Introduction and Motivations

• $F$ a non-archimedian local field, e.g. $\mathbb{Q}_ p, \mathbb{F}_ p((t))$;
• Weil group $W_ F\subseteq \Gamma_ F:=\mathrm{Gal}(\overline{F}/F)$;
• $G$ a connected reductive group over $F$.

Roughly speaking, classical local Langlands predicts a bijection $$ {\text{smooth irreducible representations of $G(F)$}}\cong {\text{L-parameters $\varphi:W_ F\rightarrow {}^L G$ up to $\hat{G}$-conjugacy}}, $$ where $\hat{G}$ is the Langlands dual of $G$ and ${}^LG:=\hat{G}\rtimes \Gamma_ {\widetilde{F}}$.

For $\mathrm{char}(F)>0$, this is proved by Laumon-Rapoport-Stuhler. For $\mathrm{char}(F)=0$, this is proved by Harris-Taylor and Henniart.

For $G\neq \mathrm{GL}_ n$, this is still widely open and the precise formulation is very complicated.

Kazhdan-Lusztig showed that if $G$ is split, then ${\text{smooth irr reps of $G(F)$ with Iwahori fixed vectors}}\hookrightarrow {(\varphi,r)}$, where

This suggests that at least we need to add some extra data $r$ to guarantee injectivity.

Now we notice that $$ \begin{split} \mathrm{Rep}(Z_ {\hat{G}}(\varphi))&=\mathrm{Coh}(\text{pt}/Z_ {\hat{G}}(\varphi))\ &=\mathrm{Coh}(\hat{G}\text{-orbit of }\varphi/\hat{G}). \end{split} $$ It leads us to investigate the stack ${\varphi: W_ F\rightarrow \hat{G}}/\hat{G}=:\mathrm{Loc}_ {\hat{G}}$. This is a reasonable algebraic stack and we can talk about $\mathrm{Coh}(\mathrm{Loc}_ {\hat{G}})$. (Then a pair $(\varphi,r)$ is remembered as a coherent sheaf supported on a $\hat{G}$-orbit.)

To get surjectivity, one also needs to enlarge the left-hand side.

Classically (after Vogan, ……) one should consider representations of all (pure) inner forms of $G$ altogether.

There are reasons to enlarge the above even further. Namely, one should consider $G$, and extended pure inner forms of all its Levi subgroups.

Such consideration appears in Kotwitz’s mod $p$ Shimura varieties but now we want to record a motivation more related to this course.

Recall that Deligne-Lusztig theory studies representations of a finite group of Lie type via geometry. This can be viewed as a toy model (finite field counterpart) of local Langlands.

Let $\kappa$ be a finite field and $H$ be a reductive group over $\kappa$. Set $q:=\# \kappa$ and $\mathrm{Fr}:H\rightarrow H$ to be $q$-Frobenius.

Note that

• $\mathrm{Rep}(H(\kappa))=\mathrm{Shv}(\text{pt}/H(\kappa))$, where by $\mathrm{Shv}$ we use $\ell$-adic sheaf theory. Then we have $\mathrm{Shv}(\text{pt}/H(\kappa))\cong \mathrm{Shv}(H/\mathrm{Ad}_ {\mathrm{Fr}}H)$ by Lang’s isogeny theorem, where $\mathrm{Ad}_ {\mathrm{Fr}}$ denotes the Frobenius-twisted conjugation action, i.e. $h_ 1\cdot h_ 2:=h_ 1^{-1}h_ 2F(h_ 1)$.
• Consider morphism $\pi: H/\mathrm{Ad}_ {\mathrm{Fr}}B_ H\rightarrow H/\mathrm{Ad}_ {\mathrm{Fr}}H$, whose fibers are ismorphic to $H/B_ H$ the flag variety of $H$. In particular, $\pi$ is proper and $\pi_ !=\pi_ *$.
• We obtain a functor $$ \pi_ *=\pi_ !: \mathrm{Shv}(H/\mathrm{Ad}_ {\mathrm{Fr}}B_ H)\rightarrow \mathrm{Shv}(H/\mathrm{Ad}_ {\mathrm{Fr}}H). $$

The meaning point of the above considerations is that $H/\mathrm{Ad}_ {\mathrm{Fr}}H$ has too sterile geometry to exploit while $H/\mathrm{Ad}_ {\mathrm{Fr}}B_ H$ has richer geometry for us to take advantage of.

Recall that we have Bruhat decomposition $H=\sqcup_ {w\in W_ H}B_ HwB_ H$. Let $i_ w: B_ HwB_ H\hookrightarrow H$ denote the inclusion of strata. Note this stratum is $\mathrm{Ad}_ {\mathrm{Fr}}$-stable, since $B$ is defined over $\mathbb{F}_ q$ (we choose such a Borel at the very beginning using the fact that a reductive group over a finite field is always quasi-split). Then consider $$ R_ w:=\pi_ *(i_ w)_ *\overline{\mathbb{Q}_ \ell}_ {B_ HwB_ H/\mathrm{Ad}_ {\mathrm{Fr}}B_ H}\in D(\mathrm{Rep}(H(\kappa))), $$ which are exactly Deligne-Lusztig characters.

Back to our original arithmetic setting $G/F$.

• We define $LG$ to be loop group of $G$, defined over $\kappa$, where $\kappa$ is the residue field of $F$, such that $$LG(\kappa)=G(F).$$ We have Iwahori group $I_ w\subset LG$, which can be viewed as the counterpart of Borel $B_ H$ in the finite field setting.
• Informally, we want to consider the affine analog of Deligne-Lusztig theory $$ \pi_ *: \mathrm{Shv}(LG/\mathrm{Ad}_ {\mathrm{Fr}}I_ w)\rightarrow\mathrm{Shv}(LG/\mathrm{Ad}_ {\mathrm{Fr}}LG). $$ However, $|LG/\mathrm{Ad}_ {\mathrm{Fr}}LG|$ is not a point anymore because Lang’s isogeny fails in this case. For $b\in |LG/\mathrm{Ad}_ {\mathrm{Fr}}LG|$, set $J_ b:=Z_ {LG}(b)$, whose $F$-points is an extended inner form of a Levi of $G$.