Lusztig's Conjecture

Conjecture 1 (Lusztig,2021,Type A). Let $G$ be type A. For every involution $w\in W$ such that $w^{2}=1$, there exists a character $M_ {w}$ of $G(\mathbb{F}_ {p})$ over $\overline{\mathbb{F}_ {p}}$, such that for any $\rho$ a complex irreducible unipotent representation of $G(\mathbb{F}_ {p})$, then $$\underline{\rho}=\sum_ {w\in C_ {\rho},w^{2}=1}M_ {w},$$ for $c_ {\rho}\subseteq W$ be the the two-sided cell attached to $\rho$.

Further, the $M_ {w}$ satisfy some nice properties:

  1. For any involution $w$, $M_ {w}$ is a linear combination of $\overline{V}_ {\lambda}$, for $\lambda$ “very close” to $(p-1)\omega_ {I(w)}$.
  2. We can define a “dimension polynomial” which is some $d_ {w}(x)$ such that $d_ {w}(p)=\text{dim} M_ {w}$.
  3. THese dimension polynomials satisfy some nice symmetry: for any involution $w$, there exists some involution $w’$, such that $x^{k}d_ {w}(1/x)=d_ {w’}(x)$.
  4. $M_ {w}$ are positive.

Definition 2. For any $w\in W$, let $I(w)\subseteq S$ be defined by $$I(w)=\{s\in S: \ell(ws)<\ell(w)\}.$$ Let $\overline{I(w)}:=S-I(w)$.

Definition 3. For $s\in S$,$\omega_ {s}$ is the corresponding fundamental weight. For $S’\subseteq S$, we define $$\omega_ {S’}:=\sum_ {s\in S’}\omega_ {s}.$$

Example 4. In type $A_ {3}$, we have that

  • $M_ {1}=V_ {0,0,0}$,
  • $M_ {s_ {1}}=V_ {p-1,0,0}$,
  • $M_ {s_ {2}}=V_ {0,p-1,0}$,
  • $M_ {s_ {3}}=V_ {0,0,p-1}$,
  • $M_ {s_ {1}s_ {3}}=V_ {p-1,0,p-1}-V_ {p-2,0,p-2}=L_ {p-1,0,p-1}$,
  • $M_ {s_ {2}s_ {1}s_ {3}s_ {2}}=V_ {0,p-1,0}+V_ {0,p-3,0}$,
  • $M_ {s_ {1}s_ {2}s_ {1}}=V_ {0,p-1,0}+V_ {0,p-3,0}$,
  • $M_ {s_ {1}s_ {2}s_ {1}}=V_ {p-1,p-1,0}$,
  • $M_ {s_ {1}s_ {3}s_ {2}s_ {3}s_ {1}}=V_ {p-1,0,p-1}+V_ {p-2,0,p-2}$,
  • $M_ {s_ {2}s_ {3}s_ {2}}=V_ {0,p-1,p-1}$,
  • $M_ {w_ {0}}=V_ {p-1,p-1,p-1}$.

There are five cells in type $A_ {3}$ given 4, (3,1), (2,2), (2,1,1), (1,1,1,1).

Theorem 5 ([Bezrukavnikov-Finkelberg-Kazhdan-Morton-Ferguson]).

  • Lusztig’s elements $M_ {w}$ exist. There is an explicit formula for them.
  • We can explicitly write any $\underline{\rho}$ as a linear combination of Weyl characters.
  • However, properties 3 (dimension polynomial reciprocity) and 4 (positivity) are false. We know why they should be false.

Goal: Reduce the construction of the elements $M_ {w}$ to construction of a “nice” basis of $\mathbb{C}[T]$ over $\mathbb{C}[T]^{W}$.

Let $[p]: T\rightarrow T$ be given by $z\mapsto z^{p}$.

Let $\pi: T\rightarrow T//W$ be the projection.

Let $\overline{[p]}:T//W\rightarrow T//W$ be the unique morphism such that $\pi\circ [p]=\overline{[p]}\circ \pi$.

Let $(T//W)_ {p}$ be the fixed point scheme of $\overline{[p]}$, which can be proved to be reduced scheme.

Let $\iota_ {p}: (T//W)_ {p}\rightarrow T//W$ be the inclusion.

The sheaf $\pi_ {\ast}\mathscr{O}_ {T}$ carries an action of $W$ while $\iota^{\ast}\pi_ {\ast}\mathscr{O}_ {T}$ carries an automorphism $\phi$ defined by $\phi(f)=[p]^{\ast}f$.

The semi-simple conjugacy classes are in bijection with closed points of $T//W$. The semisimple conjugacy classes of $G(\mathbb{F}_ {p})$ are in bijection with closed points of $(T//W)_ {p}$.

The Brauer character map $\beta$ is a map to $\mathscr{O}((T//W)_ {p})$.

Proposition 6 (Jantzen,1986). Let $\rho_ {\chi}$ be the complex unipotent irreducible representable of $G(\mathbb{F}_ {p})$ associated to irreducible representable $\chi$ of $W$. Then $$\beta(\underline{\rho}_ {\chi})=\frac{1}{\text{dim}\chi}\text{tr}(\phi,[\iota_ {p}^{\ast}\pi_ {\ast}\mathscr{O}_ {T}:\chi]),$$

Corollary 7. Suppose that $\widetilde {\phi} $ is some automorphism of $\pi_ {\ast}\mathscr{O}_ {T}$ commuting with the action of $W$, such that $\iota_ {p}^{\ast}\widetilde {\phi}=\phi$. Then let $$\widetilde {\rho}_ {\chi}=\frac{1}{\text{dim}\chi} \text{tr}(\widetilde {\phi}, [\pi_ {\ast}\mathscr{O}_ {T}:\chi] ) $$ which is a character of algebraic group $G$ over $\overline{\mathbb{F}_ {p}}$. Then $\underline{\rho}_ {\chi} = \widetilde {\rho}_ {\chi}|_ {G(\mathbb{F}_ {q})} .$

Remark 8. The choice of $\widetilde {\phi} $ is equivalent to a choice of basis of $\mathscr{O}(T)$ over $\mathscr{O}(T)^{W}$.

Reasons If $\{f_ {w}\}_ {w\in W}$ is a basis, then define $\widetilde {\phi}(f_ {w})=[p]^{\ast}f_ {w}$. Then extend $\mathscr{O}(T)^{W}$-linearly.

If $[\pi_ {\ast}\mathscr{O}_ {T}:\chi]=$span of some subset of basis elements, then RHS is $$\sum_ {w\in C}\langle [p]^{\ast}f_ {w},f_ {w}^{\ast} \rangle,$$ and $M_ {w}:=\langle [p]^{\ast}f_ {w},f_ {w}^{\ast} \rangle.$