# Algebraic representations and
constructible sheaves

Geordie Williamson  Research Institute for Mathematical Sciences, Kyoto
University, Kyoto 606-8502, JAPAN.

###### Abstract.

These are notes for my Takagi lecture at the University of Tokyo in November,
2016. I survey what is known about simple modules for reductive algebraic
groups. The emphasis is on characteristic $p>0$ and Lusztig’s character
formula. I explain ideas connecting representations and constructible sheaves
(Finkelberg-Mirković conjecture) in the spirit of the Kazhdan-Lusztig
conjecture. I also discuss a conjecture with S. Riche (a theorem for
$\mathrm{GL}_{n}$) which should eventually make computations more feasible.

## Introduction

Let $G$ denote an algebraic group over an algebraically closed field $\Bbbk$.
A representation of $G$ is a $\Bbbk$-vector space $V$ and a homomorphism
$G\rightarrow\mathrm{GL}(V)$ of algebraic groups. In this article we discuss
various approaches to the representation theory of reductive algebraic groups
(like $\mathrm{GL}_{n},\mathrm{Sp}_{2n},\dots,\mathrm{E}_{8}$) via
constructible sheaves.

Studying the representation theory of $G$ can be thought of as “harmonic
analysis in algebraic geometry”. Over fields of characteristic zero the theory
is well understood and extremely useful. It parallels the theory of compact
Lie groups. Much research over the last five decades has focused on the case
of characteristic $p>0$. Here the theory is highly developed, however several
fundamental questions remain unsolved.

The deepest result in the field (at least on the level of characters) is
Lusztig’s formula. It gives character formulas for certain simple modules,
from which the characters of all simple modules can be deduced.111If our
characteristic $p$ is not too small. Such subtleties will be ignored in the
introduction. If we fix the root system of our group and let $p$ vary, then we
know that Lusztig’s character formula holds if $p$ is very large. However only
in very few cases (e.g.
$\mathrm{SL}_{2},\mathrm{SL}_{3},\mathrm{SL}_{4},\mathrm{Sp}_{4},\mathrm{G}_{2}$)
do we know precisely when it holds! We also don’t understand well what happens
when it fails.

Lusztig’s character formula was motivated by the Kazhdan-Lusztig conjecture,
which gives the characters of simple highest weight representations of complex
semi-simple Lie algebras. The Kazhdan-Lusztig conjecture was first proved by
establishing a bridge to constructible sheaves on the flag variety. Once one
has traversed such a bridge, deep theorems concerning constructible sheaves
(e.g. the decomposition theorem, the Weil conjectures, …) can be used to
deduce the Kazhdan-Lusztig conjecture.222In the words of Bernstein [Ber]: “The
amazing feature of the proof is that it does not try to solve the problem but
just keeps translating it in languages of different areas of mathematics
(further and further away from the original problem) until it runs into
Deligne’s method of weight filtrations which is capable to solve it.”

By analogy with the Kazhdan-Lusztig conjecture one would like to build a
bridge between representations of $G$ and constructible sheaves. The goal
being to better understand Lusztig’s character formula (amongst other things).
Building such a bridge turns out to be much harder in this setting. The most
satisfactory such statements are the geometric Satake equivalence and the
Finkelberg-Mirković conjecture333The reader is warned that the Finkelberg-
Mirković conjecture is still a conjecture. However it is very useful as a
guiding principle. Furthermore, recent work of Achar, Mautner, Riche and Rider
seems to bring us close to a proof.. Both results purport an equivalence
between the representation theory of $G$ and a category of perverse sheaves on
the affine Grassmannian ${\mathcal{G}r^{\vee}}$ associated to the (complex)
Langlands dual group. Under both such equivalences the base field of the
representation theory corresponds to the coefficients of the perverse sheaves.
The space ${\mathcal{G}r^{\vee}}$, however, is fixed.

The Finkelberg-Mirković conjecture is easily seen to imply Lusztig’s character
formula for large $p$. It also gives a character formula for all $p$ in terms
of the Euler characteristic of the stalks of intersection cohomology complexes
with $\Bbbk$-coefficients in. In this way, deciding for which $p$ Lusztig’s
character formula holds becomes a question about controlling torsion in
certain local integral intersection cohomology groups. Roughly speaking, it
was by producing many unexpected torsion classes that the author was recently
able to show that Lusztig’s character formula cannot hold with the hoped-for
bounds.

The Finkelberg-Mirković conjecture provides a very conceptually satisfying
“constructible picture” of representations. However it seems unlikely that it
will help with computations (at least with current tools). Indeed, the
calculation of (the Euler characteristics of) the stalks of intersection
cohomology complexes with coefficients in a field of characteristic $p>0$ is
notoriously difficult.

In practice it is often easier to calculate the stalks of parity sheaves.
(These are analogues of intersection cohomology complexes whose stalks satisfy
a parity vanishing property. In this setting they only really become
interesting with coefficients of positive characteristic.) Thus one is led to
try to find a character formula in which the stalks of parity sheaves appear.
Such a conjecture has recently been formulated by Riche and the author, and
proved for $G=\mathrm{GL}_{n}$. The result is a character formula for tilting
modules in terms of the $p$-canonical basis. This conjecture should be related
via Koszul duality to the Finkelberg-Mirković conjecture.

Due to limitations (both of time and the author’s competence) we do not
discuss closely related categories of coherent sheaves. One can regard
algebraic representations as $G$-equivariant coherent sheaves on a point. From
this point of view most of the results of this paper can be viewed as special
cases of coherent / constructible equivalences appearing in the geometric
Langlands program. It was in this context that characteristic zero analogues
of the results we discuss were often first proved [ABG04, AB09, Bez16].
Another glaring omission is that we do not discuss the infinitesimal group
schemes (Frobenius kernels etc.) which appear naturally in the theory. Thus we
do not discuss Lie algebra representations, nor the Bezrukavnikov-Mirković-
Rumynin theory of localisation in positive characteristic. This theory is the
natural extension to positive characteristic of the original proof of the
Kazhdan-Lusztig conjectures via $D$-modules.

###  0.1. Structure of this paper

This paper consists of two sections:

  1. §1.

_Algebraic representations:_ We review the fundamentals of the theory of
representations of algebraic groups: classification of simple modules, induced
modules, Weyl modules, tilting modules, Steinberg’s tensor product theorem,
the translation and linkage principles. Our goal is to give all results needed
to understand the statement of Lusztig’s character formula. We survey what is
known and not known regarding Lusztig’s formula. Finally, we explain an
observation of Lusztig which predicts the values at 1 of certain affine
Kazhdan-Lusztig polynomials.

  2. §2.

_Constructible sheaves_ : After reviewing the basics of perverse and parity
sheaves we define the Hecke category. We then discuss the geometric Satake
equivalence and Finkelberg-Mirković conjecture. We explain why the Finkelberg-
Mirković conjecture implies character formulas for algebraic groups in terms
of stalks of intersection cohomology complexes, and why the presence of
torsion can be used to deduce that Lusztig’s character formula does not hold
for certain primes. Finally, we outline a conjectural link between tilting
modules and parity sheaves.

We conclude the paper with a list of frequently used notation.

###  0.2. Acknowledgements

I would like to thank P. Achar, H. H. Andersen, R. Bezrukavnikov, C. Bonnafé,
S. Donkin, P. Fiebig, J. C. Jantzen, D. Juteau, X. He, A. Henderson, S. Kumar,
G. Lonergan, G. Lusztig, S. Makisumi, C. Mautner, I. Mirković, S. Riche, L.
Rider, R. Rouquier, W. Soergel and K. Vilonen for useful discussions and
observations on the subject of this paper. I am very grateful to D. Juteau, M.
Kaneda and S. Riche for feedback on a first draft.

###  0.3. Conventions

If we write $G\;\lefttorightarrow\;X$ we mean that the group $G$ acts on $X$.

Given an abelian category ${\mathcal{A}}$ we let $[{\mathcal{A}}]$ denote the
Grothendieck group of ${\mathcal{A}}$. If ${\mathcal{A}}$ is an additive
category, its split Grothendieck group, denoted $[{\mathcal{A}}]_{\oplus}$, is
the quotient of the free module on symbols $[M]$ for all objects
$M\in{\mathcal{A}}$ modulo the relations
$[M]=[M^{\prime}]\oplus[M^{\prime\prime}]$ if $M\cong M^{\prime}\oplus
M^{\prime\prime}$. In both settings the class of $M\in{\mathcal{A}}$ is
denoted $[M]$. If ${\mathcal{A}}$ is addition graded (i.e. equipped with an
equivalence $M\mapsto M[1]$) we view $[{\mathcal{A}}]_{\oplus}$ as a
${\mathbb{Z}}[v,v^{\pm 1}]$-module via $v^{\pm 1}[M]:=[M[\pm 1]]$.

##  1\. Algebraic representations

###  1.1. Root data and the group

We fix a reduced root datum $(X,\Phi,X^{\vee},\Phi^{\vee})$ with $X$ the
character lattice, $\Phi\subset X$ the roots, $X^{\vee}$ the cocharacter
lattice and $\Phi^{\vee}\subset X^{\vee}$ the coroots. To our root datum we
may associate a split connected reductive “Chevalley” group scheme
$G_{\mathbb{Z}}$ over ${\mathbb{Z}}$. For any field $k$, extension of scalars
yields an algebraic group over $k$ which is split, connected, reductive and
has the above root data. Throughout:

$\displaystyle\text{$\Bbbk$ denotes an algebraically closed field of
characteristic $p\geq 0$};$ $G_{\Bbbk}$ denotes the reductive algebraic group
over $\Bbbk$ deduced from $G_{\mathbb{Z}}$.

We will make the following assumption:

$\begin{array}[]{c}\text{Our root system (and thus our group $G_{\Bbbk}$) is
semi-simple}\\\ \text{ and simply-connected:
${\mathbb{Z}}\Phi^{\vee}=X^{\vee}$.}\end{array}$

(This assumption is not essential for most of the theory discussed below.
However including it simplifies the exposition.) Let us fix a system of
positive roots and coroots

$\Phi_{+}\subset\Phi\quad\text{and}\quad\Phi_{+}^{\vee}\subset\Phi^{\vee}.$

We let $T_{\mathbb{Z}}\subset B_{\mathbb{Z}}\subset G_{\mathbb{Z}}$ denote
“maximal torus” and “Borel” subgroup schemes which arise in the construction
of $G_{\mathbb{Z}}$; their extension of scalars yield a maximal torus
$T_{\Bbbk}$ and Borel subgroup $B_{\Bbbk}$ of $G_{\Bbbk}$ for any $\Bbbk$. We
assume that our Borel subgroup is chosen such that:

the positive roots $\Phi_{+}$ are the weights that appear in
$\operatorname{Lie}G/\operatorname{Lie}B$.

Thus the roots occurring in $\operatorname{Lie}B$ are the _negative_ roots
$-\Phi^{+}$. We denote by

$\displaystyle X_{+}:=\\{\lambda\in
X\;|\;\langle\alpha^{\vee},\lambda\rangle\geq 0\text{ for all
$\alpha^{\vee}\in\Phi^{\vee}_{+}$}\\},$ $\displaystyle
X^{\vee}_{+}:=\\{\gamma\in X^{\vee}\;|\;\langle\gamma,\alpha\rangle\geq
0\text{ for all $\alpha\in\Phi_{+}$}\\}$

the subsets of dominant weights and coweights.

###  1.2. Representations and simple modules

In the next three sections we recall some fundamentals about representations
of reductive algebraic groups. The results are standard and we do not give
detailed references; excellent sources include [Jan86, Jan03].

Given a linear algebraic group $H$ defined over $\Bbbk$ we denote by
$\operatorname{\mathrm{Rep}}H$ its category of _finite-dimensional_
algebraic444It is traditional to call algebraic representations of $G_{\Bbbk}$
“rational”. I am avoiding this terminology as it seems a reliable source of
confusion for mathematicians from other fields. representations. What a
representation is was defined in the introduction; alternatively we could
define $\operatorname{\mathrm{Rep}}H$ to be the abelian category of finite-
dimensional $\Bbbk[H]$-comodules, where $\Bbbk[H]$ denotes the regular
functions on $H$. We denote by $\operatorname{Irr}H$ the set of isomorphism
classes of simple modules in $\operatorname{\mathrm{Rep}}H$.

We will almost exclusively study representations of our semi-simple group
$G_{\Bbbk}$. If the context is clear we will often abbreviate:

$\operatorname{\mathrm{Rep}}:=\operatorname{\mathrm{Rep}}G_{\Bbbk}.$

To any $B_{\Bbbk}$-module $V$ we may associate the trivial vector bundle
$G_{\Bbbk}\times V$ on $G_{\Bbbk}$. The quotient for the $B_{\Bbbk}$-action
$b\cdot(g,v):=(gb^{-1},bv)$ exists and yields a vector bundle
${\mathcal{L}}_{V}$ on $G_{\Bbbk}/B_{\Bbbk}$. Taking global sections of this
vector bundle gives rise to the induction functor:

$\operatorname{ind}_{B_{\Bbbk}}^{G_{\Bbbk}}:\operatorname{\mathrm{Rep}}B_{\Bbbk}\rightarrow\operatorname{\mathrm{Rep}}G_{\Bbbk}:V\mapsto\Gamma(G_{\Bbbk}/B_{\Bbbk},{\mathcal{L}}_{V}).$

This functor preserves finite-dimensional modules because
$G_{\Bbbk}/B_{\Bbbk}$ is complete.

In particular for any character $\lambda\in X$ of $T_{\Bbbk}$ we can inflate
via $B_{\Bbbk}\twoheadrightarrow B_{\Bbbk}/[B_{\Bbbk},B_{\Bbbk}]=T_{\Bbbk}$ to
obtain a $B_{\Bbbk}$-module $\Bbbk_{\lambda}$ and then induce (we set
${\mathcal{L}}_{\lambda}:={\mathcal{L}}_{\Bbbk_{\lambda}}$)

$\operatorname{ind}_{B_{\Bbbk}}^{G_{\Bbbk}}\Bbbk_{\lambda}=\Gamma(G_{\Bbbk}/B_{\Bbbk},{\mathcal{L}}_{\lambda}).$

It turns out that
$\operatorname{ind}_{B_{\Bbbk}}^{G_{\Bbbk}}\Bbbk_{\lambda}\neq 0$ if and only
if $\lambda\in X_{+}$. Thus, for $\lambda\in X_{+}$ we set

$\nabla_{\lambda}:=\operatorname{ind}_{B_{\Bbbk}}^{G_{\Bbbk}}\Bbbk_{\lambda}\in\operatorname{\mathrm{Rep}}.$

We call $\nabla_{\lambda}$ an _induced module_. If $p=0$ then each
$\nabla_{\lambda}$ is simple. In general each $\nabla_{\lambda}$ has simple
socle. We set

$L_{\lambda}:=\operatorname{socle}(\nabla_{\lambda}).$

The following gives the classification of the simple $G_{\Bbbk}$-modules:

######  Theorem 1.1.

We have a bijection:

$\displaystyle X_{+}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{Irr}G_{\Bbbk}$
$\displaystyle\lambda$ $\displaystyle\mapsto L_{\lambda}.$

We denote by $\sigma$ a Chevalley involution on $G_{\Bbbk}$ and consider the
contravariant functor ${\mathbb{D}}$ given by

$\displaystyle V\mapsto(V^{*})^{\sigma}$

where $(-)^{\sigma}$ denotes twisting by the Chevalley involution. Then
${\mathbb{D}}$ is a duality on $\operatorname{\mathrm{Rep}}$ (i.e.
${\mathbb{D}}^{2}\cong\operatorname{id}$). The twist by $\sigma$ is to ensure

(1.1) $\displaystyle{\mathbb{D}}(L_{\lambda})=L_{\lambda}.$

We set

$\Delta_{\lambda}:={\mathbb{D}}(\nabla_{\lambda})$

and call it a _Weyl module_. We could alternatively have defined $L_{\lambda}$
as the simple head of $\Delta_{\lambda}$. For any $\lambda\in X_{+}$ we have
maps (unique up to scalar):

(1.2) $\Delta_{\lambda}\twoheadrightarrow
L_{\lambda}\hookrightarrow\nabla_{\lambda}.$

######  Example 1.2.

If $G_{\Bbbk}=\mathrm{SL}_{2}$ then we can identify $X={\mathbb{Z}}$,
$\Phi=\\{\pm 2\\}$, $X_{+}={\mathbb{Z}}_{\geq 0}$. We have
$G_{\Bbbk}/B_{\Bbbk}={\mathbb{P}}^{1}$ and $\mathcal{L}_{n}={\mathcal{O}}(n)$
all $n\in X$. We have $\Gamma({\mathbb{P}}^{1},{\mathcal{O}}(n))\neq 0$ if and
only if $n\geq 0$. If $V=\Bbbk x\oplus\Bbbk y$ denotes the natural module of
$G_{\Bbbk}$ then
$\nabla_{n}=\Gamma({\mathbb{P}}^{1},{\mathcal{O}}(n))=S^{n}(V)$ and
$\Delta_{n}=\nabla_{n}^{*}$ for all $n\geq 0$. If $p=0$ then all $\nabla_{n}$
are simple. If $p\geq 0$ then $\nabla_{0},\dots,\nabla_{p-1}$ are simple but
$\nabla_{p}$ is not: $L_{p}=\Bbbk x^{p}\oplus\Bbbk y^{p}\subset\nabla_{p}$ is
a non-trivial submodule.

###  1.3. Characters

Any $M$ in $\operatorname{\mathrm{Rep}}T_{\Bbbk}$ is semi-simple and
$\operatorname{Irr}T_{\Bbbk}=X$. Hence we have a canonical isomorphism

(1.3) $[\operatorname{\mathrm{Rep}}T_{\Bbbk}]={\mathbb{Z}}[X].$

We identify both sides of (1.3) and write elements as (finite) sums
$\sum_{\lambda\in X}m_{\lambda}e^{\lambda}$. Given any
$M\in\operatorname{\mathrm{Rep}}G_{\Bbbk}$ its _character_

$\operatorname{ch}M\in{\mathbb{Z}}[X]$

is the class of the restriction of $M$ to $T_{\Bbbk}$ in
$[\operatorname{\mathrm{Rep}}T_{\Bbbk}]$. Concretely,

$\operatorname{ch}M=\sum_{\lambda\in X}(\dim M(\lambda))e^{\lambda}$

where $M(\lambda)\subset M$ denotes the $\lambda$ weight space of $T_{\Bbbk}$.

Let $\Delta\subset\Phi^{+}$ denote the simple roots corresponding to our
choice of positive roots. Set

$\rho:=\frac{1}{2}\sum_{\alpha\in\Phi_{+}}\alpha.$

Let $W$ denote the Weyl group with simple reflections
$S=\\{s_{\alpha}\;|\;\alpha\in\Delta\\}$. We denote by
$x\mapsto\varepsilon_{x}$ the sign character of $W$. The _dot action_ of $W$
on $X$ is given by

$x\bullet\lambda:=x(\lambda+\rho)-\rho.$

For any $\lambda\in X^{+}$ consider the _Weyl character_

$\chi_{\lambda}:=\frac{\sum_{x\in
W}\varepsilon_{x}e^{x\bullet\lambda}}{\sum_{x\in W}\varepsilon_{x}e^{x\bullet
0}}\in{\mathbb{Z}}[X]^{W}.$

If $p=0$ then for any $\lambda\in X_{+}$ we have
$\Delta_{\lambda}=L_{\lambda}=\nabla_{\lambda}$ and

$\operatorname{ch}L_{\lambda}=\chi_{\lambda}.$

If $p>0$ then this is no longer true in general, as we have already seen for
$SL_{2}$. However it is still true (a consequence of Kempf vanishing:
$H^{i}(G_{\Bbbk}/B_{\Bbbk},\mathcal{L}_{\lambda})=0$ for $\lambda\in X_{+}$
and $i>0$) that

(1.4)
$\displaystyle\operatorname{ch}\Delta_{\lambda}=\operatorname{ch}\nabla_{\lambda}=\chi_{\lambda}.$

The basic problem which motivates this survey is:

(1.5) Determine $\operatorname{ch}L_{\lambda}$ for all $\lambda\in X_{+}$.

As explained above, the answer is known if $p=0$. Thus $p>0$ is the case of
interest for this survey. By considerations of highest weight

$\\{[\Delta_{\lambda}]\;|\;\lambda\in
X_{+}\\},\quad\\{[L_{\lambda}]\;|\;\lambda\in
X_{+}\\}\quad\text{and}\quad\\{[\nabla_{\lambda}]\;|\lambda\in X_{+}\\}$

are all bases for $[\operatorname{\mathrm{Rep}}]$ (of course
$[\Delta_{\lambda}]=[\nabla_{\lambda}]$ by (1.1)). It turns out to be
convenient to rephrase our basic problem as follows:

(1.6) Find expressions $[L_{\lambda}]=\sum_{\mu}m_{\mu,\lambda}[\Delta_{\mu}]$
for all $\lambda\in X_{+}$.

This is equivalent to writing $\operatorname{ch}L_{\lambda}$ in terms of Weyl
characters which, in turn, is equivalent to (1.5).

###  1.4. Steinberg’s theorems

Assume that $p>0$. Recall that our group $G_{\Bbbk}$ arises by extension of
scalars from a group scheme $G_{\mathbb{Z}}$ defined over the integers. In
particular, it arises via extension of scalars from a group over
${\mathbb{F}}_{p}$ and hence has a natural ${\mathbb{F}}_{p}$-rational
structure. We denote by

$\mathrm{Fr}:G_{\Bbbk}\rightarrow G_{\Bbbk}$

the Frobenius map. (Concretely, because $G_{\Bbbk}$ has an
${\mathbb{F}}_{p}$-rational structure, it can be defined as a closed subgroup
of some $GL_{N}$ by equations with coefficients in ${\mathbb{F}}_{p}$; the
Frobenius map $\mathrm{Fr}$ is given by the $p^{th}$-power on coordinates in
any such embedding.) Precomposing by $\mathrm{Fr}$ defines the functor of
_Frobenius twist_ on $\operatorname{\mathrm{Rep}}$:

$M\mapsto M^{\mathrm{Fr}}.$

If $\operatorname{ch}M=\sum m_{\lambda}e^{\lambda}$ then
$\operatorname{ch}(M^{\mathrm{Fr}})=(\operatorname{ch}M)^{\mathrm{Fr}}:=\sum
m_{\lambda}e^{p\lambda}$. We denote the iterates of $\mathrm{Fr}$ by

$M\mapsto M^{\mathrm{Fr}^{m}}.$

It is easy to see that if $M$ is simple, then so is $M^{\mathrm{Fr}}$. (If
$\Bbbk$ is perfect then, as representations of abstract groups, we are simply
twisting by an automorphism.) However much more is true. For any $\ell\geq 0$
consider the set of _$\ell$ -restricted weights:_

$X_{1}^{\ell}=\\{\lambda\in
X_{+}\;|\;\langle\alpha^{\vee},\lambda\rangle<\ell\text{ for all
$\alpha\in\Delta$}\\}.$

######  Theorem 1.3 (Steinberg).

If $\lambda\in X_{1}^{p}$ and $\gamma\in X_{+}$ then $L_{\lambda}\otimes
L_{\gamma}^{\mathrm{Fr}}$ is simple.

By our assumption that our root system is simply connected there exist
fundamental weights $\\{\varpi_{\alpha}\;|\;\alpha\in\Delta\\}\subset X_{+}$
(i.e. $\langle\alpha^{\vee},\varpi_{\beta}\rangle=\delta_{\alpha,\beta}$ for
all $\alpha,\beta\in\Delta$). We can rewrite $X_{1}^{\ell}$ in these
coordinates as

$X_{1}^{\ell}=\\{\sum_{\alpha\in\Delta}a_{\alpha}\varpi_{\alpha}\;|\;0\leq
a_{\alpha}<\ell\text{ for all $\alpha\in\Delta$}\\}.$

Given any $\lambda\in X_{+}$ we can consider its _$p$ -adic expansion_

$\lambda=\sum_{i=1}^{m}\lambda_{i}p^{i}\quad\text{with $\lambda_{i}\in
X_{1}^{p}$.}$

It follows immediately from Steinberg’s theorem and Theorem 1.1 that:

(1.7) $\displaystyle L_{\lambda}:=L_{\lambda_{0}}\otimes
L_{\lambda_{1}}^{\mathrm{Fr}}\otimes\dots\otimes
L_{\lambda_{m}}^{\mathrm{Fr}^{m}}.$

######  Example 1.4.

We continue Example 1.2 with $G_{\Bbbk}=\mathrm{SL}_{2}$. We have

$\operatorname{ch}\Delta_{n}=\operatorname{ch}\nabla_{n}=\chi_{n}=\frac{e^{n}-e^{-n-2}}{e^{0}-e^{-2}}=e^{n}+e^{n-2}+\dots+e^{-n}\quad\text{for
all $n\geq 0$.}$

Moreover, $\operatorname{ch}L_{n}=\operatorname{ch}\Delta_{n}$ if $n<p$ (i.e.
if $n$ is $p$-restricted). For general $n$ we consider its $p$-adic expansion
$n=\sum_{i=0}^{m}n_{i}p^{i}$. By Steinberg’s theorem:

$\operatorname{ch}L_{n}=(e^{n_{0}}+e^{n_{0}-2}+\dots+e^{-n_{0}})(e^{n_{1}}+e^{n_{1}-2}+\dots+e^{-n_{1}})^{\mathrm{Fr}}\dots(e^{n_{m}}+\dots+e^{-n_{m}})^{\mathrm{Fr}^{m}}$

Thus Steinberg’s theorem solves our basic question (1.5) for
$\mathrm{SL}_{2}$. However $\mathrm{SL}_{2}$ is essentially the only case
where Steinberg’s theorem gives the complete answer.

We now briefly recall the Steinberg restriction theorem. Logically it is
irrelevant for the rest of this survey, however it is such a beautiful theorem
that it would be criminal not to mention it. Let us temporarily denote by $G$
the split form of our group over ${\mathbb{F}}_{p}$. Everything that we have
done in the previous sections can be done over ${\mathbb{F}}_{p}$. Hence we
obtain representations $\Delta_{\lambda},L_{\lambda},\nabla_{\lambda}$ for all
$\lambda\in X_{+}$ of the group scheme $G$. Taking rational points we obtain
representations of the finite group of Lie type $G({\mathbb{F}}_{q})$ for any
$q=p^{\ell}$. We denote these representations by the same symbols.

######  Theorem 1.5 (Steinberg restriction theorem).

The set $\\{L_{\lambda}\;|\;\lambda\in X_{1}^{q}\\}$ is a set of
representatives for the isomorphism classes of simple $\Bbbk
G({\mathbb{F}}_{q})$-modules.

In particular a solution to the basic question (1.5) would yield considerable
information about the irreducible representations of $\Bbbk
G({\mathbb{F}}_{q})$ for all $q$.

######  Remark 1.6.

Steinberg’s restriction theorem gives a remarkably tight connection on the
level of simple modules. One further beautiful connection is given by the
theory of generic cohomology [CPSvdK77, Par87]. However on the level of
categories the finite and algebraic groups appear quite different. At present
we know much more about the category of algebraic representations than of
$\Bbbk G({\mathbb{F}}_{q})$ (e.g. compare the induction theorems of [ABG04,
HKS16, AR16c] with the solution of Broué’s conjecture for
$SL_{2}({\mathbb{F}}_{q})$ [Chu01, Oku00]).

###  1.5. Tilting modules

We briefly recall the theory of tilting modules. Excellent sources for this
material include the paper of Donkin [Don93], the surveys of Andersen [And01]
and Mathieu [Mat00] as well as [Jan03, Chapter E].

The starting point is the fundamental vanishing theorem:

(1.8)
$\displaystyle\operatorname{Ext}^{i}(\Delta_{\lambda},\nabla_{\mu})=\begin{cases}\Bbbk&\text{if
$i=0$ and $\lambda=\mu$,}\\\ 0&\text{otherwise}.\end{cases}$

Let us define $\operatorname{\mathrm{Rep}}_{\Delta}$ (resp.
$\operatorname{\mathrm{Rep}}_{\nabla}$) to be the full subcategory of
$\operatorname{\mathrm{Rep}}$ consisting of modules which admit a filtration
whose successive quotients are isomorphic to $\Delta_{\mu}$ (resp.
$\nabla_{\mu}$) for some $\mu\in X_{+}$. We will call such a filtration a
_Weyl_ (resp. _good_) filtration. We say that a module is _tilting_ if it
belongs to both $\operatorname{\mathrm{Rep}}_{\Delta}$ and
$\operatorname{\mathrm{Rep}}_{\nabla}$, that is, if it possesses both a Weyl
and a good filtration. We denote by $\operatorname{\mathrm{Tilt}}G_{\Bbbk}$
(or $\operatorname{\mathrm{Tilt}}$ if the context is clear) the full
subcategory of tilting modules. (Note that $\operatorname{\mathrm{Tilt}}$ is
additive but almost never abelian.)

Given a tilting module $M$ we denote by $(M:\Delta_{\lambda})$ (resp.
$(M:\nabla_{\lambda})$ the multiplicity of $\Delta_{\lambda}$ in a Weyl (resp.
good) filtration of $M$. This number is well defined because

$\operatorname{ch}M=\sum_{\lambda\in
X_{+}}(M:\Delta_{\lambda})\chi_{\lambda}=\sum_{\lambda\in
X_{+}}(M:\nabla_{\lambda})\chi_{\lambda}.$

######  Theorem 1.7 ([Rin91], [Don93]).

For each $\lambda\in X_{+}$ there exists an indecomposable tilting module
$T_{\lambda}$ with highest weight $\lambda$. Moreover $\dim
T_{\lambda}(\lambda)=1$ and we have a bijection:

$\displaystyle X_{+}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\left\\{\begin{array}[]{c}\text{indecomposable}\\\
\text{tilting modules}\end{array}\right\\}/\cong$ $\displaystyle\lambda$
$\displaystyle\mapsto T_{\lambda}.$

Note that ${\mathbb{D}}$ exchanges $\operatorname{\mathrm{Rep}}_{\Delta}$ and
$\operatorname{\mathrm{Rep}}_{\nabla}$ and thus preserves
$\operatorname{\mathrm{Tilt}}$. By highest weight considerations we deduce
that indecomposable tilting modules are self-dual:

(1.9) ${\mathbb{D}}T_{\lambda}\cong T_{\lambda}.$

If we write

$[T_{\lambda}]=\sum m_{\mu,\lambda}[\Delta_{\mu}]$

then $m_{\lambda,\lambda}=1$ and $m_{\mu,\lambda}=0$ if $\mu\not\leq\lambda$
(again by highest weight considerations). In particular, the elements
$[T_{\lambda}]$ are upper-triangular in the basis $\\{[\Delta_{\lambda}]\\}$
of $[\operatorname{\mathrm{Rep}}]$ and thus also provide a basis.

Another fundamental theorem concerning tilting modules is:

######  Theorem 1.8.

  1. (1)

If $M,M^{\prime}$ are tilting modules, then so is $M\otimes M^{\prime}$.

  2. (2)

If $M$ is a tilting module and $L\subset G_{\Bbbk}$ is a Levi subgroup, then
the restriction of $M$ to $L$ is tilting.

######  Remark 1.9.

The proof of Theorem 1.7 is not difficult. On the other hand, Theorem 1.8
seems to be difficult. The first proof was given by Wang [Wan82] in type $A$
and large characteristics for other groups, then Donkin [Don85] gave a
different proof which covered almost all cases (he had to exclude $p=2$ for
$E_{7},E_{8}$). The first uniform proof is due to Mathieu and uses Frobenius
splitting [Mat90] (see also [Mat00]). For other approaches to the theorem see
[Lit92, Pol89, Par94, Kan98]. We will discuss an approach to Theorem 1.8 via
the affine Grassmannian in §2.4.

######  Remark 1.10.

Tilting modules provide powerful tools in the study of
$\operatorname{\mathrm{Rep}}$.

  1. (1)

Let $[\operatorname{\mathrm{Tilt}}]_{\oplus}$ denote its split Grothendieck
group. Theorem 1.8 implies that $\operatorname{\mathrm{Tilt}}$ is a monoidal
category, and thus $[\operatorname{\mathrm{Tilt}}]_{\oplus}$ is a ring. The
inclusion
$\operatorname{\mathrm{Tilt}}\hookrightarrow\operatorname{\mathrm{Rep}}$
induces an isomorphism of rings555In the words of Donkin [Don93]: “Perhaps the
main point of tilting modules is that they provide a section of the character
map.”

$[\operatorname{\mathrm{Tilt}}]_{\oplus}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[\operatorname{\mathrm{Rep}}].$

Moreover, the classes $\\{[T_{\lambda}]\;|\;\lambda\in X_{+}\\}$ give a basis
with strong positivity properties: it has positive coefficients when written
in the basis $[\Delta_{\lambda}]$; and, it has positive structure constants.

  2. (2)

From (1.8) it follows immediately that if
$M,M^{\prime}\in\operatorname{\mathrm{Tilt}}$ then

$\operatorname{Ext}^{i}(M,M^{\prime})=0\quad\text{for $i>0$.}$

Moreover, $\operatorname{\mathrm{Tilt}}$ is easily seen to generate the
derived category $D^{b}(\operatorname{\mathrm{Rep}})$. Tilting theory
guarantees that the inclusion
$\operatorname{\mathrm{Tilt}}\subset\operatorname{\mathrm{Rep}}$ induces an
equivalence of triangulated categories

$K^{b}(\operatorname{\mathrm{Tilt}})\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D^{b}(\operatorname{\mathrm{Rep}}).$

Thus tilting modules and morphisms between them provide a “homological
skeleton” of $\operatorname{\mathrm{Rep}}$. In this sense, tilting modules are
somewhat analogous to projective or injective objects. (Note that neither
injective nor projective objects exist in $\operatorname{\mathrm{Rep}}$:
injective (resp. projective) objects only exist after passage to the ind-
(resp. pro-) completion of $\operatorname{\mathrm{Rep}}$.)

###  1.6. Tilting characters

As well as the basic problem (1.5) of determining the character of the simple
modules, another problem which motivates this survey is:

(1.10) Determine $\operatorname{ch}T_{\lambda}$ for all $\lambda\in X_{+}$.

As earlier it is convenient to reformulate the problem as follows:

(1.11) Find expressions
$[T_{\lambda}]=\sum_{\mu}n_{\mu,\lambda}[\Delta_{\mu}]$ for all $\lambda\in
X_{+}$.

This appears to be a difficult problem. At the end of this survey we will
outline an approach to this problem via the Hecke category. Here is a brief
overview of what is known:

  1. (1)

As for simple modules, there is a kind of tensor product theorem for tilting
modules: if $M$ is indecomposable tilting and $\lambda$ belongs to the set
$(p-1)\rho+X_{1}^{p}$ then

$T(\lambda)\otimes M^{\mathrm{Fr}}$

is indecomposable tilting. (More precisely, this is a theorem if $p\geq 2h-2$
by [Don93, Proposition 2.1] and Example 1 following it, and would follow for
all $p$ from [Don93, Conjecture 2.2]). This allows one (under mild
restrictions on $p$) to determine the characters of all tilting modules from
the knowledge of the characters of $T(\lambda)$ belonging to the two sets

$\displaystyle\\{\lambda\;|\;\lambda\in(p-1)\rho+X_{1}^{p}\\}\quad\text{and}$
$\displaystyle\\{\lambda=\sum_{\alpha\in\Delta}a_{\alpha}\varpi_{\alpha}\;|\;0\leq
a_{\alpha}<p\text{ for some $\alpha\in\Delta$}\\}.$

Note, however that the second set is infinite in all types other than
(products of) $\mathrm{SL}_{2}$. For $\mathrm{SL}_{2}$ this formula can be
used to determine all tilting characters, see [Don93, §2, Example 2]. See
[LW15] for a description of the characters that may be obtained in this way
for a general root system.

  2. (2)

By an observation of Andersen [And98] (see also [RW15, §1.8]), knowledge of a
finite set of tilting characters determines all simple characters if $p\geq
2h-2$.

  3. (3)

By a result of Erdmann [Erd94] the determination of the characters of
indecomposable tilting modules for $\mathrm{GL}_{n}$ (or equivalently
$\mathrm{SL}_{n}$) in characteristic $p$ is equivalent to determining all
decomposition numbers for representations of all symmetric groups which are
indexed by partitions with less than or equal to $n$ parts. This is an
unsolved problem for $n\geq 3$, reflecting the fact that the characters of
indecomposable tilting modules are unknown for $\mathrm{SL}_{n}$ for $n\geq
3$.

  4. (4)

By results of Donkin on the Ringel self-duality of the Schur algebra one can
rephrase the question of determining the characters of a fixed tilting module
for $\mathrm{GL}_{n}$ in terms of certain simple characters for
$\mathrm{GL}_{N}$ for some (usually much larger) $N$. Thus knowledge of all
tilting characters for some $\mathrm{GL}_{n}$ and fixed characteristic $p$
would yield some information about simple characters for $\mathrm{GL}_{N}$ for
large $N$ (and thus for $p$ “small” relative to $N$).

  5. (5)

Andersen [And97, And00] and Andersen-Kulkarni [AK08] have proved a sum formula
for tilting modules (this formula was inspired by Jantzen’s sum formula for
Weyl modules). Like Jantzen’s formula it does not give complete information,
but is very useful in small rank. Jensen [Jen00] (see also Parker [Par08]) has
used this formula to determine some new tilting characters for
$\mathrm{SL}_{3}$.

  6. (6)

In the analogous setting of quantum groups at a root of unity the
determination of the characters of the indecomposable tilting modules was
solved by Soergel [Soe97b, Soe97a].

###  1.7. The (extended) affine Weyl group

Here we briefly discuss the (extended) affine Weyl group. A very clear
treatment of this material can be found in [IM65].

Let $X_{\mathbb{R}}:=X\otimes_{{\mathbb{Z}}}{\mathbb{R}}$. The affine Weyl
group $\mathcal{W}$ is the subgroup of affine transformations of
$X_{\mathbb{R}}$ generated by $W$ (acting linearly) and ${\mathbb{Z}}\Phi$
(acting by translation). In formulas:

$\mathcal{W}:=W\ltimes{\mathbb{Z}}\Phi\;\lefttorightarrow\;X_{\mathbb{R}}.$

Given $\lambda\in{\mathbb{Z}}\Phi$ we denote by $t_{\lambda}\in\mathcal{W}$
the corresponding translation.

The group $\mathcal{W}$ is also generated by the affine reflections

$s_{\alpha,m}(\lambda):=\lambda-\langle\alpha^{\vee},\lambda\rangle\alpha+m\alpha$

in the hyperplanes

$H_{\alpha,m}:=\\{\lambda\in
X_{\mathbb{R}}\;|\;\langle\alpha^{\vee},\lambda\rangle=m\\}$

for all $\alpha\in\Phi^{+}$ and $m\in{\mathbb{Z}}$. The set

$C_{-}:=\\{\lambda\in
X_{\mathbb{R}}\;|\;-1\leq\langle\alpha^{\vee},\lambda\rangle\leq 0\;\text{for
all $\alpha\in\Phi^{+}$}\;\\}\subset X_{\mathbb{R}}$

is a fundamental domain for the action of $\mathcal{W}$ on $X_{\mathbb{R}}$
[Bou68, Ch. V, §3].

Consider the set $\mathcal{S}$ of reflections in those hyperplanes
$H_{\alpha,m}$ which intersect $C_{-}$ in codimension one (the _walls_ of
$C_{-}$). Then $\mathcal{S}$ generates $\mathcal{W}$. Moreover, equipped with
these generators $\mathcal{W}$ is a Coxeter group [Bou68, Ch. V, §3].
Throughout, whenever we view $\mathcal{W}$ as a Coxeter group, it will always
be with respect to the generators $\mathcal{S}$. We denote by $\ell$ the
length function on $\mathcal{W}$ with respect to the generating set
$\mathcal{S}$ and by $\leq$ the Bruhat order on $\mathcal{W}$.

######  Warning 1.11.

Most authors (for example [Bou68, IM65]) define the affine Weyl group to be
the semi-direct product of $W$ with the _co_ root lattice. Thus, our
$\mathcal{W}$ is what is usually referred to as the affine Weyl group of the
dual root system $\Phi^{\vee}$. In particular, to determine the Coxeter type
of $\mathcal{W}$, one should consider the extended Dynkin diagram of the
_dual_ root system $\Phi^{\vee}$. The convention we adopt here is better
adapted to the combinatorics of representations of algebraic groups. It can
also be seen as a shadow of Langlands duality, as should become clearer in §2.

The _extended affine Weyl group_ is the subgroup of affine transformations of
$X_{\mathbb{R}}$ generated by $W$ and the weight lattice $X$:

$\displaystyle\mathcal{W}^{\mathrm{ext}}:=W\ltimes
X\;\lefttorightarrow\;X_{\mathbb{R}}.$

As above, given $\lambda\in X$ we denote by
$t_{\lambda}\in\mathcal{W}^{\mathrm{ext}}$ the translation by $\lambda$. The
extended affine Weyl group has a length function
$\ell:\mathcal{W}^{\mathrm{ext}}\rightarrow{\mathbb{Z}}_{\geq 0}$ given by

$\displaystyle\ell(x)=\left|\left\\{\begin{array}[]{c}\text{hyperplanes
$H_{\alpha,n}$ separating the }\\\ \text{interior of $C_{-}$ from that of
$x(C_{-})$}\end{array}\right\\}\right|.$

Because ${\mathbb{Z}}\Phi\subset X$, the affine Weyl group $\mathcal{W}$ is a
subgroup of $\mathcal{W}^{\mathrm{ext}}$. The length function $\ell$ restricts
to the standard (Coxeter) length function on $\mathcal{W}$.

If we consider the subset of _length zero elements_

$\Omega:=\\{\omega\in\mathcal{W}^{\mathrm{ext}}\;|\;\ell(\omega)=0\\}=\\{\omega\in\mathcal{W}^{\mathrm{ext}}\;|\;\omega(C_{-})=C_{-}\\}$

then $\Omega\cong X/{\mathbb{Z}}\Phi$, $\Omega$ acts via conjugation as
automorphisms of the Coxeter system $(\mathcal{W},\mathcal{S})$, and
$\mathcal{W}^{\mathrm{ext}}$ is the semi-direct product

$\mathcal{W}^{\mathrm{ext}}=\Omega\ltimes\mathcal{W}.$

We extend the Bruhat order to $\mathcal{W}^{\mathrm{ext}}$ by declaring that
$\omega_{1}x_{1}\leq\omega_{2}x_{2}$ for $\omega_{1},\omega_{2}\in\Omega$,
$x_{1},x_{2}\in\mathcal{W}$ if $\omega_{1}=\omega_{2}$ and $x_{1}\leq x_{2}$.

Of course, both $\mathcal{W}$ and $\mathcal{W}^{\mathrm{ext}}$ preserve the
character lattice $X\subset X_{\mathbb{R}}$.

###  1.8. The (extended) affine Hecke algebra and Kazhdan-Lusztig basis

The _extended affine Hecke algebra_ $\mathrm{H}^{\mathrm{ext}}$ is the
$\mathbb{Z}[v^{\pm 1}]$-algebra generated by symbols
$\\{h_{w}\;|\;w\in\mathcal{W}^{\mathrm{ext}}\\}$ subject to the relations

$\displaystyle h_{w}h_{w^{\prime}}=h_{ww^{\prime}}\quad\text{if
$\ell(ww^{\prime})=\ell(w)+\ell(w^{\prime})$, and}$ $\displaystyle
h_{s}^{2}=(v^{-1}-v)h_{s}+h_{\operatorname{id}}\quad\text{for
$s\in\mathcal{S}$}.$

It is an associative unital (with unit $1:=h_{\operatorname{id}}$) algebra.
The set $\\{h_{x}\;|\;x\in\mathcal{W}^{\mathrm{ext}}\\}$ is a
$\mathbb{Z}[v^{\pm 1}]$-basis for $\mathrm{H}^{\mathrm{ext}}$ called the
standard basis. The basis elements $\\{h_{x}\;|\;x\in\mathcal{W}\\}$ (resp.
$\\{h_{x}\;|\;x\in W\\}$) span a subalgebra
$\mathrm{H}\subset\mathrm{H}^{\mathrm{ext}}$, the _affine Hecke algebra_
(resp. _finite Hecke algebra_). The affine Hecke algebra is isomorphic to the
Hecke algebra of the Coxeter system $(\mathcal{W},\mathcal{S})$.

Each standard basis element $h_{x}$ is invertible. The Kazhdan-Lusztig
involution is the algebra involution $h\mapsto\overline{h}$ on
$\mathrm{H}^{\mathrm{ext}}$ determined by $h_{x}\mapsto h_{x^{-1}}^{-1}$ and
$v\mapsto v^{-1}$. The following is a classical theorem of Kazhdan-Lusztig:

######  Theorem 1.12 (Kazhdan-Lusztig [KL79, Lus83]).

For all $x\in\mathcal{W}^{\mathrm{ext}}$ there exists a unique element
$\underline{h}_{x}$ such that:

  1. (1)

(“self-duality”) $\overline{\underline{h}_{x}}=\underline{h}_{x}$;

  2. (2)

(“Bruhat upper-triangularity”) we have

$\underline{h}_{x}=\sum_{y\leq x}h_{y,x}h_{y}$

for polynomials $h_{y,x}\in\mathbb{Z}[v]$ with $h_{x,x}=1$ and $h_{y,x}\in
v{\mathbb{Z}}[v]$ for all $y<x$.

By property (2) the set $\\{\underline{h}_{x}\\}$ is a basis for
$\mathrm{H}^{\mathrm{ext}}$, the _Kazhdan-Lusztig basis_. The polynomials
$h_{y,x}$ are the _Kazhdan-Lusztig polynomials_. We extend their definition to
all pairs $y,x$ by setting $h_{y,x}:=0$ if $y\not\leq x$.

######  Remark 1.13.

For an excellent introduction to Kazhdan-Lusztig polynomials the reader is
referred to [Soe97b].

######  Remark 1.14.

For $\omega\in\Omega$ and $x\in\mathcal{W}$ we have
$h_{\omega^{-1}}\underline{h}_{\omega x}=\underline{h}_{x}$ and hence
$h_{\omega y,\omega x}=h_{y,x}$ for all $y\in\mathcal{W}^{\mathrm{ext}}$. Thus
all Kazhdan-Lusztig polynomials may be calculated in $\mathrm{H}$ and have
non-negative coefficients (see Theorem 2.2).

###  1.9. The linkage principle

From now on we assume that $p>0$. In classical highest weight representation
theory it is often necessary to shift the origin to $-\rho$ and consider the
_dot action_ of $W$ on $X_{\mathbb{R}}^{*}$:

$x\bullet\mu:=x(\mu+\rho)-\rho.$

In the representation theory of $G_{\Bbbk}$ in characteristic $p$ it is
necessary to dilate the action of the (extended) affine Weyl group by $p$ and
shift the origin to $-\rho$. In this way we are led to the _$p$ -dilated dot
action_:

$\mathcal{W}\bullet_{p}\;\lefttorightarrow\;X_{\mathbb{R}}^{*}\quad(\text{resp.
}\mathcal{W}^{\mathrm{ext}}\bullet_{p}\;\lefttorightarrow\;X_{\mathbb{R}}^{*}\;)$

defined via

$\displaystyle x\bullet_{p}\mu$ $\displaystyle:=x\bullet\mu\quad\text{for
$x\in W$ and}$ $\displaystyle t_{\lambda}\bullet_{p}\mu$
$\displaystyle:=\mu+p\lambda\quad\text{for $\lambda\in{\mathbb{Z}}\Phi$ (resp.
$\lambda\in X$).}$

We say that $\lambda\in X_{\mathbb{R}}^{*}$ is _$p$ -regular_ if its
stabiliser under the $p$-dilated dot action is trivial. After $p$-dilation and
$-\rho$ shift, a fundamental domain for the $\bullet_{p}$-action is

$C_{-}^{p}:=\\{\mu\in
X_{\mathbb{R}}^{*}\;|\;-p\leq\langle\alpha^{\vee},\mu+\rho\rangle\leq
0\;\text{for all $\alpha\in\Phi^{+}$}\;\\}.$

For $\mu\in C_{-}^{p}$, let $\operatorname{\mathrm{Rep}}_{\mu}$ denote the
full subcategory of all algebraic representations of $G_{\Bbbk}$ whose
composition factors are simple modules with highest weight belonging to the
$\mathcal{W}$ orbit of $\mu$ under the $p$-dilated dot action:

$\operatorname{\mathrm{Rep}}_{\mu}:=\langle L_{\lambda}\;|\;\lambda\in
W\bullet_{p}\mu\cap X_{+}\rangle\subset\operatorname{\mathrm{Rep}}.$

The linkage principle asserts that we have a direct sum decomposition of
abelian categories:

(1.12) $\operatorname{\mathrm{Rep}}=\bigoplus_{\mu\in
C_{-}^{p}}\operatorname{\mathrm{Rep}}_{\mu}.$

In other words, any indecomposable module belongs to some
$\operatorname{\mathrm{Rep}}_{\mu}$ and if
$M\in\operatorname{\mathrm{Rep}}_{\mu},M^{\prime}\in\operatorname{\mathrm{Rep}}_{\mu^{\prime}}$
then $\operatorname{Hom}(M,M^{\prime})=0$ unless $\mu=\mu^{\prime}$. Abusing
language, we will refer to each $\operatorname{\mathrm{Rep}}_{\mu}$ as a
_block_ of $\operatorname{\mathrm{Rep}}$.

######  Remark 1.15.

This is an abuse of language because the decomposition (1.12) is not the
finest possible, and hence does not give the block decomposition in the usual
meaning of the term. To understand the true block decomposition one needs to
consider analogues of the $p$-dilated dot action for higher powers of $p$
[Don80]. However below we will assume that $p$ is greater than the Coxeter
number in which case $\operatorname{\mathrm{Rep}}_{\mu}$ is indecomposable as
an abelian category for “most” $\mu\in C_{-}^{p}$.666More precisely,
$\operatorname{\mathrm{Rep}}_{\mu}$ is indecomposable if and only if there
exists $\alpha\in\Phi_{+}$ such that $\langle\alpha^{\vee},\mu+\rho\rangle$ is
not divisible by $p$ [Jan03, §II.7.2].

Consider the set

$C_{+}^{p}:=\\{\lambda\in X_{+}\;|\;\langle\alpha^{\vee},\mu+\rho\rangle\leq
p\;\text{for all $\alpha\in\Phi_{+}$}\\}.$

It is a consequence of the linkage principle that

(1.13) $\Delta_{\lambda}\text{ is simple, if }\lambda\in C_{+}^{p}.$

###  1.10. The (extended) principal block

Of particular importance is the _principal block_

$\operatorname{\mathrm{Rep}}_{0}:=\langle
L_{\lambda}\;|\;\lambda\in\mathcal{W}\bullet_{p}0\cap X_{+}\rangle$

and the _extended principal block_

$\mathrm{Rep}_{0}^{\mathrm{ext}}:=\langle
L_{\lambda}\;|\;\lambda\in\mathcal{W}^{\mathrm{ext}}\bullet_{p}0\cap
X_{+}\rangle.$

Note that the trivial module $L_{0}$ belongs to
$\operatorname{\mathrm{Rep}}_{0}$ (which explains the name principal block).
Because $0=w_{0}\bullet_{p}(-2\rho)$, in the above notation we have

$\operatorname{\mathrm{Rep}}_{0}=\operatorname{\mathrm{Rep}}_{-2\rho}.$

Similarly, if $\Omega\subset\mathcal{W}^{\mathrm{ext}}$ denotes the subset of
length zero elements defined in the previous section one has

$\mathrm{Rep}_{0}^{\mathrm{ext}}=\bigoplus_{\omega\in\Omega}\operatorname{\mathrm{Rep}}_{\omega\bullet_{p}(-2\rho)}.$

The importance of the extended principal block is the following. Consider the
Frobenius twist functor

$(-)^{\mathrm{Fr}}:\operatorname{\mathrm{Rep}}\rightarrow\operatorname{\mathrm{Rep}}.$

It sends a simple module of highest weight $\lambda$ to a simple module of
highest weight $p\lambda$. In particular its image lands in the extended
principal block, and we may view Frobenius twist as a functor:

$(-)^{\mathrm{Fr}}:\operatorname{\mathrm{Rep}}\rightarrow\mathrm{Rep}_{0}^{\mathrm{ext}}.$

In fact we can say a little more. For any weight $\lambda\in X$ we can write
$\lambda=\lambda_{0}+p\lambda_{1}$ with $\lambda_{0}\in X_{1}^{p}$ and

$L_{\lambda}=L_{\lambda_{0}}\otimes L_{\lambda_{1}}^{\mathrm{Fr}}.$

Hence

$L_{\lambda}\otimes
L_{\gamma}^{\mathrm{Fr}}=L_{\lambda_{0}}\otimes(L_{\lambda_{1}}\otimes
L_{\gamma})^{\mathrm{Fr}}.$

Thus if $L_{\lambda}\in\operatorname{\mathrm{Rep}}_{0}$ then so is
$L_{\lambda}\otimes
L_{\gamma}^{\mathrm{Fr}}\in\mathrm{Rep}_{0}^{\mathrm{ext}}$. In other words,
the bifunctor

$\displaystyle\mathrm{Rep}_{0}^{\mathrm{ext}}\times\operatorname{\mathrm{Rep}}$
$\displaystyle\rightarrow\mathrm{Rep}_{0}^{\mathrm{ext}}$ $\displaystyle(V,M)$
$\displaystyle\mapsto V\otimes M^{\mathrm{Fr}}$

makes $\mathrm{Rep}_{0}^{\mathrm{ext}}$ a (right) module category over
$\operatorname{\mathrm{Rep}}$.

######  Remark 1.16.

More generally, any “extended” block

$\operatorname{\mathrm{Rep}}_{\mu}^{\textrm{ext}}:=\langle
L_{\lambda}\;|\;\lambda\in\mathcal{W}^{\mathrm{ext}}\bullet_{p}\mu\cap
X_{+}\rangle$

is a module category over $\operatorname{\mathrm{Rep}}$ via the Frobenius
twist.

###  1.11. Translation functors

The linkage principle implies that in order to understand
$\operatorname{\mathrm{Rep}}$ as an abelian category it is enough to
understand each block $\operatorname{\mathrm{Rep}}_{\mu}$. Translation
functors can be used to relate these blocks and often reduce questions to the
study of the principal block $\operatorname{\mathrm{Rep}}_{0}$.

For any $\lambda\in C_{-}^{p}$ let ${\mathrm{inc}}_{\lambda}$ (resp.
${\mathrm{pr}}_{\lambda}$) denote the inclusion (resp. projection) functor to
the block
$\operatorname{\mathrm{Rep}}_{\lambda}\subset\operatorname{\mathrm{Rep}}$. Fix
$\lambda,\mu\in C_{-}^{p}$. We define the _translation functor_

$T_{\lambda}^{\mu}:\operatorname{\mathrm{Rep}}_{\lambda}\rightarrow\operatorname{\mathrm{Rep}}_{\mu}$

via

$T_{\lambda}^{\mu}:={\mathrm{pr}}_{\mu}(V\otimes({\mathrm{inc}}_{\lambda}(-)))$

where $V$ is any module whose extremal weights are
$W(\mu-\lambda)$.777Different choices of module yield isomorphic functors
[Jan03, §7.6, Remark 1]. Thus with the definition above $T_{\lambda}^{\mu}$ is
only defined up to isomorphism. For example, if $\nu$ is the unique element of
$W(\mu-\lambda)\cap X_{+}$ we could take $V:=L_{\nu}$ or $V:=\Delta_{\nu}$.
One may think of a translation functor as a certain “matrix coefficient of the
functor $V\otimes(-)$”. Because $V\otimes(-)$ and $V^{*}\otimes(-)$ are
biadjoint one easily deduces that $T_{\lambda}^{\mu}$ and $T_{\mu}^{\lambda}$
are biadjoint.

To describe the effect of translation functors on blocks we need a little more
notation. For any $x\in C_{-}$ its stabiliser in $\mathcal{W}$ is generated by
those $s\in\mathcal{S}$ which fix it. A _facet_ of $C_{-}$ is a non-empty
subset consisting of all points with a fixed stabiliser in $\mathcal{W}$. Each
facet is locally closed (i.e. open in its closure) and $C_{-}$ is the union of
its facets. The same statements and definitions apply verbatim for $C_{-}^{p}$
if instead we consider the $p$-dilated dot action of $\mathcal{W}$.

The two most useful properties of translation functors, often called
_Jantzen’s translation principles_ , are the following:

  1. (1)

If $\lambda,\mu\in C_{-}^{p}$ belong to the same facet then
$T_{\lambda}^{\mu}:\operatorname{\mathrm{Rep}}_{\lambda}\rightarrow\operatorname{\mathrm{Rep}}_{\mu}$
is an equivalence of abelian categories [Jan03, Proposition II.7.8] preserving
standard modules.

  2. (2)

If $\lambda,\mu\in C_{-}^{p}$ and $\mu$ belongs to the closure of the facet
containing $\lambda$ then
$T_{\lambda}^{\mu}:\operatorname{\mathrm{Rep}}_{\lambda}\rightarrow\operatorname{\mathrm{Rep}}_{\mu}$
sends each simple (resp. standard) module to a simple (resp. standard) module
or zero. We refer the reader to [Jan03, §§II.7.11-15] for the precise
statements.

A consequence of these two properties is that, if $\mu$ belongs to the closure
of the facet containing $\lambda$, and if one knows character formulas

(1.14) $\operatorname{ch}L_{x\bullet_{p}\lambda}=\sum
a_{y,x}\chi_{y\bullet_{p}\lambda}$

for all $x\bullet_{p}\lambda\in X_{+}$, then one may easily deduce similar
character formulas for $L_{x\bullet_{p}\mu}$ for all $x\bullet_{p}\mu\in
X_{+}$.

The interior of $C_{-}^{p}$ constitutes the unique open facet of $C_{-}^{p}$.
The following are equivalent:

  1. (1)

there exists a $p$-regular weight $\lambda\in X$;

  2. (2)

$C_{-}^{p}$ contains a point of $X$ in its interior;

  3. (3)

$C_{-}^{p}$ contains $-2\rho$ in its interior;

  4. (4)

$p\geq h$ where $h$ is the _Coxeter number_ 888Warning: $h$ will usually
disagree with the classical definition of the Coxeter number if our root
system is decomposable.:

$h=\max_{\alpha\in\Phi_{+}}(\langle\alpha^{\vee},\rho\rangle+1).$

Thus, if $p\geq h$ and we know expressions (1.14) for all simple modules in
$\operatorname{\mathrm{Rep}}_{0}$, then we may deduce character formulas for
all simple modules in $\operatorname{\mathrm{Rep}}$.

###  1.12. Lusztig’s character formula

We keep the notation from previous sections.

######  Conjecture 1.17 (Lusztig conjecture [Lus80], original version).

Fix a $p$-regular weight $\mu\in C_{-}^{p}$ and $x\in\mathcal{W}$ such that
$x\bullet_{p}\mu\in X_{+}$. Suppose that $p\geq h$ and that
$\langle\alpha^{\vee},x\bullet_{p}\mu+\rho\rangle\leq p(p-h+2)$ for all
$\alpha\in\Phi_{+}$ (“Jantzen’s condition”). Then

(LCF) $[L_{x\bullet_{p}\mu}]=\sum_{y\leq x\atop y\bullet_{p}\mu\in
X^{+}}\varepsilon_{yx}h_{y,x}(1)[\Delta_{y\bullet_{p}\mu}].$

######  Remark 1.18.

Some remarks concerning Lusztig’s conjecture:

  1. (1)

Lusztig’s original formulation fixed the choice $\mu=-2\rho$. It is equivalent
to the above formulation by Jantzen’s translation principle. In this way one
can also see that (LCF) implies a similar formula where $x$ and $y$ are
allowed to belong to the extended affine Weyl group
$\mathcal{W}^{\mathrm{ext}}$ rather than $\mathcal{W}$ (but still satisfy the
other conditions).

  2. (2)

If $\mu$ is not $p$-regular then using translation functors one can deduce
from (LCF) an identical expression for $[L_{x\bullet_{p}\mu}]$ as long as one
assumes that $x$ is of minimal length amongst all such
$x^{\prime}\in\mathcal{W}$ with $x\bullet_{p}\mu=x^{\prime}\bullet_{p}\mu$.

  3. (3)

One of the remarkable aspects of (LCF) is that it predicts that part of the
representation theory of $G_{\Bbbk}$ is “independent of $p$”: if we use the
$p$-dilated dot action to parametrise our highest weights, the coefficients
expression simple modules in terms of standard modules are independent of $p$!

  4. (4)

Let us try to explain the meaning of Jantzen’s condition. Consider the
$p$-adic expansion of our highest weight

$x\bullet_{p}\mu=\sum\lambda_{i}p^{i}\quad\text{with $\lambda_{i}\in
X_{1}^{p}$.}$

Jantzen noticed (see [Jan08, §4.4] and [Jan08, §8.22]) that a necessary
condition for independence of $p$ is that

(1.15) $\lambda_{i}$ is zero for $i\geq 2$ and $\lambda_{1}\in C_{p}^{+}$.

It is easy to prove that if $x\bullet_{p}\mu$ satisfies Jantzen’s condition
then it satisfies (1.15). However in general there will be weights satisfying
(1.15) which do not satisfy Jantzen’s condition. As far as we can tell,
Jantzen’s condition provides an easily defined and large set on which (1.15)
holds, and has no significance beyond that.

  5. (5)

Let us try to give a rough idea why (1.15) is necessary for independence of
$p$. Write

$x\bullet_{p}\mu=\lambda_{0}+p\lambda_{1}$

with $\lambda_{0}\in X_{1}^{p}$. Then $\lambda_{1}$ is independent of $p$.
Consider the module

$\widetilde{L_{x\bullet_{p}\mu}}:=L_{\lambda_{0}}\otimes\Delta_{\lambda_{1}}^{\mathrm{Fr}}.$

It follows from Lusztig’s conjecture for quantum groups (a theorem) that

$[\widetilde{L_{x\bullet_{p}\mu}}]=\sum_{y\leq x\atop y\bullet_{p}\mu\in
X^{+}}\varepsilon_{yx}h_{y,x}(1)[\Delta_{y\bullet_{p}\mu}].$

(The important point is that this expression is independent of $p$.) If $p$ is
large enough then $\lambda_{1}$ belongs to $C_{p}^{+}$ and
$\Delta_{\lambda_{1}}$ is simple, and thus we have equality
$\widetilde{L_{x\bullet_{p}\mu}}=L_{x\bullet_{p}\mu}$ by Steinberg’s tensor
product theorem. In particular, if we can write $[L_{x\bullet_{p}\mu}]$ in
terms of standard modules in a manner which is independent of $p$ then we must
have $\widetilde{L_{x\bullet_{p}\mu}}=L_{x\bullet_{p}\mu}$ and hence
$\Delta_{\lambda_{1}}$ must be simple. This is ensured by (1.15).

  6. (6)

For other discussions of Lusztig’s conjecture see [And87], [Don98], [Sco98]
and [Hum06].

###  1.13. Lusztig’s conjecture and the Steinberg tensor product theorem

Suppose that $\mu$ is $p$-regular and $x\bullet_{p}\mu$ is dominant. Consider
the $p$-adic expansion of our highest weight

$x\bullet_{p}\mu=\sum\lambda_{i}p^{i}\quad\text{with $\lambda_{i}\in
X_{1}^{p}$.}$

If we can apply (LCF) to $L_{x\bullet_{p}\mu}$ then $\lambda_{i}=0$ for $i\geq
2$ and $\lambda_{1}\in C_{p}^{+}$ (see Remark 1.18(4)). Hence,
$\Delta_{\lambda_{1}}$ is simple and by Steinberg’s tensor product theorem:

$L_{x\bullet_{p}\mu}=L_{\lambda_{0}}\otimes\Delta_{\lambda_{1}}^{\mathrm{Fr}}.$

If $\lambda_{1}\neq 0$ (and assuming (LCF)) there are two ways to calculate
the character of this module:

  1. (1)

We can apply (LCF) to $L_{x\bullet_{p}\mu}$ directly;

  2. (2)

We can apply (LCF) to calculate $L_{\lambda_{0}}$ and then multiply it with
the Frobenius twist of the character of $\Delta_{\lambda_{1}}$. (Note that
$\lambda_{0}$ might no longer be $p$-regular, in which case we apply Remark
1.18(2).)

Thus the following is reassuring:

######  Theorem 1.19 (Kato [Kat85]).

Lusztig’s conjecture is consistent with the Steinberg tensor product theorem;
that is, both of the above approaches give the same answer.

For any fixed $p$, Lusztig’s conjecture provides us with only finitely many
characters. However, by Steinberg’s tensor product theorem we can calculate
the characters of all $L_{\lambda}$ for $\lambda\in X_{+}$ if we know the
characters of $L_{\lambda}$ for all $\lambda\in X_{1}^{p}$. All weights in
$X_{1}^{p}$ satisfy the Jantzen condition if and only if

$\langle\alpha^{\vee},(p-1)\rho+\rho\rangle\leq p(p-h+2)\quad\text{for all
$\alpha\in\Phi_{+}$,}$

or in other words if

$h-1\leq p-h+2\;\Leftrightarrow\;p\geq 2h-3.$

Thus Lusztig’s conjecture provides a complete conjectural answer if $p\geq
2h-3$.

Since Kato’s result several authors came to regard the following stronger
version of Lusztig’s conjecture as realistic:999Scott [Sco98] refers to it as
“Kato’s extension of the Lusztig conjecture”. Jantzen [Jan03, Jan08] says that
it “seems to be a realistic conjecture”.

######  Conjecture 1.20 (Lusztig conjecture, revised version).

Suppose that $p\geq h$. Then (LCF) holds if $x\bullet_{p}\mu\in X_{1}^{p}$.

######  Remark 1.21.

As we will discuss below, the bound both in the original and revised version
of Lusztig’s conjecture are much too optimistic. However Kato’s theorem and
the revised version are important for (at least) the following reason: as $p$
varies, the number of weights for which one needs to check the original
formulation of Lusztig’s conjecture gets larger and larger. However it is not
difficult to see that the set

$\\{w\in\mathcal{W}\;|\;w\bullet_{p}(-2\rho)\in X_{1}^{p}\\}$

is independent of $p$, as long as $p\geq h$. In this way Lusztig’s conjecture
becomes a finite problem and one might hope to settle it for “all primes at
once”. We will have more to say about this in the second part.

######  Remark 1.22.

As we have mentioned above, combining Steinberg’s tensor product theorem and
(LCF) yields a character formula for all highest weights $\lambda\in X^{+}$.
However for general $\lambda$ this is rather indirect: one needs to apply
(LCF) once for each $p$-adic digit of $\lambda$. Haboush [Hab80] and Humphreys
[Hum06, §3.12] advocate the consideration of different Weyl groups for each
power of $p$. This intriguing idea appears not yet to have borne fruit.

######  Remark 1.23.

Recently Lusztig [Lus15] defined characters

$E_{\lambda}^{0},E_{\lambda}^{1},E_{\lambda}^{2},\dots,E_{\lambda}^{\infty}\in({\mathbb{Z}}X)^{W}$

for fixed $p$ and any highest weight $\lambda$. They are approximations to the
character of $L_{\lambda}$ in the following sense: $E_{\lambda}^{0}$ is the
character of the simple highest weight module in characteristic $0$;
$E_{\lambda}^{1}$ is the character of the simple highest weight module for a
quantum group at a $p^{th}$-root of unity; $E_{\lambda}^{n}$ is obtained from
$E_{\lambda}^{n-1}$ by a formula involving Kazhdan-Lusztig polynomials; and
for fixed $\lambda$ as $n\rightarrow\infty$ the $E_{\lambda}^{n}$ stabilise to
the character $E_{\lambda}^{\infty}$ predicted by (LCF) and Steinberg’s tensor
product theorem. Thus $E^{\infty}_{\lambda}$ gives the character of
$L_{\lambda}$ for large $p$. One might hope that $E_{\lambda}^{n}$ is the
character of a simple highest weight module for a quantum group like object
which has an “$n$-step Steinberg tensor product theorem”. For
$\mathfrak{sl}_{2}$ such an object (for any $n$) has recently been proposed by
Angiono [Ang16].

###  1.14. Lusztig’s character formula and weight multiplicity

After stating his conjecture, Lusztig noticed that it implied an interesting
property of certain Kazhdan-Lusztig polynomials attached to the (extended)
affine Weyl group. Namely, their value at 1 gives the dimension of a weight
space in a simple finite dimensional representation of the Langlands dual
group (see (1.17) below).

Lusztig’s idea was to interpret what his character formula predicts for
Frobenius twists of simple modules. Let us first introduce some notation.
Given $\mu\in X_{+}$ set

$I(\mu)=\\{s\in S\;|\;s(\mu)=\mu\\}$

and let $W^{\mu}$ denote the set of minimal coset representatives for
$W/W_{I(\mu)}$. Given $\mu\in X_{+}$ we define

$\sigma^{\mu}=\sum_{x\in W^{\mu}}e^{x\mu}.$

We will need the identity

(1.16) $\sigma^{\mu}=\sum_{x\in W^{\mu}}\varepsilon_{x}\chi_{\mu-\rho+x\rho}$

which follows (after a little thought) from Weyl’s character formula.

Now fix $\lambda\in X_{+}$ and suppose that $p$ is large enough so that
$L_{\lambda}=\Delta_{\lambda}$ is simple and $p\lambda$ satisfies Jantzen’s
condition. Thus the character

$\operatorname{ch}L_{\lambda}=\sum_{\mu\in
X_{+}}(\dim\Delta_{\lambda}(\mu))\sigma^{\mu}$

is given by Weyl’s character formula. We also have

$L_{p\lambda}=\Delta_{\lambda}^{\mathrm{Fr}}$

and hence

$\displaystyle\operatorname{ch}L_{p\lambda}$ $\displaystyle=\sum_{\mu\in
X_{+}}(\dim\Delta_{\lambda}(\mu))\sigma^{p\mu}$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:WCI}}}{{=}}\sum_{\mu\in
X_{+}}(\dim\Delta_{\lambda}(\mu))\sum_{x\in
W^{\mu}}\varepsilon_{x}\chi_{p\mu-\rho+x\rho}$

We can rewrite this in terms of the $p$-dilated dot action of
$\mathcal{W}^{\mathrm{ext}}$ as

$\displaystyle\operatorname{ch}L_{t_{\lambda}w_{0}\bullet_{p}(-2\rho)}$
$\displaystyle=\sum_{\mu\in X_{+}}(\dim\Delta_{\lambda}(\mu))\sum_{x\in
W^{\mu}}\varepsilon_{x}\operatorname{ch}\Delta_{t_{\mu}xw_{0}\bullet_{p}(-2\rho)}.$

Comparing this with (LCF) we arrive at the prediction:101010We use the version
of (LCF) with the extended affine Weyl group, see Remark 1.18(1).

(1.17) $h_{t_{\mu}w_{0},t_{\lambda}w_{0}}(1)=\dim\Delta_{\lambda}(\mu).$

In other words, (LCF) predicts that the dimensions of weight spaces of Weyl
modules occur as values at 1 of Kazhdan-Lusztig polynomials for
$\mathcal{W}^{\mathrm{ext}}$. This fact was proven by Lusztig [Lus83]
(independently of his conjecture).

######  Remark 1.24.

Because Kazhdan-Lusztig polynomials have non-negative coefficients this gives
a refinement of the dimension of the weight space. A representation theoretic
interpretation of this refinement was given by Brylinski [Bry89] in terms of
what is nowadays called the Brylinski-Kostant filtration of the weight space.

###  1.15. Status of Lusztig’s character formula

We give a brief summary of what is known about Lusztig’s conjecture:

  1. (1)

Lusztig formulated his conjecture in analogy to the Kazhdan-Lusztig conjecture
[KL79]. At the time the case of $\mathrm{SL}_{2}$ was known and Jantzen had
determined the characters of all simple modules for
$\mathrm{SL}_{3},\mathrm{Sp}_{4},\mathrm{G}_{2}$ and $\mathrm{SL}_{4}$ using
his sum formula [Jan77]. Jantzen had also noticed that a character formula for
large $p$ would also determine the characters of simple highest weight modules
in characteristic zero [Jan79, Corollar im Anhang]. The fact that his
conjecture implied the Kazhdan-Lusztig conjecture was also noticed by Lusztig.

  2. (2)

The first proof of the Lusztig conjecture for $p\gg 0$ was obtained by
combining works of Kashiwara-Tanisaki [KT95, KT96] (relating Kazhdan-Lusztig
polynomials and affine Lie algebras), Kazhdan-Lusztig [KL93, KL94a, KL94b]
(relating affine Lie algebras and quantum groups at a root of unity), Lusztig
[Lus94, Lus95] (handling the non-simply-laced case) and Andersen-Jantzen-
Soergel [AJS94] (relating quantum groups and modular representations of the
Lie algebra). (These steps followed a program outlined by Lusztig in [Lus90a,
Lus90b].) The main result of Andersen-Jantzen-Soergel is the existence of a
finitely generated ${\mathbb{Z}}$-algebra $B$ whose base change to a field of
characteristic $p>h$ controls the principal block of restricted Lie algebra
representations and whose base change to ${\mathbb{C}}$ controls
representations of the small quantum group at a root of unity. It is then
possible to deduce Lusztig’s conjecture for algebraic groups in characteristic
$p\gg 0$ from the case of the quantum group. The algebra $B$ is not explicit
and this approach did not yield any reasonable bounds on $p$. Over a decade
later, a more direct route between perverse sheaves and the quantum group was
provided by Arkhipov-Bezrukavnikov-Ginzburg [ABG04].

  3. (3)

More recently, Fiebig used his theory of moment graphs to provide a new proof
of Lusztig’s conjecture [Fie11]. The idea is to give a functor from a
combinatorial category of “moment graph sheaves” associated to the affine
Grassmannian to a combinatorial category constructed by Andersen-Jantzen-
Soergel controlling Lie algebra representations. Lusztig’s conjecture is then
deduced from the decomposition theorem applied to intersection cohomology
complexes on the affine Grassmannian. In essence, Fiebig’s approach simplifies
the original proof discussed above by providing a direct link between
intersection cohomology complexes and the work of Andersen-Jantzen-Soergel. By
a careful analysis of the combinatorics of moment graph sheaves Fiebig was
able to give an explicit (enormous) bound above which Lusztig’s conjecture
holds [Fie12] and establish the multiplicity one case [Fie10b]. Using recent
work of Elias and the author establishing Soergel’s conjecture [EW14] and its
local version [Wil16a], all of the arguments used by Fiebig can be made
entirely algebraic. Using related ideas, an algorithm using Soergel bimodules
to produce the “bad primes” for Lusztig’s conjecture was discovered by
Libedinsky [Lib15].

  4. (4)

The localisation theorem [BMR08, BMR06] of Bezrukavnikov-Mirković-Rumynin,
provides a completely different approach to Lusztig’s conjecture, which is
closer to the original ($D$-module) proof of the Kazhdan-Lusztig conjecture.
Working in the broader setting of Lie algebra representations the authors
establish an equivalence of derived categories with coherent sheaves on
Springer fibres. Roughly speaking, these categories are modules over the
affine Hecke category [BR13] and one can use an alternative realisation of
this category [Bez16] to deduce Lusztig’s conjectures for Lie algebra
representations for large $p$ [BM13]. These are known to imply Lusztig’s
character formula for $G_{\Bbbk}$ [Fie10a].

  5. (5)

Recently the author (building on joint work with Elias [EW13] and He [HW15])
discovered many counter-examples to the expected bounds in Lusztig’s
conjecture [Wil16c, Wil16b]. The upshot is that the above bounds (like $p\geq
h$ or $p\geq 2h-3$) are much too optimistic. In fact, recent advances in
number theory imply that there is no polynomial bound in the Coxeter number
for the validity of Lusztig’s conjecture (see the appendix to [Wil16c] by
Kontorovich, McNamara and the author). We will discuss these results in more
detail in the next section.

##  2\. Constructible Sheaves

###  2.1. Notation

Let $X$ denote a complex algebraic variety acted on by an algebraic group $H$.
For simplicity we assume:

  1. (1)

$H$ has finitely many orbits on $X$;

  2. (2)

each orbit is simply connected;

  3. (3)

each orbit is equivariantly simply connected; i.e. the stabiliser of any point
in $H$ is connected.

More generally, we will also allow $X$ to be an ind-variety with a compatible
action of a pro-algebraic group $H$, such that each finite dimensional
approximation satisfies the above conditions (see [JMW14, §2.7]).

For a fixed ring of coefficients $\Bbbk$ we consider:

$\displaystyle D_{H}^{b}(X;\Bbbk):\begin{array}[]{c}\text{the equivariant
derived category}\\\ \text{of constructible sheaves of $\Bbbk$-modules on
$X$,}\end{array}$ $\displaystyle
D_{(H)}^{b}(X;\Bbbk):\begin{array}[]{c}\text{the derived category of
sheaves}\\\ \text{ constructible with respect to the $H$-orbits on
$X$.}\end{array}$

(In the ind-variety case these categories are defined as the direct limits
under extension by zero of the corresponding finite dimensional
approximations. In particular, any object has finite dimensional support.) We
will sometimes ignore the coefficients and instead write $D_{H}^{b}(X)$ and
$D^{b}_{(H)}(X)$ if the context is clear. We denote by
$\operatorname{For}:D_{H}^{b}(X)\rightarrow D^{b}_{(H)}(X)$ the functor of
forgetting the equivariance. The full subcategories of perverse sheaves are
denoted

$\displaystyle{\mathbf{P}}_{H}(X)\subset D^{b}_{H}(X),$
$\displaystyle{\mathbf{P}}_{(H)}(X)\subset D^{b}_{(H)}(X).$

Consider the decomposition of $X$ into $H$-orbits:

$X=\bigsqcup_{\lambda\in\Lambda}X_{\lambda}.$

Given $\lambda\in\Lambda$ we denote by $j_{\lambda}:X_{\lambda}\hookrightarrow
X$ the inclusion, $d_{\lambda}$ the (complex) dimension of $X_{\lambda}$ and
by $\underline{\Bbbk}_{X_{\lambda}}$ the constant sheaf on $X_{\lambda}$. We
have the perverse sheaves

$\displaystyle\Delta^{\Bbbk}_{\lambda}:={}^{p}j_{\lambda!}(\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}]),\quad\mathbf{IC}^{\Bbbk}_{\lambda}:={}^{p}j_{\lambda!*}(\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}])\quad\nabla^{\Bbbk}_{\lambda}:={}^{p}j_{\lambda*}(\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}])$

which by abuse of notation we regard as objects both of ${\mathbf{P}}_{H}(X)$
and ${\mathbf{P}}_{(H)}(X)$ (it will be clear from the context which object we
mean below). As above, we will sometimes drop the superscript indicating the
coefficients if it is clear from the context. Note that if $j_{\lambda}$ is an
affine morphism (in particular if $X_{\lambda}$ is affine) then

$\Delta^{\Bbbk}_{\lambda}=j_{\lambda!}(\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}]),\quad\nabla^{\Bbbk}_{\lambda}=j_{\lambda*}(\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}]).$

Our assumptions on $X$ and $H$ guarantee that if $\Bbbk$ is a field then the
set

$\\{\mathbf{IC}_{x}^{\Bbbk}\;|\;x\in\Lambda\\}$

is a complete set of representatives for the simple objects in
${\mathbf{P}}_{H}(X;\Bbbk)$ (resp. ${\mathbf{P}}_{(H)}(X;\Bbbk)$).

Sometimes it will be useful to take integral coefficients below. Consider the
functor of extension of scalars

$\displaystyle(-)\otimes^{L}_{{\mathbb{Z}}}\Bbbk:D_{(H)}(X;{\mathbb{Z}})\rightarrow
D_{(H)}(X;\Bbbk).$

For any $\lambda$ as above the object
$\mathbf{IC}_{\lambda}^{\mathbb{Z}}\otimes_{\mathbb{Z}}^{L}\Bbbk$ is perverse
but not simple in general. However if we fix $\lambda$ and allow $\Bbbk$ to
vary then $\mathbf{IC}_{\lambda}^{\mathbb{Z}}\otimes_{\mathbb{Z}}^{L}\Bbbk$
will fail to be simple in only finitely many (positive) characteristics. For
background on decomposition numbers for perverse sheaves the reader is
referred to [Jut09].

In addition let us assume

$H^{j}(X_{\lambda};{\mathbb{Z}})=H^{j}_{H}(X_{\lambda};{\mathbb{Z}})=0\quad\text{for
$j$ odd, and all $\lambda\in\Lambda$.}$

For any field $\Bbbk$ and $\lambda\in\Lambda$ we denote by

${\mathcal{E}}_{\lambda}^{\Bbbk}\in D^{b}_{H}(X)\quad\text{(resp. $\in
D^{b}_{(H)}(X)$)}$

the indecomposable parity sheaf (if it exists). Recall that
${\mathcal{E}}_{\lambda}^{\Bbbk}$ is characterised up to isomorphism by the
properties: it is supported on $\overline{X_{\lambda}}$; its restriction to
$X_{\lambda}$ is isomorphic to $\underline{\Bbbk}_{X_{\lambda}}[d_{\lambda}]$;
and, its stalks and costalks vanish in degrees of parity differing from that
of $d_{\lambda}$. We denote by

$\operatorname{\mathrm{Par}}_{H}(X)\subset
D^{b}_{H}(X)\quad\text{and}\quad\operatorname{\mathrm{Par}}_{(H)}(X)\subset
D^{b}_{(H)}(X)$

the full subcategories of parity complexes.

###  2.2. The affine Grassmannian and flag variety

Let $G^{\vee}$ denote a complex reductive group whose root datum is dual to
the root datum of $G_{\Bbbk}$. Let $T^{\vee}\subset G^{\vee}$ denote the dual
torus. Because $G_{\Bbbk}$ is assumed simply connected, $G^{\vee}$ is of
adjoint type.

To $G^{\vee}$ we associate its algebraic loop group $G^{\vee}((t))$. Let
$K=G^{\vee}[[t]]$ denote a maximal compact subgroup. We consider the
evaluation at $t=0$ map

$\mathrm{ev}:K=G^{\vee}[[t]]\stackrel{{\scriptstyle
t=0}}{{\longrightarrow}}G^{\vee}$

and consider the Iwahori subgroup

$\mathrm{Iw}:=\mathrm{ev}^{-1}(B^{\vee})$

where $B^{\vee}$ is the Borel subgroup in $G^{\vee}$ containing $T^{\vee}$ and
whose Lie algebra contains all characters in $\Phi_{+}^{\vee}$.

The _affine flag variety_ and _affine Grassmannian_ are the spaces

$\displaystyle{\mathcal{F}l^{\vee}}:=G^{\vee}((t))/\mathrm{Iw},$
$\displaystyle{\mathcal{G}r^{\vee}}:=G^{\vee}((t))/K.$

Both are ${\mathbb{C}}$-schemes (of infinite type) in a natural way. In fact,
they are both ind-projective varieties. The natural projection

(2.1) $\displaystyle p:{\mathcal{F}l^{\vee}}\rightarrow{\mathcal{G}r^{\vee}}$

realises ${\mathcal{F}l^{\vee}}$ as a $G^{\vee}/B^{\vee}$-bundle over
${\mathcal{G}r^{\vee}}$.

We have

$\displaystyle\pi_{0}({\mathcal{F}l^{\vee}})=\pi_{0}({\mathcal{G}r^{\vee}})=\Omega=X_{*}/{\mathbb{Z}}\Phi^{\vee}$

where $\Omega$ is the set of length-zero elements introduced in §1.7.

######  Remark 2.1.

All connected components of ${\mathcal{F}l^{\vee}}$ and
${\mathcal{G}r^{\vee}}$ are isomorphic as ind-varieties (even as
$\mathrm{Iw}$-varieties). Each connected component is isomorphic to the Kac-
Moody flag variety associated to the extended Cartan matrix of $G^{\vee}$.

The $\mathrm{Iw}$-orbits on ${\mathcal{F}l^{\vee}}$ yields the Bruhat
decomposition

(2.2)
$\displaystyle{\mathcal{F}l^{\vee}}=\bigsqcup_{x\in\mathcal{W}^{\mathrm{ext}}}{\mathcal{F}l_{x}^{\vee}}\quad\text{where}\quad{\mathcal{F}l_{x}^{\vee}}:=\mathrm{Iw}\cdot
x\mathrm{Iw}/\mathrm{Iw}$

and each $\mathrm{Iw}$-orbit is an affine space of dimension

(2.3) $\displaystyle\dim{\mathcal{F}l_{x}^{\vee}}=\ell(x).$

Similarly, the $\mathrm{Iw}$-orbits on ${\mathcal{G}r^{\vee}}$ give the Bruhat
decomposition

(2.4)
$\displaystyle{\mathcal{G}r^{\vee}}=\bigsqcup_{x\in\mathcal{W}^{\mathrm{ext}}/W}{\mathcal{G}r_{x}^{\vee}}\quad\text{where}\quad{\mathcal{G}r_{x}^{\vee}}:=\mathrm{Iw}\cdot
x\mathrm{Iw}/K$

and each cell in the Bruhat decomposition is isomorphic to an affine space of
dimension

(2.5)
$\displaystyle\dim{\mathcal{G}r_{x}^{\vee}}=\ell(x_{-})=\ell(x_{+})-\ell(w_{0})$

where $x_{-}$ (resp. $x_{+}$) is the minimal (resp. maximal) element in the
coset $xW$.

Recall that any element of $X$ is a coweight of $G^{\vee}$ which we can regard
as a point $t^{\lambda}\in G^{\vee}[t,t^{-1}]\subset G^{\vee}((t))$. These
points feature in the decomposition of ${\mathcal{G}r^{\vee}}$ into
$K$-orbits:

(2.6) $\displaystyle{\mathcal{G}r^{\vee}}=\bigsqcup_{\lambda\in
X_{+}}{\mathcal{G}r_{\lambda}^{\vee}}\quad\text{where}\quad{\mathcal{G}r_{\lambda}^{\vee}}:=K\cdot
t^{\lambda}K/K.$

Each cell is of dimension

(2.7) $\displaystyle\dim K\cdot t^{\lambda}K/K=2\langle\lambda,\rho\rangle.$

Moreover, each cell is an affine space bundle over a partial flag variety and
in particular is simply connected.

Let $\Bbbk$ denote a fixed field of coefficients. With a slight variation on
the notation of the previous section denote by

$\mathbf{IC}^{\Bbbk}_{x,\mathrm{Iw}}\in
D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}}),\;\mathbf{IC}^{\Bbbk}_{x,K}\in
D^{b}_{\mathrm{Iw}}({\mathcal{G}r^{\vee}})\text{ and
}\mathbf{IC}^{\Bbbk}_{\lambda}\in D^{b}_{K}({\mathcal{G}r^{\vee}})$

the simple perverse sheaves.

Recall the following classical theorem of Kazhdan and Lusztig:

######  Theorem 2.2 (Kazhdan-Lusztig [KL80]).

For $x,y\in\mathcal{W}^{\mathrm{ext}}$ we have

$h_{y,x}=\sum_{i\in{\mathbb{Z}}}\dim
H^{-i}((\mathbf{IC}^{\mathbb{Q}}_{x,\mathrm{Iw}})_{y})v^{i-\ell(y)}.$

Because $h_{y,x}\in v^{\ell(x)-\ell(y)}{\mathbb{Z}}[v^{2}]$, setting $v=-1$ we
deduce that

$\varepsilon_{x}\varepsilon_{y}(h_{y,x}(1))=\varepsilon_{y}\chi((\mathbf{IC}_{x,\mathrm{Iw}}^{{\mathbb{Q}}})_{y})$

or in other words

(2.8) $\chi((\mathbf{IC}_{x}^{\mathbb{Q}})_{y})=\varepsilon_{x}h_{y,x}(1).$

We now deduce a similar formula for the affine Grassmannian. Fix
$x,y\in\mathcal{W}^{\mathrm{ext}}$ and assume that $x\in xW$ and $y\in yW$ are
maximal. Because $p:{\mathcal{F}l^{\vee}}\rightarrow{\mathcal{G}r^{\vee}}$ is
a smooth fibration with fibre $G^{\vee}/B^{\vee}$ and
${\mathcal{F}l_{x}^{\vee}}\subset p^{-1}{\mathcal{G}r_{x}^{\vee}}$ is open and
dense we have

$p^{*}\mathbf{IC}_{x,K}[\ell(w_{0})]\cong\mathbf{IC}_{x,\mathrm{Iw}}.$

Thus

(2.9)
$\chi((\mathbf{IC}_{x,K}^{\mathbb{Q}})_{y})=\varepsilon_{w_{0}}\chi((\mathbf{IC}_{x,\mathrm{Iw}}^{\mathbb{Q}})_{y}=\varepsilon_{w_{0}}\varepsilon_{x}h_{y,x}(1).$

###  2.3. The (extended) affine Hecke category

Recall the affine Hecke algebra $\mathrm{H}$ and the extended affine Hecke
algebra $\mathrm{H}^{\mathrm{ext}}$ from §1.8. In this section we will
describe categorifications of $\mathrm{H}$ and $\mathrm{H}^{\mathrm{ext}}$ via
sheaves on ${\mathcal{F}l^{\vee}}$.

The category $D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})$ is equipped with the
structure of a monoidal category:

$\displaystyle*:D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})\times
D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})\rightarrow
D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})$

defined via

$\displaystyle{\mathcal{F}}*{\mathcal{G}}:=m_{*}(\widetilde{{\mathcal{F}}{\mathcal{G}}})$

where
$m:G^{\vee}((t))\times_{\mathrm{Iw}}{\mathcal{F}l^{\vee}}\rightarrow{\mathcal{F}l^{\vee}}$
is induced by the multiplication map on $G^{\vee}((t))$ and, in a shorthand
notation,

$\widetilde{{\mathcal{F}}{\mathcal{G}}}:={\mathrm{res}}_{\mathrm{Iw}^{4}}^{\mathrm{Iw}^{3}}({\mathcal{F}}\boxtimes{\mathcal{G}}).$

In more detail, to construct $\widetilde{{\mathcal{F}}{\mathcal{G}}}$, we:

  1. (1)

use the quotient equivalence
$D^{b}_{\mathrm{Iw}\times\mathrm{Iw}}(G^{\vee}((t)))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})$
to view the exterior tensor product ${\mathcal{F}}\boxtimes{\mathcal{G}}\in
D^{b}_{\mathrm{Iw}^{4}}(G^{\vee}((t))\times G^{\vee}((t)))$;

  2. (2)

restrict along the map
$\mathrm{Iw}^{3}\hookrightarrow\mathrm{Iw}^{4}:(a,b,c)\mapsto(a,b^{-1},b,c)$
to obtain an object
${\mathrm{res}}_{\mathrm{Iw}^{4}}^{\mathrm{Iw}^{3}}({\mathcal{F}}\boxtimes{\mathcal{G}})\in
D^{b}_{\mathrm{Iw}^{3}}(G^{\vee}((t))\times G^{\vee}((t)))$;

  3. (3)

use the equivalence $D^{b}_{\mathrm{Iw}^{3}}(G^{\vee}((t))\times
G^{\vee}((t)))\stackrel{{\scriptstyle\sim}}{{\rightarrow}}D^{b}_{\mathrm{Iw}}(G^{\vee}((t))\times_{\mathrm{Iw}}{\mathcal{F}l^{\vee}})$
to view
${\mathrm{res}}_{\mathrm{Iw}^{4}}^{\mathrm{Iw}^{3}}({\mathcal{F}}\boxtimes{\mathcal{G}})$
as an object $\widetilde{{\mathcal{F}}{\mathcal{G}}}\in
D^{b}_{\mathrm{Iw}}(G^{\vee}((t))\times_{\mathrm{Iw}}{\mathcal{F}l^{\vee}})$.

######  Remark 2.3.

The above definition of convolution is mimicking convolution of
$H$-biinvariant functions on a finite group $G$ (for $H\subset G$ a subgroup).
It was in the context of Grothendieck’s function-sheaf dictionary that this
definition was first made [Spr82].

######  Remark 2.4.

The above definition makes sense if we regard $G^{\vee}((t))$ and
$\mathrm{Iw}$ as (infinite-dimensional) topological groups. However if one
wishes to work in a more algebraic category (for example to apply the
decomposition theorem) then more care is needed, see [Nad05, §2.2 and §3.3].

######  Remark 2.5.

Below an important property of convolution is that it preserves parity
complexes [JMW14, Theorem 4.8]. In its basic form, this observation goes back
to Springer, Brylinski and MacPherson [Spr82]. Its importance for modular
representation theory was emphasised by Soergel in [Soe00].

Given a collection of objects $\\{{\mathcal{F}}_{i}\\}_{i\in I}$ in
$D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})$ let
$\langle{\mathcal{F}}_{i}\rangle_{*,\oplus,[{\mathbb{Z}}]}$ denote the
additive, graded, monoidal envelope of $\\{{\mathcal{F}}_{i}\\}_{i\in I}$: its
objects are the direct sums of shifts of tensor products

${\mathcal{F}}_{\textbf{i}}:={\mathcal{F}}_{i}*{\mathcal{F}}_{j}*\dots*{\mathcal{F}}_{k}$

for any finite sequence ${\textbf{i}}=(i,j,\dots,k)$ of elements of $I$. (We
allow the empty sequence and set
${\mathcal{F}}_{\emptyset}:={\mathcal{E}}_{\operatorname{id}}$.) We denote by
$\langle{\mathcal{F}}_{i}\rangle_{*,\oplus,[{\mathbb{Z}}],\mathrm{Kar}}$ the
category obtained from
$\langle{\mathcal{F}}_{i}\rangle_{*,\oplus,[{\mathbb{Z}}]}$ by adjoining all
direct summands of objects.

######  Remark 2.6.

The notation is intended to remind us that there is a formal procedure
(“Karoubi envelope”) which allows us to produce (a category equivalent to)
$\langle{\mathcal{F}}\rangle_{*,\oplus,[{\mathbb{Z}}],\mathrm{Kar}}$ starting
from $\langle{\mathcal{F}}_{i}\rangle_{*,\oplus,[{\mathbb{Z}}]}$.

We set

$\displaystyle{\mathcal{H}}_{\mathrm{BS}}:=\langle{\mathcal{E}}_{s}\;|\;s\in
S\rangle_{*,\oplus,[{\mathbb{Z}}]},\quad{\mathcal{H}}^{\mathrm{ext}}_{\mathrm{BS}}:=\langle\\{{\mathcal{E}}_{s}\;|\;s\in
S\\}\cup\\{{\mathcal{E}}_{\omega}\;|\;\omega\in\Omega\\}\rangle_{*,\oplus,[{\mathbb{Z}}]},$
$\displaystyle{\mathcal{H}}:=\langle{\mathcal{E}}_{s}\;|\;s\in
S\rangle_{*,\oplus,[{\mathbb{Z}}],\mathrm{Kar}},\quad{\mathcal{H}}^{\mathrm{ext}}:=\langle{\mathcal{E}}_{s}\;|\;s\in
S\\}\cup\\{{\mathcal{E}}_{\omega}\;|\;\omega\in\Omega\\}\rangle_{*,\oplus,[{\mathbb{Z}}],\mathrm{Kar}}.$

We call ${\mathcal{H}}$ and ${\mathcal{H}}^{\mathrm{ext}}$ the _(extended)
affine Hecke category_. One may check that ${\mathcal{H}}$ is the full
subcategory of ${\mathcal{H}}^{\mathrm{ext}}$ consisting of objects supported
on the identity component of ${\mathcal{F}l^{\vee}}$.

It follows from Remark 2.5 that every object of ${\mathcal{H}}^{\mathrm{ext}}$
is parity. In fact, one has equalities

$\displaystyle{\mathcal{H}}^{\mathrm{ext}}=\operatorname{\mathrm{Par}}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})\quad\text{and}\quad{\mathcal{H}}^{\mathrm{ext}}=\operatorname{\mathrm{Par}}_{\mathrm{Iw}}(({\mathcal{F}l^{\vee}})_{0})$

where $({\mathcal{F}l^{\vee}})_{0}$ denotes the identity component of
${\mathcal{F}l^{\vee}}$. The following theorem explains the name of
${\mathcal{H}}$ and ${\mathcal{H}}^{\mathrm{ext}}$.

######  Theorem 2.7.

There exists a unique isomorphism

$\mathrm{H}^{\mathrm{ext}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[{\mathcal{H}}^{\mathrm{ext}}]_{\oplus}$

such that $\underline{h}_{s}\mapsto[{\mathcal{E}}_{s}]$ for all
$s\in\mathcal{S}$ and $\omega\mapsto{\mathcal{E}}_{\omega}$ for all
$\omega\in\Omega$. This isomorphism induces an isomorphism
$\mathrm{H}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[{\mathcal{H}}]_{\oplus}$.

(Recall that $[{\mathcal{H}}^{\mathrm{ext}}]_{\oplus}$ denotes the split
Grothendieck group of ${\mathcal{H}}^{\mathrm{ext}}$, see §0.3.) We denote by

$\operatorname{ch}:[{\mathcal{H}}^{\mathrm{ext}}_{\oplus}]\rightarrow\mathrm{H}^{\mathrm{ext}}$

the inverse to the isomorphism of the theorem. It may be described explicitly
via:

$\operatorname{ch}({\mathcal{F}})=\sum_{x\in\mathcal{W}^{\mathrm{ext}}}\left(\sum_{i\in{\mathbb{Z}}}\dim
H^{-i}({\mathcal{F}}_{y})v^{i-\ell(y)}\right)h_{x}.$

We define elements ${}^{p}\underline{h}_{x}$ via

${}^{p}\underline{h}_{x}:=\operatorname{ch}({\mathcal{E}}_{x}).$

Then $\\{{}^{p}\underline{h}_{x}\\}$ is a basis for
$\mathrm{H}^{\mathrm{ext}}$ called the _$p$ -canonical basis_. (It only
depends on the characteristic of $\Bbbk$.) If we write

${}^{p}\underline{h}_{x}:=\sum_{y\leq x}{}^{p}h_{y,x}h_{y}$

then the polynomials ${}^{p}h_{h,x}$ are the _$p$ -Kazhdan-Lusztig
polynomials_. Here are some basic properties:

  1. (1)

For fixed $x\in\mathcal{W}^{\mathrm{ext}}$ and $p\gg 0$ we have
${}^{p}\underline{h}_{x}=\underline{h}_{x}$.

  2. (2)

If we write

${}^{p}\underline{h}_{x}=\sum{}^{p}a_{y,x}\cdot{}^{p}\underline{h}_{y}$

then ${}^{p}a_{y,x}\in{\mathbb{Z}}_{\geq 0}[v,v^{-1}]$ and
$\overline{{}^{p}a_{y,x}}={}^{p}a_{y,x}$. In particular, ${}^{p}h_{y,x}$ have
$\geq 0$ coefficients.

  3. (3)

If we write

${}^{p}\underline{h}_{x}{}^{p}\underline{h}_{y}=\sum{}^{p}\mu^{z}_{y,x}\cdot{}^{p}\underline{h}_{z}$

then ${}^{p}\mu^{z}_{y,x}\in{\mathbb{Z}}_{\geq 0}[v,v^{-1}]$ and
$\overline{{}^{p}\mu^{z}_{y,x}}={}^{p}\mu^{z}_{y,x}$.

For further discussion and examples the reader is referred to [JW15].

######  Remark 2.8.

Kazhdan-Lusztig polynomials depend only on the underlying Coxeter system. This
is no longer true for $p$-Kazhdan-Lusztig polynomials (see [JW15]).

######  Remark 2.9.

It is important for computations that ${\mathcal{H}}$ and
${\mathcal{H}}^{\mathrm{ext}}$ have an alternative diagrammatic / algebraic
presentation via generators and relations [EW13, RW15].

###  2.4. Geometric Satake equivalence

As for $D^{b}_{\mathrm{Iw}}({\mathcal{F}l^{\vee}})$, we can equip
$D^{b}_{K}({\mathcal{G}r^{\vee}})$ with the structure of a monoidal category
via

${\mathcal{F}}*{\mathcal{G}}:=m_{*}\widetilde{{\mathcal{F}}{\mathcal{G}}}$

where
$m:G^{\vee}((t))\times_{K}{\mathcal{G}r^{\vee}}\rightarrow{\mathcal{G}r^{\vee}}$
is induced by the multiplication on $G^{\vee}((t))$ and

$\widetilde{{\mathcal{F}}{\mathcal{G}}}:={\mathrm{res}}_{K^{4}}^{K^{3}}({\mathcal{F}}\boxtimes{\mathcal{G}}).$

is defined by mimicking the construction in §2.3, with $K$ in place of
$\mathrm{Iw}$.

In this setting two miracles occur:

  1. (1)

The convolution preserves ${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})$: if
${\mathcal{F}},{\mathcal{G}}\in{\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})$ then
so is ${\mathcal{F}}*{\mathcal{G}}$;

  2. (2)

The convolution $*$ is symmetric: we have a canonical isomorphism
${\mathcal{F}}*{\mathcal{G}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathcal{G}}*{\mathcal{F}}$
equipping ${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})$ with the structure of
symmetric tensor category.

Recall our semi-simple algebraic group $G_{\Bbbk}$ over $\Bbbk$ whose root
system is dual to that of $G^{\vee}$.

######  Theorem 2.10 (Geometric Satake [MV07]).

There is an equivalence of monoidal categories

${\mathcal{S}}:({\mathbf{P}}_{K}({\mathcal{G}r^{\vee}};\Bbbk),*)\stackrel{{\scriptstyle\sim}}{{\rightarrow}}(\operatorname{\mathrm{Rep}}G_{\Bbbk},\otimes_{\Bbbk}).$

More generally, this theorem is true with coefficients in any commutative
ring.111111In [MV07] the result is proved under the additional assumption that
the ring of coefficients if Noetherian and of finite global dimension. However
this is for simplicity only (see the discussion at the bottom of pg. 100 of
[MV07]).

######  Remark 2.11.

Some remarks on the geometric Satake equivalence:

  1. (1)

A remarkable aspect of the proof of geometric Satake is that it does not
construct a functor in either direction! Instead, one uses the Tannakian
formalism to deduce that
${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}};{\mathbb{Z}})$ is equivalent to the
representations of some group scheme over ${\mathbb{Z}}$. After considerable
work, one manages to identify this group scheme with the Chevalley group
scheme. Thus the equivalence can actually be seen as providing a
_construction_ of the dual group. In the above notation, $G_{\mathbb{Z}}$ is
constructed starting from $G^{\vee}$. As far as the author is aware, geometric
Satake is the only known construction of the Langlands dual group.

  2. (2)

There is no proof of geometric Satake which works directly with coefficients
in characteristic $p$. At present, the case of coefficients of characteristic
$p$ is deduced by reduction modulo $p$ from the corresponding statement over
${\mathbb{Z}}$.

  3. (3)

Recent work of Mautner-Riche [MR13] would yield a new proof that
${\mathcal{S}}$ is an equivalence of abelian categories if one could prove
that ${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}};k)$ is a highest weight category
without using geometric Satake.

  4. (4)

Most of the basic theorems concerning the representation theory of $G_{\Bbbk}$
have no proof on the side of perverse sheaves. For example, at present there
is no geometric proof of either the Steinberg tensor product theorem or the
linkage principle.

  5. (5)

If the characteristic $p$ of $\Bbbk$ is good then the parity sheaves
${\mathcal{E}}_{\lambda}\in D^{b}_{K}({\mathcal{G}r^{\vee}};\Bbbk)$ are
perverse and correspond under the geometric Satake equivalence to tilting
modules [JMW16, MR13]. This gives a proof via constructible sheaves of Theorem
1.8 (at least in good characteristic). This result was used by Achar and Rider
to show that the stalks of $\Delta_{\lambda}^{\mathbb{Z}}$ are free of
$p$-torsion if $p$ is good for $G$ (“Mirković-Vilonen conjecture” [Jut08])
[AR15].

###  2.5. The Finkelberg-Mirković conjecture

The geometric Satake equivalence is not so useful for studying character
formulas, because the linkage principle is not visible. (Indeed, as we have
tried to explain, it is more often the case that theorems on representations
of $G_{\Bbbk}$ predict remarkable behaviour on the constructible side of
geometric Satake which have no geometric explanation at present.) In this
section we explain a conjecture of Finkelberg-Mirković which gives a geometric
realisation of the extended principal block. Because it already incorporates
the linkage principle, it is much more useful for understanding characters
(amongst other things).

Recall that $\operatorname{\mathrm{Rep}}$ denotes the category of finite-
dimensional algebraic representations of our semi-simple group $G_{\Bbbk}$,
which is defined over a field $\Bbbk$. Also recall the extended principal
block $\mathrm{Rep}_{0}^{\mathrm{ext}}\subset\operatorname{\mathrm{Rep}}$. In
§1.10 we explained that Frobenius twist makes
$\mathrm{Rep}_{0}^{\mathrm{ext}}$ a right module category over
$\operatorname{\mathrm{Rep}}$ via $(V,M)\mapsto V\otimes M^{\mathrm{Fr}}$.

By imitating the convolution product on
${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})$ one can define a convolution product

$*:D^{b}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}})\times
D^{b}_{K}({\mathcal{G}r^{\vee}})\rightarrow D^{b}_{K}({\mathcal{G}r^{\vee}})$

which again (miraculously) descends to a convolution product

$*:{\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}})\times{\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})\rightarrow{\mathbf{P}}_{K}({\mathcal{G}r^{\vee}}),$

making ${\mathbf{P}}_{(\mathrm{Iw})}$ a right module category over
${\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})$.

######  Conjecture 2.12 (Finkelberg-Mirković [FM99]).

There is an equivalence

${\mathcal{Q}}:\mathrm{Rep}_{0}^{\mathrm{ext}}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}{\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}},k)$

mapping

$\displaystyle L_{x\bullet_{p}(-2\rho)}\mapsto\mathbf{IC}_{x^{-1}}$
$\displaystyle\Delta_{x\bullet_{p}(-2\rho)}\mapsto\Delta_{x^{-1}}$

for all $x\in\mathcal{W}^{\mathrm{ext}}$ such that $x\bullet_{p}(-2\rho)\in
X_{+}$.

Moreover, this equivalence is compatible with geometric Satake and Frobenius
twist. That is, the following diagram is commutative up to natural
isomorphism:

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
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######  Remark 2.13.

Some remarks concerning the Finkelberg-Mirković conjecture:

  1. (1)

At first sight the natural inclusions on both sides of this equivalence appear
to go in opposite (“wrong”) directions: on the representation theory side we
have $\mathrm{Rep}_{0}^{\mathrm{ext}}\subset\operatorname{\mathrm{Rep}}$;
whereas on the perverse sheaf side forgetting $K$-equivariance defines a
fully-faithful embedding
$\operatorname{For}_{K}:{\mathbf{P}}_{K}({\mathcal{G}r^{\vee}})\hookrightarrow{\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}})$.
This “contradiction” is resolved by Frobenius twist. By acting on the right on
the trivial representation, the commutativity of (2.10) implies that we have a
commutative diagram up to natural isomorphism:

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In fact, [FM99] ask only for the commutativity of (2.11).

  2. (2)

The Finkelberg-Mirković conjecture is still a conjecture. Recently, Achar and
Riche have come very close to a proof [AR16c]. Building on work of Achar-Riche
[AR16a, AR16b, AR14a], Achar-Rider [AR15, AR14b] and Mautner-Riche [MR13] they
prove that a certain “mixed” version of the category
${\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}})$ provides a graded
enhancement of $\mathrm{Rep}_{0}^{\mathrm{ext}}$, and check compatibility with
geometric Satake. The remaining difficulty is to construct a “forgetting the
mixed structure” functor to
${\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}})$.

  3. (3)

The Finkelberg-Mirković conjecture predicts that
${\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}},{\mathbb{Z}})$ provides an
abelian category over ${\mathbb{Z}}$ from which the principal block of
$G_{\Bbbk}$ in any characteristic $p\geq h$ may be deduced by “reduction
modulo $p$”. This provides a simple explanation for many independence of $p$
results, as we hope will become clear. The existence of such an integral form
seems mysterious from an algebraic point of view.

Let us explain how the Finkelberg-Mirković conjecture may be used to deduce
character formulas for $\operatorname{\mathrm{Rep}}_{0}$ in terms of the
geometry of the affine Grassmannian. As usual, we start by writing

$[L_{x\bullet_{p}(-2\rho)}]=\sum a_{y,x}[\Delta_{y\bullet_{p}(-2\rho)}]$

for certain $a_{y,x}\in{\mathbb{Z}}$. Applying the equivalence ${\mathcal{Q}}$
from the Finkelberg-Mirković conjecture we deduce

$[\mathbf{IC}_{x^{-1}}]=\sum a_{y,x}[\Delta_{y^{-1}}].$

Taking the Euler characteristic on both sides at a point of the stratum
$\mathcal{G}r_{y^{-1}}^{\vee}$ yields121212In case the reader wants to worry
about signs: The dimension of $\mathcal{G}r_{y^{-1}}^{\vee}$ is
$\ell(y^{-1})-\ell(w_{0})$, because $y\in Wy$ is maximal. This is where the
$\varepsilon_{w_{0}}\varepsilon_{y^{-1}}$ comes from.

(2.12)
$a_{y,x}=\varepsilon_{w_{0}}\varepsilon_{y^{-1}}\chi((\mathbf{IC}_{x^{-1}})_{y^{-1}}).$

Thus the $a_{y,x}$ are (up to sign) simply the Euler characteristics of the
stalks of the intersection cohomology complexes on ${\mathcal{G}r^{\vee}}$!

Moreover, if the characteristic $p$ of $\Bbbk$ is large enough then,

$\chi((\mathbf{IC}^{\Bbbk}_{x^{-1},K})_{y^{-1}})=\chi((\mathbf{IC}^{\mathbb{Q}}_{x^{-1},K})_{y^{-1}})\stackrel{{\scriptstyle\eqref{eq:ICKchi}}}{{=}}\varepsilon_{x^{-1}}\varepsilon_{w_{0}}h_{y^{-1},x^{-1}}(1)$

or in other words (using that $h_{y^{-1},x^{-1}}=h_{y,x}$ and
$\varepsilon_{x^{-1}}=\varepsilon_{x},\varepsilon_{y^{-1}}=\varepsilon_{y}$)

(2.13) $a_{y,x}=\varepsilon_{yx}h_{y,x}(1)\quad\quad\text{for $p$ large}.$

This is the prediction made by Lusztig’s conjecture. Recall that in order to
confirm Lusztig’s conjecture we only need to check (2.13) for finitely many
$x$ and $y$ (see Remark 1.21). Thus the Finkelberg-Mirković conjecture implies
Lusztig’s conjecture for large $p$, and helps us have some picture about what
might “go wrong”.

###  2.6. Lusztig’s conjecture and torsion

In the previous section we have explained why the Finkelberg-Mirković
conjecture implies character formulas in terms of stalks of intersection
cohomology complexes with coefficients in $\Bbbk$. In this section we explain
(still assuming the Finkelberg-Mirković conjecture) that torsion in local
integral intersection cohomology controls Lusztig’s conjecture.

Fix $p\geq h$ and consider the subsets of $\mathcal{W}$ defined as follows:

$\displaystyle
J_{p}:=\\{x\in\mathcal{W}\;|\;\langle\alpha^{\vee},x^{-1}\bullet_{p}(-2\rho)+\rho\rangle\leq
p(p-h+2)\text{ for all $\alpha\in\Phi_{+}$}\\},$ $\displaystyle
R:=\\{x\in\mathcal{W}\;|\;x^{-1}\bullet_{p}(-2\rho)\in X_{1}^{p}\\}.$

The set $R$ is independent of $p$.

######  Theorem 2.14.

Assume $p\geq h$ and the Finkelberg-Mirković conjecture.

  1. (1)

The original form of Lusztig’s conjecture (Conjecture 1.17) holds if and only
if $\mathbf{IC}_{x,K}^{\mathbb{Z}}$ has no $p$-torsion in its stalks and
costalks, for all $x$ in $J_{p}$.

  2. (2)

The revised form of Lusztig’s conjecture (Conjecture 1.20) holds if and only
if $\mathbf{IC}_{x,K}^{\mathbb{Z}}$ has no $p$-torsion in its stalks and
costalks, for all $x$ in $R$.

######  Remark 2.15.

We have explained in §1.13 why Kato’s theorem implies that the equivalent
conditions in (2) imply the equivalent conditions in (1) if $p\geq 2h-3$. From
a geometric perspective this is rather surprising: for large $p$ there are
many more Schubert varieties in ${\mathcal{G}r^{\vee}}$ parametrised by
$J_{p}$ than by $R$.

######  Remark 2.16.

Taking into account Remark 1.18(1), identical arguments to those below show
that Lusztig’s conjecture is equivalent to the absence of $p$-torsion in the
stalks or costalks of $\mathbf{IC}_{x,K}^{\mathbb{Z}}$ for any $x$ belonging
to the set

$J^{\textrm{ext}}_{p}:=\\{x\in\mathcal{W}^{\mathrm{ext}}\;|\;\langle\alpha^{\vee},x^{-1}\bullet_{p}(-2\rho)+\rho\rangle\leq
p(p-h+2)\text{ for all $\alpha\in\Phi_{+}$}\\}$

The following is an immediate consequence of (2) and the above remark:

######  Corollary 2.17.

Let $\kappa\geq 2h-3$ and assume the Finkelberg-Mirković conjecture. Suppose
that all stalks and costalks of $\mathbf{IC}_{x,K}^{\mathbb{Z}}$ are free of
$p$-torsion, for all $p\geq\kappa$ and all $x\in R$. Then (both formulations
of) Lusztig’s conjecture hold in all characteristics $p\geq\kappa$.

Let us explain why Theorem 2.14 holds. Suppose first that the stalks and
costalks of $\mathbf{IC}_{x,K}^{{\mathbb{Z}}}$ are free of $p$-torsion, for
all $x\in J_{p}$ (resp. $R$). Then

$\mathbf{IC}_{x,K}^{{\mathbb{Z}}}\otimes_{{\mathbb{Z}}}^{L}\Bbbk=\mathbf{IC}_{x,K}^{\Bbbk}$

and hence

$\chi((\mathbf{IC}_{x,K}^{{\mathbb{Q}}})_{y})=\chi((\mathbf{IC}_{x,K}^{{\mathbb{Z}}}\otimes_{{\mathbb{Z}}}^{L}\Bbbk)_{y})=\chi((\mathbf{IC}_{x,K}^{\Bbbk})_{y})$

for all $x\in J_{p}$ (resp. $x\in R$) and all $y$. This implies the original
(for $x\in J_{p}$) and revised (for $x\in R$) forms of Lusztig’s conjecture,
as we have explained in the previous section.

The other direction is a little more involved. Assume that the stalks and
costalks of $\mathbf{IC}_{x}^{{\mathbb{Z}}}$ are not free of $p$-torsion, for
some $x\in J_{p}$ (resp. $x\in X$). Hence
${}^{p}\underline{h}_{x}\neq\underline{h}_{x}$ (see [WB12, Corollary 3.13]).
If we assume in addition that $x\in J_{p}$ (resp. $X$) is minimal such that
${}^{p}\underline{h}_{x}\neq\underline{h}_{x}$, then
${}^{p}a_{y,x}\in{\mathbb{Z}}$ for all $y\leq x$ (see §2.3 for the notation
$a_{y,x}$). Now, by [Wil15, Proposition 2.3] it follows that
$\mathbf{IC}_{x,K}^{{\mathbb{Z}}}\otimes_{\mathbb{Z}}^{L}\Bbbk$ has a non-
trivial decomposition number; in other words that

$[\mathbf{IC}_{x,K}^{{\mathbb{Z}}}\otimes_{\mathbb{Z}}^{L}\Bbbk]\neq[\mathbf{IC}_{x,K}^{\Bbbk}]\quad\text{in
$[{\mathbf{P}}_{(\mathrm{Iw})}({\mathcal{G}r^{\vee}},\Bbbk)]$.}$

Thus there exists a $y$ such that

$\chi((\mathbf{IC}_{x,K}^{{\mathbb{Q}}})_{y})=\chi((\mathbf{IC}_{x,K}^{{\mathbb{Z}}}\otimes_{{\mathbb{Z}}}^{L}\Bbbk)_{y})\neq\chi((\mathbf{IC}_{x,K}^{\Bbbk})_{y})$

and (LCF) cannot hold for the simple module $L_{x^{-1}\bullet_{p}(-2\rho)}$,
as we explained in the previous section.

######  Remark 2.18.

The implication $\Leftarrow$ of Theorem 2.14(2) is a theorem of Fiebig [Fie11]
(see also [FW14]). It is a key ingredient in his proof of Lusztig’s conjecture
for large $p$, as well as his bound [Fie12].

######  Remark 2.19.

One needs to be careful when dealing with the set $R$ because its image in
$\mathcal{W}/W$ is usually not closed in the Bruhat order. (That is, there
exist Bruhat cells ${\mathcal{G}r_{x}^{\vee}}$ for $x\in R$ whose closures
contain cells $\mathcal{G}r_{y}^{\vee}$ with $y\notin R$.)

###  2.7. Counter-examples to expected bounds

In this section we describe the results of [Wil16c, Wil16b] in the context of
the Finkelberg-Mirković conjecture.

In the previous section we explained (assuming the Finkelberg-Mirković
conjecture) that torsion in the stalks and costalks of
$\mathbf{IC}_{x,K}^{\mathbb{Z}}$ for certain $x\in\mathcal{W}$ controls
Lusztig’s conjecture. However it seems very difficult to compute or understand
the torsion in the stalks of $\mathbf{IC}_{x,K}^{\mathbb{Z}}$. Soergel
suggested that it might be worthwhile to study the finite flag variety as a
“toy model” for the geometric study of Lusztig’s conjecture.131313In the words
of Soergel [Soe00]: “The goal of this article is to forward this problem
[Lusztig’s conjecture] to the topologists or geometers.”

To explain the relevance of the finite flag variety we need to recall the
notion of smooth equivalence. A _singularity_ is a pair $(X,x)$ where $X$ is
an algebraic variety and $x\in X$ is a point. Two singularities $(X,x)$ and
$(Y,y)$ are _smoothly equivalent_ if there exists another singularity $(Z,z)$
and smooth maps $X\stackrel{{\scriptstyle
f}}{{\leftarrow}}Z\stackrel{{\scriptstyle g}}{{\rightarrow}}Y$ with $x=f(z)$
and $y=g(z)$. If $(X,x)$ and $(Y,x)$ are smoothly equivalent then small
Euclidean neighbourhoods of $x$ and $y$ are analytically isomorphic, up to
taking a product with a smooth variety. In particular, the stalks
$(\mathbf{IC}_{X}^{\mathbb{Z}})_{x}$ and $(\mathbf{IC}_{Y}^{\mathbb{Z}})_{y}$
are isomorphic up to shift (see e.g. [Jut09, Proposition 3.8]).

Consider the finite flag variety $X^{\vee}:=G^{\vee}/B^{\vee}$ and its Bruhat
decomposition

$X^{\vee}=\bigoplus_{x\in W}X^{\vee}_{x}\quad\text{where
$X^{\vee}_{x}:=B^{\vee}\cdot xB^{\vee}/B^{\vee}.$}$

We denote by $\mathbf{IC}_{x,B^{\vee}}^{\mathbb{Z}}$ the integral intersection
complex of the Schubert variety $\overline{X^{\vee}_{x}}$.

Suppose that $z\in\mathcal{W}^{\mathrm{ext}}$ is minimal in its coset $Wz$ and
choose elements $x=x^{\prime}z$ and $y=y^{\prime}z$ with
$x^{\prime},y^{\prime}\in W$ (so $\ell(x)=\ell(x^{\prime})+\ell(z)$,
$\ell(y)=\ell(y^{\prime})+\ell(z)$). We have:141414A sketch: the map
$gB^{\vee}\mapsto g\cdot z\mathrm{Iw}/\mathrm{Iw}$ defines an embedding
$X^{\vee}\hookrightarrow{\mathcal{F}l^{\vee}}$. Consider $E=\bigsqcup_{x\in
Wz}{\mathcal{F}l_{x}^{\vee}}$. There exists a morphism $E\rightarrow X^{\vee}$
making $E$ into an affine space bundle over $X^{\vee}$. This map is compatible
with the $\mathrm{Iw}$ (resp. $B^{\vee}$) orbits on $E$ and $X^{\vee}$ and
induces a smooth morphism $\overline{{\mathcal{F}l_{x}^{\vee}}}\cap
E\rightarrow\overline{X^{\vee}_{x^{\prime}}}$.

The singularities $(\overline{{\mathcal{F}l_{x}^{\vee}}},y)$ and
$(\overline{X^{\vee}_{x^{\prime}}},y^{\prime})$ are smoothly equivalent.

Moreover, if $x$ and $y$ are maximal in their cosets $xW$ and $yW$
respectively then (by considering the smooth map
$p:{\mathcal{F}l^{\vee}}\rightarrow{\mathcal{G}r^{\vee}}$) we conclude:

The singularities $(\overline{{\mathcal{G}r_{x}^{\vee}}},y)$,
$(\overline{{\mathcal{F}l_{x}^{\vee}}},y)$ and
$(\overline{X^{\vee}_{x^{\prime}}},y^{\prime})$ are smoothly equivalent.

The upshot is that if $x$ and $y$ belong to the same right $W$-coset then
torsion in the stalk of $(\mathbf{IC}_{x,K}^{\mathbb{Z}})$ at
$yK/K\in{\mathcal{G}r^{\vee}}$ can be calculated on the finite flag variety.

Consider the coset $Wt_{-\rho}w_{0}\subset\mathcal{W}^{\mathrm{ext}}$. For
$x\in W$ we have

$(xt_{-\rho}w_{0})^{-1}\bullet_{p}(-2\rho)=w_{0}(p(-\rho)+x^{-1}\bullet(-2\rho))=p\rho+w_{0}x^{-1}\bullet(-2\rho).$

We conclude that $Wt_{-\rho}w_{0}\subset J_{p}^{\textrm{ext}}$ (see Remark
2.16) if and only if

$\langle p\rho+\rho,\alpha^{\vee}\rangle\leq p(p-h+2)\text{ for all
$\alpha\in\Phi_{+}$}.$

One may check that this is the case if $p>2h-3$. From the results of the
previous section we conclude:

######  Theorem 2.20.

If Lusztig’s character formula is true for all $p\geq\kappa>2h-3$ then there
is no $p$-torsion in the stalks or costalks of the integral intersection
cohomology complexes $\mathbf{IC}_{x,B^{\vee}}^{\mathbb{Z}}$ for all $x\in W$
and $p\geq\kappa$.

######  Remark 2.21.

The discussion above deduced the Theorem 2.20 above from the Finkelberg-
Mirković conjecture. This is ahistorical, and the above theorem is known
independently of the Finkelberg-Mirković conjecture. In [Soe00] Soergel proves
a theorem very similar to the the above formulation. The exact formulation
above may be deduced by combining Soergel’s results with the theory of parity
sheaves [JMW14].

For $m\geq 1$ define $T(m)$ to be the maximal prime number $p$ which occurs as
torsion in the stalk or costalk of some integral intersection cohomology
complex on the flag variety of $\mathrm{GL}_{m}({\mathbb{C}})$ (if there is no
torsion we set $T(m)=1$). Because Schubert varieties for $\mathrm{GL}_{m}$ are
also Schubert varieties for $\mathrm{GL}_{m+1}$ our function $T$ is
monotonically increasing. Here is a table of some known values of our function

$m$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---
$T(m)$ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | $\geq 2$ | $\geq 2$ | $\geq 3$ | …

(The values of $T(m)$ for $m\geq 9$ are due to Braden and the author: for
$T(m)$ with $m\leq 8$ see [WB12], the value $T(9)=2$ is unpublished. The value
$T(12)\geq 3$ is due to Polo (unpublished) who showed more generally that
$T(4p)\geq p$ for all primes $p$.)

Given the above (admittedly rather limited) data the following is surprising:

######  Theorem 2.22.

The function $T(m)$ grows at least exponentially in $m$.

Combining this with the above results one obtains:

######  Corollary 2.23.

Suppose that $\kappa(h)$ is a function of the Coxeter number, such that
Lusztig’s character formula holds for any $G_{\Bbbk}$ in characteristic
$p\geq\kappa(h)$. Then $\kappa(h)$ grows at least exponential in $h$.

The main idea of [Wil16c] is that certain structure constants occurring in
Schubert calculus for the cohomology ring $H^{*}(X^{\vee};{\mathbb{Z}})$ also
occur as torsion in local integral intersection cohomology in much higher rank
groups. Using these ideas it is shown, for example, that any prime number
dividing any entry of a word of length $\ell$ in the semi-group

${\small\left\langle\left(\begin{matrix}1&1\\\
0&1\end{matrix}\right),\left(\begin{matrix}1&0\\\
1&1\end{matrix}\right)\right\rangle}\subset\mathrm{SL}_{2}({\mathbb{Z}})$

occurs as torsion in the stalks or costalks of some
$\mathbf{IC}_{x,B^{\vee}}^{\mathbb{Z}}$ on the flag variety of
$\mathrm{GL}_{5+3\ell}({\mathbb{C}})$. Some non-trivial number theory (which
relies on recent advances in “thin groups”) gives the above results on torsion
growth.

######  Remark 2.24.

The main result of [Wil16c] uses a formula for certain entries of intersection
forms obtained by the author and He [HW15]. This result uses the theory of
generators and relations for Soergel bimodules [EW13] in a crucial way. A
purely geometric proof [Wil16b] of the main result of [Wil16c] was discovered
later.

###  2.8. Tilting modules and the Hecke category

In this section we give a brief description of the conjectures and results of
[RW15]. The goal is to describe $\operatorname{\mathrm{Rep}}_{0}$ (or more
precisely its tilting modules) via the Hecke category. For more detail on any
of the material below, the reader is referred to [RW15].

Recall that ${}^{f}\mathcal{W}^{\mathrm{ext}}$ denotes the set of minimal
coset representatives for $W\setminus\mathcal{W}^{\mathrm{ext}}$. It will be
convenient to simplify notation as follows:

$L_{x}:=L_{x\bullet_{p}0},\quad\Delta_{x}:=\Delta_{x\bullet_{p}0},\quad\nabla_{x}:=\nabla_{x\bullet_{p}0},\quad
T_{x}:=T_{x\bullet_{p}0}\quad\in\operatorname{\mathrm{Rep}}_{0}.$

Throughout we assume that $p>h$, where $h$ is the Coxeter number. This choice
guarantees that for all $s\in\mathcal{S}$ we may fix a weight $\mu_{s}\in
C_{-}^{p}$ whose stabiliser under the $p$-dilated dot action is precisely
$\langle s\rangle\subset\mathcal{W}$ (see [Jan03, §6.3(1)]). We define the
_wall-crossing functor_ associated to $s\in\mathcal{S}$ as

$\Theta_{s}:=T_{\mu_{s}}^{-2\rho}\circ
T_{-2\rho}^{\mu_{s}}:\operatorname{\mathrm{Rep}}_{0}\rightarrow\operatorname{\mathrm{Rep}}_{0}.$

It will be convenient to view wall-crossing functors as acting on the right.

Consider the _anti-spherical module_

$\mathrm{AS}:={\mathrm{sgn}}\otimes_{{\mathbb{Z}}W}{\mathbb{Z}}\mathcal{W}=\bigoplus_{x\in{}^{f}\mathcal{W}}{\mathbb{Z}}\varepsilon\otimes
x$

obtained by inducing the sign representation
${\mathrm{sgn}}={\mathbb{Z}}\varepsilon$ of the finite Weyl group to the
affine Weyl group. Because the classes $[\Delta_{x}]$ for
$x\in{}^{f}\mathcal{W}$ span the Grothendieck group we have an isomorphism:

$\displaystyle\mathrm{AS}$
$\displaystyle\stackrel{{\scriptstyle\sim}}{{\rightarrow}}[\operatorname{\mathrm{Rep}}_{0}],$
$\displaystyle\varepsilon\otimes x$
$\displaystyle\mapsto[\Delta_{x}]\quad\text{for all $x\in{}^{f}\mathcal{W}$.}$

Moreover, it is an easy consequence of [Jan03, Chapter 7] that we can upgrade
to an isomorphism of right ${\mathbb{Z}}\mathcal{W}$-modules if we make
$[\operatorname{\mathrm{Rep}}_{0}]$ into a $\mathcal{W}$-module via

$[M]\cdot(1+s):=[M\Theta_{s}]\quad\text{for all $s\in\mathcal{S}$.}$

Thus the action of wall-crossing functors on the principal block categorifies
the anti-spherical module.

The main conjecture of [RW15] is that this action of the affine Weyl group on
the Grothendieck group can be lifted to the Hecke category:

######  Conjecture 2.25.

$\operatorname{\mathrm{Rep}}_{0}$ is a right module category over
${\mathcal{H}}$, with ${\mathcal{E}}_{s}$ acting via $\Theta_{s}$.

######  Remark 2.26.

Actually, this is a slight simplification of the conjecture, which nonetheless
captures its spirit. (The version in [RW15] requires that certain generating
morphisms in ${\mathcal{H}}$ arise from adjunctions between translation
functors, see [RW15, §5.1].)

######  Remark 2.27.

In [RW15] the above conjecture is proved for $\mathrm{GL}_{n}$. The proof uses
the Chuang-Khovanov-Lauda-Rouquier theory of categorification of Lie algebras
[CR08, Rou08, KL09, KL11]. (This theory is only available at present in type
$A$.) It also makes essential use of a recent theorem of Brundan [Bru16].

The main point of [RW15] is that the above conjecture has strong structural
and numerical consequences for $\operatorname{\mathrm{Rep}}_{0}$. To discuss
this we need to explain another categorification of the anti-spherical module.

The anti-spherical module is quantized via the right $\mathrm{H}$-module

$\mathrm{AS}_{v}:={\mathrm{sgn}}_{v}\otimes_{\mathrm{H}_{f}}\mathrm{H}=\bigoplus_{x\in{}^{f}\mathcal{W}}{\mathbb{Z}}[v^{\pm
1}]n_{x}$

where ${\mathrm{sgn}}_{v}$ denotes the sign representation of the finite Hecke
algebra $\mathrm{H}_{f}$ given by $h_{s}\mapsto-v$ for all $s\in S$ and
$n_{x}:=1\otimes h_{x}$ for $x\in{}^{f}\mathcal{W}$. The module
$\mathrm{AS}_{v}$ has a canonical basis
$\\{\underline{n}_{x}\;|\;x\in{}^{f}\mathcal{W}\\}$ constructed by Deodhar
[Deo87] (see also [Soe97b]). We have:

$\displaystyle
n_{\operatorname{id}}\cdot\underline{h}_{x}:=\begin{cases}\underline{n}_{x}&\text{if
$x\in{}^{f}\mathcal{W}$,}\\\ 0&\text{otherwise.}\end{cases}$

It is not difficult to see that $\mathrm{AS}_{v}$ has alternative descriptions
as

$\mathrm{AS}_{v}=\mathrm{H}/\langle\underline{h}_{s}\mathrm{H}\;|s\in
S\rangle=\mathrm{H}/(\bigoplus_{x\notin{}^{f}\mathcal{W}}{\mathbb{Z}}[v^{\pm
1}]\underline{h}_{x}).$

Thus it is natural to try to categorify the anti-spherical quotient as a
quotient of additive categories:

$\mathcal{AS}:={\mathcal{H}}/\langle{\mathcal{E}}_{x}\;|\;x\not\in{}^{f}\mathcal{W}\rangle_{\oplus,[{\mathbb{Z}}]}.$

(That is, $\mathcal{AS}$ is defined to be the quotient of ${\mathcal{H}}$ by
the ideal of morphisms factoring through any direct sum of shifts of
${\mathcal{E}}_{x}$, for some $x\not\in{}^{f}\mathcal{W}$.)

It is not difficult to see that $\mathcal{AS}$ is a right module category over
${\mathcal{H}}$ and that the identification
$\mathrm{H}=[{\mathcal{H}}]_{\oplus}$ induces a canonical identification

$\mathrm{AS}_{v}=[\mathcal{AS}]_{\oplus}$

of right $\mathrm{H}$-modules. The image of ${\mathcal{E}}_{x}$ in
$\mathcal{AS}$ is indecomposable if $x\in{}^{f}\mathcal{W}$ and is zero
otherwise. Its class

${}^{p}\underline{n}_{x}:=[{\mathcal{E}}_{x}]\in\mathrm{AS}_{v}$

defines the _$p$ -canonical basis_ in the anti-spherical module. The
$p$-canonical basis gives rise to the _anti-spherical $p$-Kazhdan-Lusztig
polynomials_ ${}^{p}n_{y,x}$ via

${}^{p}\underline{n}_{x}:=\sum_{y\in{}^{f}\mathcal{W}}{}^{p}n_{y,x}n_{y}.$

The basis $\\{{}^{p}\underline{n}_{x}\\}_{x\in{}^{f}\mathcal{W}}$ enjoys
positivity properties analogous to those of the $p$-canonical basis (see
§2.3).

Recall that $\mathcal{AS}$ is an additive graded category. Let us denote by
$\mathcal{AS}_{/[{\mathbb{Z}}]}$ the category obtained by “forgetting the
grading”: it has the same objects as $\mathcal{AS}$ and morphisms are given by

$\operatorname{Hom}_{\mathcal{AS}_{/[{\mathbb{Z}}]}}({\mathcal{E}},{\mathcal{E}}^{\prime}):=\bigoplus_{m\in{\mathbb{Z}}}\operatorname{Hom}_{\mathcal{AS}}({\mathcal{E}},{\mathcal{E}}^{\prime}[m]).$

Let $\operatorname{\mathrm{Tilt}}_{0}\subset\operatorname{\mathrm{Rep}}_{0}$
denote the full subcategory of tilting modules. Note that
$\operatorname{\mathrm{Tilt}}_{0}$ is preserved by wall-crossing
functors.151515This follows because translation functors preserve the
categories of modules with good or Weyl filtration. Alternatively, one may
appeal to Theorem 1.8 and the fact that we may choose tilting modules to
define our translation functors. Thus if Conjecture 2.25 holds then
$\operatorname{\mathrm{Tilt}}_{0}$ is preserved by the action of the Hecke
category.

######  Theorem 2.28.

Assume Conjecture 2.25 holds:

  1. (1)

We have an equivalence

$\mathcal{AS}_{/[{\mathbb{Z}}]}\stackrel{{\scriptstyle\sim}}{{\rightarrow}}\operatorname{\mathrm{Tilt}}_{0}$

of ${\mathcal{H}}$-module categories.

  2. (2)

For all $x,y\in{}^{f}\mathcal{W}$ we have:

$(T_{x}:\Delta_{y})={}^{p}n_{y,x}(1).$

######  Remark 2.29.

Some remarks concerning the theorem (assuming Conjecture 2.25):

  1. (1)

Part (1) of the theorem implies that $\operatorname{\mathrm{Tilt}}_{0}$ admits
a grading (given by $\mathcal{AS}$). In [RW15] it is explained how this
grading can be used to produce a grading on $\operatorname{\mathrm{Rep}}_{0}$.
Another grading on $\operatorname{\mathrm{Rep}}_{0}$ is constructed in
[AR16c]. These two gradings should be related by Koszul duality.

  2. (2)

Part (2) of the theorem can be seen as evidence for the philosophy that
Kazhdan-Lusztig polynomials should be replaced by $p$-Kazhdan-Lusztig
polynomials in modular representation theory.

  3. (3)

In the analogous setting of quantum groups at a root of unity part (2) of the
theorem (with $p$-Kazhdan-Lusztig polynomials replaced by ordinary Kazhdan-
Lusztig polynomials) is a theorem of Soergel [Soe97b, Soe97a].

  4. (4)

As we discussed in §1.6, if $p\geq 2h-2$ then a small part of the knowledge of
tilting characters can be used to obtain the simple characters. Thus the above
theorem implies a (rather complicated) formula for the simple characters in
terms of anti-spherical $p$-Kazhdan-Lusztig polynomials. It is not difficult
to see that this formula implies Lusztig’s conjecture for large $p$. However,
this formula is _not_ simply the Lusztig character formula (LCF) with Kazhdan-
Lusztig polynomials replaced by $p$-Kazhdan-Lusztig polynomials.

## List of notation

Here is a list of frequently used notation, in order of appearance:

$X,X^{\vee}$ | character lattice, cocharacter lattice, §1.1
---|---
$\Phi,\Phi^{\vee}$ | roots, coroots, §1.1
$G_{\mathbb{Z}}$ | Chevalley group scheme corresponding to our root datum, §1.1
$\Bbbk$, $p$ | an algebraically closed field, its characteristic, §1.1
$G_{\Bbbk}$ | our connected, semi-simple and simply connected group over $\Bbbk$, §1.1
$T_{\Bbbk},B_{\Bbbk}$ | maximal torus, Borel subgroup in $G_{\Bbbk}$, §1.1
$\Phi_{+},\Phi_{+}^{\vee}$ | positive roots, positive coroots, §1.1
$X_{+},X_{+}^{\vee}$ | dominant weights and coweights, §1.1
$\operatorname{\mathrm{Rep}}H$ | abelian category of algebraic representations of $H$, §1.2
$\operatorname{Irr}H$ | isomorphism classes of simple $H$-modules, §1.2
$\operatorname{\mathrm{Rep}}$ | algebraic representations of $G_{\Bbbk}$, §1.2
$L_{\lambda}$ | simple module with highest weight $\lambda\in X$, §1.2