsaw, 1983), 403–414, Warsaw, 1984.
  * [14] ———, Quantum groups and their primitive ideals, Springer, 1995.
  * [15] A. A. Kucherov, O. A. Pikhtilkova, and S. A. Pikhtilkov, On primitive Lie algebras, J. Math. Sci., 186(4) (2012), 651–654.
  * [16] J. M. Osborn, Modules for Novikov Algebras, in Second International Conference on Algebra (Barnaul, 1991), 327–338, Contemp. Math., 184, Amer. Math. Soc., Providence, RI, 1995.
  * [17] ———, Novikov algebras, Nova J. Algebra Geom., 1(1) (1992), 1–13.
  * [18] ———, Infinite-dimensional Novikov algebras of characteristic $0$, J. Algebra, 167(1) (1994), 146–167.
  * [19] ———, Modules for Novikov algebras of characteristic 0, Comm. Algebra, 23 (1995), 3627–3640.
  * [20] A. S. Panasenko, Semiprime Novikov algebras. Internat. J. Algebra Comput., 32(7) (2022), 1369–1378.
  * [21] L. H. Rowen, Ring theory, vol. I, Academic Press, Inc., 1988.
  * [22] I. Shestakov and Z. Zhang, Solvability and nilpotency of Novikov algebras, Comm. Algebra, 48(12) (2020), 5412–5420.
  * [23] X. Xu, On simple Novikov algebras and their irreducible modules, J. Algebra, 185 (1996), 905–934.
  * [24] X. Xu, Classification of simple Novikov algebras and their irreducible modules of characteristic $0$, J. Algebra, 246 (2001), 673–707.
  * [25] E. I. Zel’manov, On a class of local translation invariant Lie algebras, Sov. Math. Dokl., 35(6) (1987), 216–218.


# On the dimension of some union of affine Deligne-Lusztig varieties

Arghya Sadhukhan  Department of Mathematics, University of Maryland, College
Park, MD 20742<EMAIL_ADDRESS>
###### Abstract.

In this paper, we consider certain union $X(\mu,b)$ of affine Deligne-Lusztig
varieties in the affine flag variety that arises in the study of mod-$p$
reduction of Rapoport-Zink spaces and moduli spaces of shtukas associated to a
connected reductive group. Under a mild hypothesis on $\mu$, but no further
restrictions on the group, we compute its dimension in the case where $b$ is
the maximal neutrally acceptable element.

###### Key words and phrases:

Affine Deligne-Lusztig variety, Newton stratification, dimension formula,
Kottwitz set, affine Weyl group

###### 2010 Mathematics Subject Classification:

20G25,11G25,20F55

##  1\. Introduction

###  1.1. Motivation

A central tool in understanding the reduction modulo $p$ of Shimura varieties,
and therefore in the study of arithmetic properties encoded in them, is the
Newton stratification. Roughly speaking, at least for the Shimura varieties of
Hodge type that admits a moduli interpretation, this stratification on the
special fiber (of a suitable integral model) is defined by the loci where the
$p$-divisible groups attached to the corresponding abelian varieties have
fixed isogeny type. Investigation into the unique closed Newton stratum, the
so-called basic locus, in the last decades has found fruitful applications in
the realm of the Langlands program. We mention the work toward Zhang’s
arithmetic fundamental lemma [RTZ13], as well as the theory of non-archimedean
uniformization of Shimura varieties, first pioneered by Cerednik and Drinfeld,
and further developed by Rapoport–Zink [RZ96], Fargues [Far04], and
Howard–Pappas [HP17].

The natural range of the index set for the Newton stratification corresponding
to a Shimura variety associated to the group-theoretic data
$(\mathbf{G},\\{\mu\\})$ is the set $B(\mathbf{G},\mu)$ of neutral acceptable
elements. Using the classification of the Kottwitz set obtained in [Kot85] and
[Kot97], $B(\mathbf{G},\mu)$ can be viewed as a subset of the dominant
rational (relative) Weyl chamber. Thus it becomes a partially ordered set with
respect to the usual dominance relation. It is easy to see from the definition
that $B(\mathbf{G},\mu)$ has a unique minimal element; the basic locus
corresponds to this minimal element. A nontrivial result due to He and Nie
[HN18] asserts that there is a unique maximal element in $B(\mathbf{G},\mu)$;
to give an example, when the $\mu$-ordinary locus is nonempty, it is the
unique maximal Newton stratum, and then the maximal element of
$B(\mathbf{G},\mu)$ is $\mu^{\diamond}$, the twisted Galois-average (see
section 2.4.3) of a dominant representative in $\\{\mu\\}$. However, the
description of this maximal element is rather mysterious in general.

The primary goal of this article is to investigate the dimension of a certain
union of affine Deligne-Lusztig varieties associated to this maximal element.
Such a geometric object is the group-theoretic model for the maximal Newton
stratum. We remark that in the situations where the He-Rapoport axioms in
[HR17] hold, this sought-after dimension is equal to the dimension of the
maximal Newton stratum minus the dimension of the corresponding central leaf,
see [He16, §2.12]. The first quantity is $\langle 2\rho,\mu\rangle$, where
$2\rho$ is the sum of the (relative) positive roots, so in that case, we
essentially want to compute the latter dimension. The said (union of) affine
Deligne-Lusztig varieties can be defined purely in terms of group-theoretic
data without any reference to Shimura varieties, which we briefly do in the
following subsection. In the remainder of the article, we resort to this
latter setup.

###  1.2. Main theorem

Let $\mathbf{G}$ be a connected reductive group over a non-archimedean local
field $F$. Denote by $\sigma$ the Frobenius of $\breve{F}$ over $F$, where
$\breve{F}$ is the completion of the maximal unramified extension of $F$. We
choose a $\sigma$-invariant Iwahori subgroup $\breve{I}$ of
$\mathbf{G}(\breve{F})$ . Fix a geometric conjugacy class $\\{\mu\\}$ of
cocharacters of $\mathbf{G}$ over the separable closure $\bar{F}$ and choose a
dominant representative $\mu$ in it. We denote by $\operatorname{Adm}(\mu)$
the $\mu$-admissible set, a finite subset of the Iwahori-Weyl group
$\widetilde{W}$, see section 2 for details. Then for an element
$b\in\mathbf{G}(\breve{F})$, we define following Rapoport [Rap05] the union of
affine Deligne-Lusztig varieties

$X(\mu,b)=\\{g\breve{}\mathcal{I}\in\mathbf{G}(\breve{F})/\breve{}\mathcal{I};g^{-1}b\sigma(g)\in\breve{}\mathcal{I}\operatorname{Adm}(\mu)\breve{}\mathcal{I}\\}.$

This is the set of geometric points of a subscheme, locally of finite type, of
the affine flag variety attached to $\breve{F}$ (in the usual sense in equal
characteristic; in the sense of Zhu [Zhu17] and Bhatt-Scholze [BS17] in mixed
characteristics). The interest in such a set comes from the fact that when $F$
is $p$-adic and $\mu$ is minuscule, sets of this form arise as the set of
geometric points of the underlying reduced set of a Rapoport-Zink formal
moduli space of $p$-divisible groups, see [RV14]. Something analogous holds in
the function field case for formal moduli spaces of shtukas, where the
minuscule hypothesis can be dropped, cf. [Vie18]. In either case, by He’s work
on the Kottwitz-Rapoport conjecture in [He16a], the set $B(\mathbf{G},\mu)$
can be characterized as the set of those $\sigma$-conjugacy classes $[b]$ such
that $X(\mu,b)\neq\emptyset$. Our main result is then the following.

######  Theorem 1.1.

Assume that the image $\underline{\mu}_{\text{ad}}$ of $\mu$ in
$X_{*}(T_{\text{ad}})_{\Gamma_{0}}$ has depth at least $2$ in every
$\bar{F}$-simple factor of $\mathbf{G}_{\text{ad}}$. Let $b=b_{\max}$ be the
maximal element of $B(\mathbf{G},\mu)$. Then

$\dim
X(\mu,b)=\text{rk}_{F}^{\text{ss}}\mathbf{G}^{*}-\text{rk}_{F}^{\text{ss}}\mathbf{G},$
(1.1)

where $\mathbf{G}^{*}$ is the unique (up to isomorphism) quasi-split inner
form of $\mathbf{G}$, and $\text{rk}_{F}^{\text{ss}}$ stands for the
semisimple $F$-rank.

Here depth of a dominant cocharacter is an estimate of its distance from the
walls of the dominant Weyl chamber, see section 2.3 for a precise definition.
We remark that, in particular, $\mu$ cannot be minuscule because of this
restriction on its depth. Hence, by way of application in arithmetic geometry,
this result is only of interest in the function field setup.

Let us point out that the quantity on the RHS of eq. 1.1 is a non-negative
integer since every torus in $\mathbf{G}$ transfers to $\mathbf{G}^{*}$, cf.
[Lan89, lemma 2.1], and it is zero precisely when $\mathbf{G}$ is quasi-split.
On the other hand, by applying Chai’s description of $B(\mathbf{G},\mu)$ in
[Cha00, §7.1], [HN18, lemma 2.5], we can see that the maximal element
$b_{\max}$ has Newton point $\mu^{\diamond}$ exactly when $\mathbf{G}$ is
quasi-split; also cf. [GHN20, proposition 5.5] for a different argument. As a
result, the dimension on the LHS above can be directly checked to be zero in
the quasi-split case. Hence, eq. 1.1 indeed recovers this trivial case (which
is valid even in the absence of the depth hypothesis), and as such this
theorem also serves as a geometric way to measure how far $\mathbf{G}$ is from
being quasi-split.

###  1.3. Context of the result and outline of the proof

Let us put theorem 1.1 into context. For a quasi-split group $\mathbf{G}$, a
dimension formula on $X(\mu,b)$ for general $b$ was first obtained by He and
Yu [HY21] under a superregularity hypothesis on the depth of $\mu$. In [Sad21,
theorem C], the same formula was shown to be valid under the weaker assumption
that $\mu$ is regular, i.e., $\text{depth}(\mu)\geq 1$. Let us now place
ourselves in the context of a general group $\mathbf{G}$, and let $b_{\min}$
be the unique minimal (equivalently, basic) element of $B(\mathbf{G},\mu)$.
Then the work of Görtz, He and Rapoport in [GHR22] gives characterizations of
when $X(\mu,b_{\min})$ can be of minimal dimension zero, or of maximal
dimension $\langle 2\rho,\mu\rangle$; the model cases of such phenomena are
the Lubin-Tate case and the Drinfeld case, respectively. Let us also remark
that in [GHN20], Görtz, He and Nie provide a lower bound for the dimension of
$X(\mu,b_{\min})$, and they are also able to classify completely when this
becomes an equality. Besides these results, as far as the author is aware,
there is no dimension formula available for $X(\mu,b)$, nor is there any
conjectural description. Standing at this juncture, it is therefore natural to
investigate the dimension problem for the other extremal element of
$B(\mathbf{G},\mu)$, namely $b_{\max}$.

It is worth pointing out that our dimension formula can also be obtained more
directly, using explicit knowledge of the element $b_{\max}$ as discussed in
section 5. Our method, on the other hand, has the merit of producing all the
elements, at least conjecturally, in the Iwahori-Weyl group whose associated
Iwahori cells contribute to the maximal Newton stratum; such description would
be useful to analyze the (top dimensional) irreducible components of
$X(\mu,b_{\max})$ in the non quasi-split setup.

We now briefly sketch the strategy of the proof. We first identify $b_{\max}$
as the maximum of generic $\sigma$-conjugacy classes associated with the
maximal translation elements in the $\mu$-admissible set. Then we proceed to
express such generic $\sigma$-conjugacy classes in two distinct ways, first in
terms of $\sigma$-twisted Demazure power and then via a reduction to the
quasi-split case. Such considerations allow us to express the dimension of
individual affine Deligne-Lusztig varieties associated to such generic
$\sigma$-conjugacy classes in terms of certain statistics on the quantum
Bruhat graph introduced by Brenti, Fomin and Postnikov [BFP99]. Further
analysis in section 3.5 shows that $\dim X(\mu,b_{\max})$ lies between the
length and the reflection length of certain minimal length elements of some
Frobenius-twisted conjugacy classes in the finite Weyl group. We then show
that such minimal length elements are, in fact, partial Coxeter elements via a
case-by-case calculation in section 4, and hence their length (which is equal
to their reflection length) matches the asserted dimension in theorem 1.1,
thereby finishing the proof.

######  Acknowledgement.

The author thanks Xuhua He for his suggestion to work on this problem and for
many useful remarks throughout the course of this work. He would also like to
thank Michael Rapoport for his interest in this work and his suggestions on a
preliminary draft, and Mita Banik for her help with SageMath computations.

##  2\. Preliminaries

###  2.1. Notations

Recall that $\mathbf{G}$ is a connected reductive group over a non-archimedean
local field $F$. Let $\breve{F}$ be the completion of the maximal unramified
extension of $F$ and $\sigma$ be the Frobenius morphism of $\breve{F}/F$. The
residue field of $F$ is a finite field $\mathbb{F}_{q}$ and the residue field
of $\breve{F}$ is the algebraically closed field $\bar{\mathbb{F}}_{q}$. We
write $\breve{G}$ for $\mathbf{G}(\breve{F})$. We use the same symbol $\sigma$
for the induced Frobenius morphism on $\breve{G}$. Let $S$ be a maximal
$\breve{F}$-split torus of $\mathbf{G}$ defined over $F$, which contains a
maximal $F$-split torus. Let $\mathcal{A}$ be the apartment of
$\mathbf{G}_{\breve{F}}$ corresponding to $S_{\breve{F}}$. We fix a
$\sigma$-stable alcove $\mathfrak{a}$ in $\mathcal{A}$, and let
$\breve{}\mathcal{I}\subset\breve{G}$ be the Iwahori subgroup corresponding to
$\mathfrak{a}$. Then $\breve{}\mathcal{I}$ is $\sigma$-stable.

Let $T$ be the centralizer of $S$ in $\mathbf{G}$. Then $T$ is a maximal
torus. We denote by $N$ the normalizer of $T$ in $\mathbf{G}$. The
_Iwahori–Weyl group_ (associated to $S$) is defined as

$\widetilde{W}=N(\breve{F})/T(\breve{F})\cap\breve{}\mathcal{I}.$

For any $w\in\widetilde{W}$, we choose a representative $\dot{w}$ in
$N(\breve{F})$; however if there is no possibility of confusion we will call
the lift $w$ too. The action $\sigma$ on $\breve{G}$ induces a natural action
of $\sigma$ on $\widetilde{W}$, which we still denote by $\sigma$.

We denote by $\ell$ the length function on $\tilde{W}$ determined by the base
alcove $\mathfrak{a}$ and denote by $\tilde{}\mathbb{S}$ the set of simple
reflections in $\tilde{W}$. Let $W_{\text{aff}}$ be the subgroup of
$\widetilde{W}$ generated by $\tilde{}\mathbb{S}$. Then $W_{\text{aff}}$ is an
affine Weyl group. Let $\Omega\subset\widetilde{W}$ be the subgroup of length-
zero elements in $\widetilde{W}$. Then

$\widetilde{W}=W_{\text{aff}}\rtimes\Omega.$

Since the length function is compatible with the $\sigma$-action, the semi-
direct product decomposition $\widetilde{W}=W_{\text{aff}}\rtimes\Omega$ is
also stable under the action of $\sigma$.

Let $W=N(\breve{F})/T(\breve{F})$ be the relative Weyl group. We denote by
$\mathbb{S}$ the subset of $\widetilde{\mathbb{S}}$ consisting of simple
reflections generating $W$. We let $\Phi$ (resp. $\Delta$) denote the set of
roots (resp. simple roots) for $W$. We write $\Gamma$ for
$\operatorname{Gal}(\bar{F}/F)$, and write $\Gamma_{0}$ for the inertia
subgroup of $\Gamma$. Then fixing a special vertex of the base alcove
$\mathfrak{a}$, we have the splitting

$\widetilde{W}=X_{*}(T)_{\Gamma_{0}}\rtimes
W=\\{t^{\underline{\lambda}}w;\underline{\lambda}\in
X_{*}(T)_{\Gamma_{0}},w\in W\\}.$

Note that if $\mathbf{G}$ is not quasi-split over $F$, then there does not
exist a $\sigma$-stable special vertex in $\mathfrak{a}$ and thus the
splitting $\widetilde{W}=X_{*}(T)_{\Gamma_{0}}\rtimes W$ is not
$\sigma$-stable.

For an irreducible Weyl group $W$ of rank $n$, we follow the labeling of roots
as in [Bou02] and we usually write $s_{i}$ instead of $s_{\alpha_{i}}$, where
$\Delta=\\{\alpha_{i}:1\leq i\leq n\\}$. Let $w_{0}$ be the longest element in
$W$, and for $i\in[1,n]$ we let $w_{i,0}$ be the longest element of the
parabolic subgroup of $W$ corresponding to $\Delta\setminus\\{\alpha_{i}\\}$.
Let $\rho$ be the dominant weight with $\langle\alpha^{\vee},\rho\rangle=1$
for any $\alpha\in\Delta$. Let $\\{\varpi_{i}^{\vee}:1\leq i\leq n\\}$ be the
set of fundamental coweights. If $\varpi^{\vee}_{i}$ is minuscule, we denote
the image of $t^{\underline{\varpi}_{i}^{\vee}}$ under the projection
$\widetilde{W}\rightarrow\Omega$ by $\tau_{i}$; then conjugation by $\tau_{i}$
is a length preserving automorphism of $\widetilde{W}$ which we denote by
$\text{Ad}(\tau_{i})$.

###  2.2. Quantum Bruhat graphs

We recall the quantum Bruhat graph introduced by Brenti, Fomin, and Postnikov
in the context of quantum cohomology ring of complex flag variety, see
[BFP99], also [FGP97]. By definition, a _quantum Bruhat graph_ $\Gamma_{\Phi}$
is a directed graph with

  * •

vertices given by the elements of $W$;

  * •

upward edges $w\rightharpoonup ws_{\alpha}$ for some $\alpha\in\Phi^{+}$ with
$\ell(ws_{\alpha})=\ell(w)+1$;

  * •

downward edges $w\rightharpoondown ws_{\alpha}$ for some $\alpha\in\Phi^{+}$
with $\ell(ws_{\alpha})=\ell(w)-\langle 2\rho,\alpha^{\vee}\rangle+1$.

The _weight of a path_ in $\Gamma_{\Phi}$ is defined to be the sum of weights
of the edges in the path. For any $x,y\in W$, we denote by $d(x,y)$ the
minimal length among all paths in $\Gamma_{\Phi}$ from $x$ to $y$. Any path
between $x$ and $y$ affording $d(x,y)$ as its length is called a shortest path
between them. Then the following lemma defines a function $\text{wt}:W\times
W\rightarrow\mathbb{Z}_{\geq 0}\Phi^{\vee}$.

######  Lemma 2.1.

[Pos05, Lemma 1], [BFP99, Lemma 6.7] Let $x,y\in W$.

  1. (1)

There exists a directed path (consisting of possibly both upward and downward
edges) in $\Gamma_{\Phi}$ from $x$ to $y$.

  2. (2)

Any two shortest paths in $\Gamma_{\Phi}$ from $x$ to $y$ have the same
weight, which we denote by $\text{wt}(x,y)$.

  3. (3)

Any path in $\Gamma_{\Phi}$ from $x$ to $y$ has weight
$\geqslant\text{wt}(x,y)$.

We need the following results, taken from [HY21, §4.2] and [HN21, §2.5].

######  Lemma 2.2.

Let $x,y\in W$. Then

  1. (1)

We have $\langle\text{wt}(a,b),\alpha\rangle\leq 2$, for any simple root
$\alpha$.

  2. (2)

$\ell(y)-\ell(x)=d(x,y)-\langle 2\rho,\text{wt}(x,y)\rangle$.

###  2.3. Demazure product

We follow [He09, §1]. For any $x,y\in\widetilde{W}$, the subset
$\\{xy^{\prime};y^{\prime}\leqslant y\\}$ (resp.
$\\{x^{\prime}y;x^{\prime}\leqslant x\\}$,
$\\{x^{\prime}y^{\prime};x^{\prime}\leqslant x,y^{\prime}\leqslant y\\}$) of
$\widetilde{W}$ contains a unique maximal element with respect to the Bruhat
order. Moreover, we have

$\max\\{xy^{\prime};y^{\prime}\leqslant
y\\}=\max\\{x^{\prime}y;x^{\prime}\leqslant
x\\}=\max\\{x^{\prime}y^{\prime};x^{\prime}\leqslant x,y^{\prime}\leqslant
y\\}.$

We denote this element by $x\ast y$ and we call it the Demazure product of $x$
and $y$. Then $(\widetilde{W},\ast)$ is a monoid and the Demazure product is
determined by the following two rules

  * •

$x\ast y=xy$ if $x,y\in\widetilde{W}$ such that $\ell(xy)=\ell(x)+\ell(y)$;

  * •

$s\ast w=w$ if $s\in\tilde{}\mathbb{S}$, $w\in\widetilde{W}$ such that $sw<w$.

The Demazure product occurs naturally in the convolution product of Iwahori
cells in the sense that for any two elements $x,y\in\widetilde{W}$, we have
$\breve{I}\dot{x}\breve{I}\ast\breve{I}\dot{y}\breve{I}=\breve{I}(\dot{x}\ast\dot{y})\breve{I}$.
By such connection, it has also found applications in the study of generic
Newton points, cf. theorem 3.1.

An interesting connection between the Demazure product on $\widetilde{W}$ and
the quantum Bruhat graph of $\Phi_{W}$ is established in [HN21]. For any
dominant $\underline{\lambda}\in X_{*}(T)_{\Gamma_{0}}$, we define the depth
of $\underline{\lambda}$ to be
$\text{depth}(\underline{\lambda})=\min\\{\langle\alpha,\underline{\lambda}\rangle:\alpha\in\Delta\\}$.

######  Theorem 2.3.

[HN21, proposition 3.3] Let
$x_{1}t^{\underline{\mu}_{1}}y_{1},x_{2}t^{\underline{\mu}_{2}}y_{2}$ be two
elements of $\widetilde{W}$ such that the depth of
$\underline{\mu}_{1},\underline{\mu}_{2}$ is at least $2$. Then
$\underline{\mu}_{1}+\underline{\mu}_{2}-\text{wt}(y_{1}^{-1},x_{2})$ is
dominant, and

$x_{1}t^{\underline{\mu}_{1}}y_{1}\ast
x_{2}t^{\underline{\mu}_{2}}y_{2}=x_{1}t^{\underline{\mu}_{1}+\underline{\mu}_{2}-\text{wt}(y_{1}^{-1},x_{2})}y_{2}.$

###  2.4. The $\sigma$-conjugacy classes of $\breve{G}$

We say that two elements $b,b^{\prime}\in\breve{G}$ are $\sigma$-conjugate if
there is some $g\in\breve{G}$ such that $b^{\prime}=gb\sigma(g)^{-1}$. Let
$B(\mathbf{G})$ be the set of $\sigma$-conjugacy classes on $\breve{G}$. By
the work of Kottwitz in [Kot85] and [Kot97], any $\sigma$-conjugacy class
$[b]$ is determined by two invariants:

  * •

The Kottwitz point $\kappa([b])\in\pi_{1}(\mathbf{G})_{\Gamma}$, the set of
$\Gamma$-coinvariants of the Borovoi fundamental group $\pi_{1}(\mathbf{G})$;

  * •

The Newton point
$\nu([b])\in((X_{*}(T)_{\Gamma_{0},\mathbb{Q}})^{+})^{\langle\sigma\rangle}$,
the set of $\langle\sigma\rangle$-invariants of the intersection of
$X_{*}(T)_{\Gamma_{0}}\otimes\mathbb{Q}=X_{*}(T)^{\Gamma_{0}}\otimes\mathbb{Q}$
with the set $X_{*}(T)_{\mathbb{Q}}^{+}$ of dominant elements in
$X_{*}(T)_{\mathbb{Q}}$.

We denote by $\leqslant$ the dominance order on $X_{*}(T)_{\mathbb{Q}}^{+}$,
i.e., for $\nu,\nu^{\prime}\in X_{*}(T)_{\mathbb{Q}}^{+}$, we have
$\nu\leqslant\nu^{\prime}$ if and only if $\nu^{\prime}-\nu$ is a non-negative
(rational) linear combination of positive coroots over $\breve{F}$. The
dominance order on $X_{*}(T)_{\mathbb{Q}}^{+}$ extends to a partial order on
$B(\mathbf{G})$. Namely, for $[b],[b^{\prime}]\in B(\mathbf{G})$, we say that
$[b]\leqslant[b^{\prime}]$ if and only if $\kappa([b])=\kappa([b^{\prime}])$
and $\nu([b])\leqslant\nu([b^{\prime}])$.

Denote by ${\bf J}_{b}$ the $\sigma$-centralizer group of $b$; this is an
algebraic group over $F$ with $F$-rational points given by

${\bf J}_{b}(F)=\\{g\in\breve{G}\mid g^{-1}b\sigma(g)=b\\}.$

For any reductive group $\mathbf{H}$ over $F$, we denote by
$\text{rk}_{F}^{\text{ss}}\mathbf{H}$ the semisimple $F$-rank of $\mathbf{H}$.
The following result is implicit in [Kot06, §1.9].

######  Proposition 2.4.

Let $\mathbf{G}$ be quasi-simple over $F$ and assume $\tau\in\Omega$. Then

$\text{rk}_{F}^{\text{ss}}\mathbf{J}_{\dot{\tau}}=\mid\text{Ad}\tau\circ\sigma\text{
orbits on }\widetilde{\mathbb{S}}\mid-1.$

####  2.4.1. The straight $\sigma$-conjugacy classes

Note that the action of $\sigma$ on $\breve{G}$ gives rise to an action on
$\widetilde{W}$, still denoted by $\sigma$. The set of $\sigma$-conjugacy
classes in $\widetilde{W}$ is denoted by $B(\widetilde{W},\sigma)$.

Let $w\in\widetilde{W},n\in\mathbb{N}$. The $n$-th $\sigma$-twisted power of
$w$ is defined by $w^{\sigma,n}=w\sigma(w)\sigma^{2}(w)\cdots\sigma^{n-1}(w)$.
Then by definition, $w$ is called a $\sigma$-straight element if
$\ell(w^{\sigma,n})=n\ell(w)$, for all $n\in\mathbb{N}$. A $\sigma$-conjugacy
class of $\widetilde{W}$ is called straight if it contains a straight element,
and we denote the collection of such straight $\sigma$-conjugacy classes by
$B(\widetilde{W},\sigma)_{\text{str}}$. We have a map
$\Psi:B(\mathbf{G})\rightarrow B(\widetilde{W},\sigma)$, coming from the
assignment $w\rightarrow\dot{w}$. The importance of the straight
$\sigma$-conjugacy class comes from the following result.

######  Theorem 2.5.

[He14, theorem 3.7] The restriction of $\Psi$ induces a bijective map
$\Psi:B(\widetilde{W})_{\text{str}}B(\mathbf{G})$. Moreover, We have the
following commutative diagram

$B(\widetilde{W})_{\text{str}}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
82.48047pt\raise 5.39166pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{\Psi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 161.5495pt\raise 0.0pt\hbox{\hbox{\kern
0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
25.3959pt\raise-27.37555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{(\kappa,\nu)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
55.68344pt\raise-28.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}{\hbox{\kern
79.27475pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 161.5495pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{B(\mathbf{G})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
132.54832pt\raise-23.37555pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{(\kappa,\nu)}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
112.71567pt\raise-28.2511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
27.0pt\raise-41.7511pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\pi_{1}(\mathbf{G})_{\sigma}\times((X_{*}(T)_{\Gamma_{0},\mathbb{Q}})^{+})^{\langle\sigma\rangle}}$}}}}}}}{\hbox{\kern
173.40541pt\raise-41.7511pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces$

####  2.4.2. The Newton poset

Given $w\in\widetilde{W}$, define

$B(\mathbf{G})_{w}:=\\{[b]\in
B(\mathbf{G}):[b]\cap\breve{I}w\breve{I}\neq\emptyset\\}.$

It is easy to see that $B(\mathbf{G})_{w}$ has an unique maximal element
$b_{w}$, which coincides with the generic $\sigma$-conjugacy class in
$\breve{I}w\breve{I}$. In the case of a split group, an explicit formula for
the Newton point of this maximal element was first obtained in [Mil21] under
certain superregularity hypothesis on the dominant cocharacter associated with
$w$. We need the following strengthened formula for the elements in shrunken
chambers in the context of quasi-split groups, obtained in [HN21]. We will
write $\nu_{w}$ to denote $\nu(b_{w})$.

######  Theorem 2.6.

[HN21, proposition 3.1] Suppose that $\mathbf{G}$ is quasi-split and adjoint
over $F$. Let $x,y\in W$ and $\underline{\mu}\in X_{*}(T)_{\Gamma_{0}}^{+}$ be
such that $\text{depth}(\underline{\mu})\geq 2$. Then
$\nu_{xt^{\underline{\mu}}y}$ is the average of $\sigma$-orbit of
$\underline{\mu}-\text{wt}(y^{-1},\sigma(x))$.

####  2.4.3. Neutrally acceptable elements

Let $\\{\mu\\}$ be a conjugacy class of cocharacters over $\bar{F}$. Choose
$\mu$ be a dominant representative of $\\{\mu\\}$ and denote by
$\underline{\mu}$ its image in $X_{*}(T)_{\Gamma_{0}}$. We define the
$\mu$-admissible set

$\operatorname{Adm}(\mu)=\\{w\in\widetilde{W};\ w\leqslant
t^{x(\underline{\mu})}\text{ for some }x\in W\\}.$

We also have the set of neutrally acceptable elements

$B(\mathbf{G},\mu)=\\{[b]\in
B(\mathbf{G})\mid\kappa([b])=\mu^{\natural},\nu([b])\leq\mu^{\diamond}\\}.$

Here $\mu^{\natural}$ denotes the common image of $\mu\in\\{\mu\\}$ in
$\pi_{1}(\mathbf{G})_{\sigma}$, and $\mu^{\diamond}$ denotes the average of
the $\sigma_{0}$-orbit of $\underline{\mu}$. Here $\sigma_{0}$ denotes the
$L$-action of $\sigma$, see [GHN20, definition 2.1] for details. The set
$B(\mathbf{G},\mu)$ inherits a partial order from $B(\mathbf{G})$. Since the
Kottwitz map $\kappa$ is constant on $B(\mathbf{G},\mu)$, we may view it as a
subset of $X_{*}(T)_{\Gamma_{0},\mathbb{Q}}$ via the Newton map. The following
description of $B(\mathbf{G},\mu)$ is obtained in [HN18, theorem 1.1 & lemma
2.5]. To state the result, for each $\sigma_{0}$ orbit $\mathcal{O}$ on
$\mathbb{S}$, we set
$\varpi_{\mathcal{O}}=\sum\limits_{i\in\mathcal{O}}\varpi_{i}$.

######  Theorem 2.7.

[HN18] (1) Let $v\in X_{*}(T)_{\Gamma_{0},\mathbb{Q}}$. Then $v\in
B(\mathbf{G},\mu)$ if and only if $\sigma_{0}(v)=v$ is dominant, and for any
$\sigma_{0}$-orbit $\mathcal{O}$ on $\mathbb{S}$ with $\langle
v,\alpha_{i}\rangle\neq 0$ for each (or equivalently, some) $i\in\mathcal{O}$,
we have that $\langle\mu+\sigma(0)-v,\varpi_{\mathcal{O}}\rangle\in\mathbb{Z}$
and $\langle\mu-v,\varpi_{\mathcal{O}}\rangle\geq 0$.

(2) The set $B(\mathbf{G},\mu)$ contains a unique maximal element.

###  2.5. Affine Deligne-Lusztig varieties

The _affine Deligne-Lusztig variety_ associated to $w\in\tilde{W}$ and
$b\in\breve{G}$ is a locally closed subscheme of the affine flag variety of
$\bf G$, locally of finite type over $\overline{\mathbb{F}}_{p}$ and of finite
dimension, with the set of geometric points given by

$X_{w}(b)(\overline{\mathbb{F}}_{p})=\\{g\breve{I}\in\breve{G}/\breve{I}\mid
g^{-1}b\sigma(g)\in\breve{I}w\breve{I}\\}.$

If $F$ is of equal characteristic, then by affine flag variety we mean the
“usual” affine flag variety; in the case of mixed characteristic, this notion
should be understood in the sense of perfect schemes, as developed by Zhu
[Zhu17] and by Bhatt and Scholze [BS17].

We now define

$X(\mu,b)=\bigcup\limits_{w\in\operatorname{Adm}(\mu)}X_{w}(b).$

Settling the Kottwitz-Rapoport conjecture made in [KR03] and [Rap05] about the
non-emptiness pattern for $X(\mu,b)$, He proves the following result in
[He16].

######  Theorem 2.8.

[He16a, theorem A] $X(\mu,b)\neq\emptyset$ if and only if $[b]\in
B(\mathbf{G},\mu)$.

Note that by theorem 2.8, we have

$B(\mathbf{G},\mu)=\bigcup\limits_{w\in\operatorname{Adm}(\mu)}B(\mathbf{G})_{w},$

hence $b_{\max}=\max\limits_{w\in\operatorname{Adm}(\mu)}b_{w}$. Since
$w[b_{w}]$ is an increasing function $\widetilde{W}\rightarrow
X_{*}(T)_{\Gamma_{0},\mathbb{Q}}$, we conclude that

$b_{\max}=\max\limits_{x\in W}b_{t^{x(\mu)}}.$

##  3\. Explicating $b_{\max}$ via generic Newton point

###  3.1. Dimension of a generic ADLV

Following [He21], we define the $n$-th $\sigma$-twisted Demazure power of $w$
by setting

$w^{\sigma,n}=w*\sigma(w)*\sigma^{2}(w)*\cdots*\sigma^{n-1}(w).$

We need the following result, which on the one hand describes the dimension of
a single affine Deligne-Lusztig variety associated with generic
$\sigma$-conjugacy class, and on the other hand quantifies such generic
$\sigma$-conjugacy class in terms of $\sigma$-twisted Demazure power. For the
first equality below, see [He16, theorem 2.23] and [MV20, lemma 3.2], whereas
for the second equality, see [He21, theorem 0.1].

######  Theorem 3.1.

Let $w\in\widetilde{W}$. Then $\dim X_{w}(b_{w})=\ell(w)-\langle
2\rho,\nu_{w}\rangle=\ell(w)-\lim\limits_{n\rightarrow\infty}\frac{\ell(w^{*_{\sigma},n})}{n}$.

Note that there can be elements $w\in\widetilde{W}$ that are not pure
translations but still satisfy $b_{w}=b_{\max}$; this can be seen e.g. using
the Deligne-Lusztig reduction method, see [He14, proposition 4.2]. However, if
$w$ contributes to top dimensional components - i.e., $\dim
X(\mu,b_{\max})=\dim X_{w}(b_{\max})$ \- then $w$ is necessarily a translation
element of the form $t^{x\underline{\mu}}$ for some $x\in W$, by the first
equality in theorem 3.1.

Now, again by an application of the first equality in theorem 3.1 we see that
$\dim X_{t^{x\underline{\mu}}}(b_{t^{x\underline{\mu}}})$ is a decreasing
function of $b_{t^{x\underline{\mu}}}$; therefore, finding top dimensional
$X_{w}(b_{\max})$ inside $X(\mu,b_{\max})$ boils down to understanding
elements $x\in W$ such that $\dim
X_{t^{x\underline{\mu}}}(b_{t^{x\underline{\mu}}})$ is minimized, and this
minimum value of the dimension is indeed the dimension of $X(\mu,b_{\max})$.
We approach the problem of computing $\dim
X_{t^{x\underline{\mu}}}(b_{t^{x\underline{\mu}}})$ below in two different
ways. In the rest of the paper, we omit the underline for simplicity and write
$\mu$ instead of $\underline{\mu}$ throughout.

###  3.2. A standard reduction

Let $\mathbf{G}$ be a connected reductive group over $F$, and let
$\mathbf{G}_{\text{ad}}$ be its adjoint group. Let $T_{\text{ad}}$ be the
image of $T$ in $\mathbf{G}_{\text{ad}}$, and denote by $\mu_{\text{ad}}$ the
image of $\mu$ in $X_{*}(T_{\text{ad}})_{\Gamma_{0}}$. Similarly for any
$b\in\breve{G}$, we denote by $b_{\text{ad}}$ its image in
$\breve{G}_{\text{ad}}$. By [Kot97, proposition 4.10], we then have an
isomorphism of posets

$B(\mathbf{G},\mu)\rightarrow
B(\mathbf{G}_{\text{ad}},\mu_{\text{ad}}),\text{~{}via~{}}[b][b_{\text{ad}}].$

Next, we have a decomposition
$\mathbf{G}_{\text{ad}}\simeq\mathbf{G}_{1}\times\cdots\times\mathbf{G}_{r}$,
where each $\mathbf{G}_{i}$ is adjoint and simple over F. Write
$\mu=(\mu_{1},\cdots,\mu_{r})$, where $\mu_{i}$ is a dominant coweight of
$\mathbf{G}_{i}$. This way we can identify

$B(\mathbf{G}_{\text{ad}},\mu)=\prod B(\mathbf{G}_{i},\mu_{i}).$

We also have

$\dim X^{\mathbf{G}}(\mu,b)=\dim
X^{\mathbf{G}_{\text{ad}}}(\mu_{\text{ad}},b_{\text{ad}})=\sum\dim
X^{\mathbf{G}_{i}}(\mu_{\text{ad},i},b_{\text{ad},i}).$

Hence, it suffices to focus on adjoint $F$-simple groups. We work with the
absolute local Dynkin diagram (i.e., the affine Dynkin diagram attached to
$\mathbf{G}$ over $\breve{F}$), together with the diagram automorphism induced
by Frobenius.

We can describe such groups in terms of tuples $(\widetilde{W},\sigma)$, where
$\widetilde{W}$ is the associated Iwahori-Weyl group with its finite root
datum being irreducible of adjoint type, and
$\sigma=\text{Ad}(\tau\sigma_{0})$ is a length preserving automorphism of
$\widetilde{W}$ with $\tau\in\Omega,\sigma_{0}(\mathbb{S})=\mathbb{S}$. Thus
$\text{Ad}\tau$ (resp. $\text{Ad}\sigma_{0}$) corresponds to a symmetry of the
affine (resp. finite) Dynkin diagram. More concretely, $\tau$ can be taken to
be $\tau_{i}=t^{\varpi_{i}^{\vee}}w_{i,0}w_{0}$ whenever $\varpi_{i}^{\vee}$
is a minuscule fundamental coweight, and $\sigma_{0}$ can be nontrivial only
when $W$ is of type $A_{n},D_{n}(n\geq 5)$ and $E_{6}$. Throughout section 3
and section 5, we denote by $\zeta$ the finite Weyl group part of $\tau$, i.e.
$\zeta=\zeta_{i}:=w_{i,0}w_{0}$ for some minuscule fundamental coweight
$\varpi_{i}^{\vee}$ \- often suppressing the label $i$ as well. We henceforth
assume that $\text{depth}(\mu)\geq 2$.

###  3.3. via Demazure power

Here we express $\langle 2\rho,\nu_{t^{x(\mu)}}\rangle$ explicitly using
Demazure power of $t^{x(\mu)}$.

#### 3.3.1.

To ease notational burden, we separately discuss the case with trivial
$\sigma_{0}$ first. Suppose that $o(\zeta)=m$. Then we have

$\displaystyle(t^{x\mu})^{*_{\sigma},m+1}$
$\displaystyle=(xt^{\mu}x^{-1})*(\zeta
xt^{\mu}x^{-1}\zeta^{-1})*\cdots*(\zeta^{m-1}xt^{\mu}x^{-1}\zeta^{1-m})*(xt^{\mu}x^{-1})$
$\displaystyle=xt^{\mu_{1}}x^{-1},\text{~{}with~{}}\mu_{1}:=(m+1)\mu-\sum\limits_{i=0}^{m-1}\text{wt}(\zeta^{i}x,\zeta^{i+1}x).$

Here the second equality follows from a repeated application of theorem 2.3.
It is now easy to see that for any positive integer $k$, we have

$(t^{x\mu})^{*_{\sigma},km+1}=xt^{\mu_{k}}x^{-1},\text{~{}with~{}}\mu_{k}:=(km+1)\mu-k\sum\limits_{j=0}^{m-1}\text{wt}(\zeta^{j}x,\zeta^{j+1}x).$

Note that $\mu_{k}$ is dominant under the depth hypothesis on $\mu$ and we
have $\frac{1}{km+1}\ell((t^{x\mu})^{*_{\sigma},km+1})=\langle
2\rho,\mu\rangle-\frac{k}{km+1}\langle
2\rho,\sum\limits_{j=0}^{m-1}\text{wt}(\zeta^{j}x,\zeta^{j+1}x)\rangle$. Since
the limit in theorem 3.1 exists, we deduce

$\lim\limits_{n\rightarrow\infty}\frac{1}{n}\ell((t^{x\mu})^{*_{\sigma},n})=\lim\limits_{k\rightarrow\infty}\frac{1}{km+1}\ell((t^{x\mu})^{*_{\sigma},km+1})=\langle
2\rho,\mu\rangle-\frac{1}{m}\langle
2\rho,\sum\limits_{j=0}^{m-1}\text{wt}(\zeta^{j}x,\zeta^{j+1}x)\rangle.$

Thus by theorem 3.1, $\dim X_{t^{x\mu}}(b_{t^{x\mu}})=\frac{1}{m}\langle
2\rho,\sum\limits_{j=0}^{m-1}\text{wt}(\zeta^{j}x,\zeta^{j+1}x)\rangle$.

#### 3.3.2.

Now we consider the general case, where $\sigma_{0}$ may be nontrivial. Then

$\sigma^{j}(t^{x\mu})=\text{Ad}((\zeta\sigma_{0})^{j})(t^{x\mu}))=\text{Ad}(\zeta^{\sigma_{0},j})(t^{\sigma_{0}^{j}(x\mu)}),$

where we recall
$\zeta^{\sigma_{0},j}=\zeta\sigma_{0}(\zeta)\sigma_{0}^{2}(\zeta)\cdots\sigma_{0}^{j-1}(\zeta)$.
Let $m^{\prime}$ denote the order of $\zeta\sigma_{0}$ inside
$W\rtimes\langle\sigma_{0}\rangle\subset\operatorname{Aut}(W)$; since the
group is adjoint, this is also the order of $\sigma$ as an automorphism of the
subset of translation elements in $\widetilde{W}$.

Then a similar calculation as above shows for any positive integer $k$,

$(t^{x\mu})^{*_{\sigma},km^{\prime}+1}=xt^{\mu_{k}^{\prime}}x^{-1},\text{~{}with~{}}\mu_{k}^{\prime}=\mu+k\sum\limits_{j=0}^{m^{\prime}-1}\sigma_{0}^{j}(\mu)-k\sum\limits_{j=0}^{m^{\prime}-1}\text{wt}(\zeta^{\sigma_{0},j}\sigma_{0}^{j}(x),\zeta^{\sigma_{0},j+1}\sigma_{0}^{j+1}(x)).$

In other words, the dominant translation part of
$(t^{x\mu})^{*_{\sigma},km^{\prime}+1}$ is

$\mu_{k}^{\prime}:=\mu+km^{\prime}\mu^{\diamond}-k\sum\limits_{j=0}^{m^{\prime}-1}\text{wt}(\zeta\sigma_{0})^{j}(x),(\zeta\sigma_{0})^{j+1}(x)).$
(3.1)

Noting that $\langle 2\rho,\mu\rangle=\langle
2\rho,\sigma_{0}^{j}(\mu)\rangle$ as $2\rho$ is
$\langle\sigma_{0}\rangle$-invariant, we then proceed in the same way and
conclude $\dim X_{t^{x\mu}}(b_{t^{x\mu}})=\frac{1}{m^{\prime}}\langle
2\rho,\sum\limits_{j=0}^{m^{\prime}-1}\text{wt}(\zeta\sigma_{0})^{j}(x),(\zeta\sigma_{0})^{j+1}(x))\rangle$.

Rewriting the dimension formula obtained above using lemma 2.2, we deduce the
following.

######  Proposition 3.2.

Let $\mathbf{G}$ be an adjoint $F$-simple group and $x\in W$ be an element in
its Weyl group. Suppose that $\sigma=\text{Ad}(\tau\sigma_{0})$ is the
Frobenius, with $\tau\in\Omega$ and $\sigma_{0}(\mathbb{S})=\mathbb{S}$. Let
$\zeta$ be the finite Weyl group component of $\tau$, and assume that
$\text{depth}(\mu)\geq 2$. Finally, let $o_{\text{tr}}(\sigma)$ denote the
order of $\sigma$ as an element of
$\operatorname{Aut}(X_{*}(T)_{\Gamma_{0}})$. Then

$\dim
X_{t^{x\mu}}(b_{t^{x\mu}})=\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}d(\zeta\sigma_{0})^{j}(x),(\zeta\sigma_{0})^{j+1}(x)).$

###  3.4. via reduction to quasi-split case

Recall that $b_{\max}=\max\limits_{x\in W}b_{t^{x(\mu)}}$. For computation of
the generic Newton point, we can further pass on to a quasi-split group in the
following way. For an adjoint $F$-simple group $\mathbf{G}$ as in the previous
subsection, let $\mathbf{G}^{*}$ be its quasi-split inner form; then the
Frobenius action for $\mathbf{G}^{*}$ is given by
$\sigma_{\mathbf{G}^{*}}=\sigma_{0}$. By [GHN16, §2.4], we can identify
$B(\mathbf{G})=B(\mathbf{G}^{*})$ via $[b]\rightarrow[b\dot{\tau}]$, and this
bijection preserves the partial order defined via the closure relations in
$\breve{G}=\breve{G}^{*}$. Hence, restricting this map to a natural bijection
$B(\mathbf{G})_{w}=B(\mathbf{G}^{*})_{w\tau}$ for any
$w\in\widetilde{W}(\mathbf{G})=\widetilde{W}(\mathbf{G}^{*})$, we then get
$\nu^{\mathbf{G}}([b_{w}])=\nu^{\mathbf{G}^{*}}([b_{w}\dot{\tau}])=\nu^{\mathbf{G}^{*}}([b_{w\tau}])$.

We now give alternative (in fact, simpler) formula for the dimension by
relating it to the quasi-split case.

######  Proposition 3.3.

In the setup of proposition 3.2, we have $\dim
X_{t^{x\mu}}(b_{t^{x\mu}})=d(\zeta^{-1}x,\sigma_{0}(x))$.

###### Proof.

Assume first that $\text{depth}(\mu)\geq 3$. Note that
$t^{x\mu}\tau=t^{x\mu}t^{\varpi_{i}^{\vee}}w_{i,0}w_{0}=xt^{\mu+x^{-1}\varpi_{i}^{\vee}}x^{-1}w_{i,0}w_{0}$.
Here $\mu+x^{-1}\varpi_{i}^{\vee}$ is dominant with depth at least $2$: for
any simple root $\alpha$, we have

$\langle\mu+x^{-1}\varpi_{i}^{\vee},\alpha\rangle=\langle\mu,\alpha\rangle+\langle\varpi_{i}^{\vee},x\alpha\rangle\geq\langle\mu,\alpha\rangle+\langle\varpi_{i}^{\vee},-\theta\rangle=\langle\mu,\alpha\rangle-1\geq
2.$

Hence, by theorem 2.6 we have
$\nu^{\mathbf{G}}(b_{t^{x\mu}})=\nu^{\mathbf{G}^{*}}([xt^{\mu+x^{-1}\varpi_{i}^{\vee}}x^{-1}\zeta])=(\mu+x^{-1}\varpi_{i}^{\vee}-\text{wt}(\zeta^{-1}x,\sigma_{0}(x)))^{\diamond}$,
giving us

$\dim X_{t^{x\mu}}(b_{t^{x\mu}})=\langle
2\rho,\mu-\nu(b_{t^{x(\mu)}})\rangle=\langle
2\rho,\text{wt}(\zeta^{-1}x,\sigma_{0}(x))-x^{-1}\varpi_{i}\rangle.$ (3.2)

Note that we used $\langle\sigma_{0}\rangle$-invariance of $2\rho$ above. Now
using the length formula, we get

$\langle
2\rho,\mu\rangle=\ell(t^{x(\mu)})=\ell(t^{x(\mu)}\tau)=\ell(xt^{\mu+x^{-1}\varpi_{i}^{\vee}}x^{-1}\zeta)=\ell(x)+\langle
2\rho,\mu+x^{-1}\varpi_{i}^{\vee}\rangle-\ell(x^{-1}\zeta),$

giving us $\langle
2\rho,x^{-1}\varpi_{i}^{\vee}\rangle=\ell(x^{-1}\zeta)-\ell(x)$. Plugging this
in eq. 3.2 and applying lemma 2.2, we then get

$\dim X_{t^{x\mu}}(b_{t^{x\mu}})=\langle
2\rho,\text{wt}(\zeta^{-1}x,\sigma_{0}(x))\rangle-\ell(x^{-1}\zeta)+\ell(x)=d(\zeta^{-1}x,\sigma_{0}(x)).$

Combining corollary 3.4 and proposition 3.2, we see that this last formula is
valid even under the weaker hypothesis that $\text{depth}(\mu)\geq 2$. Hence,
we are done. ∎

Let us record an immediate corollary of proposition 3.3 and proposition 3.2.

######  Corollary 3.4.

Let $x\in W$. Suppose that $\sigma_{0}$ is an automorphism of $W$ with
$\sigma_{0}(\mathbb{S})=\mathbb{S}$, and $\zeta=\zeta_{i}=w_{i,0}w_{0}$ for
some minuscule fundamental coweight $\varpi^{\vee}_{i}$. Finally, let
$o_{\text{tr}}(\sigma)=\min\\{j:\zeta^{\sigma_{0},j}=1\\}$. Then

$\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}d(\zeta\sigma_{0})^{j}(x),(\zeta\sigma_{0})^{j+1}(x))=d(\zeta^{-1}x,\sigma_{0}(x)).$

###  3.5. Finishing the proof

As an consequence of the dimension formulas in the preceeding sections, we
obtain the following estimate.

######  Proposition 3.5.

In the setup of proposition 3.2, we have

$\ell(x^{-1}\zeta\sigma_{0}(x))\geq\dim
X_{t^{x\mu}}(b_{t^{x\mu}})\geq\ell_{R}(x^{-1}\zeta\sigma_{0}(x)).$

###### Proof.

We can prove it using the dimension formula established in either proposition
3.2 or proposition 3.3, but we choose to resort to the latter for notational
ease. Then the assertion follows from the easy observation:

(a) for any two elements $a,b\in W$, we have $\ell(a^{-1}b)\geq
d(a,b)\geq\ell_{R}(a^{-1}b)$.

Taking any reduced expression for $a^{-1}b=s_{i_{1}}\cdots s_{i_{k}}$ and
following the edges determined by the simple roots
$\alpha_{i_{k}},\cdots,\alpha_{i_{1}}$ in order, we obtain a path from $a$ to
$b$ which has exactly $\ell(a^{-1}b)$ edges. This proves the first inequality.
For the other one, let $d=d(a,b)$ and choose a shortest path $a\rightarrow
ar_{1}\rightarrow ar_{1}r_{2}\cdots\rightarrow ar_{1}\cdots r_{d}=b$, where
$r_{i}$’s are not necessarily simple roots. Then $a^{-1}b=r_{1}\cdots r_{d}$
is a decomposition of $a^{-1}b$ into reflections, proving the other
inequality. ∎

The reflection length of $w\in W$ is the smallest non-negative integer $l$
such that $w$ can be written as a product of $l$ reflections in $W$. We denote
by $\ell_{R}(w)$ the reflection length of $w$. Let $\xi\in W$. Then we denote
the $\sigma_{0}$ conjugacy class of $\xi$ in $W$ by
$\mathcal{O}_{\sigma_{0}}(\xi)$. We define
$\ell_{R}(\mathcal{O}_{\sigma_{0}}(\xi))=\min\\{\ell_{R}(w):w\in\mathcal{O}_{\sigma_{0}}(\xi)\\}$.
We can define $\ell(\mathcal{O}_{\sigma_{0}}(\xi))$ similarly.

###### Proof of theorem 1.1.

Since $\dim X(\mu,b_{\max})=\min\limits_{x\in W}\dim
X_{t^{x\mu}}(b_{t^{x\mu}})$, we see from proposition 3.5

$\ell(\mathcal{O}_{\sigma_{0}}(\zeta))\geq\dim
X(\mu,b_{\max})\geq\ell_{R}(\mathcal{O}_{\sigma_{0}}(\zeta)).$

By lemma 4.1 proved in section 4, we have
$\ell(\mathcal{O}_{\sigma_{0}}(\zeta))=\ell_{R}(\mathcal{O}_{\sigma_{0}}(\zeta))=|\sigma_{0}\backslash\widetilde{\mathbb{S}}|-|\sigma\backslash\widetilde{\mathbb{S}}|$,
hence it equals the dimension. By proposition 2.4,
$|\sigma_{0}\backslash\widetilde{\mathbb{S}}|-|\sigma\backslash\widetilde{\mathbb{S}}|=\text{rk}_{F}^{\text{ss}}\mathbf{G}^{*}-\text{rk}_{F}^{\text{ss}}\mathbf{G}$,
and we are done. ∎

######  Remark 3.6.

As a byproduct of our discussion, we obtain that

$\ell_{R}(\mathcal{(}O_{\sigma_{0}}(\zeta))=\min\limits_{x\in
W}d(\zeta^{-1}x,\sigma_{0}(x))$

for the element $\zeta=w_{i,0}w_{0}$, whenever $\varpi_{i}^{\vee}$ is
minuscule. Contrast this with theorem 5.1 in [HY21] that says

$\ell_{R}(\mathcal{O}_{\sigma_{0}}(w))=\min\limits_{x\in
W}d(x,\sigma_{0}(x)w)$

for the element $w=w_{0}$. It would be an interesting problem to explore
connection between reflection length and distance in the quantum Bruhat graph
for arbitrary elements of $W$.

##  4\. Minimal length elements in certain $\sigma_{0}$-conjugacy class

For simplicity, in the rest of the paper, we will say $\sigma_{0}$-orbits in
$\widetilde{\mathbb{S}}$ instead of $\text{Ad}(\sigma_{0})$-orbits, likewise
for $\text{Ad}(\sigma)$ etc. The goal of this section is to prove the
following result via a case-by-case analysis.

######  Lemma 4.1.

Suppose that $(W,\mathbb{S})$ is an irreducible finite Coxeter system
associated to a Weyl group $W$ and let $\widetilde{\mathbb{S}}$ be the
associated set of affine simple roots. Let $\sigma_{0}$ be a diagram
automorphism of $(W,\mathbb{S})$ and
$\tau=t^{\varpi_{i}^{\vee}}w_{i,0}w_{0}\in\widetilde{W}$, where
$\varpi_{i}^{\vee}$ is a minuscule fundamental coweight. Then there exists a
subset $J\subset\mathbb{S}$ of cardinality
$|\sigma_{0}\backslash\widetilde{\mathbb{S}}|-|\tau\sigma_{0}\backslash\widetilde{\mathbb{S}}|$
such that the set $\mathcal{O}_{\min,\sigma_{0}}(w_{i,0}w_{0})$ of minimal
length elements in $\mathcal{O}_{\sigma_{0}}(w_{i,0}w_{0})$ consists of
Coxeter elements of $W_{J}$. This subset $J$ is unique, unless
$(W,\varpi_{i}^{\vee})$ is of the form $(^{2}D_{2n},\varpi_{2n-1}^{\vee})$ or
$(^{2}D_{2n},\varpi_{2n}^{\vee})$.

###### Proof.

As our calculation below shows, the set $J$ is either empty or $W_{J}$ has
only one Coxeter element in the cases where $\sigma_{0}$ is nontrivial. As for
the cases where $\sigma_{0}$ is trivial, we only have to show some Coxeter
element from asserted $W_{J}$ belongs to
$\mathcal{O}_{\min,\sigma_{0}}(w_{i,0}w_{0})$ \- since all such Coxeter
elements are conjugate to each other.

We follow the labelling of [Bou02]. To simplify notation, in the exceptional
types we may simply write $s_{ij\cdots}$ for $s_{i}s_{j}\cdots$. For $1\leq
a\leq b\leq n$, we abbreviate $s_{[a,b]}=s_{a}s_{a+1}\cdots s_{b}$. Below we
group together the cases according to their absolute local Dynkin diagram.

  1. (1)

Type $A_{n}$: Here all the fundamental coweights are minuscule and $\tau_{i}$
acts on $\widetilde{}\mathbb{S}$ by $i$-step rotations, thus the number of
$\tau_{i}$-orbits on $\widetilde{S}$ are
$\frac{|\widetilde{}\mathbb{S}|}{|o(\text{Ad~{}}\tau_{i})|}=\frac{n+1}{\gcd(n+1,i)}$.

Here $\zeta_{1}=s_{[1,n]}$ is a Coxeter element for $W$ and
$\zeta_{i}=\zeta_{1}^{i}$ for $i\geq 2$. Set
$\kappa_{i}=\frac{n+1}{\gcd(n+1,i)}$ and let $m$ be the largest integer
smaller than $\frac{n}{\kappa_{i}}$, i.e.
$m=\lfloor\frac{n}{\kappa_{i}}\rfloor$. Then we have

$J_{i}=\\{1,\cdots,\kappa_{i}-1\\}\cup\\{\kappa_{i+1},\cdots,2\kappa_{i}-1\\}\cup\cdots\cup\\{m\kappa_{i}+1,\cdots,n\\}.$

  2. (2)

Type ${}^{2}A_{n}$: The nontrivial diagram automorphism $\sigma_{0}$ of
$\mathbb{S}$ is conjugation by $w_{0}$, hence minimal length elements in
$\mathcal{O}_{\sigma_{0}}(w_{i,0}w_{0})$ are maximal length elements in
$\mathcal{O}_{\text{id}}(w_{i,0})$ multiplied with $w_{0}$ on the right. The
orbits of $\sigma_{0}$ on $\widetilde{\mathbb{S}}$ are

$\begin{cases}\\{0\\},\\{i,n+1-i\\}\text{~{}for~{}}1\leq
i\leq\frac{n}{2}&,\text{if $n$ is even;}\\\
\\{0\\},\\{\frac{n+1}{2}\\},\\{i,n+1-i\\}\text{~{}for~{}}1\leq
i\leq\frac{n-1}{2}&,\text{if $n$ is odd.}\end{cases}$

In this case, $\tau_{i}\sigma_{0}$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,i\\},\\{j,i-j\\}$ for $0<j<i$, and $\\{i+j,n-j\\}$ for $0<j<n-i$. We
have that $J=\emptyset$ if $n$ is even or $i$ is even, and
$J=\\{\frac{n+1}{2}\\}$ if both $n,i$ are odd.

  3. (3)

Type $B_{n}$: Here the only minuscule coweight is $\varpi_{1}^{\vee}$, so
$\tau=\tau_{1}$; its orbits are $\\{0,1\\},\\{i\\}$ for $2\leq i\leq n$. We
have $\zeta_{1}=s_{[1,n-1]}s_{n}s_{[1,n-1]}^{-1}$, thus $J=\\{n\\}.$

  4. (4)

Type $C_{n}$: Here the only minuscule coweight is $\varpi_{n}^{\vee}$, so
$\tau=\tau_{n}$; its orbits are

$\begin{cases}\\{i,n+1-i\\},\text{~{}for~{}}0\leq i\leq\frac{n-1}{2}&,\text{if
$n$ is odd;}\\\ \\{\frac{n}{2}\\},\\{i,n+1-i\\},\text{~{}for~{}}0\leq
i\leq\frac{n}{2}-1&,\text{if $n$ is even.}\end{cases}$

We have $\zeta_{n}=s_{n}s_{[n-1,n]}\cdots s_{[2,n]}s_{[1,n]}$ and
$J=\\{i:1\leq i\leq n,i\text{~{}is odd}\\}$.

  5. (5)

Type $D_{n}$: Here we have three minuscule coweights:
$\varpi_{1}^{\vee},\varpi_{n-1}^{\vee},\varpi_{n}^{\vee}$. There is an outer
diagram automorphism of $D_{n}$ permuting the last two coweights. Thus it
suffices to consider the case where $\tau=\tau_{1}$ or $\tau_{n}$.

Case 1: $\tau=\tau_{1}$. The $\tau$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,1\\},\\{n-1,n\\}$ and $\\{i\\}$ for $2\leq i\leq n-2$. We have
$\zeta=\zeta_{1}=s_{[1,n-2]}s_{[1,n]}^{-1}$ and $J=\\{n-1,n\\}$.

Case 2: $\tau=\tau_{n}$; then there are two sub-cases.

     * •

$n$ is odd: The $\tau$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,n,1,n-1\\}$ and $\\{i,n-i\\}$ for $2\leq i\leq\frac{n-1}{2}$. Here
$\zeta=\zeta_{n}=s_{n}s_{[n-2,n-1]}s_{[n-3,n-2]}s_{n}\cdots
s_{n}s_{[3,n-1]}s_{[2,n-2]}s_{n}s_{[1,n-1]}$ and $J=\\{i:1\leq i\leq
n-2,i\text{~{}is odd}\\}\cup\\{n-1,n\\}$.

     * •

$n$ is even: The $\tau$-orbits on $\widetilde{\mathbb{S}}$ are $\\{i,n-i\\}$
for $0\leq i\leq\frac{n}{2}$. Here
$\zeta=\zeta_{n}=s_{n}s_{[n-2,n-1]}s_{[n-3,n-2]}s_{n}$ $\cdots
s_{n}s_{[2,n-1]}s_{[1,n-2]}s_{n}$ and

$J=\begin{cases}\\{i:1\leq i\leq n,i\text{~{}is odd}\\},&,\text{if $n$ is
congruent to $0$ mod $4$;}\\\ \\{i:1\leq i\leq n-2,i\text{~{} is
odd}\\}\cup\\{n\\}&,\text{if $n$ is congruent to $2$ mod $4$.}\end{cases}$

  6. (6)

Type ${}^{2}D_{n}$: The nontrivial diagram automorphism $\sigma_{0}$ of
$\mathbb{S}$ here swaps the vertices labeled $n-1,n$; the orbits on
$\widetilde{\mathbb{S}}$ are therefore $\\{n-1,n\\},\\{i\\}$ for $0\leq i\leq
n-2$. As explained before, we have the following three cases.

Case 1: $\tau=\tau_{1}$. The $\tau\sigma_{0}$-orbits on
$\widetilde{\mathbb{S}}$ are $\\{0,1\\}$ and $\\{i\\}$ for $2\leq i\leq n$.
One may directly check that $1\in\mathcal{O}_{\sigma_{0}}(\zeta_{1})$ and
hence $J=\emptyset$.

Case 2: $\tau=\tau_{n}$; then there are two sub-cases.

     * •

$n$ is odd: The $\tau\sigma_{0}$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{i,n-i\\}$ for $0\leq i\leq\frac{n-1}{2}$, and $J=\\{i:1\leq i\leq
n-2,i\text{~{}is odd}\\}$.

     * •

$n$ is even: The $\tau\sigma_{0}$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,1,n-1,n\\}$ and $\\{i,n-i\\}$ for $2\leq i\leq\frac{n}{2}$, and the
choices for $J$ are $\\{i:1\leq i\leq n,i\text{~{}is odd}\\}$ and $\\{i:1\leq
i\leq n-2,i\text{~{}is odd}\\}\cup\\{n\\}$.

  7. (7)

Type ${}^{3}D_{4}$: Without loss of generality, we may assume that
$\sigma_{0}$ is the outer diagram automorphism on $D_{4}$ sending
$1\rightarrow 3\rightarrow 4\rightarrow 1$. As $\langle\sigma_{0}\rangle$ acts
transitively on $\\{1,3,4\\}$, it suffices to consider the case where
$\tau=\tau_{1}$. In that case, the orbits of $\tau\sigma_{0}$ on
$\widetilde{\mathbb{S}}$ are $\\{0,1,4\\},\\{2\\},\\{3\\}$. Finally, one
directly checks that $1\in\mathcal{O}_{\sigma_{0}}(\zeta_{1})$, whence
$J=\emptyset$.

  8. (8)

Type $E_{6}$: There are two minuscule coweights:
$\varpi_{1}^{\vee},\varpi_{6}^{\vee}$. The unique outer diagram automorphism
of $E_{6}$ permutes these two coweights. Thus it suffices to consider the case
where $\tau=\tau_{1}$; its orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,1,6\\},\\{2,3,5\\},\\{4\\}$. We have
$\zeta=\zeta_{1}=s_{1345624534132456}$, and $J=\\{1,3,5,6\\}$.

  9. (9)

Type ${}^{2}E_{6}$: As explained above, it suffices to consider the case where
$\tau=\tau_{1}$. Here $\sigma_{0}$ acts on $\widetilde{\mathbb{S}}$ by
swapping $1\leftrightarrow 6,3\leftrightarrow 5$, so its orbit on
$\widetilde{\mathbb{S}}$ are $\\{0\\},\\{1,6\\},\\{2\\},\\{3,5\\},\\{4\\}$.
Also, the orbits of $\tau\sigma_{0}$ on $\widetilde{\mathbb{S}}$ are
$\\{0,1\\},\\{2,3\\},\\{4\\},\\{5\\},\\{6\\}$. Finally, one directly checks
that $1\in\mathcal{O}_{\sigma_{0}}(\zeta_{1})$, whence $J=\emptyset$.

  10. (10)

Type $E_{7}$: There is a unique minuscule coweight: $\varpi_{7}^{\vee}$, so
$\tau=\tau_{7}$. In this case, $\tau$-orbits on $\widetilde{\mathbb{S}}$ are
$\\{0,7\\},\\{1,6\\},\\{3,5\\},\\{2\\},\\{4\\}$. Here
$\zeta=\zeta_{7}=\zeta=s_{765432456713456245341324567}$, and $J=\\{2,5,7\\}$.

∎

######  Remark 4.2.

In type $A_{n}$, the elements $\zeta_{i}$ are regular elements of the Weyl
group. In that case, the asserted conjugacy relations follow more conceptually
from [Spr74, theorem 4.2(4)].

##  5\. Explicit description of the Newton point of $b_{\max}$

Note that the existence of $b_{\max}$ as in theorem 2.7 was asserted in [HN18]
via reduction to a similar statement on the associated Iwahori-Weyl group, and
then they further reduce it to treat the superbasic case and conclude it from
there. However, this is somewhat indirect and does not give an explicit
formula for the maximal element. Based on our calculation, in this section we
list out $\nu(b_{\max})$ for the cases treated in lemma 4.1.

### 5.1.

Recall that the calculation in section 3 combined with theorem 3.1 gives us

$\langle 2\rho,\nu_{t^{x\mu}}\rangle=\langle
2\rho,\mu^{\diamond}-\text{av}_{\sigma}(x)\rangle,$ (5.1)

where we set
$\text{av}_{\sigma}(x):=\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{i=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}((\zeta\sigma_{0})^{i}(x),(\zeta\sigma_{0})^{i+1}(x))$
However, one can upgrade this.

######  Proposition 5.1.

Suppose that we are in the setup of proposition 3.2. Let us further assume
that $\nu_{t^{x\mu}}$ is regular. Then we have

$\nu_{t^{x\mu}}=\mu^{\diamond}-\text{av}_{\sigma}(x).$

######  Lemma 5.2.

Let $w\in\widetilde{W}$. Then there is a straight element
$w_{s}\in\widetilde{W}$ such that $w_{s}\leq w$ and $\nu(w_{s})=\nu_{w}$.

###### Proof.

This essentially follows from the arguments in [He21, section 2.2]. ∎

###### Proof of proposition 5.1.

Let $w^{\prime}=t^{x(\mu)}$ and choose a straight element $w_{s}$
corresponding to $w:=w^{\prime}\tau$ as given in lemma 5.2. By properties of
Demazure product and by definition of straight elements,

$w_{s}^{\sigma_{0},n-1}\leq w_{s}^{\sigma_{0},n}=w_{s}^{*_{\sigma_{0}},n}\leq
w^{*_{\sigma_{0}},n}\text{ for all }n\in\mathbb{N}.$ (5.2)

Note that $w^{\prime*_{\sigma},n}=w^{*_{\sigma_{0}},n}$ whenever
$(\tau\sigma_{0})^{n}=1$, i.e. $o_{\text{tr}}(\sigma)\mid n$. Furthermore, for
such $n$ we have $\sigma^{n}(t^{x\mu})=t^{x\mu},\sigma_{0}^{n}=1$, and hence

$w^{*_{\sigma_{0}},n+1}=(w^{\prime}\tau)^{*_{\sigma_{0}},n}*w^{\prime}\tau=(t^{x\mu})^{*_{\sigma},n}*t^{x\mu}\tau=(t^{x\mu})^{*_{\sigma},n+1}\tau.$
(5.3)

Let us recall the following well-known fact.

(a) Let $\text{pr}:\widetilde{W}\rightarrow X_{*}(T)_{\Gamma_{0}}$ be the
projection map, which sends any element $w$ to the unique dominant coweight
$\lambda$ with $w\in Wt^{\lambda}W$. Assume that $w_{1},w_{2}\in\widetilde{W}$
such that $w_{1}\leq w_{2}$ in Bruhat order. Then
$(\text{pr}(w_{1}))^{+}\leq(\text{pr}(w_{2}))^{+}$ in the dominance order.

Recall now from section 3.3 that $o_{\text{tr}}(\sigma)\mid o(\sigma)$; let
$o(\sigma)=\Xi o_{\text{tr}}(\sigma)$. Then the calculation in section 3.3,
combined with eq. 5.3, shows that for $n=ko(\sigma)+1$ (with
$k\in\mathbb{Z}_{\geq 0}$), we have

$w^{*_{\sigma_{0}},n}=xt^{\mu_{k}^{\prime\prime}}x^{-1}\tau=x^{\mu_{k}^{\prime\prime}+x^{-1}\varpi_{i}^{\vee}}x^{-1}\zeta,\text{
with
}\mu_{k}^{\prime\prime}=\mu+ko(\sigma)\mu^{\diamond}-k\Xi\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}(\zeta\sigma_{0})^{i}(x),(\zeta\sigma_{0})^{i+1}(x)).$

Since $\nu(w_{s})$ is assumed to be regular, we have by [He14, proposition
2.4] that $w_{s}=yt^{\gamma}\sigma_{0}(y)^{-1}$ for some $y\in W$. Using this
in eq. 5.2 with $n=ko(\sigma)+1$ and then applying fact (a), we thus obtain

$ko(\sigma)\nu(w_{s})\leq\mu+x^{-1}\varpi^{\vee}+ko(\sigma)\mu^{\diamond}-k\Xi\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}(\zeta\sigma_{0})^{i}(x),(\zeta\sigma_{0})^{i+1}(x)).$

This gives

$\nu_{w}=\nu(w_{s})\leq\frac{1}{ko(\sigma)}(\mu+x^{-1}\varpi^{\vee})+\mu^{\diamond}-\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}(\zeta\sigma_{0})^{i}(x),(\zeta\sigma_{0})^{i+1}(x)).$

Letting $k$ to be sufficiently large and noting that Newton points have
bounded denominator, we obtain
$\nu_{w}\leq\mu^{\diamond}-\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}(\zeta\sigma_{0})^{i}(x),(\zeta\sigma_{0})^{i+1}(x)).$
Hence by eq. 5.1, we are done.

∎

### 5.2.

To describe $\nu(b_{\max})=\max\limits_{x\in W}\nu_{t^{x\mu}}$, it suffices by
proposition 5.1 to tabulate

$\mu^{\diamond}-\nu(b_{\max})=\min\limits_{x\in W}\text{av}_{\sigma}(x),$
(5.4)

for the cases discussed in lemma 4.1. Note that
$(\zeta\sigma_{0})^{j}(x))^{-1}(\zeta\sigma_{0})^{j+1}(x)=\sigma_{0}^{j}(x^{-1}\zeta\sigma_{0}(x))$.
By a combination of [Sad21, corollary 3.3] and [Sad21, lemma 3.1], we have
$\text{wt}(\zeta\sigma_{0})^{j}(x),(\zeta\sigma_{0})^{j+1}(x))\leq\text{wt}(\sigma_{0}^{j}(x^{-1}\zeta\sigma_{0}(x)),1)$,
whence

$\mu^{\diamond}-\nu(b_{\max})\leq\min\limits_{x\in
W}\frac{1}{o_{\text{tr}}(\sigma)}\sum\limits_{j=0}^{o_{\text{tr}}(\sigma)-1}\text{wt}(\sigma_{0}^{j}(x^{-1}\zeta\sigma_{0}(x)),1).$

Thus $\mu^{\diamond}=\nu(b_{\max})$ whenever
$1\in\mathcal{O}_{\min,\sigma_{0}}(\zeta)$. One can read off such cases from
the list given in the proof of lemma 4.1; those cases can also be directly
seen to be corresponding to quasi-split groups. For the remaining cases, the
proof of theorem 1.1 shows that the minimum in eq. 5.4 is realized at some/any
element $x\in W$ such that $x^{-1}\zeta\sigma_{0}(x)$ is a partial Coxeter
element as described in lemma 4.1. This way, one can evaluate
$\text{av}_{\sigma}(x)$ at such an element $x$ to get the answer.

Alternatively, we can describe this minimum by applying the criterion in
theorem 2.7. Let us call $\Xi_{\sigma}=\min\limits_{x\in
W}\text{av}_{\sigma}(x)=\mu^{\diamond}-\nu(b_{\max})$. Note that, when
$\sigma=\text{Ad}(\tau_{i})$, the condition in theorem 2.7 translates to

$\Xi_{\sigma}=\min\\{\xi:\langle\varpi_{i}^{\vee}+\xi,\varpi_{j}\rangle\in\mathbb{Z},\langle\xi,\varpi_{j}\rangle\geq
0,\text{ for all }j\\}.$

Thus, we can simply use expressions of fundamental coweights in terms of
simple coroots coming from the inverse of the Cartan matrix and find this
quantity. For a real number $m$, we set $\\{m\\}=m-\lfloor m\rfloor$.

  1. (1)

Type $A_{n}$: For $1\leq i\leq n$, we have
$\varpi_{i}^{\vee}=\sum\limits_{j=1}^{n}a_{ij}\alpha_{j}^{\vee}$, where

$a_{ij}=\min(i,j)-\frac{ij}{n+1}=\min(i,j)-\lfloor\frac{ij}{n+1}\rfloor-\\{\frac{ij}{n+1}\\}.$
(5.5)

Hence,
$\Xi_{\sigma}=\sum\limits_{j=1}^{n}\\{\frac{ij}{n+1}\\}\alpha_{j}^{\vee}$.

  2. (2)

Type $B_{n}$: Here
$\varpi_{1}^{\vee}=\sum\limits_{j=1}^{n-1}\alpha_{j}^{\vee}+\frac{1}{2}\alpha_{n}^{\vee}$,
whence $\Xi_{\sigma}=\frac{1}{2}\alpha_{n}^{\vee}$.

  3. (3)

Type $C_{n}$: Here
$\varpi_{n}^{\vee}=\sum\limits_{j=1}^{n}\frac{j}{2}\alpha_{j}^{\vee}$, whence
$\Xi_{\sigma}=\sum\limits_{j\text{ odd},1\leq j\leq
n}\frac{1}{2}\alpha_{j}^{\vee}$.

  4. (4)

Type $D_{n}$: Here the relevant coweights are $\varpi_{1}^{\vee}$ and
$\varpi_{n}^{\vee}$.

We have
$\varpi_{1}^{\vee}=\sum\limits_{j=1}^{n-2}\alpha_{j}^{\vee}+\frac{1}{2}(\alpha_{n-1}^{\vee}+\alpha_{n}^{\vee})$,
thus $\Xi_{\sigma}=\frac{1}{2}(\alpha_{n-1}^{\vee}+\alpha_{n}^{\vee})$.

On the other hand,
$\varpi_{n}^{\vee}=\sum\limits_{j=1}^{n-2}\frac{j}{2}\alpha_{j}^{\vee}+\frac{n-2}{4}\alpha_{n-1}^{\vee}+\frac{n}{4}\alpha_{n}^{\vee}$,
whence

$\Xi_{\sigma}=\begin{cases}&\sum\limits_{j\text{ odd},1\leq j\leq
n-2}\frac{1}{2}\alpha_{j}^{\vee}+(1-\\{\frac{n-2}{4}\\})\alpha_{n-1}^{\vee}+(1-\\{\frac{n}{4}\\})\alpha_{n}^{\vee},\text{
if $n$ is odd;}\\\ &\sum\limits_{j\text{ odd},1\leq j\leq
n-2}\frac{1}{2}\alpha_{j}^{\vee}+(1-\\{\frac{n-2}{4}\\})\alpha_{n-1}^{\vee},\text{
if $n$ is congruent to $0$ mod $4$;}\\\ &\sum\limits_{j\text{ odd},1\leq j\leq
n-2}\frac{1}{2}\alpha_{j}^{\vee}+(1-\\{\frac{n}{4}\\})\alpha_{n}^{\vee},\text{
if $n$ is congruent to $2$ mod $4$.}\end{cases}$

  5. (5)

Type $E_{6}$: Here
$\varpi_{1}^{\vee}=\frac{1}{3}(4\alpha_{1}^{\vee}+3\alpha_{2}^{\vee}+5\alpha_{3}^{\vee}+6\alpha_{4}^{\vee}+4\alpha_{5}^{\vee}+2\alpha_{6}^{\vee})$,
thus
$\Xi_{\sigma}=\frac{1}{3}(2\alpha_{1}^{\vee}+\alpha_{3}^{\vee}+2\alpha_{5}^{\vee}+4\alpha_{6}^{\vee})$.

  6. (6)

Type $E_{7}$: Here
$\varpi_{7}^{\vee}=\frac{1}{2}(2\alpha_{1}^{\vee}+3\alpha_{2}^{\vee}+4\alpha_{3}^{\vee}+6\alpha_{4}^{\vee}+5\alpha_{5}^{\vee}+4\alpha_{6}^{\vee}+3\alpha_{7}^{\vee})$,
thus
$\Xi_{\sigma}=\frac{1}{2}(\alpha_{2}^{\vee}+\alpha_{5}^{\vee}+\alpha_{7}^{\vee})$.

Finally, we have the following remaining cases where $\sigma_{0}$ is non-
trivial:

  1. (1)

Type ${}^{2}A_{n}$: As per our discussion above, we can assume that both $n,i$
are odd. It is easy to see using eq. 5.5 that
$\varpi_{j}+\varpi_{n+1-j}\in\mathbb{Z}_{\geq 0}\Delta$ for all $1\leq
j<\frac{n+1}{2}$. Then by theorem 2.7,
$\Xi_{\sigma}=\min\\{\xi:\langle\varpi_{i}^{\vee}+\xi,\varpi_{\frac{n+1}{2}}\rangle\in\mathbb{Z},\langle\xi,\varpi_{\frac{n+1}{2}}\rangle\geq
0,\sigma_{0}(\xi)=\xi\\}$, whence again by eq. 5.5 applied to
$j=\frac{n+1}{2}$ we get
$\Xi_{\sigma}=\frac{1}{2}\alpha_{\frac{n+1}{2}}^{\vee}$.

  2. (2)

Type ${}^{2}D_{n}$: As per our discussion above, we only need to deal with the
case when $\tau=\tau_{n}$. Considerations similar to the case of type $D_{n}$
shows that if $\Xi_{\sigma}=\sum\limits_{j=1}^{n}x_{j}\alpha_{j}^{\vee}$, then
$x_{j}=\frac{1}{2}$ for $1\leq j\leq n-2$; to find the rest of the
coefficients, we have to use the conditions:

$\sigma_{0}(\Xi_{\sigma})=\Xi_{\sigma},\langle\Xi_{\sigma},\varpi_{n-1}+\varpi_{n}\rangle\geq
0,\langle\varpi_{n}^{\vee}+\Xi_{\sigma},\varpi_{n-1}+\varpi_{n}\rangle\in\mathbb{Z}.$

This in turn yields $x_{n-1}=x_{n},x_{n-1}+x_{n}\geq 0$ and
$x_{n-1}+x_{n}+\frac{2n-2}{4}\in\mathbb{Z}$. From this, we deduce

$\Xi_{\sigma}=\begin{cases}&\sum\limits_{j\text{ odd},1\leq j\leq
n-2}\frac{1}{2}\alpha_{j}^{\vee},\text{ if $n$ is odd;}\\\
&\sum\limits_{j\text{ odd},1\leq j\leq
n-2}\frac{1}{2}\alpha_{j}^{\vee}+\frac{1}{4}(\alpha_{n-1}^{\vee}+\alpha_{n}^{\vee}),\text{
if $n$ is even.}\end{cases}$

## References

  * [BFP99] Francesco Brenti, Sergey Fomin and Alexander Postnikov  “Mixed Bruhat operators and Yang-Baxter equations for Weyl groups”  In _Internat. Math. Res. Notices_ , 1999, pp. 419–441
  * [Bou02] Nicolas Bourbaki  “Lie groups and Lie algebras. Chapters 4–6” Translated from the 1968 French original by Andrew Pressley, Elements of Mathematics (Berlin)  Springer-Verlag, Berlin, 2002, pp. xii+300
  * [BS17] Bhargav Bhatt and Peter Scholze  “Projectivity of the Witt vector affine Grassmannian”  In _Invent. Math._ 209.2, 2017, pp. 329–423
  * [Cha00] Ching-Li Chai  “Newton polygons as lattice points”  In _Amer. J. Math._ 122.5, 2000, pp. 967–990  URL: http://muse.jhu.edu.proxy-um.researchport.umd.edu/journals/american_journal_of_mathematics/v122/122.5chai.pdf
  * [Far04] Laurent Fargues  “Cohomologie des espaces de modules de groupes $p$-divisibles et correspondances de Langlands locales” Variétés de Shimura, espaces de Rapoport-Zink et correspondances de Langlands locales  In _Astérisque_ , 2004, pp. 1–199
  * [FGP97] Sergey Fomin, Sergei Gelfand and Alexander Postnikov  “Quantum Schubert polynomials”  In _J. Amer. Math. Soc._ 10.3, 1997, pp. 565–596
  * [GHN16] Ulrich Goertz, Xuhua He and Sian Nie  “Fully Hodge-Newton decomposable Shimura varieties”  arXiv, 2016  URL: https://arxiv.org/abs/1610.05381
  * [GHN20] Ulrich Görtz, Xuhua He and Sian Nie  “Basic loci of Coxeter type with arbitrary parahoric level”  arXiv, 2020  URL: https://arxiv.org/abs/2006.08838
  * [GHR22] Ulrich Görtz, Xuhua He and Michael Rapoport  “Extremal cases of Rapoport-Zink spaces”  In _J. Inst. Math. Jussieu_ 21.5, 2022, pp. 1727–1782
  * [He09] Xuhua He  “A subalgebra of 0-Hecke algebra”  In _J. Algebra_ 322.11, 2009, pp. 4030–4039
  * [He14] Xuhua He  “Geometric and homological properties of affine Deligne-Lusztig varieties”  In _Ann. of Math. (2)_ 179.1, 2014, pp. 367–404
  * [He16] Xuhua He  “Hecke algebras and $p$-adic groups”  In _Current developments in mathematics 2015_ Int. Press, Somerville, MA, 2016, pp. 73–135
  * [He16a] Xuhua He  “Kottwitz-Rapoport conjecture on unions of affine Deligne-Lusztig varieties”  In _Ann. Sci. Éc. Norm. Supér. (4)_ 49.5, 2016, pp. 1125–1141
  * [He21] Xuhua He  “Affine Deligne-Lusztig varieties associated with generic Newton points”  arXiv, 2021  URL: https://arxiv.org/abs/2107.14461
  * [HN18] Xuhua He and Sian Nie  “On the acceptable elements”  In _Int. Math. Res. Not. IMRN_ , 2018, pp. 907–931
  * [HN21] Xuhua He and Sian Nie  “Demazure product of the affine Weyl groups”  arXiv, 2021  URL: https://arxiv.org/abs/2112.06376
  * [HP17] Benjamin Howard and Georgios Pappas  “Rapoport-Zink spaces for spinor groups”  In _Compos. Math._ 153.5, 2017, pp. 1050–1118
  * [HR17] X. He and M. Rapoport  “Stratifications in the reduction of Shimura varieties”  In _Manuscripta Math._ 152.3-4, 2017, pp. 317–343  DOI: 10.1007/s00229-016-0863-x
  * [HY21] Xuhua He and Qingchao Yu  “Dimension formula for the affine Deligne-Lusztig variety $X(\mu,b)$”  In _Math. Ann._ 379.3-4, 2021, pp. 1747–1765
  * [Kot06] Robert E. Kottwitz  “Dimensions of Newton strata in the adjoint quotient of reductive groups”  In _Pure Appl. Math. Q._ 2.3, Special Issue: In honor of Robert D. MacPherson. Part 1, 2006, pp. 817–836
  * [Kot85] Robert E. Kottwitz  “Isocrystals with additional structure”  In _Compositio Math._ 56.2, 1985, pp. 201–220
  * [Kot97] Robert E. Kottwitz  “Isocrystals with additional structure. II”  In _Compositio Math._ 109.3, 1997, pp. 255–339
  * [KR03] R. Kottwitz and M. Rapoport  “On the existence of $F$-crystals”  In _Comment. Math. Helv._ 78.1, 2003, pp. 153–184
  * [Lan89] R.. Langlands  “On the classification of irreducible representations of real algebraic groups”  In _Representation theory and harmonic analysis on semisimple Lie groups_ 31, Math. Surveys Monogr.  Amer. Math. Soc., Providence, RI, 1989, pp. 101–170
  * [Mil21] Elizabeth Milićević  “Maximal Newton points and the quantum Bruhat graph”  In _Michigan Math. J._ 70.3, 2021, pp. 451–502
  * [MV20] Elizabeth Milićević and Eva Viehmann  “Generic Newton points and the Newton poset in Iwahori-double cosets”  In _Forum Math. Sigma_ 8, 2020, pp. Paper No. e5018
  * [Pos05] Alexander Postnikov  “Quantum Bruhat graph and Schubert polynomials”  In _Proc. Amer. Math. Soc._ 133.3, 2005, pp. 699–709
  * [Rap05] Michael Rapoport  “A guide to the reduction modulo $p$ of Shimura varieties” Automorphic forms. I  In _Astérisque_ , 2005, pp. 271–318
  * [RTZ13] Michael Rapoport, Ulrich Terstiege and Wei Zhang  “On the arithmetic fundamental lemma in the minuscule case”  In _Compos. Math._ 149.10, 2013, pp. 1631–1666
  * [RV14] Michael Rapoport and Eva Viehmann  “Towards a theory of local Shimura varieties”  In _Münster J. Math._ 7.1, 2014, pp. 273–326
  * [RZ96] M. Rapoport and Th. Zink  “Period spaces for $p$-divisible groups” 141, Annals of Mathematics Studies  Princeton University Press, Princeton, NJ, 1996, pp. xxii+324
  * [Sad21] Arghya Sadhukhan  “Affine Deligne-Lusztig Varieties and Quantum Bruhat Graph” accepted in Mathematische Zeitschrift, available at https://arxiv.org/pdf/2110.02172.pdf., 2021
  * [Spr74] T.. Springer  “Regular elements of finite reflection groups”  In _Invent. Math._ 25, 1974, pp. 159–198
  * [Vie18] Eva Viehmann  “Moduli spaces of local ${\bf G}$-shtukas”  In _Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures_ World Sci. Publ., Hackensack, NJ, 2018, pp. 1425–1445
  * [Zhu17] Xinwen Zhu  “Affine Grassmannians and the geometric Satake in mixed characteristic”  In _Ann. of Math. (2)_ 185.2, 2017, pp. 403–492