Proof of Finkelberg-Mirkovic Conjecture
Reference:
Assume that we have a functor $$F: D^{b}\mathrm{Coh}^{\check{G}\times \mathbb{G}_ {m}}(\widetilde {N}) \rightarrow D^{b}\mathrm{Rep}_ {0}(G)$$ and $$\Phi: D^{b}\mathrm{Coh}^{\check{G}\times \mathbb{G}_ {m}}(\widetilde {N}) \rightarrow D_ {I^{u}}^{\text{mix}}(\mathrm{Gr}’,\mathbb{k}),$$ as in Lecture 10, and
- $F\circ \langle 1 \rangle [1]\cong F$,
- $\Phi\circ \langle 1 \rangle \langle 1 \rangle [-1]\Phi$.
Theorem 1 ([Achar-Makisumi-Riche-Williamson]). There is an equivalence of categories (Koszul duality) $$\Psi: D_ {I^{u}}^{\text{mix}}(\mathrm{Gr}’,\mathbb{k})\cong D^{\text{mix}}_ {(I^{u},\psi)}(\mathrm{FL}),$$ mapping \begin{align} \triangle_ {w} & \mapsto \triangle_ {w} \\ \Delta_ {w} & \mapsto \Delta_ {w} \end{align} and $\Psi\circ \langle 1 \rangle = \langle -1 \rangle [1]\circ \Psi$, and mapping indecomposable tilting perverse sheaves to indecomposable parity complexes.
This implies that multiplicities of standard objects in indecomposable tilting objects $D_ {I^{u}}^{\text{mix}}(\mathrm{Gr}’,\mathbb{k})$m and hence we reduce to dimensions of stalks of parity complexes in $D^{b}_ {(I^{u},\psi)}(\mathrm{FL},\mathbb{k})$, which is known to be given by $p$-Kazhdan-Lusztig polynomials.
Then we have $$D_ {I^{u}}^{\text{mix}}(\mathrm{Gr},\mathbb{k})\xrightarrow{\cong} D^{b}\mathrm{Coh}^{\check{G}\times \mathbb{G}_ {m}}(\widetilde {N} ) \rightarrow D^{b}_ {\text{Stein}}(B)\xrightarrow{\mathrm{RInd}_ {B}^{G}} D^{b}\mathrm{Rep}_ {0}(G),$$ where $D^{b}_ {\text{Stein}}(B)$ is the derived category of complexes of $B$-modules whose cohomology is trivial on $B_ {1}\subseteq B$.
- The formality theory gives that $F’:D^{b}\mathrm{Coh}^{\check{G}\times \mathbb{G}_ {m}}(\widetilde {N})$ is a graded version of $D^{b}_ {\text{Stein}}(B)$.
- Induction theorem says that $\mathrm{RInd}_ {B}^{G}$ is an equivalence of categories. In more details,
- $F’$ is a degrading functor with respect to $\langle 1 \rangle [1]$ such that for any $V\in \mathrm{Rep}(\mathbb{G}^{(1)})$, $F’(\mathscr{F}\otimes V)\cong F(\mathscr{F})\otimes \mathrm{Fr}^{\ast}(V)$.
- $\mathrm{RInd}_ {B}^{G}$ is an equivalence and for any $V\in \mathrm{Rep}(\mathbb{G}^{(1)})$, one has that $\mathrm{RInd}_ {B}^{G}(M\otimes \mathrm{Fr}^{\ast}(V))\cong \mathrm{RInd}_ {B}^{G}(M)\otimes \mathrm{Fr}^{\ast}(N).$
Alternative proof via “Smith-Trevmann thoery”
- Geometric Satake and Koszul duality gives an equivalence $$\mathrm{Rep}(\check{G})\xrightarrow{\cong}\mathrm{Perv}_ {(I^{u},\psi)}(\mathrm{Gr}’,\mathbb{k}).$$
- Smith-Trevman theory gives a localization functor which relates sheaves on affine Grassmannian to sheaves on fixed points under the group of $p$-th root of unity in $\mathbb{F}$ via loop rotation. (also see Riche.)
- These fixed points identify with a disjoint union of partial flag varieties of some $p$-dilated loop groups of $G$.
- This localization functor is fully faithful on tilting modules.
- This allows us to compute dimensions of morphisms between tiltings.
- Riche, Exercise 7.10 tells that if we understand these dimensions, we understand multiplicities of standards/costandards in tilting.
Geometric counter-example to Lusztig’s conjecture
Let’s suppose that $G$ is defined over $\mathbb{Z}$ and $\text{char}(\mathbb{k})=p$.
Archar-Riche, Fiebig: Lusztig’s conjecture for $G_ {\mathbb{k}}$ is equivalent to absence of $p$-torsion in the stalks and costalks of $\mathrm{IC}(\overline{\mathrm{Gr}_ {x}},\mathbb{Z})$ for any $x\in {}^{f}W_ {\text{aff}}$ satisfying Jantzen’s condition.
How does this torsion show up?
Let $X= \{ x=\begin{pmatrix}c & -a \\ b & -c \end{pmatrix}\in \mathfrak{sl}_ {2}(\mathbb{C})\text{ such that $x$ is nilpotent} \}= \operatorname{Spec}[a,b,c]/(ab-c^{2})\cong \mathrm{Spec}\mathbb{C}[x^{2},xy,y^{2}]$. Then $X$ is a cone and $X=X^{\text{reg}}\cup\{0\}$.
Suppose that $\mathscr{F}$ is a perverse sheaf on $X$ with respect to this given stratification. Where can $\mathscr{H}^{i}(\mathscr{F}|_ {X’})$ be nonzero, where $X’\in \{X^{\text{reg}}, \{0\}\}$.
| -3 | -2 | -1 | 0 | 1 | |
|---|---|---|---|---|---|
| $X_ {\text{reg}}$ | 0 | $\ast$ | 0 | 0 | 0 |
| $\{0\}$ | 0 | $\ast$ | $\ast$ | $\ast$ | 0 |
To compute $$\mathrm{IC}(X,\mathbb{k})=\mathrm{IC}(\overline{X}_ {\text{reg},\mathbb{k}}),$$ we use Deligne’s formula $\mathrm{IC}(X,\mathbb{k})=\tau_ {<0}j_ {\ast}\underline{\mathbb{k}}_ {X^{\text{reg}}}[2]$, where $j: X^{\text{reg}}\rightarrow X$.
Computation.
First compute $(j_ {\ast}\underline{\mathbb{k}}_ {X^{\text{reg}}}[2])_ {0}=\operatorname{lim}_ {\epsilon\rightarrow 0} H^{\ast+2}(B(0,\epsilon)\cap X^{\text{reg}},\mathbb{k})$, but $B(0,\epsilon)\cap X^{\text{reg}}$ is homeomorphic to $S^{3}/\pm 1\cong \mathbb{RP}^{3}$. Since $$ H^{n}(\mathbb{RP}^{3}m\mathbb{Z})= \begin{cases} \mathbb{Z} & \text{if }n=0, \\ \mathbb{Z}/2\mathbb{Z} & \text{if } n=2, \\ \mathbb{Z} & \text{if }n=3, \\ 0 & & \text{otherwise}. \end{cases} $$ Then by universal coefficient theorem, one gets that $H^{\ast}(\mathbb{R}\mathbb{P}^{3},\mathbb{k})\cong \mathbb{k}\oplus \mathbb{k}_ {2}[-1] \oplus \mathbb{k}_ {2}[-2]\oplus \mathbb{k}[-3],$ where $\mathbb{k}_ {2}=\mathbb{k}$ if $2=0$ in $\mathbb{k}$, and $0$ otherwise.
Then stalks of $j_ {\ast}\mathbb{k}_ {X^{\text{reg}}}[2]$ are
| -2 | -1 | 0 | 1 | |
|---|---|---|---|---|
| $X_ {\text{reg}}$ | $\mathbb{k}$ | 0 | 0 | 0 |
| $\{0\}$ | $\mathbb{k}$ | $\mathbb{k}_ {2}$ | $\mathbb{k}_ {2}$ | $\mathbb{k}$ |
Therefore, the stalks of $\mathrm{IC}(X,\mathbb{k})$ are
| -2 | -1 | 0 | |
|---|---|---|---|
| $X_ {\text{reg}}$ | $\mathbb{k}$ | 0 | 0 |
| $\{0\}$ | $\mathbb{k}$ | $\mathbb{k}_ {2}$ | 0 |
Similarly, one has that $\mathrm{IC}(X,\mathbb{Z})= \mathbb{Z}[2]$. Then stalks of $\mathrm{IC}(X,\mathbb{Z})$ are
| -2 | -1 | 0 | |
|---|---|---|---|
| $X_ {\text{reg}}$ | $\mathbb{Z}$ | 0 | 0 |
| $\{0\}$ | $\mathbb{Z}$ | 0 | 0 |
and stalks of $\mathbb{D}(\mathrm{IC}(X,\mathbb{Z}))$ are
| -2 | -1 | 0 | |
|---|---|---|---|
| $X_ {\text{reg}}$ | $\mathbb{Z}$ | 0 | 0 |
| $\{0\}$ | $\mathbb{Z}$ | 0 | $\mathbb{Z}/2\mathbb{Z}$ |
Therefore, $\mathrm{IC}(X,\mathbb{Z})\otimes^{\mathbb{L}}\mathbb{k}$ is simple if the characteristic of $\mathbb{k}$ is odd, otherwise it has composition factors $\mathrm{IC}(X,\mathbb{k})$ and $\mathrm{IC}(0,\mathbb{k})$.
Overview
Now let $G$ be defined over $\mathbb{Z}$. We could study
- Representations of $G_ {\mathbb{C}}$ over $\mathbb{C}$.
- Representations of $G_ {\mathbb{k}}$ over $\overline{\mathbb{F}_ {p}}$. “modular representation theory”
- Representations of $G(\mathbb{F}_ {q})$ over $\mathbb{C}$.
- Naively, it is just a finite group. (character theory in general finite group representation theory.)
- Deligne-Lusztig theory in 70s.
- Representations of $G(\mathbb{F}_ {q})$ over $\overline{\mathbb{F}_ {p}}$.
- Work of Jantzen.
- (relation with 2) Consider representations obtained by restriction of $G_ {\mathbb{F}_ {p}}$ to $\mathbb{F}_ {p}$-points $V\rightarrow \overline{V}.$
- (relation with 3) Brauer-Nesbitt 1940s modular reduction $\rho\mapsto \underline{\rho}$.
Lusztig in 2021 (building off of much older work) conjectured that if you start a nice $\rho$ in 3, then $\underline{\rho}$ in 4 can be written as $\underline{\rho} = \sum_ {\lambda\in ?} c_ {\lambda} \overline{V}_ {\lambda} = \sum_ {\lambda\in ?} d_ {\lambda} \overline{L}_ {\lambda}$, for $V_ {\lambda}$ a Weyl module with highest weight $\lambda$ and irreducibles $L_ {\lambda}$.