Representation of $G(\mathbb{F}_ {q})$

This week we care about unipotent irreducible representations of $G(\mathbb{F}_ {q})$, where $G=\mathrm{GL}_ {n}$ or $\mathrm{SL}_ {n}$.

Recall: We are going to explain a conjecture by Lusztig about how to write $$\underline{\rho}=\sum_ {\lambda}c_ {\rho,\lambda}\overline{V_ {\lambda}},$$ where $\rho$ is a unipotent irreducible representation of $G(\mathbb{F}_ {q})$ over $\mathbb{C}$. In type $A$, unipotent irreducible representations is easy and Deligne-Lusztig theory is needed beyond type $A$.

Unipotent irreducible representations

First we work over $\mathbb{C}$.

In type $A$, we only need to check split tori and so the definition above agrees with the general definition in the remark above.

Now we just write $\mathbb{B}=B(\mathbb{F}_ {q})$ and similarly for $G$ and $T$ for convenience. Note that $V^{\mathbb{B}}\neq 0$ if and only if $\mathrm{Hom}_ {B}(V,\mathbb{C})\neq 0$. However, $\mathrm{Hom}_ {\mathbb{B}}(V,\mathbb{B})\cong (V^{\ast})^{\mathbb{B}}\cong (V^{\mathbb{B}})^{\ast}\cong \mathrm{Hom}_ {\mathbb{G}}(V,\mathbb{C}[\mathbb{B}\backslash \mathbb{G}])$. In paricular, we have a bijection between $V\in\mathrm{Irr}(\mathbb{G})$ in $\mathbb{C}[\mathbb{B}\backslash\mathbb{G}]$ and $\mathrm{Irr}(\operatorname{End}_ {\mathbb{G}}(\mathbb{C}[\mathbb{B}\backslash \mathbb{G}]))$.

Now let $\mathbb{C}[\mathbb{B}\backslash\mathbb{G}]^{\mathbb{B}}$ be the $\mathbb{B}$-invariants.Define a convolution product $$\mathbb{C}[\mathbb{B}\backslash \mathbb{G}]^{\mathbb{B}}\otimes \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]\rightarrow \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]$$ by $$(F\ast f)(g)=\frac{1}{|\mathbb{B}|}\sum_ {h\in \mathbb{G}}F(h)f(h^{-1}g).$$ Then $F\ast(-): \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]\rightarrow \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]$ is $\mathbb{G}$-equivariant. So $ \mathbb{C}[\mathbb{B}\backslash\mathbb{G}/\mathbb{B}]\cong \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]^{\mathbb{B}}$ is an associative algebra with respect to convolution product, acting on $ \mathbb{C}[\mathbb{B}\backslash\mathbb{G}]$ by $\mathbb{G}$-equivariant endomorphisms. This gives a map $$ \mathbb{C}[\mathbb{B}\backslash\mathbb{G}/\mathbb{B}]\rightarrow \operatorname{End}_ {\mathbb{G}}(\mathbb{C}[\mathbb{B}\backslash\mathbb{G}]).$$

Write Bruhat decomposition $G = \sqcup_ {w\in W}BwB$. Set $T_ {w}=\chi_ {BwB}$.

In particular, $H_ {p}\cong \mathbb{C}[\mathbb{B}\backslash \mathbb{G}/\mathbb{B}]$ local Hecke algebra.

We know that $H_ {v}$ deforms $\mathbb{C}[W]$ as $H_ {1}=\mathbb{C}[w]$. Then irreducible representations are in bijection with irreducible representations of $\mathbb{C}[W]$.

Kazhdan-Lusztig cells

Combinatorial theory which will allow us to understand how irreducible representations live in left regular representation $H_ {v}$ action on $H_ {v}$.

Recall that $A=\mathbb{C}[v,v^{-1}]$. Then $H_ {v}$ is an $A$-algebra.

Let $T_ {w}$ be such that $(T_ {s}-v)(T_ {s}+v^{-1})=0$ for any simple reflection $s\in S$.

Let $a\mapsto \overline{a}$ be the $\mathbb{C}$-algebra involution $A\rightarrow A$ defined by $v\mapsto v^{-1}$ and there is an $(A,\overline{(-)})$-semilinear ring homomorphism $H_ {v}\rightarrow H_ {v}$, $x\mapsto \overline{x}$ determined by $T_ {s}\mapsto T_ {s}^{-1}$ and $T_ {w}\mapsto T_ {w^{-1}}^{-1}$.

Note that $r_ {w,w}=1$.

For $n\in\mathbb{Z}$m define $$A_ {\leq n}:=\bigoplus_ {m\leq n} \mathbb{C}v^{m}$$ and $A_ {\leq n}$, $A_ {<n}$, $A_ {>n}$ similarly. Define $H_ {\leq 0}:=\oplus_ {w}A_ {\leq 0} T_ {w}$ and $H_ {<0}$ similarly.

Existence and uniqueness of Kazhdan-Lusztig basis

Fix $w\in W$. For any $x\leq w$, we will construct $u_ {x}\in A_ {\leq0}$, such that

1. $u_ {w}=1$,
2. for any $x<w$, $u_ {x}\in A_ {<0}$, and $$\overline{u_ {x}}-u_ {x}=\sum_ {x<y\leq w}r_ {x,y}u_ {y}.$$

Then we define $C_ {w}:=\sum_ {y\leq w}u_ {y}T_ {y}\in H_ {\leq 0}$. Then \begin{align} \overline{C}_ {w} &= \sum_ {y\leq w} \overline{u}_ {y}\overline{T}_ {y} \\ &= \sum_ {y\leq w} \underline{u_ {y}}\sum_ {x\leq y}\overline{r}_ {x,y}T_ {x}\\ &=\sum_ {x\leq w}(\sum_ {x\leq y\leq w}\overline{r}_ {x,y}\overline{u_ {y}})T_ {x}\\ &=\sum_ {x\leq w}(\overline{a}_ {x}+u_ {x})T_ {x}\\ &=\sum_ {x\leq w}u_ {x}T_ {x}\\ &=C_ {w}. \end{align}

Now we investigate the uniqueness.

We claim that if $h\in H_ {<0}$, satisfies $\overline{h}=h$, then $h=0$.

Cells and cell representations

See examples for affine Weyl groups.

This is $I_ {w,L}$ in this specific case, and it is based ideal.

We have that $H_ {v}\cong \oplus_ {\mathscr{C}}L_ {\mathscr{C}}$.

In type $A$,by Robinson–Schensted correspondence, $S_ {n}$ corresponds to a pair standard Young tableux with same shape (numbering with increasing along rows and columns). For example,

Then left cell corresponds to fix a SYT on the left, and two-sided cells correspond to pick a shape.