Consruction of $M_ {w}$

Construction of $M_ {w}$

Definition 1. In each Kazhdan-Lusztig cell, there is a unique almost involution of minimal length, called the Duflo involution.

Definition 2. If $w$ and $w^{-1}$ lie in the same left cell, we call $w$ a almost involution. We call the set of almost involutions $\mathcal{J}$.

Definition 3. For any $w\in \mathcal{J}$, let $M_ {w}:=\langle [p]^{\ast}f_ {w_ {0}d},f_ {w_ {0}w}^{\ast} \rangle$, where $d$ is the Duflo involution in the same left cell as $w$.

Lemma 4. For any $w\in \mathcal{J}$, there exists some subset $\Lambda_ {w}\subseteq \Lambda=\mathbb{X}^{\ast}$ independent of $p$ and some coefficients $r_ {\lambda}\in \mathbb{Z}$, such that $$M_ {w}=\sum_ {\lambda\in \Lambda_ {w}}r_ {\lambda}V_ {(p-1)\omega_ {I(w)}+\lambda}.$$

Proof.

Let $w\in \mathcal{J}$, let $d$ be the Duflo involution in the same left cell. First note that $[p]^{\ast}f_ {w_ {0}d}$ is a linear combination of monomials from the $W$-orbit of $\operatorname{exp}(p\omega_ {I(w)})$, since $\overline{I(w_ {0}d)}=I(d)=I(w)$. The paring of such monomials $\operatorname{exp}(\nu)$ with another $\operatorname{exp}(\mu)$ appearing with non-zero coefficients in $f_ {w_ {0}w}^{\ast}$ equals to $V_ {y(\nu+\mu)-\rho}$ for $y\in W$, such that $y(\nu+\mu)$ is dominant. We have $V_ {y(\nu+\mu)-\rho}=V_ {(p-1)\omega_ {I(w)}+\lambda}$ for some $\lambda$.

We can write $$[p]^{\ast}f_ {w_ {0}d}=\sum_ {\nu\in W\cdot_ {p}\omega_ {I(w)}}a_ {\nu}\operatorname{exp}(\nu)$$ and $$f_ {w_ {0}w}^{\ast}=\sum_ {\mu\in \Lambda_ {w}‘}b_ {\mu}\operatorname{exp}(\mu),$$ where $\Lambda_ {w}’$ is the set of $\mu$ such that $\operatorname{exp}(\mu)$ has non-zero coefficient in $f_ {w_ {0}w}^{\ast}$ (independent of $p$).

Therefore, the desired formula holds where $\Lambda_ {w}$ is the set of all $\lambda$ such that $\mu+\nu$ and $(p-1)\omega_ {I(w)}+p+\lambda $ lie in the same $W$-orbit for $\mu\in w\cdot_ {p}\omega_ {I(w)}$ and $\nu\in \Lambda_ {w}’$. For every such $\lambda$, $r_ {\lambda}=\sum_ {\nu,\mu}\pm a_ {\nu}b_ {\mu}.$

Recall that $\{C_ {w}\}_ {w\in W}$ is the Kazhdan-Lusztig basis of $H_ {v}$ or of $\mathbb{C}[W]$.

Definition 5. Let $h_ {x,y,z}\in \mathbb{Z}[v,v^{-1}]$ be the structure constant of multiplication of $\{C_ {w}\}_ {w\in W}$ in $H_ {v}$ for $x,y,z\in W$, i.e. $C_ {x}C_ {y}=\sum_ {z\in W}h_ {x,y,z}C_ {z}$. We define $r_ {x,y,z}$ to be the top degree coefficient of $h_ {x,y,z}$.

Definition 6. Let $J$ be the free abelian group generated by $\{t_ {w}\}_ {w\in W}$ equipped with ring structure given by $t_ {x}t_ {y}=\sum_ {w\in W}r_ {x,y,z}t_ {z}$. This multiplication is associative. The identity of multiplication is given by $1_ {J}:=\sum_ {d\text{ Duflo involution}}t_ {d}$.

Given any two-sided cell $c\in W$, let $$J_ {c}\operatorname{span}\{t_ {w}:w\in c\}.$$ There is a direct sum decomposition $J=\bigoplus_ {c}J_ {c}$, where the unit of each $J_ {c}$ is $1_ {J_ {c}}:=\sum_ {d\in c\text{ Duflo}}t_ {d}$.

There is a natural correspondence between irreducible modules over $H_ {v}$, $\mathbb{C}[W]$ and $J$.

Proposition 7. For a two-sided cell $c$ of $W$, there is a $(\mathbb{Z}[W]_ {c},J_ {c})$-bimodule $B_ {c}$ with basis $b_ {w},w\in c$. It is defined so that \begin{align} \mathbb{Z}[W]_ {c}&\rightarrow B_ {c}\\ C_ {w} &\mapsto b_ {w} \end{align} and \begin{align} J_ {c} &\rightarrow B_ {c} \\ t_ {w} & \mapsto b_ {w} \end{align} are isomorphisms as left $\mathbb{Z}[W]_ {c}$-modules and right $J_ {c}$-modules, respectively. The action of the algebras $\mathbb{Z}[W]_ {c}$ and $J_ {c}$ on $B_ {c}$ are mutual centralizers for one another.

Notation:

  • subscript: two sided cell piece with respect to $C_ {w}$.
  • superscript: two sided cell piece with respect to $C_ {w}’$.

Recall that $A:=\mathbb{C}[T]^{W}$ and $N=\mathbb{C}[T]$, and $$\xi: A[W]^{c}\xrightarrow{\cong}N_ {\mathbb{Z}}^{c}.$$

Corollary 8. The map $$\eta: A[B_ {c}]\rightarrow N^{w_ {0}c}$$ given by $b_ {w}\mapsto f_ {w_ {0}w}$ is an isomorphism of $A$-modules intertwining with $W$-action (after twisting by signs).

As a result, there is a well-defined action of $A[J_ {c}]$ on $N^{w_ {0}c}$.

Proposition 9.

  1. For any two-sided cell $c$, the endomorphism $\widetilde {\phi}|_ {\text{gr}_ {c}(N)}$ coincides with the right action of some element $h\in A\otimes_ {\mathbb{Z}}J_ {c}$.
  2. $h=\sum_ {w\in c}M_ {w}t_ {w}$ “Intrinsic definition of $M_ {w}$”.
Proof.

Since $\sum_ {\text{Duflo }d\in c}t_ {d}$ is the identity of $J_ {c}$, we have that \begin{align} h &=(\eta^{-1}\widetilde {\phi}\eta )(\sum_ {d\in C}t_ {d}) \\ & = (\eta^{-1}\widetilde {\phi})(\sum_ {d\in C}f_ {w_ {0}d})\\ & = \eta^{-1}(\sum_ {d\in c}\sum_ {w\sim_ {L}d}\langle [p]^{\ast}f_ {w_ {0}d},f_ {w_ {0}w^{\ast}} \rangle f_ {w_ {0}w})\\ & = \sum_ {d\in c}\sum_ {w\sim_ {L}d} M_ {w}\eta^{-1}(f_ {w_ {0}w}) \\ & = \sum_ {d\in c}\sum_ {w\sim_ {L}d} M_ {w}t_ {w}\\ & = \sum _ {w\in c}M_ {w}t_ {w}. \end{align}

Non-abelian Fourier Transform

For any $\chi\in \operatorname{Irr}(W)$, there exists a “principal series module” $\mathrm{U}_ {\chi}\in \operatorname{Irr}(G(\mathbb{F}_ {p}))$ lies in $\mathrm{Ind}_ {B(\mathbb{F}_ {p})}^{G(\mathbb{F}_ {p})}(\operatorname{triv})$.

Also there is a virtual representation $V_ {\chi}$ associated to a character sheaf on $G$.

There exists an involution on $\operatorname{Irr}(G(\mathbb{F}_ {p}))$ called the non-abelian Fourier transform sending $V_ {\chi}\leftrightarrow \mathrm{U}_ {\chi}$.

For any $y\in W$, there exists some virtual characters $R_ {y}^{1}$ (unipotent Deligne-Lusztig character) comes from geometry.

Definition 10. A unipotent irreducible representation $\rho$ of $G(\mathbb{F}_ {p})$ is an irreducible representation such that $(\rho, R_ {y}^{1})\neq 0$ for some $y\in W$.

For any $w\in \mathcal{J}$, let $$R_ {\alpha_ {w}}:=\frac{1}{|W|}\sum_ {\chi\in \operatorname{Irr}(W),y\in W}C_ {w,\chi}\text{tr}(y|\chi)R_ {y}^{1},$$

where $c_ {w,\chi}$ is the trace of $t_ {w}$ in the representation of $J$ associated to $\chi$.

By Lusztig, \begin{align} R_ {\alpha_ {w}} &= \sum_ {\chi\in\operatorname{Irr}(W)}C_ {w,\chi}V_ {\chi}\\ & = \mathrm{FT}(\sum_ {\chi\in\operatorname{Irr}(W)}C_ {w,\chi}\mathrm{U}_ {\chi}). \end{align}

Lusztig’s actual conjecture

Conjecture 11. For every $w\in \mathcal{J}$, there is a non-zero character $M_ {w}$ of $G(\mathbb{F}_ {p})$ over $\overline{\mathbb{F}_ {p}}$ such that for any irreducible unipotent representation $\rho$, we have that $$\underline{\rho}=\sum_ {w\in \mathcal{J}} (\rho: R_ {\alpha_ {w}})M_ {w}.$$

Corollary to our intrinsic definition of $M_ {w}$:

For any $\chi\in\operatorname{Irr}(W)$, one has that $$\underline{V_ {\chi}}=\sum_ {w\in \mathcal{J}}(V_ {\chi}:R_ {\alpha_ {w}})M_ {w}.$$

Proof.

By Jantzen’s theorem and our nice choice of basis, $$\underline{V}_ {\chi}=\widetilde {V}_ {\chi}|_ {G(\mathbb{F}_ {p})},$$ where $\widetilde {V}_ {\chi} = \frac{1}{\text{dim}\chi}\operatorname{tr}(\widetilde {\phi}|(\text{gr}_ {w}N:\chi)).$ Then by intrinsic definition of $M_ {w}$, one has \begin{align} \widetilde {V}_ {\chi}|_ {G(\mathbb{F}_ {p})} & = \frac{1}{\text{dim}\chi}\operatorname{tr}(\sum_ {w\in c}M_ {w}t_ {w}|(\mathbb{C}[W]_ {c}:\chi)) \\ & = \frac{1}{\text{dim}\chi}\sum_ {w\in c}\operatorname{tr}(\sum_ {w\in c}t_ {w}|(\mathbb{C}[W]_ {c}:\chi))M_ {w}\\ & = \sum_ {w\in c}C_ {w,\chi}M_ {w}\\ & = \sum_ {w\in \mathcal{J}}(V_ {\chi}:R_ {\alpha_ {w}})M_ {w}. \end{align}