Introduction

Our main reference will be Lecture notes by Simon Riche.

Representations of reductive groups

Let $G$ be a reductive group over an algebraically closed field $\mathbb{k}$.

1950’s by Chevalley: Classification of simple representations.

Let $B$ be a Borel subgroup and $T\subseteq B$ a maximal torus.

Definition 1. Let $\mathrm{Rep}(G)$ denote the category of finite dimensional algebraic $G$-modules. For any $\lambda\in \mathbb{X}^{\ast}(T)$, we have a $G$-module $N(\lambda):=\mathrm{Ind}_ {B}^{G}(\lambda)$ called Weyl module, whose character is given by Weyl character formula.

Fact:

  1. $N(\lambda)=0$ unless $\lambda\in\mathbb{X}^{\ast}(T)_ {+}$.
  2. If $\lambda\in\mathbb{X}^{\ast}(T)_ {+}$, then $N(\lambda)$ contains a unique simple submodule $L(\lambda)$.
  3. The assignments $\lambda\mapsto L(\lambda)$ is a bijection between $\mathbb{X}^{\ast}(T)_ {+}$ and isomorphic classes of simple objects in $\mathrm{Rep}(G)$.
  4. If $\text{char}(\mathbb{k})=0$, then $L(\lambda)=N(\lambda)$.
  5. In general, we will re-index to get $L(w)$ and $N(w)$ for $w\in W_ {\text{aff}}$, and want to understand $\text{char}(L(w))=\sum_ {y\in W_ {\text{aff}}}\boxed{?}\,\,\text{char}(N(y))$.

Lusztig conjecture

Lusztig conjecture gives a first approximation and was conjectured for “large primes”.

Lusztig’s conjecture was originally expected to be true for $p>h$ Coxeter number.

In 2013, Geordie Williamson proved that for $\mathrm{GL}_ {n}$, Lusztig’s formula can not be true under the assumption of the form $p\geq P(n)$ for any fixed polynomial $P$.

Before this,Jantzen-Anderson 1970s, Lusztig 1980s, Kazhdan-Lusztig, Kashiwara-Tanisaki, Anderson-Jantzen-Soegel 1990s, proved that Lusztig’s conjecture is true for large $p$.

The picture is: Kazhdan-Lusztig polynomial (relatied to geometry of $G/B$ and affine flag variety) is replaced by $p$-Kazhdan Lusztig polynomials (tilting modules and parity sheaves).

Representations of $G(\mathbb{F}_ {q})$ over $\overline{\mathbb{F}_ {p}}$

  • What do representations of $G(\mathbb{F}_ {p})$ over $\overline{\mathbb{F}_ {p}}$ look like? (Jantzen’s work)
  • How do we reduce representations of $G(\mathbb{F}_ {p})$ over $\mathbb{C}$ by modulo $p$ to get modular representations? (Brauer-Nesbitt 1940s)
  • Formulas for modular reduction of $\mathrm{IrrRep}(G(\mathbb{F}_ {p}),\mathbb{C})$ after reducing mod $p$,$$\text{char}(\overline{\rho}) = \sum_ {\lambda}c_ {\rho,\lambda}\text{char}(N(\lambda))$$ where $\rho$ is an irreducible representation over $\mathbb{C}$. Lusztig conjectured such formula but it is not quite correct.
  • What geometric representation theory input can help explain these representation theory.

Representations

  • Now let $\mathbb{k}$ be an algebraically closed field of $\text{char}(\mathbb{k})=p$. Let $G,B,T$ be as before
  • We call elements of $\mathbb{X}:=\mathbb{X}^{\ast}(T)$ weights.
  • Let $U\subseteq B$ be the unipotent radical and we have $T\ltimes U\xrightarrow{\cong} B$ under multiplication.Then $\lambda\in \mathbb{X}^{\ast}(T)$ extends to $B\rightarrow \mathbb{G}_ {m}$.
  • Let $B^{+}$ and $U^{+}$ be the opposite Borel and its unipotent radical.
  • Denote by $R\subseteq \mathbb{X}$ the roots.
  • $R^{+}\subseteq R$ the subset of positive roots, i.e. $T$-weights in $\mathrm{Lie}(U^{+})$.
  • $R^{s}\subseteq R^{+}$ simple roots.
  • $\mathbb{X}^{\vee}:=\mathbb{X}_ {\ast}(T)$ cocharacters.
  • $R^{\vee} \subseteq \mathbb{X}^{\vee}$ corrots. We have $R\xrightarrow{\cong} R^{\vee}$ under $\alpha\mapsto \alpha^{\vee}$.
  • $\mathbb{X}^{\ast}(T)_ {+}=\{\lambda\in \mathbb{X}: \forall \alpha\in R^{+}, \langle \lambda,\alpha^{\vee} \rangle\geq 0\}$.
  • Let $W= N_ {G}(T)/T$ and $S\subseteq W$ simple reflections, $w_ {0}$ the longest element in $W$.
  • $(W,S)$ is a Coxeter system.

For any $\mathbb{k}$-algebraic group $H$, denote by $\mathrm{Rep}(H)$ the category of finite dimensional algebraic $H$-modules and $\mathrm{Rep}^{\infty}(H)$ the category of all algebraic $H$-modules.

For any $V\in \mathrm{Rep}(H)$, its dual $V^{\ast}$ is defined by $(h\cdot f)(v):=f(h^{-1}\cdot v)$ for any $h\in H,f\in V^{\ast},v\in V$.

For any algebraic subgroup $K\subseteq H$, there is an induction functor $\mathrm{Ind}_ {K}^{H}: \mathrm{Rep}^{\infty}(K)\rightarrow \mathrm{Rep}^{\infty}(H)$, which sends $(M,\rho)$ where $\rho: K\rightarrow \mathrm{GL}(M)$ to space of algebraic functions $\{f:H\rightarrow M: f(hk)=\rho(k^{-1})\cdot f(h),\forall h\in H, k\in K\}.$

Definition 2. For $\lambda\in \mathbb{X}$, the induced module $N(\lambda)$ is defined as $$N(\lambda):=\mathrm{Ind}_ {B}^{G}\mathbb{k}_ {\lambda}=\{f\in\mathscr{O}_ {G}: \forall b\in B, g\in G, f(gb)=\lambda(b)^{-1}f(g)\}.$$

Example 3. Take $\lambda=0$, one get $N(0)=\mathscr{O}(G/B)=\mathbb{k}$.

Fact: $N(\lambda)$ is finite dimensional for any $\lambda\in \mathbb{X}$.

Definition 4. For $\lambda\in\mathbb{X}$, the Weyl module $M(\lambda)$ is defined as $$M(\lambda):=N(-w_ {0}\lambda)^{\ast}.$$

For any algebraic $T$-module $M$, we have weight decomposition $$M=\bigoplus_ {\lambda\in \mathbb{X}}M_ {\lambda}.$$

Definition 5. We set $\text{wt}(M):=\{\lambda\in\mathbb{X}: M_ {\lambda}\neq 0\}\subseteq \mathbb{X}$.

Definition 6. If $\text{dim}(M_ {\lambda})<\infty$ for any $\lambda\in\text{wt}(M)$, we define $$\text{ch}(M):=\sum_ {\lambda\in\mathbb{X}}\text{dim}(M_ {\lambda})e^{\lambda}\in \mathbb{Z}[\mathbb{X}].$$

Lemma 7. For $M\in\mathrm{Rep}(G)$, $\text{wt}(M)\subseteq \mathbb{X}$ is invariant under the action of $W$. In fact, $\text{ch}(M)$ is invariant under $W$.

Therefore, $\text{ch}: \mathrm{Rep}(G)\rightarrow \mathbb{Z}[\mathbb{X}]^{W}$.

Lemma 8. For any $\lambda\in\mathbb{X}$, $\mu\in \text{wt}(N(\lambda))$ if and only if $\mu\leq \lambda$, i.e. $\mu-\lambda$ is a $\mathbb{Z}_ {>0}$-linear combination of positive roots. If $N(\lambda)\neq 0$, then $N(\lambda)_ {\lambda}$ has dimension 1.