Spectral Deligne-Lusztig Induction via Categorical Trace

🔗General Sheaf Theory

Category of Correspondences

Let $\mathcal{C}$ be a category closed under finite limits. Let $V$ and $H$ be two classes of morphisms in $\mathcal{C}$, both satisfying the following conditions:

1. contains isomorphisms;
2. stable under equivalence;
3. stable under base change;
4. stable under compositions.

Then we can define $\mathrm{Corr}(\mathcal{C})_ {V,H}$ the category of correspondences:

• Objects: same as $\mathrm{Ob}(\mathcal{C})$;
• Morphisms: $\mathrm{Corr}(\mathcal{C})_ {V,H}(Z,X)=\{ X \xleftarrow{f\in V} Y \xrightarrow{g\in H} Z \}$.

Then $\mathrm{Corr}(\mathcal{C})_ {V,H}$ is a symmetric monoidal category with $X\otimes Y:=X\times Y$.

Convolution Products

Assume that $X\xrightarrow{f}Y\in V$ and $X\xrightarrow{\Delta}X\times X\in H$. Then $X\times_ Y X$ is an algebra object in $\mathrm{Corr}(\mathcal{C})_ {V,H}$, given by

We call the multiplication map in $\mathrm{Corr}_ {V,H}$ $(X\times_ YX)\otimes (X\times_ YX)\rightarrow X\times_ YX$ the convolution product.

Sheaf Theory

This is an efficient way to package the functor equalities. In particular, given a correspondence

we get a functor $$f_ +g^\star:\mathcal{D}(Z)\rightarrow \mathcal{D}(X).$$ Moreover, we have a map (exterior product) $$\boxtimes:\mathcal{D}(X)\otimes_ \Lambda\mathcal{D}(Y)\rightarrow\mathcal{D}(X\times Y).$$ We also have

• Base change theorem: where $f\in V$ and $g\in H$, then $g^\star\circ f_ +\cong (f')_ +\circ (g')^\star$.
• Projection formula: $(f:X\rightarrow Y)\in V\cap H$, and $\Delta_ X,\pi_ X\in H$, then $f_ +(\mathcal{F}\otimes f^\star\mathcal{G})\cong f_ +\mathcal{F}\otimes \mathcal{G}$, where $\mathcal{F}\in\mathcal{D}(X)$ and $\mathcal{G}\in\mathcal{D}(Y)$.

The monoidal structure is given by

In this case, $\mathcal{D}(X\times_ Y X)$ is only monoidal. If $(\pi_ X:X\rightarrow\mathrm{pt})\in H$, $(\Delta_ {X/Y}:X\rightarrow X\times_ YX)\in V$, then we have a unit given by $(\Delta_ {X/Y})_ +\pi_ X^\star\Lambda$.

Fact: if $(\pi: X\rightarrow\textrm{pt})\in H$ and $(\Delta_ X:X\rightarrow X\times X)\in H$, then $X$ is a commutative algebra object in $\mathrm{Corr}(\mathcal{C})_ {V,H}$. The correspondence $\pi$ gives the unit and $\Delta$ gives the multiplication law. In this case $\mathcal{D}(X)$ has a natural symmetric monoidal structure, with a unit given by $$\mathrm{Mod}_ \Lambda\rightarrow \mathcal{D}(\mathrm{pt})\xrightarrow{(\pi_ X)^\star}\mathcal{D}(X)$$ and tensor product given by $$\mathcal{D}(X)\otimes\mathcal{D}(X)\rightarrow\mathcal{D}(X\times X)\xrightarrow{(\Delta_ X)^\star}\mathcal{D}(X).$$

Define $\mathcal{C}_ H$ (resp. $\mathcal{C}_ V$) to be the subcategory of $\mathcal{C}$ with the same objects but $\mathcal{C}_ H(X,Y):={f\in\mathcal{C}(X,Y):f\in H}$ (similarly for $\mathcal{C}_ V$). Then $(\mathcal{C}_ H)^{\textrm{op}}\hookrightarrow\mathrm{Corr}(\mathcal{C})_ {V,H}$ is a symmetric monoidal functor.

Abstract Deligne-Lusztig Induction

Let $f:X\rightarrow Y$ be a morphism in $\mathcal{C}$ and $\phi_ x:X\rightarrow X$, $\phi_ y:Y\rightarrow Y$. Suppose that $\phi_ x\circ f\cong f\circ \phi_ y$. Recall our correspondence:

By our construction, if $\phi_ x\in H$, then $\delta\in H$ and $\delta^\star$ is defined. If $f\in V$, then $\pi\in V$, and then $\pi_ +$ is defined.

For general $\phi_ x,\phi_ y$, a similar form of the lemma above holds.

Recall that for $\mathcal{A}$ an associative algebra over $\Lambda$, and $f: \mathcal{A}\rightarrow V$ a $\Lambda$-linear map such that $f(ab)=f(ba)$, then $f$ uniquely factors through $A/[A,A]=A\otimes_ {A\otimes A^{\textrm{rev}}}A$.

This motivates us to study Deligne-Lusztig induction via categorical trace.

🔗Categorical Trace

Recall that for $\mathcal{A}$ an algebra in $\mathrm{LinCat}_ \Lambda$ and $\mathcal{F}$ an $\mathcal{A}$-bimodule, we define $\mathrm{tr}(\mathcal{A},\mathcal{F})=\mathcal{A}\otimes_ {A\otimes A^{\textrm{rev}}}\mathcal{F}$.

This can be viewed as a trace construction in a 3-category $\mathrm{Morita}(\mathrm{LinCat}_ \Lambda)$. Informally, $\mathrm{Morita}(\mathrm{LinCat}_ \Lambda)$ is defined by

• Objects: $\mathrm{Alg}(\mathrm{LinCat}_ \Lambda)$.
• Morphisms: $\mathrm{Maps}(A,B):={}_ B\mathrm{Mod}_ A$.

Then we have bar resolution

🔗Abstract DL Induction via Categorical Trace

A key observation is that Spectral Deligne-Lusztig induction can be understood via categorical trace construction.

Let $A$ be an algebra object in $\mathrm{LinCat}_ \Lambda$ as before. Assume that $\phi:\mathcal{A}\rightarrow \mathcal{A}$ is an algebra automorphism. Then we define $\mathcal{A}$-bimodule ${}^\phi\mathcal{A}$ by

• the underlying category of ${}^\phi\mathcal{A}$ is same as $\mathcal{A}$;
• ${}^\phi\mathcal{A}$ is a right-$\mathcal{A}$ module by the usual multiplication of $\mathcal{A}$ on $\mathcal{A}$ from the right;
• ${}^\phi\mathcal{A}$ is a left-$\mathcal{A}$ module by the $\phi$-twsited multiplication of $\mathcal{A}$ on $\mathcal{A}$ from the left.

Then we have its categorical trace $\mathrm{tr}(\mathcal{A},{}^\phi\mathcal{A})$.

As in Cor. 4.1, $\mathcal{D}(X\times_ YX)$ is a monoidal category. As in the setting of Deligne-Lusztig induction (Def 4.2), $X\times_ YX\in\mathrm{Corr}(\mathcal{C})_ {V,H}$ has an algebra automorphism induced by $\phi_ x$, and hence an algebra automorphism $\phi:\mathcal{D}(X\times_ YX)\rightarrow\mathcal{D}(X\times_ YX)$.

Therefore, we can form the trace construction $$\mathrm{tr}:\mathcal{D}(X\times_ YX)\rightarrow \mathrm{tr}(\mathcal{D}(X\times_ YX),{}^\phi\mathcal{D}(X\times_ YX)).$$

In favorable cases, $\gamma$ will be fully faithful. We now give a condition for this to hold.

Assumption A: We assume that there exists a subclass of morphisms $VR\subseteq V$,

1. $VR$ is stable under base change, composition, and equivalence and contains isomorphisms;
2. for any $f\in VR$, the right adjoint $f^+:=(f_ +)^R$ is continuous;
3. (Beck-Chevalley isomorphism) for any Cartesian square with $f\in VR, g\in H$, the natural morphism $(g')^\star f^+\xrightarrow{\cong}(f')^+\circ g^\star$ is an equivalence, and for any Cartesian square with $f\in VR, g\in V$, the natural morphism $g'_ +(f')^+\xrightarrow{\cong} f^+g_ +$ is an equivalence.
4. (projection formula) For any $(f:Y\rightarrow X)\in VR$, the natural map $f_ +(\mathcal{F}\otimes f^+\mathcal{G})\xrightarrow{\cong}(f_ +\mathcal{F})\otimes\mathcal{G}$ is an isomorphism, for $\mathcal{F}\in \mathcal{D}(Y),\mathcal{G}\in\mathcal{D}(X)$.

Assumption B: We assume that there exists a subclass of morphisms $HR\subseteq H$,

1. $HR$ is stable under base change, composition, equivalence and contains isomorphisms;
2. for any $f\in HR$, the right adjoint $f_ \star:=(f^\star)^R$ is continuous;
3. (Beck-Chevalley isomorphism) for any Cartesian square with $f\in V, g\in HR$, the natural morphism $f_ + g’_ \star\xrightarrow{\cong} g_ \star f’_ +$ is an equivalence, and for any Cartesian square with $f\in H, g\in HR$, the natural morphism $f^\star g_ \star\xrightarrow{\cong} g’_ \star(f’)^\star$ is an equivalence.
4. (projection formula) For any $(f:Y\rightarrow X)\in VR$, the natural map $f_ \star(\mathcal{F}\otimes f^\star\mathcal{G})\xrightarrow{\cong}(f_ \star\mathcal{F})\otimes\mathcal{G}$ is an isomorphism, for $\mathcal{F}\in \mathcal{D}(Y),\mathcal{G}\in\mathcal{D}(X)$.

🔗Ind-Coh Sheaves

Our desired sheaf theory is as below.

Now let’s delve into a little more details. Consider $\mathrm{Sch}_ \Lambda^{\textrm{qc},\textrm{afp}}$ the category of quasi-compact, almost of finite presentation schemes over an ordinary commutative ring $\Lambda$.

We define $\mathrm{Coh}(X)$ to be the subcategory of $\mathrm{QCoh}(X)$, consisting of those complexes $\mathcal{F}$, satisfying $H^i(\mathcal{F})=0$ for all but finitely many $i$, and $H^i(F)\in \mathrm{Coh}(X)^\heartsuit$. (Classically the corresponding triangulated 1-category is denoted by $D_ \mathrm{Coh}^b(X)$, and for nice schemes the derived category of coherent sheaves coincides with the subcategory of $D^b_ {\mathrm{QCoh}}(X)$ with coherent cohomology sheaves.)

For $f: X\rightarrow Y$ a morphism in $\mathrm{Sch}_ \Lambda^{\textrm{qc},\textrm{afp}}$, we always have a pushforward functor $$f_ *: \mathrm{QCoh}(X)^+\rightarrow \mathrm{QCoh}(Y)^+,$$ preserving bounded below complexes, which restricts to $$f_ *: \mathrm{Coh}(X)\rightarrow \mathrm{QCoh}(Y)^+\hookrightarrow\mathrm{IndCoh}(Y).$$ Then we extend this functor to get

To sum up, we can define $f_ *^{\mathrm{Ind}}$ for unconditional $f\in \mathrm{Mor}(\mathrm{Sch}^{\textrm{qc},afp})_ \Lambda$ and $f_ !$ when $f$ is proper or $f$ has finite Tor dimension.

So, we get a sheaf theory $$\mathrm{IndCoh}:\mathrm{Corr}(\mathrm{Sch}_ {\Lambda}^{\textrm{qc,aft}})_ {V=\textrm{all},H=\textrm{isomorphisms}}\rightarrow \mathrm{LinCat}_ \Lambda.$$ Moreover,

• $VR$=proper morphisms;
• $VL$=open embeddings.

Now using Nagata compactification, for any quasi-compact quasi-separated morphism $f:X\rightarrow Y$, we have

So we get a sheaf theory $$\mathrm{IndCoh}:\mathrm{Corr}(\mathrm{Sch}_ {\Lambda}^{\textrm{qc,aft}})_ {V=\textrm{all},H=\textrm{sep}}\rightarrow \mathrm{LinCat}_ \Lambda.$$ Moreover,

• $VR$=proper morphisms;
• $HR\supseteq$ morphisms of finite Tor dimensions.

Now we define the sheaf theory for general stacks.

🔗Applications to Spectral DL Induction

Now a lift of Frobenius $\phi$ acts on $X=\mathcal{X}_ {{}^cB,\breve{F}}^{\textrm{tame}}\hookrightarrow \hat{B}\overline{\tau}/\hat{B}$ and $Y=\mathcal{X}_ {{}^cG,\breve{F}}^{\textrm{tame}}\hookrightarrow \hat{G}\overline{\tau}/\hat{G}$.

We have the magical correspondence

We get the spectral Deligne-Lusztig induction $$\mathrm{Ind}^{\mathrm{DL}}:=\pi_ *^{\mathrm{Ind}}\delta^!: \mathrm{IndCoh}(S_ {\hat{G}\overline{\tau}})\rightarrow\mathrm{IndCoh}(\mathcal{X}_ {{}^cG,F}^{\textrm{tame}}).$$ Now we can state our main theorem.

When $\overline{\tau}=1$, $X^u=\hat{U}/\hat{B}$. We get $$\mathrm{tr}(\mathcal{A}^{\textrm{unip}},{}^\phi\mathcal{A}^{\textrm{unip}})\cong \mathrm{IndCoh}(\mathcal{X}_ {{}^cG,F}^{\textrm{unip}}).$$

🔗Coherent Springer Sheaf

For simplicity, let’s assume that $\overline{\tau}=1$. Then we have

where $S_ {\hat{G}}’$ and $S_ {\hat{G}}$ are classical and $S_ {\hat{G}}^{\textrm{unip}}$ is derived. Recall that $S_ {\hat{G},w}$ is an irreducible component of dimension 0.

Warning: In general $S_ {\hat{G},w}‘$ is not irreducible, and we have $$[S_ {\hat{G},w}’]=\sum_ {v^{-1}\leq w^{-1}}a_ {wv}[S_ {\hat{G},v}’]$$ as cycles. We mentioned the following two facts before (over $\Lambda=\overline{\mathbb{Q}_ {\ell}}$).