Tilting character formula

Our goals are

  • Sketch of geometric proof for character formula for tilting modules.
  • “Torsion explosion”: geometric reason for failure of Lusztig’s original conjecture.

Tilting character formula and Anti-spherical $p$-Kazhdan-Lusztig polynomials

Let ${}^{f}W_ {\text{aff}}\subseteq W_ {\text{aff}}$ be elements in $W_ {\text{aff}}$ minimal in their right coset relative to $W\subset W_ {\text{aff}}$.

For $y,w\in {}^{f}W_ {\text{aff}}$, we define ${}^{p}n_ {y,w} = \sum_ {z\in W}(-v)^{\ell(z)}\cdot{}^{p}h_ {zy,w}$ called “anti-spherical p-Kazhdan-Lusztig polynomials”.

Conjecture 1. Assume that $p\geq h$, $\lambda\in C\cap \mathbb{X}$ where $C$ is the fundamental box. For any $y,w\in W_ {\text{aff}}$, we have that $$(T( w\cdot_ {p}\lambda ):N(y\cdot_ {p}\lambda)) = {}^{p}n_ {y,w}(1).$$ We can assume $\lambda=0$ by certain reduction process.

Finkelberg-Mirkovic conjecture

There are multiple proofs of this conjecture. We are now going to discuss its geometric proof. Recall that geometric Satake gives a monoidal equivalence $$\mathrm{Sat}: \mathrm{Perv}_ {L^{+G}}(\mathrm{Gr}_ {G},\mathbb{k})\xrightarrow{\cong} \mathrm{Rep}( \check{G} )$$ where $\mathbb{k}$ is an algebraically closed field of characteristic $p$, $\check{G}$ is a split connected reductive group over $\mathbb{k}$ with maximal torus $\check{T}$ over $\mathbb{k}$. So let’s suppose we choose data such that $\mathbb{G}^{(1)}\cong \check{G}$, $\mathbb{B}^{(1)}\cong \check{B}$, $\mathbb{T}^{( 1 )}\cong \check{T}$. Then we denote by $\mathbf{X}:=\mathbb{X}^{\ast}(\mathbb{T})\cong \mathbb{X}^{\ast}(T^{(1)})\cong \mathbb{X}_ {\ast}(T)$, such that the pullback $\mathbb{T}\rightarrow \mathbb{T}^{(1)}$ corresponds to $\lambda\mapsto p\lambda$.

We denote $\mathrm{Gr}’:=L^{+}G\backslash LG$ which is isomorphic to $\mathrm{Gr}=LG/L^{+}G$ via $g\mapsto g^{-1}$.

Let $I^{u}$ denote the pro-unipotent radical of $I$. Let $\mathrm{Perv}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})$, which is

  • highest weight category with respect to weight poset ${}^{f}W_ {\text{ext}}$.
  • $\nabla_ {w},\triangle_ {w},\mathrm{IC}_ {w}$ standard, costandard, and simple for $w\in {}^{f}W_ {\text{ext}}$.

Then we have an action $D^{b}_ {L^{+}G}(\mathrm{Gr},\mathbb{k})\times D^{b}_ {I^{u}}(\mathrm{Gr}‘.\mathbb{k})\rightarrow D^{b}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})$.

This is $t$-exact so it gives an action of $(\mathrm{Perv}_ {L^{+}G}(\mathrm{Gr},\mathbb{k}),\ast)$ on $\mathrm{Perv}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})$, proved by Gaitsgory.

Conjecture 2 (Finkelberg-Mirkovic conjecture). Assume $p\geq h$, there is an equivalence of categories $$\mathrm{FM}: \mathrm{Perv}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})\xrightarrow{\cong}\mathrm{Rep}_ {0}(\mathbb{G}),$$ which satisfies:

  • for any $w\in {}^{f}W_ {\text{aff}}$ we have that \begin{gather} \mathrm{FM}(\mathrm{IC}_ {w})\cong L(w\cdot_ {p}\lambda)\\ \mathrm{FM}(\triangle_ {w})\cong M(w\cot_ {p}\lambda)\\ \mathrm{FM}(\nabla_ {w})\cong N(w\cdot_ {p}\lambda) \end{gather}
  • For $\mathscr{F}\in \mathrm{Perv}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})$ and $\mathscr{G}\in\mathrm{Perv}_ {L^{+G}}(\mathrm{Gr},\mathbb{k})$, we have that $\mathrm{FM}(\mathscr{G}\ast \mathscr{F})\cong \mathrm{FM}(\mathscr{F})\otimes \mathrm{Fr}^{\ast}(\mathrm{Sat}(\mathscr{G}))$.

Impact on characters

Suppose the conjecture is true. Then on $K$-groups one has $$K_ {0}(\mathrm{Perv}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})) \xrightarrow{\cong} K_ {0}(\mathrm{Rep}_ {0}(\mathbb{G})),$$ in left hand side $[\mathscr{F}]=\sum_ {y\in {}^{f}W_ {\text{aff}}}(-1)^{\ell(y)} \chi_ {y}(\mathscr{F})[\nabla_ {y}]$ where $\chi_ {y}$ is taking Euler characteristic of stalks at $y$. So in particular, FM isomorphism implies that $$[L(w\cdot_ {p}\lambda)]=\sum_ {y\in {}^{f}W_ {\text{aff}}}(-1)^{\ell(y)} \chi_ {y}(\mathrm{IC}_ {w})[M(y\cdot_ {p}\lambda)].$$

Singular version

Let $\mu\in \mathbb{X}\cap \overline{C}$. There is an equivalence of categories $$\mathrm{FM}_ {\mu}: \mathrm{Perv}_ {(I^{u}_ {A},\chi_ {A})}(\mathrm{Gr}’,\mathbb{k})\cong \mathrm{Rep}_ {\mu}(G)$$ satisfying similar properties. Here $A\subseteq S_ {\text{aff}}$ is the subset of elements fixing $\mu$, then $I^{u}_ {A}$ pro-unipotent of the parahoric, some local system $\chi_ {A}$.

The Iwahori-Whittaker model of the Satake category

What if $\mu=-\rho$? Then $A=S_ {\text{aff}}$. Then the theorem says $\mathrm{FM}_ {-\rho}: \mathrm{Perv}_ {(I^{u}_ {S_ {\text{aff}}},\chi_ {S_ {\text{aff}}})}(\mathrm{Gr}’,\mathbb{k})\cong \mathrm{Rep}_ {-rho}(G)$.

What is $\chi_ {S_ {\text{aff}}}$?

Let $\psi_ {I}:I^{u}\rightarrow \mathbb{G}_ {a}$ be given by $\psi_ {I}(gg^{\leq 0}) = \Psi(g)$ for $g\in I^{u}\cap N$, $g^{\leq 0}\in I^{u}\cap B^{-}$, and $\Psi(n)=\mathrm{Res}(\frac{dt}{t}\sum_ {\alpha\text{ simple}}U_ {\alpha}(n))$.

Consider $(I^{u},\psi_ {I})$-equivariant objects in $D^{b}(\mathrm{Gr}‘,\mathbb{k})$. There is an equivalence $$\mathrm{Perv}_ {L^{+}G}(\mathrm{Gr},\mathbb{k})\xrightarrow{\cong} \mathrm{Perv}_ {(I^{u},\psi_ {I})}(\mathrm{Gr}’,\mathbb{k})$$ called “spherical-antispherical isomorphism”.

Iwahori-Whittaker should be thought as Koszul dual of equivalence category.

Proof of tilting character via Koszul duality

The idea is to rephrase the question in terms of functors $D^{b}\mathrm{Rep}_ {0}G\xleftarrow {F} D^{b}\mathrm{Coh}^{\check{G}\times\mathbb{G}_ {m}}(\widetilde {\mathscr{N}})\xrightarrow{\cong} D_ {(I^{u},\psi)}^{\text{mix}}(\mathrm{FL},\mathbb{k})$, where

  • the right isomorphism is called Arkhipov-Bezrukavnikov equivalence,
  • the variety $\widetilde {\mathscr{N}} $ is the Springer resolution of $\check{G_ {\mathbb{k}}}\cong \mathbb{G}^{(1)}$, i.e. cotangent bundle of $\check{G}/\check{B}$, with $\mathbb{G}_ {m}$-action by dilation along the cotangent direction by weight $-2$.

There is an equivalence given by “shift” $$\langle 1 \rangle:D^{b}\mathrm{Coh}^{\check{G}\times\mathbb{G}_ {m}}(\widetilde {\mathscr{N}})\xrightarrow{\cong}D^{b}\mathrm{Coh}^{\check{G}\times\mathbb{G}_ {m}}(\widetilde {\mathscr{N}}).$$

The functor $F$ is not an equivalence but it is “close”: there is a canonical isomorphism $F\circ \langle 1 \rangle [1] \cong F$ such that

  • for any $\mathscr{F}, \mathscr{G}\in D^{b}\mathrm{Coh}^{\check{G}\times\mathbb{G}_ {m}}(\widetilde {\mathscr{N}})$, $F$ induces an isomorphism $\oplus_ {n\in \mathbb{Z}}\mathrm{Hom}(\mathscr{F},\mathscr{G}\langle n \rangle [n])\xrightarrow{\cong}\mathrm{Hom}(F(\mathscr{F}),F(\mathscr{G})) $,
  • the essential image of $F$ generates $D^{b}\mathrm{Rep}_ {0}(G)$ as a triangulated category.
  • For $\mathscr{F}\in D^{b}\mathrm{Coh}^{\check{G}\times \mathbb{G}_ {m}}(\widetilde {\mathscr{N}} )$, $V\in \mathrm{Rep}(\check{G})$, we have that $F(\mathscr{F}\otimes V)\cong F(F)\otimes \mathrm{Fr}^{\ast}_ {G}(V)$.

Other ingredient:

  • The category $D_ {I^{u}}^{\text{mix}}(\mathrm{Gr}‘,\mathbb{k})$ is the mixed derived category of $I^{u}$-equivalent sheaves on $\mathrm{Gr}’$, which has a perverse $t$-structure and has heart $\mathrm{Perv}^{\text{mix}}_ {I^{u}}(\mathrm{Gr}’,\mathbb{k})$ and a Tate twist auto-equivalence $\langle 1 \rangle$.
  • There is an equivalence of triangulated categories $$\Phi:D^{b}\mathrm{Coh}^{\check{G}\times\mathbb{G}_ {m}}(\widetilde {\mathscr{N}})\xrightarrow{\cong} D_ {I^{u}}^{\text{mix}}(\mathrm{Gr}’,\mathbb{k})$$ such that $\Phi\circ \langle 1 \rangle \cong \langle 1 \rangle [-1]\circ \Phi,$ called spherical coherent-constructible correspondence
  • There is a bifunctorial isomorphism $$\Phi(\mathscr{F}\otimes \mathrm{Sat}(\mathscr{G}))\xrightarrow{\cong}\Phi(\mathscr{F})\ast \mathscr{G}.$$

Combining these functors, we will get a map $$D_ {I^{u}}^{\text{mix}}(\mathrm{Gr’,\mathbb{k}})\rightarrow D^{b}(\mathrm{Rep}_ {0}(G))$$ which is “degrading” with respect to $\langle 1 \rangle$,

  • it sends standard objects to Weyl modules,
  • it sends costandard objects to co-Weyl modules.

Some literature