A Sheaf Theoretic Approach to Deligne-Lusztig Theory
🔗A Sheaf Theoretic Approach to Deligne-Lusztig Theory
Now we engage ourselves into a toy model: finite field case. In this case, Deligne-Lusztig theory gives us a quite satisfactory understanding. However, for future generalizations, we want to re-study this theory from sheaf theoretic point of view.
Now let $\kappa$ be a finite field and $G$ a connected reductive group over $\kappa$. Let $B\subseteq G$ be a Borel subgroup (since $G$ is always quasi-split over $\kappa$) and $U\subseteq B$ the unipotent radical. We set $T=B/U$.
Let $k:=\overline{\kappa}$ be the algebraic closure. We set $q:=\#\kappa$, which is a power of $p$, i.e. $\kappa\cong \mathbb{F}_ q$ and $k\cong \overline{\mathbb{F}_ p}$. By abuse of notation, we will use $B$ to denote the base change to $k$, viewed as a Frobinus fixed Borel.
We set
- $\mathcal{C}:=\mathrm{AlgStk}_ {k}^{\textrm{fp}}$,
- $X=\ast/B$,
- $Y=\ast/G$,
- $\sigma$ the Frobenius morphism.
Then we have the correspondence
Remark 1. Of course, this diagram could be observed directly. But let me explain again how to observe this using the magical diagram (3.8). This works for a general automorphism $\phi$ of $G$ such that $\phi(B)=B$, and a general base field.
- $\mathcal{L}_ \phi Y=[\ast/G]\times_ {{}^{\Gamma_ \phi}[\ast/G]\times [\ast/G]{}^\Delta}[\ast/G]$: first, $[\ast/G]\times [\ast/G]=[\ast/G\times G]$; then note that as objects, only $(g_ 1,g_ 2)\in G\times G$; for isomorphisms, $(g_ 1,g_ 2)\cong (g_ 1’,g_ 2’)$ if there exists $g,\widetilde{g}\in G$, such that $(g_ 1’,g_ 2’)=(g^{-1}g_ 1\widetilde{g},\phi(g)^{-1}g_ 2\widetilde{g})$. Now we calculate its $\pi_ 0$. Note that we can always choose $\widetilde{g}=g_ 2^{-1}\phi(g)$, and get $(g_ 1,g_ 2)\sim (g^{-1} g_ 1g_ 2^{-1}\phi(g),1)$ for any $g\in G$; and we notice that actually, $(g_ 1,g_ 2)\sim (g_ 1’,g_ 2’)$ if and only if $g_ 1g_ 2^{-1}=\mathrm{Ad}_ \phi(g)(g_ 1’g_ 2’^{-1})$ for some $g\in G$. So $\pi_ 0=\pi_ 0(G/\mathrm{Ad}_ \phi G)$. Also, the automorphism groups match. So we get the desired results. Note this computation works for any reductive group $G$, automorphism $\phi$, and any coefficient.
- $X\times_ YX$: objects are $g\in G$, and isomorphisms are given by $g\cong g’$ if and only if there exists $b_ 1,b_ 2\in B$, such that $b_ 1gb_ 2=g’$. So $\pi_ 0=\pi_ 0(B\backslash G/B)$ and automorphism groups match apparently.
- $X\times_ Y\mathcal{L}_ \phi Y\cong X\times_ {Y\times Y}Y$: objects are $(g_ 1,g_ 2)\in G\times G$, and $(g_ 1,g_ 2)\sim (bg_ 1g,\phi(b)g_ 2g)$ for some $b\in B,g\in G$, so the data is same as $g\in G$, and $g\sim bg\phi(b)^{-1}$. So it is isomorphic to $G/\mathrm{Ad}_ \phi B$. All these stacks are classical.
This correspondence exists in general, not using the assumption that $\kappa$ is a finite field and $\sigma$ is Frobenius. But in this case, we have Lang’s isogeny and $$[G/_ {\mathrm{Ad}_ \sigma}G]\cong [\ast/G(\kappa)].$$ The map $\pi$ is proper representable and $\delta$ is smooth representable.
🔗5.1 Sheaf Theory: Ind-constructible $\ell$-adic Sheaves
Recall that we have $\ell$-adic sheaf theory (where $\ell\neq p$ is a prime): $$\mathrm{Shv}_ \ell: \mathrm{Corr}(\mathrm{AlgStk}_ k^{\textrm{fp}})_ {\textrm{V=representable, H=all}}\rightarrow \mathrm{LinCat}_ \Lambda,$$ where we can take $\Lambda=\mathbb{Z}_ \ell,\mathbb{Q}_ \ell$ or $\mathbb{F}_ \ell$ and
- for $f\in H$, we have $f^!$;
- for $g\in V$, we have $g_ \ast $.
Inside $\mathrm{Shv}_ \ell$, we have constructible complexes $\mathrm{Shv}_ \ell^c$. The sheaf theory we want is actually $$ \mathrm{IndShv}_ \ell^c: \mathrm{Corr}(\mathrm{AlgStk}_ k^{\textrm{fp}})_ {\textrm{V=representable, H=all}}\rightarrow \mathrm{LinCat}_ \Lambda.$$ The story of $\mathrm{IndShv}_ \ell^c$ and $\mathrm{Shv}_ \ell$ is similar to the story of $\mathrm{IndCoh}$ and $\mathrm{QCoh}$.
Remark 2. For a general algebraic stack $\mathcal{X}$, $\mathrm{Ind}(\mathrm{Shv}_ \ell^c)(\mathcal{X})\neq \mathrm{Shv}_ \ell(\mathcal{X})$. But for $\Lambda$ characteristic 0, algebraically closed field, and $X$ a finite type scheme over $X$, we have $\mathrm{Shv}_ \ell(X)=\mathrm{IndShv}_ \ell^c(X)$.
🔗Deligne-Lusztig Induction: Borels and Characters
Now we have a functor: $$\mathrm{Ind}^{\mathrm{DL}}:=\pi_ \ast \delta^!:\mathrm{IndShv}^c_ \ell(B\backslash G/B)\rightarrow \mathrm{Shv}_ \ell^c(\ast/G(\kappa)).$$
Remark 3. We have ismorphism of stacks $[B\backslash G/B]\cong [(G/B\times G/B)/G]$, where $G$ acts diagonally.
Let $i_ w: \mathcal{O}(w)\hookrightarrow B\backslash G/B$ the embedding corresponding to $w\in W$, where $W$ is the Weyl group. Then $i_ w$ is a locally closed embedding.
Define $$R_ w^\ast :=\mathrm{Ind}^{DL}(i_ w)_ \ast \Lambda[l(w)],$$ and $$R_ w^!:=\mathrm{Ind}^{DL}(i_ w)_ !\Lambda[l(w)].$$ See Appendix A for the connection to classical Deligne-Lusztig induction. More precisely, we state it as the following lemma.
Lemma 4. We have that $$R_ w^\ast \cong H_ \ast ^{BM}(X(w),\Lambda)$$ and $$ R_ w^!\cong H^\ast _ {c}(X(w),\Lambda)$$ as $G(\kappa)$-representations.
Remark 5. By general formalism (Thm 4.5), $\mathrm{tr}(\mathrm{IndShv}_ \ell^c(B\backslash G/B,\Lambda),\sigma)\hookrightarrow\mathrm{IndShv}_ \ell^c(\ast/G(\kappa))$ is fully faithful. However, it is not essentially surjective.
Set $\widetilde{\mathcal{O}_ w}=BwB$. Then $\widetilde{\mathcal{O}_ w}/\mathrm{Ad}_ \sigma B\rightarrow Tw/\mathrm{Ad}_ {\sigma}T=\ast/T^{w\sigma}$, where $T^{w\sigma}$ is a finite group.
Remark 6. Recall that Lang–Steinberg theorem states that if $\phi$ is surjective and has a finite number of fixed points, and $G$ is a connected affine algebraic group over an algebraically closed field, then $x\mapsto x^{-1}F(x)$ is surjective. Here we take $G$ to be $T$ defined over $k$, $F(t)=w(\sigma(t))$.
Remark 7. A maximal $\kappa$-rational torus in $G$ up to $G(\kappa)$-conjugacy is parameterized by $H^1(k,W)\cong W/{\mathrm{Ad}_ \sigma W}$ (see \cite[Cor.,1.14]{deligne1976representations}).
So indeed, we should view $T^{w\sigma}$ as $\kappa$-points of $T$ equipped with a new Frobenius.
Now assume $\Lambda=\overline{\mathbb{Q}_ \ell}$ for simplicity.
Let $\theta: T^{w\sigma}\rightarrow\overline{\mathbb{Q}_ \ell}^\times$ a character. Then we obtain a local system $\mathcal{L}_ \theta$ on $\widetilde{\mathcal{O}(w)}/\mathrm{Ad}_ \sigma B$.
Now we can define $$R_ {w,\theta}^\ast :=(\pi_ w)_ \ast \mathcal{L}_ \theta[l(w)],$$ and $$R_ {w,\theta}^!:=(\pi_ w)_ !\mathcal{L}_ \theta[l(w)],$$ where we define
Definition 8 (Deligne-Lusztig). We say $(w,\theta)\sim^{\textrm{geom}}(w’,\theta’)$ are geometrically conjugate, if $(Tw,\theta)$ and $(Tw’,\theta’)$ are conjugate in $G\otimes_ \kappa {k}$.
Example 9. For any $w,w’\in W$, we always have that $(w,1)\sim^{\textrm{geom}}(w’,1)$.
Remark 10. Although we define $\theta$ as a character of $Tw(\kappa)$, we can extend it to any finite extension over $\kappa$ by norm map: $Tw(\kappa’)\xrightarrow{\mathrm{Nm}}Tw(\kappa)\xrightarrow{\theta}\overline{\mathbb{Q}_ \ell}^\times$.
Theorem 11 (Deligne-Lusztig). We have the following.
- $<R_ {w,\theta}^\ast ,R_ {w’,\theta’}^\ast >=0$, unless $(w,\theta)\sim^{\textrm{geom}}(w’,\theta’)$.
- For any $\chi$ irreducible representation of $G(\kappa)$, there exists $(w,\theta)$, such that $<\chi,R_ {w,\theta}^\ast >\neq0$.
Definition 12. An irreducible representation $\chi$ of $G(\kappa)$ is called unipotent, if $<\chi,R_ {w,1}^\ast >\neq 0$ for some $w$. Equivalently, $\chi$ is in the image of $\widetilde{\pi}_ \ast \circ \delta^!:\mathrm{Shv}_ \ell^c(B\backslash G/B)\rightarrow \mathrm{Rep}_ {f.g.}(G(\kappa),\overline{\mathbb{Q}_ {\ell}})$.
Geometrically, for any $w\in W$, we choose a lift $\dot{w}\in N_ G(T)$, and consider a $T^{w\sigma}$-torsor $$Y(w)\rightarrow X(w),$$ where $Y(w):={gU\in G/U:g^{-1}\mathrm{Fr} g\in U\dot{w}U}$. Then we have $$R_ {w,\theta}^\ast =H_ \ast ^{BM}(Y(w))\otimes_ {T^{w \sigma}}\theta.$$ This leads us to consider $G/\mathrm{Ad}_ {\phi}U$ and expect essential surjectivity of spectral Deligne-Lusztig induction in this case.
An observation is that actually we only need certain monodromic sheaves instead of arbitrary sheaves to obtain all representations. Although $\widetilde{\pi}_ !$ is not defined on all sheaves, hopefully it is defined for such monodromic sheaves.
🔗Monodromic Sheaves
🔗Torus Case
Let $T$ be a torus defined over a $\kappa\cong \mathbb{F}_ q$, with $q$ a $p$-th power. Let $F$ be a local field with residue characteristic $p$ and $\hat{T}$ the dual torus over coefficient ring $\Lambda$.
Recall that $$\varprojlim_ {(n,p)=1}T^{[n]}\cong \mathbb{X}_ \ast (T)\otimes_ {\mathbb{Z}}\mathbb{Z}^p(1),$$ where $\mathbb{Z}^p:=\prod_ {(\ell,p)=1}\mathbb{Z}_ \ell$.
Then we have $${1\textrm{-dim tame local system on }T}\cong{\pi_ 1^{\textrm{\’et}}(T)\xrightarrow{\textrm{continuous}}\Lambda^\ast }\cong {\chi:I_ F^t=\mathbb{Z}^p(1)\xrightarrow{\textrm{cts}}\hat{T}(\Lambda)}.$$
Lemma 13. There exists a fully faithful and $t$-exact functor $$\mathrm{Ch}:\mathrm{IndCoh}(R_ {I_ F^t,\hat{T}})\hookrightarrow\mathrm{Shv}(T).$$
Note that $R_ {I_ F^t,\hat{T}}\cong\hat{T}^{\wedge,p}\hookrightarrow\hat{T}$. For any $\chi:\mathrm{Spec}\Lambda\rightarrow R_ {I_ F^t,\hat{T}}$, $\chi_ \ast \Lambda\in\mathrm{IndCoh}(R_ {I_ {F}^t,\hat{T}})$, and $\mathrm{Ch}_ \chi:=\mathrm{Ch}(\chi_ \ast \Lambda)$ is a local system on $T$.
Proof.
Need to check that $$\mathcal{H} om_ {\hat{T}}(\chi_ \ast \Lambda,\chi’_ \ast \Lambda)\cong \mathcal{H} om(\mathrm{Ch}_ \chi,\mathrm{Ch}_ {\chi’}).$$ For example, if $\chi=\chi’=u^{\textrm{triv}}$, then $$\mathcal{H} om_ {\hat{T}}(\delta,\delta)\cong R\Gamma(T,\Lambda)$$ using Koszul resolution, where the right hand side is exterior algebra $\Lambda^\bullet\mathbb{X}_ \ast (T)$.
Definition 14. We define the essential image of $\mathrm{Ch}$ to be the category of $T$-monoidal $\ell$-adic sheaves, denoted by $\mathrm{Shv}^{\textrm{mon}}(T)$.
We use $\mathrm{For}$ to denote the inclusion of subcategory $\mathrm{Shv}^{\textrm{mon}}(T)$ into $\mathrm{Shv}(T)$. Fact: we have an adjunction pair: $$\mathrm{For}:\mathrm{Shv}^{\textrm{mon}}(T)\hookrightarrow\mathrm{Shv}(T):\mathrm{Av}^{\textrm{mon}},$$ Moreover, both $\mathrm{Shv}^{\textrm{mon}}(T)$ and $\mathrm{Shv}(T)$ are compactly generated, and $\mathrm{For}$ preserves compact objects and is continuous.
For example, consider $1_ \ast \Lambda=\delta_ 1\in\mathrm{Shv}(T)$, $\mathrm{Ch}^R(\delta_ 1)=\mathrm{Ch}^{-1}\mathrm{Av}^{\textrm{mon}}(\delta_ 1)\cong \omega_ {R_ {I_ F^t,\hat{T}}}$.
Now let $\Lambda$ be a field. Fix $\tau$ a topological generator of $I_ F^t$. Then $$R_ {I_ F^t,\hat{T}}=\bigcup_ {\chi\in\hat{T}^{p}}\hat{T}^\wedge_ \chi.$$ Let $\hat{\chi}:=\hat{T}_ \chi^\wedge=\varinjlim_ {\alpha}Z_ \alpha$, where $Z_ \alpha\hookrightarrow \hat{T}$ is regular embedding. Then $$\omega_ {R_ {I_ F^t,\hat{T}}}|_ {\hat{\chi}}=\varinjlim_ {\alpha}\omega_ {Z_ \alpha},$$ with transition map being $\ast$-pushforward constructed as following: let $i_ {\alpha\beta}:Z_ \alpha\hookrightarrow Z_ \beta$ denote the closed immersion, and then adjunction gives $i_ {\alpha\beta,\ast}i_ {\alpha\beta}^!\rightarrow\mathrm{\textrm{id}}$, and then a canonical map $i_ {\alpha\beta,*}\omega_ {Z_ \alpha}\rightarrow\omega_ {\beta}$.
Observe that by adjunction, we have $$\omega_ {Z_ \alpha}=\mathcal{H} om_ {\hat{T}}(\mathcal{O}_ {Z_ \alpha},\omega_ {\hat{T}})$$
Proof.
$$\begin{split} \omega_ {Z_ \alpha}&\cong\mathcal{H} om_ {Z_ \alpha}(\mathcal{O}_ {Z_ \alpha},\omega_ {Z_ \alpha})\ &\cong \mathcal{H} om_ {Z_ \alpha}(\mathcal{O}_ {Z_ \alpha},i_ \alpha^!\omega_ {\hat{T}})\ &\cong \mathcal{H} om_ {\hat{T}}(i_ {\alpha,*}\mathcal{O}_ {Z_ \alpha},\omega_ {\hat{T}}). \end{split}$$
The canonical bundle $\omega_ {\hat{T}}$ is a line bundle shifted by $[d]$, and $\omega_ {Z_ \alpha}$ is a coherent sheaf in degree 0.
Example 15. Take $\hat{T}=\mathbb{G}_ m$ and $x$ a local parameter of $\chi$. Then $$Z_ n=\mathrm{Spec}\Lambda[x]/x^n$$ and $\omega_ {Z_ n}=\lambda[x]/x^n$. Then $\varprojlim\Lambda[x]/x^n=\Lambda[[x]]$ and $$\omega_ \chi=\varprojlim_ {n}\Lambda[x]/x^n$$ with transition map $\times x$. Therefore, $\omega_ \chi\cong\Lambda[x,x^{-1}]/\Lambda[x]$.
Denote $\mathrm{Ch}(\omega_ {\hat{\chi}})$ by $\mathrm{Ch}_ \chi^\textrm{mon}$.
Example 16. Let $\hat{T}=\mathbb{G}_ m$ and $\chi=u$ trivial, then $\mathrm{Ch}_ u^{\textrm{mon}}$ is the universal ind-unipotent local system on $\mathbb{G}_ m$ $$\begin{pmatrix} 1 & 1 & 0 & \dots & 0 &0 &\dots \\ 0 & 1 & 1 & \dots & 0 &0 &\dots \\ \ddots & \ddots & \ddots & \ddots &\ddots &\ddots &\ddots\\ 0 & 0 & 0 & 0 & 1 & 0 &\dots\\ \ddots & \ddots & \ddots & \ddots & \ddots &\ddots &\ddots \end{pmatrix}$$
Definition 17. For $\chi\subseteq R_ {I_ F^t,\hat{T}}$, we define $\mathrm{Shv}^{\chi\textrm{-mon}}(T)$ by the following diagram
Lemma 18. Let $T_ 1, T_ 2$ be two torus and $T=T_ 1\times T_ 2$. Then $$\boxtimes:\mathrm{Shv}(T_ 1)\otimes\mathrm{Shv}(T_ 2)\hookrightarrow\mathrm{Shv}(T)$$ is fully faithful, which induces equivalences $$\boxtimes:\mathrm{Shv}^{\textrm{mon}}(T_ 1)\otimes\mathrm{Shv}^{\textrm{mon}}(T_ 2)\xrightarrow{\cong}\mathrm{Shv}^{\textrm{mon}}(T)$$ and $$\boxtimes:\mathrm{Shv}^{\chi_ 1\textrm{-mon}}(T_ 1)\otimes\mathrm{Shv}^{\chi_ 2\textrm{-mon}}(T_ 2)\xrightarrow{\cong}\mathrm{Shv}^{(\chi_ 1,\chi_ 2)\textrm{-mon}}(T).$$
Remark 19. Under the sheaf-function dictionary, characters (monodromic sheaves) already give all the functions of $T$, and consider $\mathrm{Shv}^{\textrm{mon}}(T)\hookrightarrow\mathrm{Shv}(T)$ is enough (from Langlands/representation point of view).
Lemma 20. Let $f: T_ 1\rightarrow T_ 2$ be a homomorphism of tori and $\hat{f}:\hat{T}_ 2\rightarrow \hat{T}_ 1$ be the induced map of dual tori.
- we have functor $f^\ast :\mathrm{Shv}^{\textrm{mon}}(T_ 2)\rightarrow\mathrm{Shv}^{\textrm{mon}}(T_ 1)$, which is compatible with $\chi_ i$-monodromic subcategories and admit a continuous right adjoint $$f_ \ast ^{\textrm{mon}}:\mathrm{Shv}^{\textrm{mon}}(T_ 1)\rightarrow\mathrm{Shv}^{\textrm{mon}}(T_ 2).$$
- If $f$ is surjective, then $f_ \ast ^{\textrm{mon}}=f_ \ast |_ {\mathrm{Shv}^{\textrm{mon}}(T_ 1)}$, and similar for $\chi_ 1$-monodromic part.
- Under $\mathrm{IndCoh}(R_ {I_ {F}^t,\hat{T}})\cong\mathrm{Shv}^{\textrm{mon}}(T)$, $\hat{f}_ \ast ^{\mathrm{IndCoh}}$ corresponds to $f^\ast $, and $\hat{f}^!$ corresponds to $f_ \ast ^{\textrm{mon}}$.
- When $f$ is surjective, $f_ !$ restricts to $\mathrm{Shv}^{\textrm{mon}}(T_ 1)$ is the left adjoint to $f^\ast [\dim T_ 2-\dim T_ 1]$. Moreover, $$f_ ![\dim T_ 1-\dim T_ 2]=f_ \ast $$ on the $\mathrm{Shv}^{\textrm{mon}}(T_ 1)$.
Part (4) is ultimately due to the calculation:
- $H^\ast (\mathbb{G}_ m)=k[\epsilon]$, $|\epsilon|=1$,
- $H^k_ c(\mathbb{G}_ m)=\begin{cases} \overline{\mathbb{Q}_ {\ell}},\quad&\textrm{if }k=1,\ \overline{\mathbb{Q}_ {\ell}}(1),\quad&\textrm{if }k=2,\ 0,&\textrm{otherwise}, \end{cases}$ and factually $H^\ast _ c(\mathbb{G}_ m)=H^\ast (\mathbb{G}_ m)[-1]$.
Corollary 21. We have the following commutative diagram
Lemma 22. The adjunction pair $$\mathrm{For}:\mathrm{Shv}^{\textrm{mon}}(T)\hookrightarrow \mathrm{Shv}^{\textrm{mon}}(T): \mathrm{Av}^{\textrm{mon}}$$ preserves non-unital monoidal structure $\otimes$. Moreover, $\mathrm{Av}^{\textrm{mon}}$ preserves unit but $\mathrm{For}$ does not.
We have $$\bigoplus_ {\chi}\mathrm{Shv}^{\chi\textrm{mon}}(T)\hookrightarrow\mathrm{Shv}^{\textrm{mon}}(T).$$
🔗General Case
Let $X$ be a scheme (or stack) acted by a torus $T$. Then we define $\mathrm{Shv}^{\textrm{mon}}(X):=\mathrm{Shv}(X)\otimes_ {\mathrm{Shv}(T)}\mathrm{Shv}^{\textrm{mon}}(T)$.
The natural functor $$\mathrm{For}: \mathrm{Shv}^{\textrm{mon}}(X)\rightarrow\mathrm{Shv}(X)$$ has a right adjoint, still denoted by $\mathrm{Av}^{\textrm{mon}}$.
Example 23. Let $p: T’\rightarrow T$ be a surjection of tori. Assume that $T$ acts on $X$. Then $$\mathrm{Shv}^{T’\textrm{-mon}}(X)\xrightarrow{\cong}\mathrm{Shv}^{T\textrm{-mon}}(X).$$
Example 24. Let $\varphi: T’\rightarrow T$ be a fintie 'etale map and $D:=\ker(\varphi)\in T’$. We have the dual map $\hat{\varphi}:\hat{T}\rightarrow\hat{T’}$, and then $$\chi_ \varphi:=\ker(\hat{\varphi})=(\ker\varphi)^\wedge$$ the Pontryagin dual of $\ker(\varphi)$ and $u=\hat{\varphi}(\chi_ \varphi)\in\hat{T’}$ is the identity. Then we have
🔗Equivariant Sheaves
Now consider $\textrm{act},\textrm{pr}:T\times X\rightarrow X$.
A $T$-monodromic sheaf $\mathcal{F}$ has $\textrm{act}^\ast \mathcal{F}\cong\textrm{pr}^\ast \mathcal{F}\cong \mathcal{F}\boxtimes\mathcal{L}$ for some local system $\mathcal{L}$, but the isomorphism is not included in the data of monodromic sheaves, but treated as properties instead.
In contrast, in general, the definition $T$-equivariant sheaves encode $\textrm{act}^\ast \mathcal{F}\cong\textrm{pr}^\ast \mathcal{F}$ as part of data.
Now consider $f:X\rightarrow Y$ a $T$-equivariant maps. Then $f^!,f_ \ast $ preserves monodromic subcategories. Informally,
For any $\chi\in R_ {I_ F^t,\hat{T}}(\Lambda)$, we have $$\chi^!:\mathrm{IndCoh}(R_ {I_ F^t,\hat{T}})\rightarrow \mathrm{Mod}_ \Lambda.$$
Definition 25. Let $X$ be a scheme or a stack acted by $T$ and $\chi\in R_ {I_ F^t,\hat{T}}(\Lambda)$. Then we define $(T,\chi)$-equivariant category $$\mathrm{Shv}((T,\chi)\backslash X):=\mathrm{Mod}_ \Lambda\otimes_ {\mathrm{Shv}^{\textrm{mon}}(T)}\mathrm{Shv}^{\textrm{mon}}(X).$$
Remark 26. This is additional structure, and $\mathrm{Shv}((T,\chi)\backslash X)\not\subseteq\mathrm{Shv}(X)$.
Lemma 27. Take $\chi=u\in\hat{T}$ the trivial character, then $$\mathrm{Shv}((T,u)\backslash X)\cong \mathrm{Shv}([T\backslash X])$$ the usual equivariant (derived category of) sheaves.
Proof.
Consider the bar resolution and descent
🔗Deligne-Lusztig Induction: Monodromic Sheaves
Now let $G$ be a reductive group over $\kappa\cong\mathbb{F}_ q$ as before, $U\subseteq B\supseteq T$ be unipotent radical of a Borel containing a maximal torus. Recall that by general sheaf theory Thm. 4.5, we have
Also, we have $$\mathrm{Shv}^{T\textrm{-mon}}(U\backslash G/U)\hookrightarrow\mathrm{Shv}(U\backslash G/U)$$ is a full subcategory. There are 3 possible $T$-monodromic actions for $U\backslash G/U$ (left, right, diagonal); and they result in equivalent categories of monodromic sheaves. This fails for equivariant category, since $\mathrm{Shv}([B\backslash G/B])\neq \mathrm{Shv}(U\backslash G/B)$.
we have Deligne-Lusztig induction $$\mathrm{DL}:=\pi_ \ast \delta^!:\mathrm{Shv}^{\textrm{mon}}(U\backslash G/U)\rightarrow\mathrm{Shv}(G/\mathrm{Ad}_ \phi G).$$ For monodromic sheaves, $\pi_ !$ is defined and differs from $\pi_ \ast $ only by a shift, due to computation of $H_ c^\ast (\mathbb{G}_ m)$ and $H^\ast (\mathbb{G}_ m)$.
Then the general sheaf theory Thm. 4.5 can apply, and we have $$\mathrm{tr}(\mathrm{Shv}^{\textrm{mon}}(U\backslash G/U),\phi)\hookrightarrow \mathrm{Shv}(G/\mathrm{Ad}_ \phi G)\cong \mathrm{Rep}(G(\kappa)).$$ is fully faithful. The theorem below says this functor is also surjective.
Theorem 28. We have equivalence $$\mathrm{tr}(\mathrm{Shv}^{\textrm{mon}}(U\backslash G/U),\phi)\xrightarrow{\cong} \mathrm{Shv}(G/\mathrm{Ad}_ \phi G)\cong\mathrm{Rep}(G(\kappa)).$$
Essential surjectivity needs some concrete input, rather than merely abstract nonsense. We now proceed to its proof.
Now consider
Now we define $$\triangle_ {w,\chi}^{\textrm{mon}}:=(i_ w)_ \ast \textrm{pr}_ {\dot{w}}^!\mathrm{Ch}_ \chi^{\textrm{mon}}[\cdots],$$ and $$ \triangledown_ {w,\chi}^{\textrm{mon}}:=(i_ w)_ !\textrm{pr}_ {\dot{w}}^!\mathrm{Ch}_ \chi^{\textrm{mon}}[\cdots],$$ which are ind-local systems on torus direction and lie in $\mathrm{Shv}^{(w\chi,\chi)\textrm{-mon}}(U\backslash G/U)^\heartsuit$.
Recall that we have $$R_ {w,\theta}^!, R_ {w,\theta}^\ast $$ for $\theta: T^{w\sigma}\rightarrow \Lambda^\times$.
Remark 29. We might mess up with notations; both $\sigma$ and $\phi$ are used to denote the Frobenius map.
We have an exact sequence $$1\rightarrow T^{w\sigma}\rightarrow T\xrightarrow{\phi_ w:t\mapsto t^{-1}w(\sigma(t))} T\rightarrow 1,$$ and dually $$\chi_ {{\phi_ w}}\rightarrow \hat{T}\rightarrow \hat{T},$$ where $\chi_ {\phi_ w}$ is a finite scheme. Then $\theta: T^{w\sigma}\rightarrow \Lambda^\times$ corresponds to $\theta\in \chi_ {\phi_ w}$.
Proposition 30. Let $\Lambda$ be a field of characteristic 0. Then $$\mathrm{DL}(\triangledown_ {w,\chi}^{\textrm{mon}})= \begin{cases} R_ {w,\theta}^\ast ,\quad&\textrm{if }\chi=\theta\in \chi_ {\phi_ w};\ 0, &\textrm{if }\chi\notin\chi_ {\phi_ w}. \end{cases}$$ Similarly, $$\mathrm{DL}(\triangle_ {w,\chi}^{\textrm{mon}})= \begin{cases} R_ {w,\theta}^!,\quad&\textrm{if }\chi=\theta\in \chi_ {\phi_ w};\ 0, &\textrm{if }\chi\notin\chi_ {\phi_ w}. \end{cases}$$
Theorem 31 (Tilting Object). For every $w\in W$, there exists a unique $\mathrm{Tilt}_ w\in \mathrm{Shv}^{\textrm{mon}}(U\backslash G/U)^\heartsuit$, such that
- $\mathrm{Tilt}_ w$ is indecomposable,
- $\mathrm{Tilt}_ w$ admit standard filtration with associated graded $\triangle_ {v,\chi}^{\textrm{mon}}$, $v\leq w$;
- $\mathrm{Tilt}_ w$ admit costandard filtration with associated graded $\triangledown_ {v,\chi}^{\textrm{mon}}$, $v\leq w$;
Example 32. Take $G=\mathrm{SL}_ 2$ and then $G/B\cong \mathbb{P}^1=\mathbb{A}^1\sqcup\textrm{pt}$, corresponding to $W={s,1}$. We use $1$ to denote the point, and $j_ s:\mathbb{A}^1\hookrightarrow \mathbb{P}^1$ the open immersion. Take $\chi=$trivial monodromy, then we have short exact sequences of perverse sheaves $$0\rightarrow \delta_ 1\rightarrow \mathrm{Tilt}_ s\rightarrow (j_ s)_ \ast \Lambda[1] \rightarrow 0,$$ $$0\rightarrow (j_ s)_ !\Lambda[1]\rightarrow \mathrm{Tilt}_ s\rightarrow \delta_ 1\rightarrow 0.$$ Compare with IC sheaves: $$0\rightarrow \delta_ 1\rightarrow (j_ s)_ !\Lambda[1]\rightarrow\Lambda[1]\rightarrow 0,$$ $$0\rightarrow \Lambda[1]\rightarrow (j_ s)_ \ast \Lambda[1]\rightarrow \delta_ 1\rightarrow 0.$$
For $s\chi=\chi$ but $\chi\neq 1$, we have $$\mathrm{Tilt}_ s\cong (j_ s)_ \ast \Lambda[1]\cong (j_ s)_ !\Lambda[1]$$ a clean extension.
Since $U$ is unipotent, $$\mathcal{E} xt^1_ {U\textrm{-equivariant}}(\Lambda[1],\delta_ 1)\cong \mathcal{E} xt^1(\Lambda[1],\delta_ 1)\cong\mathcal{E} xt^0(\Lambda,\delta_ 1).$$
When $\Lambda=\overline{\mathbb{Q}_ \ell}$, $\mathrm{DL}(\triangle_ {w,\chi}^{\textrm{mon}})=R_ {w,\theta}^!=H_ c^\ast (Y_ w)\otimes_ {T^{w\sigma}}\theta$, where $$Y_ w:={gU\in G/U:g^{-1}\mathrm{Frob}(g)\in U\dot{w}U}$$ is acted by $T^{w\sigma}$ and $\theta:T^{w\sigma}\rightarrow \Lambda^\times.$ Then $\theta\in\mathrm{Spec}(\Lambda[T^{w\sigma}])\hookrightarrow\chi\in R_ {I_ F^t,\hat{T}}$.
Proposition 33. We have $$R_ {w,\theta}^!\in \mathcal{D}^{\geq 0},$$ $$R_ {w,\theta}^\ast \in\mathcal{D}^{\leq 0}.$$
Corollary 34. The tilting object $T_ w:=\mathrm{DL}(\mathrm{Tilt}^{\textrm{mon}})\in \mathcal{D}^{=0}$.
Corollary 35. When $\Lambda=\overline{\mathbb{Q}_ {\ell}}$, every irreducible representation of $G(\kappa)$ appears as a direct summand of $T_ w$ for some $w$.
Now we begin to prove the main theorem.
Proof of essential surjectivity of DL.
Since the target is a stable category, we only need to show that $\mathrm{Hom}(\mathrm{DL}(X), Y)=0$ for all $X$ will imply that $Y=0$. Our strategy is to show that $\mathrm{DL}\circ \mathrm{DL}^R$ contains $\mathrm{id}$ as a direct summand, and therefore in particular, $$\mathrm{Hom}(\mathrm{DL}\circ \mathrm{DL}^R(Y),Y)\neq 0,$$ if $Y\neq 0$. Now note that $\mathrm{DL}=\pi_ !\delta^!$, where
Now take $X=\ast/U$ and $Y=\ast/G$, then $\mathcal{L} f: U/U\rightarrow G/G$ is the springer resolution, and $\mathcal{L} f_ \ast \Lambda_ {\mathcal{L} X}=\mathrm{Spr}$ Springer sheaf contains $\delta_ 1$ as a direct summand. Note that $\delta_ 1\star(-)$ is the identity functor. Therefore, we see that $\mathrm{DL}\circ \mathrm{DL}^R(-)$ contains identity functor as a direct summand.
🔗Mackey Formula
$$\mathrm{DL}: \mathrm{Shv}^{\textrm{mon}}{U\backslash G/U}\rightarrow \mathrm{Shv}(G/\mathrm{Ad}_ \phi G).$$ $$\mathrm{Hom}(\mathrm{DL}(\mathcal{F}),\mathrm{DL}(\mathcal{G}))\cong \mathrm{Hom}(\mathcal{F},\mathrm{Res}\circ \mathrm{DL}(\mathcal{G})).$$
For more generality, let’s consider $\widetilde{U}\rightarrow U$ a finite isogeny. For example, consider
We have a big diagram
Recall that in classical theory
- Whittaker model $\mathrm{CG}_ \psi:=\mathrm{Ind}_ {U(\kappa)}^{G(\kappa)}\psi=C((U(\kappa),\psi)\backslash G(\kappa))$ satisfies multiplicity one property, where $\psi$ is a non-degenerate character of $U$.
- $\mathrm{Hom}(R_ w^\ast ,\mathrm{CG}_ \psi)$ concentrates on degree 0.
Proposition 36. $\mathcal{F}\in\mathrm{Shv}(B\backslash G/B)$ and $\mathcal{G}\in \mathrm{Shv}(\widetilde{U}\backslash G/\widetilde{U})$. Then $$\mathrm{Hom}(\pi_ \ast \delta^!\mathcal{F},(\pi^{\widetilde{U}})_ \ast (\delta^{\widetilde{U}})^!)$$ admits a filtration indexed by $W$, with associated gradings being $$\mathrm{Hom}_ {\mathrm{Shv}(\widetilde{U}\backslash G/B)}(\triangle_ w^l\star \mathcal{F},\mathcal{G}\star^{\widetilde{u}}\triangledown_ {\sigma(w)}^l),$$ where $w\in W$ and $\sigma$ denotes Frobenius action, and $\triangle_ w^l$, $\triangledown_ w^l$ denotes the pullback of $\triangle_ w,\triangledown_ w$ along $$\widetilde{U}\backslash G/B\rightarrow U\backslash G/B\rightarrow B\backslash G/B$$ respectively.
We have convolution product $$\star: \mathrm{Shv}(\widetilde{U}\backslash G/B)\times \mathrm{Shv}({B}\backslash G/B)\rightarrow\mathrm{Shv}(\widetilde{U}\backslash G/B)$$ using
Corollary 37 (Mackey’s Formula). There exists a spectral sequence $$ E_ 1^{p,q}=\oplus_ {l(w)=p}\mathrm{Ext}^{p+q}_ {\mathrm{Shv}(\widetilde{U}\backslash G/B)}(\triangle_ w^l\star \mathcal{F},\mathcal{G}\star^{\widetilde{U}}\triangledown_ {\sigma(w)}^l)\Rightarrow \mathrm{Hom}_ {\mathrm{Shv}(G/\mathrm{Ad}_ \phi G)}(\mathrm{DL}\mathcal{F},\mathrm{DL}^{\widetilde{U}}\mathcal{G}).$$
Now consider
Corollary 38. $\mathrm{Hom}_ {G(\kappa)}(R_ w^!,\mathrm{CG}_ \psi)=\Lambda[0]$.
Proof.
Take $\mathcal{F}=\triangle_ w,\mathcal{G}=1\psi$ in Cor. 5.37. Use $w_ 0$ to denote the longest element in $W$. First note that $\mathrm{Shv}(\widetilde{U}\backslash G/B)$ is decomposed under different additive characters of $\mathbb{F}_ p$ and only $\psi$-part is involved. Secondly, $$(\iota_ {w_ 0})_ \ast =(\iota_ {w_ 0})_ !:\mathrm{Shv}((U,\psi)\backslash Uw_ 0B/B)\xrightarrow{\cong}\mathrm{Shv}((U,\psi)\backslash G/B)\cong\mathrm{Mod}_ \Lambda.$$ is a clean extension. Then in the spectral sequence, only $w_ 0$-term matters, and we only need to compute $$\mathrm{Hom}_ {\mathrm{Shv}((U,\psi)\backslash G/B)}({}_ \psi\triangle_ {w_ 0}^l\star\triangle_ w, {}_ \psi\triangle_ {w_ 0})$$ where ${}_ \psi\triangle_ {w_ 0}:=1_ \psi\star^{\widetilde{U}}\triangle_ {w_ 0}^l\cong {}_ \psi\triangledown_ {w_ 0}\in\mathrm{Shv}((U,\psi)\backslash G/B)$.
Another two facts used in the computation are
- ${}_ \psi\triangle_ {w_ 0}^l\star \mathrm{IC}_ v=0$ unless $v=e$;
- ${}_ \psi\triangle_ {w_ 0}^l\star \triangle_ w={}_ \psi\triangle_ {w_ 0}^l$.
Now we consider monodromic version.
$$\mathrm{Shv}^{\textrm{mon}}(U\backslash G/\widetilde{U})=\oplus_ {\psi:\kappa\rightarrow\Lambda^\times}\mathrm{Shv}^{\textrm{mon}}(U\backslash G/(U,\chi)).$$
We have locally closed embedding $$\iota_ w: U\backslash UT\dot{w}U/\widetilde{U}\hookrightarrow U\backslash G/\widetilde{U}.$$
Note that $$U\backslash UT\dot{w}U\cong T\dot{w}\times (U\cap \mathrm{Ad}_ {\dot{w}^{-1}}U)\backslash U/\widetilde{U},$$ where $\dot{w}\in G$ is a lift of $w\in W=N_ G(T)/T$.
Let $J\subseteq \Delta$ be the subset of $\alpha\in \Delta$, such that $$U_ \alpha\xrightarrow{\cong}\mathbb{G}_ a$$ under the character. Let $W_ J$ be the corresponding parabolic subgroup of $W$. Then $$U\cap \mathrm{Ad}_ {\dot{w}^{-1}}U\subseteq \ker(U\rightarrow \mathbb{G}_ a)$$ if and only if $w$ is the longest element in $W_ J$, and in this case we have a map $$T\dot{w}\times (U\cap \mathrm{Ad}_ {\dot{w}^{-1}}U)\backslash U/\widetilde{U}\rightarrow T\dot{w}\times B\underline{\kappa}.$$
Define $$\mathrm{Wh}_ {!,w(,\psi)}^{(\chi\textrm{-})\textrm{mon}}:=\iota_ !\mathrm{pr}^!(\mathrm{Ch}^{(\chi\textrm{-})\textrm{mon}}\boxtimes\Lambda_ {(\psi)}[\kappa])[\cdots],$$ where $\Lambda[\kappa]=\oplus_ {\psi:\kappa\rightarrow\Lambda^\times}\Lambda$.
Proposition 39. Let $\mathcal{F}_ 1$ be an object in $\mathrm{Shv}^{\textrm{mon}}(U\backslash G/U)$, and $\mathcal{F}_ 2\in \mathrm{Shv}(\widetilde{U}\backslash G/\widetilde{U})$, then there exists a filtration on the complex of $\Lambda$-modules $\mathrm{Hom}_ {G(\kappa)}((\pi^U)_ \ast (\delta^U)^!\mathcal{F}_ 1,(\pi^{\widetilde{U}})_ \ast (\delta^{\widetilde{U}})^!\mathcal{F}_ 2)$, indexed by $W$, with associated graded being $$\mathrm{Hom}_ {\mathrm{Shv}^{\textrm{mon}}(U\backslash G/\widetilde{U})}(\mathrm{Av}_ s^\ast (\mathrm{Av}_ s)_ \ast \mathcal{F}_ 1\star^{U}\mathrm{Wh}_ {!,\sigma(w)}^{\textrm{mon}},\mathrm{Wh}^{\textrm{mon}}_ {*,w}\star^{\widetilde{U}}\mathcal{F}_ 2).$$
In general, $\psi$-parts are not preserved.
Let’s recall classical Deligne Lusztig theory when $\Lambda$ is an algebraically closed field.
Lemma 40. $(w,\theta)$ and $(w’,\theta’)$ are geometrically conjugate, if and only if there exists in $v\in W$, such that $$v\theta v^{-1}=\theta’$$ in $\hat{T}$.
Proposition 41. $\mathrm{Hom}_ {G(\kappa)}(R_ {w,\theta}^\ast ,R_ {w’,\theta’}^\ast )=0$, unless $(w,\theta)$ and $(w’,\theta’)$ are geometrically conjugate.
Remark 42. When $\mathrm{char}(\Lambda)=0$, one can compute the Euler characteristic.
Set $\psi=$trivial, then $$\mathrm{Shv}^{\textrm{mon}}(U\backslash G/(U,\psi))=\mathrm{Shv}^{\textrm{mon}}(U\backslash G/U),$$ and
- $\mathrm{Wh}_ {*,w}^{\chi\textrm{-mon}}=\triangle_ {w}^{\chi\textrm{-mon}}$,
- $\mathrm{Wh}_ {!,w}^{\chi\textrm{-mon}}=\triangledown_ {w}^{\chi\textrm{-mon}}$.
Now take $\psi=$trivial, $\mathcal{F}_ 1=\triangledown_ {w,\theta}^{\textrm{mon}}$, $\mathcal{F}_ w=\triangledown_ {w’,\theta’}^{\textrm{mon}}$, $\Lambda=\overline{\mathbb{Q}_ {\ell}}$, then $$\mathrm{Av}^\ast \circ \mathrm{Av}_ \ast \mathcal{F}_ 1\in \mathrm{Shv}^{(\theta,w^{-1}\theta)\textrm{-mon}}(U\backslash G/U),$$ $$\mathrm{Av}^\ast \circ \mathrm{Av}_ \ast \mathcal{F}_ 2\in \mathrm{Shv}^{(\theta’,w’^{-1}\theta’)\textrm{-mon}}(U\backslash G/U).$$ Therefore, computing $$\mathrm{Hom}(\mathrm{Av}^\ast \circ \mathrm{Av}_ \ast \mathcal{F}_ 1\star^U \triangle_ {\sigma(v)}^{\textrm{mon}},\triangledown_ {v}^{\textrm{mon}}\star^{U}\mathcal{F}_ 2),$$ we see the Hom space is zero unless $v\theta v^{-1}=\theta’$ and hence it recovers the classical result. In particular, $$\mathrm{Rep}(G(\kappa))=\oplus_ {s\in \hat{T}/W}\mathrm{Rep}^s(G(\kappa)).$$