Categorical Local Langlands

🔗Deligne-Lusztig Induction

In the magical correspondence (3.8), take $X=\mathbb{B} \mathcal{I}_ w^u\rightarrow Y=\mathbb{B} LG$. Then we get the correspondence

Then by the general sheaf theory result Thm. 4.5, we obtain the following theorem.

Theorem 1. The Frobenius categorical trace $$\mathrm{tr}(\mathrm{Shv}^{!,\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),\phi)\hookrightarrow \mathrm{Shv}^!(\mathrm{Isoc})$$ fully faithful embeds into $\mathrm{Shv}^!(\mathrm{Isoc})$.

🔗Essential Image

Now we want to characterize the essential image of $\mathrm{Nt}_ \ast ^u(\delta^u)^!$.

First, we need to study $\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u)$. Note that $S_ k:=\mathcal{I}_ w/\mathcal{I}_ w^u$ is a torus. Consider

Now we define $$\triangle_ {\dot{w}}^{\textrm{mon}}:=(\iota_ w)_ \ast\textrm{pr}_ {\dot{w}}^![\cdots]: \mathrm{Shv}^{\textrm{mon}}(S_ k)\rightarrow \mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),$$ $$\triangledown_ {\dot{w}}^{\textrm{mon}}:=(\iota_ w)_ !\textrm{pr}_ {\dot{w}}^![\cdots]: \mathrm{Shv}^{\textrm{mon}}(S_ k)\rightarrow \mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u).$$ Let $\widetilde{\mathrm{Ch}}$ be the unit of $\mathrm{Shv}^{\textrm{mon}}(S_ k)$, i.e. corresponding to $\omega$ under the isomorphism $$\mathrm{Shv}^{\textrm{mon}}(S_ k)\cong\mathrm{IndCoh}(R_ {I_ F^t,\hat{S}}).$$ Morally, this is the $\infty$-Jordan block $\begin{pmatrix} 1 & 1 & 0 & 0 & \dots\\ 0 & 1 & 1 & 0 & \dots\\\ \ddots &\ddots &\ddots &\ddots \end{pmatrix}$.

Then we define $$\begin{split} \widetilde{\triangle}_ w^{\textrm{mon}} &= \triangle_ w^{\textrm{mon}}(\widetilde{\mathrm{Ch}}),\ \widetilde{\triangledown}_ w^{\textrm{mon}} &= \triangledown_ w^{\textrm{mon}}(\widetilde{\mathrm{Ch}}), \end{split}$$

Then we define $$\widetilde{R}_ {\dot{w}}:=(\mathrm{Nt}^u)_ \ast (\delta^u)^!\widetilde{\triangledown}_ w^{\textrm{mon}}=\widetilde{\mathrm{Nt}}_ \ast \omega_ {{\mathcal{I}_ w^u\dot{w}\mathcal{I}_ w^u}/{\mathrm{Ad}_ \sigma\mathcal{I}_ w^u}}.$$ Moreover, we observe that $\textrm{pr}_ {\dot{w}}:\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u\rightarrow S_ k$ is a ``smooth’’ gerbe under some unipotent group.

Definition 2. Define tame subcategory $$\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G)={\mathcal{F}\in\mathrm{Shv}^!(\mathrm{Isoc}),\iota_ b^!\mathcal{F}\in\mathrm{Shv}^!(\mathrm{Isoc}_ b)\cong\mathrm{Rep}^{\textrm{sm}}(J_ b(F))\textrm{ belongs to the depth zero block}}.$$

Recall: Let $G$ be a $p$-adic group, then the category of depth zero representations $$\mathrm{Rep}^{\textrm{tame}}(G)=\mathrm{Rep}^0(G)\subseteq \mathrm{Rep}(G)$$ is the subcategory spanned by $\textrm{c-ind}_ P^G\sigma$, where $P$ is a parahoric and $\sigma$ is a representation of the levi quotient $L_ P$.

Theorem 3.

  1. The semi-orthogonal decomposition of $\mathrm{Shv}(\mathrm{Isoc}_ G)$ by ${\mathrm{Rep}(J_ b(F))}_ {b\in B(G)}$ restricts to a semi-orthogonal decomposition of $\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G)$ by ${ \mathrm{Rep}^{\textrm{tame}}(J_ b(F))}_ {b\in B(G)}$.
  2. The tame subcategory $\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G)$ is preserved by Bernstein-Zelevinsky duality $\mathbb{D}^{\mathrm{BZ}}$.
  3. Moreover, $\mathrm{tr}(\mathrm{Shv}^{!,\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),\phi)\cong \mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G)$.

Corollary 4. Let $b<b’$ and consider

Then $\iota_ b^\ast (j_ {b'})_ \ast $ preserves depth zero objects.
Sketch of Proof of Thm 6.3.

  1. Let $\dot{w}$ be of minimal length in its $\sigma$-conjugacy class. Now consider $\widetilde{R_ {\dot{w}}}:=\mathrm{Nt}^u_ \ast (\delta^u)^!\widetilde{\triangledown}_ w^{\textrm{mon}}\in\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G)$.

    Therefore, $$\widetilde{R_ {\dot{w}}}:=\mathrm{Nt}^u_ \ast (\delta^u)^!\widetilde{\triangledown}_ {\dot{w}}^{\textrm{mon}}=(\iota_ b)_ \ast (\textrm{c-ind}_ {P_ b}^{J_ b(F)}C^\bullet (Y^{(u)})).$$

  2. Claim: $\mathrm{tr}(\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),\phi)$ is generated by $${\widetilde{R_ {\dot{w}}}:\dot{w} \textrm{ minimal length in its }\sigma\textrm{-conjugacy class}}.$$

Proof of the Claim.

For $\dot{y}$, by some combinatorial arguments, we have that $l(s\dot{y}\sigma(s))=l(\dot{y})+2$ for some simple reflection $s$. Then $$\begin{split} \mathrm{Sht}^{\textrm{loc}}_ {\dot{w}} &\cong\mathrm{Sht}^{\textrm{loc}}_ {s,\dot{y},\sigma(s)}\xrightarrow[\textrm{partial Frob}]{\cong}\mathrm{Sht}^{\textrm{loc}}_ {\dot{y},\sigma(s),\sigma(s)}\ &\xrightarrow{\textrm{convolution}}\mathrm{Sht}^{\textrm{loc}}_ {\dot{y},\leq \sigma(s)}\xrightarrow{\textrm{convolution}}\mathrm{Sht}^{\textrm{loc}}_ {\leq \dot{y}\sigma(s)}\rightarrow \mathrm{Isoc}_ G. \end{split}$$ Now by $\mathrm{SL}_ 2$-computation, we have $$\triangledown_ e\rightarrow \triangledown_ s*\triangledown_ s\rightarrow \triangledown_ s\oplus\triangledown_ s[1]\xrightarrow{+1}$$ and hence $$\widetilde{R}_ {\dot{y}}^{\textrm{mon}}\rightarrow \widetilde{R}_ {\dot{w}}^{\textrm{mon}}\rightarrow \widetilde{R}_ {\dot{y\sigma(s)}}^{\textrm{mon}}\oplus \widetilde{R}_ {\dot{y\sigma(s)}}^{\textrm{mon}}[1].$$

So we get one direction of inclusion.

  1. Only need to prove the reverse inclusion. For any $\mathcal{F}\in\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ {G,\leq b})$, we have $$(\iota_ {<b})_ \ast \iota_ {<b}^!\mathcal{F}\rightarrow \mathcal{F}\rightarrow (\iota_ {b})_ \ast \iota_ b^!\mathcal{F},$$ and $(\iota_ {<b})_ \ast \iota_ {<b}^!\mathcal{F}\in\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ {G})$. Therefore, by induction on $b$, it suffices to prove for $(\iota_ {b})_ \ast \iota_ b^!\mathcal{F}$. Therefore, we only need to show that $$(\iota_ {b})_ \ast (\textrm{c-ind}_ {P_ b}^{J_ b(F)}\sigma)\in \mathrm{tr}(\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),\phi).$$ Since $\mathrm{Rep}(L_ {P_ b})$ is generated by $C^\bullet(Y_ {L_ {P_ b}}^{(u)})$ cohomology of finite DL variety. Therefore, it is enough to prove that $$(\iota_ {b})_ \ast (\textrm{c-ind}_ {P_ b}^{J_ b(F)}C^\bullet (Y_ {L_ {P_ b}}^{(u)}))\in \mathrm{tr}(\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u),\phi).$$ Now we have the bijection $${b\in B(G),P_ b\subseteq J_ b\textrm{ parahoric, }u\textrm{ elliptic element in }W_ {P_ b}}\cong B(\widetilde{W}).$$ Combining (1)(2), we complete the proof.

🔗Tamely Ramified Categorical Local Langlands Equivalence

Theorem 5 (Bezrukavnikov Equivalence). Let $G/F$ be an unramified reductive group, i.e. $G$ is quasi-split and split over an unramified extension (in particular, $G_ {\breve{F}}$ is split), $\Lambda=\overline{\mathbb{Q}}_ \ell$. Then there is a canonical monoidal equivalence $$\mathbb{B}^{\textrm{mon}}:\mathrm{IndCoh}(S_ {{}^cG,\breve{F}})\xrightarrow{\cong}\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u)$$ compatible with Frobenius $\phi$ action.

Recall that $\mathrm{Gal}(\breve{F}^t/\breve{F})\cong\mathbb{Z}^{(p)}(1)$ and $$S_ {{}^cG,\breve{F}}=\mathcal{X}_ {{}^cB,\breve{F}}^{\textrm{tame}}\times_ {\mathcal{X}_ {{}^cG,\breve{F}}^{\textrm{tame}}} \mathcal{X}_ {{}^cB,\breve{F}}^{\textrm{tame}}\hookrightarrow \hat{B}/\hat{B}\times_ {\hat{G}/\hat{G}} \hat{B}/\hat{B}.$$ Then take $\mathrm{tr}(-,\phi)$ of the Bezrukavnikov Equivalence, we obtain the tamely ramified categorical local Langlands equivalence $$\mathbb{L}_ G:\mathrm{IndCoh}(\mathcal{X}_ {{}^cG,F}^{\textrm{tame}})\xrightarrow{\cong}\mathrm{Shv}^{\textrm{tame}}(\mathrm{Isoc}_ G).$$

Example 6. We can check by hand that the units match.

  1. In general, the unit of $\mathrm{Coh}(X\times_ Y X)$ is given by $\Delta_ \ast \omega_ X$ and hence, the unit of $\mathrm{IndCoh}(S_ {{}^cG,\breve{F}})$ is $\Delta_ \ast \omega_ {{}^cB,\breve{F}}$.
  2. The unit of $\mathrm{Shv}^{\textrm{mon}}(\mathcal{I}_ w^u\backslash LG/\mathcal{I}_ w^u)$ is $\widetilde{\triangledown}_ e^{\textrm{mon}}=\widetilde{\triangle}_ e^{\textrm{mon}}$.
  3. In general, We have a pullback diagram
    So $p_ \ast \delta^!\Delta_ \ast \omega_ X=(\mathcal{L} f)_ \ast \omega_ {\mathcal{L}_ \phi X}$, where $p: X\times_ Y\mathcal{L}_ \phi Y\rightarrow \mathcal{L}_ \phi X$ and $\delta: X\times_ Y\mathcal{L}_ \phi Y\rightarrow X\times_ Y X$. Therefore, let $\pi: \mathcal{X}_ {{}^cB,F}^{\textrm{tame}}\rightarrow\mathcal{X}_ {{}^cG,F}^{\textrm{tame}}.$ Then $[\Delta_ \ast \omega_ {{}^cB,\breve{F}}]_ \mathrm{tr}=\pi_ \ast \omega_ {\mathcal{X}_ {{}^cB,F}^{\textrm{tame}}}$. Fact: $\omega_ {\mathcal{X}_ {{}^cB,F}^{\textrm{tame}}}\cong\mathcal{O}_ {\mathcal{X}_ {{}^cB,F}^{\textrm{tame}}}$ and $\pi_ \ast \omega_ {\mathcal{X}_ {{}^cB,F}^{\textrm{tame}}}=\mathrm{CohSpr}$.
  4. We know that $[\widetilde{\triangledown}_ e^{\textrm{mon}}]_ \mathrm{tr}=\textrm{c-ind}_ {\mathcal{I}^u}^{G(F)}\mathbb{Q}_ \ell$.

Theorem 7. Let $b$ be basic $P_ b\subseteq J_ b(F)$ be a parahoric, and $\sigma$ is a representation of $L_ P$ the levi quotient of $P_ b$. Let $\mathcal{A}_ \pi$ be the image of $(\iota_ {b})_ \ast (\textrm{c-ind}_ {P_ b}^{J_ b(F)}\sigma)$ under $\mathbb{L}_ G^{-1}$. Then $\mathcal{A}_ \pi\in\mathrm{Coh}(\mathcal{X}_ {{}^cG,F}^{\textrm{tame}})^\heartsuit$.

Idea of Proof.
  1. Realize $\sigma$ as a direct summand of finite DL induction of some tilting sheaf.
  2. A nontrivial theorem: for any $V\in \mathrm{Rep}(\hat{G})$, we have that $$\mathrm{Hom}(i_ {1*}(\textrm{c-ind}_ {\mathcal{I}^u}^{G(F)}\psi),[T_ w^{\textrm{mon}}\star \mathcal{Z}^{\mathrm{V}}]_ \mathrm{tr})\in\mathrm{Mod}^\heartsuit,$$ i,e, $$\mathrm{Hom}_ {\mathrm{Coh}}(\mathcal{O},[T_ w^{\textrm{mon}}]_ \mathrm{tr}\otimes V)\in\mathrm{Coh}^\heartsuit.$$ where $T_ w^{\textrm{mon}}$ is the monodromic tilting sheaf and $\mathcal{Z}^{\textrm{mon}}$ is the monodromic central sheaf.

Corollary 8. Same notation and assumption as in Thm. 6.6, $\mathcal{A}_ \pi$ is maximal Cohen-Macauly.

This morally is related to patching or automorphic lifting.

Proof.

By Thm. 6.6, we know that $\mathcal{A}_ \pi\in\mathrm{Coh}^\heartsuit$ and $\mathbb{D}^{\textrm{Serre}}\mathcal{A}_ \pi\in\mathrm{Coh}^\heartsuit$. Now suppose that $\pi$ is supercuspidal, i.e. $\pi=\textrm{c-ind}_ P^G\sigma$ for some cuspidal representation of a maximal parabolic $P$, and assume that $G$ is simply connected. Then $\mathrm{End}(\mathcal{A}_ \pi)\cong\mathrm{End}(\pi)\cong\overline{\mathbb{Q}_ \ell}$.