Nice Basis
Recall notation
- $A=\mathscr{O}(T//W)=\mathbb{Z}[T]^{W}$,
- $N = \mathscr{O}(T)=\mathbb{Z}[T]$, so $N$ is an $A$-module,
- $A_ {p} = \mathscr{O}((T//W)_ {p})$.
Proposition 1. Suppose that $\widetilde {\phi}$ is an automorphism of $\pi_ {\ast}(\mathscr{O}_ {T})$, commuting with the action of $W$ such that $\iota^{\ast}\widetilde {\phi}=\phi$. Let $$\widetilde {V}_ {\chi}=\frac{1}{\text{dim}\chi}\text{tr}(\widetilde {\phi}, [\pi_ {\ast}\mathscr{O}_ {T}:\chi])\in A$$ then the reduction mod $p$ of the irreducible representation of $G(\mathbb{F}_ {p})$ over $\mathbb{C}$ corresponding to irreducible representation $\chi$ of $W$ $\underline{V}_ {\chi} = \widetilde {V}_ {\chi}|_ {G(\mathbb{F}_ {p})} $.
If $\{f_ {w}\}_ {w\in W}$ is a basis for $N$ over $A$, then if we define $\widetilde {\phi}(f_ {w})=[p]^{\ast}f_ {w} $, then extend $A$-linearly, this gives some well defined $\widetilde {\phi} $. Then $\widetilde {\phi} $ is $W$-invariant if and only if the $\mathbb{C}$-span of $\{f_ {w}\}_ {w\in W}$ is $W$-invariant.
The goal is :
define some basis $\{f_ {w}\}_ {w\in W}$ of $N$ over $A$ such that
- some $W$-invariant property is satisfied,
- isotypic components of $N$ as a $W$-representation to be spanned by basis elements (two sided KL cell picture is respected).
Define $\langle -,- \rangle: \mathbb{C}[T]\times \mathbb{C}[T]\rightarrow \mathbb{C}[T]^{W}$. Then $M_ {w}:= \langle [p]^{\ast}f_ {w},f_ {w}^{\ast} \rangle$ .
Definition 2. For any $w\in W$, let $E_ {w}=w\text{exp}(\omega_ {I(w)})$.
This basis does not satisfy $W$-invariance.
We could define $E^{w}= (w\text{exp}(\omega_ {\overline{I(w)}}))\text{exp}(-\rho)$.
In small rank, $E_ {w}$ and $E^{w}$ are dual basis under $\langle -,- \rangle$.
In $\mathrm{SL}_ {4}$, $\langle E_ {w}, E^{1} \rangle\neq 0$ whenever $w$ is the product of two permutations indexing singular Schubert varieties.
The Kazhdan-Lusztig-Steinberg basis
Since our goal is to compute $\text{tr}(\widetilde {\phi}, [\pi_ {\ast}(\mathscr{O}_ {T}), \chi] )$. We can relax the condition that the $\mathbb{C}$-span of $\{f_ {w}\}_ {w\in W}$ is $W$-invariant.
Let $\leq$ be an order on $W$, which refines the partial order on the two-sided cells of $W$. For any $w\in W$, let $N_ {\leq w}$ and $N_ {<w}$ be the $A$-modules generated by $\{f_ {y}\}_ {y\leq w}$ and $\{f_ {y}\}_ {y < w} $ respectively, with $\text{gr}_ {w}(N)=N_ {\leq w}/N_ {< w}$. Write $\overline{f}_ {y}$ for the image of $f_ {y}$ in the graded pieces.
Lemma 3. Suppose the following conditions are satisfied.
- For any $w\in W$, the submodules $N_ {\leq w}$ and $N_ {<w}$ are $W$-invariant.
- For any $w\in W$, the $\mathbb{C}$-span of $\{\overline{f}_ {y}\}_ {y\leq w}$ in $\text{gr}_ {w}(N)$ is $W$-invariant.
- For every irreducible representation of $W$, there exists a unique equivalence class of $w\in W$, such that $\chi$ appears in $\text{gr}_ {w}(N)$ with non-zero multiplicity.
Then the endomorphism $\widetilde {\phi} $ preserves $N_ {\leq w}$ and $N_ {<w}$ and the induced automorphism of $\text{gr}_ {w}(N)$ commutes with $W$. Further, $$\text{tr}(\widetilde {\phi}, [\text{gr}_ {w}(N): \chi]) = \text{tr}(\widetilde {\phi},[N:\chi]).$$
Definition 4. For any $w\in W$, let $$f_ {w}:= \frac{1}{|W_ {I(w)}|}C’_ {w}(\text{exp}(\omega_ {\overline{I(w)}}))\in \mathbb{Z}[T]$$ and $$f^{w}=\frac{1}{|W_ {I(w)}|}w_ {0}C’_ {w_ {0}w}(\text{exp}(\omega_ {I(w)})),$$ where $W_ {I(w)}$ is the parabolic subgroup generated by $I(w)$.
Example 5. $C_ {s} = T_ {s} - v$, and $C’_ {s}= T_ {s}+v^{-1}$. These are Koszul dual to each other, i.e. in geometry $C_ {s}$ corresponds to $\mathrm{IC}_ {s}$ and $C_ {s}’$ corresponds to tilting $T_ {s}$.
Definition 6. Let $\langle , \rangle: N\times N \rightarrow A$ be defined by $\langle f,g \rangle = \frac{1}{\delta}(\sum_ {w\in W}(-1)^{\ell(w)}w(fg))\in A$, where $\delta$ is the Weyl denominator,. For $w\in W$, let $h(w)=(\rho,\omega_ {I(w)})\in \mathbb{Z}_ {\geq 0}$.
Lemma 7. For $w,v\in W$, with $h(v)\leq h(w)$, we have $\langle f_ {w}, f^{\vee} \rangle=\delta_ {w,v}$. In rank 2, we also have $\langle f_ {w},f^{\vee} \rangle=\delta_ {w,v}$ fpr any $w,v\in W$.
Corollary 8. The set $\{f_ {w}\}_ {w\in W}$ is a basis of $N_ {\mathbb{Z}}=\mathbb{Z}[T]$ over $A_ {\mathbb{Z}}=\mathbb{Z}[T]^{W}$.
Definition 9. For $c$ a two-sided Kazhdan-Lusztig cell, let $$N_ {\mathbb{Z}}^{\leq c}:=\mathbb{Z}[W]^{\leq c}N_ {\mathbb{Z}}=\oplus_ {y\in c’, c’\leq c}A_ {\mathbb{Z}}f_ {y}$$ and $N_ {\mathbb{Z}}^{<c}$ similarly defined. Let $N_ {\mathbb{Z}}^{c}= N_ {\mathbb{Z}}^{\leq c}/N_ {\mathbb{Z}}^{<c}\cong \oplus_ {y\in c} A_ {\mathbb{Z}}f_ {y}$.
Corollary 10. For $G=\mathrm{SL}(n)$, and any irreducible representation $\chi$ of $W=S_ {n}$, we have that $$\widetilde {V}_ {\chi} =\sum_ {y\in c} \langle [q]^{\ast}f_ {y},f_ {y}^{\ast} \rangle$$ for $c$ the two-sided cell corresponding to $\chi$.
Proof.
We have that $$\widetilde {V}_ {\chi} = \text{tr}(\widetilde {\phi}, [\text{gr}_ {w}(N):\chi] ) $$ where $w$ is some element of $c$. Further in type $A$, $\mathrm{gr}_ {w}N$ breaks up into direct sum of some copies of $\chi$ as a $\mathbb{C}[W]$-module. In fact, $\text{gr}_ {w}(N)\cong \chi^{\oplus \text{dim}\chi}$ as $W$-representations. Each has the form $\mathbb{C}[W]f_ {y}$ for some $y$ in the same KL cell as $w$.
So $\frac{1}{\text{dim}(\chi)}\text{tr}(\widetilde {\phi}, \text{gr}_ {w}N)=\frac{1}{\text{dim}(\chi)}\sum_ {y}\text{tr}(\widetilde {\phi}, \mathbb{C}[W]f_ {y})=\sum_ {y\in c }\langle [p]^{\ast}f_ {y},f_ {y}^{\ast} \rangle$.
In type $A$, this gives that $\underline{V}_ {\chi}$ is a sum of $\langle [p]^{\ast}f_ {y},f_ {y}^{\ast} \rangle$ terms and each of these is a linear combination of Weyl characters by the definition of $\langle -,- \rangle$.
Now compare :
| $\langle [p]^{\ast}f_ {w},f_ {w}^{\ast} \rangle$ | VS | $\langle [p]^{\ast}f_ {w},f^{w} \rangle$ |
|---|---|---|
| actually computes the correct trace | easier to compute | |
| counterexample in $A_ {4}$ | satisfies the conjectural symmetries among dimension polynomials | |
| non-positives (sometimes) | positivity |
To do:
- define $M_ {w}$ in general and state Lusztig’s full conjecture with proof sketch given by asympotic Hecke algebra,
- uniqueness of $M_ {w}$.