Stacks of Langlands Parameters

🔗Strongly Continuous Representations of a Locally Profinite Group

We fix

• $\Gamma$ a locally profinite group (Hausdorff topological group and identity element has a basis consisting of compact open subgroups. A locally profinite group is compact if and only if it is profinite.);
• $H$ a smooth affine group scheme over $\mathbb{Z}_ {\ell }$.

We have a functor $$ \begin{split} {}^{\text{cl}}R_ {\Gamma,H}: \mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}^\heartsuit &\rightarrow \mathrm{Sets},\ A &\mapsto \mathrm{Hom}_ {\text{cts}}(\Gamma,H(A)), \end{split} $$ where by ``cts’’, we mean the following definition.

The functor ${}^{\text{cl}}R_ {\Gamma,H}$ admits a derived enhancement $$ R_ {\Gamma,H}:\mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}\rightarrow \mathrm{Ani}. $$ A group homomorphism $\rho:\Gamma\rightarrow H(A)$ is the same thing as a morphism of cosimplicial sets $$ \rho^*: \mathbb{Z}_ {\ell }[H^\bullet]\rightarrow C(\Gamma^\bullet,A). $$ Now we put continuity condition on it. For a locally profinite set $S$, we define $C_ {\text{cts}}(S,A)$ to be the set of continuous maps from $S$ to $A$, where $A$ is equipped with ind-$\ell$-adic topology, i.e. write $A=\varinjlim_ {i}M_ i$ where $M_ i$ are finite $\mathbb{Z}_ {\ell }$-modules.

Then (strongly) continuous homomorphism $\rho:\Gamma\rightarrow H(A)$ now corresponds to $\rho^*:\mathbb{Z}_ {\ell }[H^\bullet]\rightarrow C_ {\text{cts}}(\Gamma^\bullet,A)$.

Now for a profinite set $S$, we have a functor $$ C_ {\text{cts}}(S,-):\mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}^\heartsuit\rightarrow\mathrm{CAlg}^\heartsuit, $$ which preserves sifted colimits (if $S$ is profinite). Then we can extend it to (non-abelian derived categories) $$ C_ {\text{cts}}(S,-):\mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}\rightarrow \mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}. $$ Then $C_ {\text{cts}}(\Gamma^\bullet,-):\mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}\rightarrow\mathrm{CAlg}_ {\mathbb{Z}_ {\ell }}^\Delta$, where the target is the category of cosimplicial animated rings over $\mathbb{Z}_ {\ell }$.

Facts:

• If $V$ is a representation of $H$, $V\otimes \mathbb{Z}_ {\ell }[H^\bullet]$ is a module over $\mathbb{Z}_ {\ell }[H^\bullet]$, and base change along $\rho$, we get $C_ {\text{cts}}(\Gamma^\bullet,V\otimes A)$. Where $A$ is classical, this is the cochain complex computing continuous group homology of $(\Gamma, V\otimes A)$.
• $R_ {\Gamma,H}$ admits tangent complex and $\mathbb{T}_ {\rho}R_ {\Gamma,H}/H=C_ {\text{cts}}(\Gamma, \mathrm{Ad}_ \rho)[-1]$, where $\mathrm{Ad}_ \rho$ is $\Gamma\xrightarrow{\rho}H$ acts on $\mathrm{Lie}(H)$ via Ad action. This is a theorem requiring some effort to prove.
• $R_ {\Gamma,H}$ is represented by an ind-scheme over $\mathbb{Z}_ {\ell }$ in general.

🔗The Stack of Langlands Parameters

Fix

• $F$ a non-archimedean local field,
• $G$ reductive group over $F$, with $(\hat{G}/\mathbb{Z}_ {\ell }, \hat{B},\hat{T},\hat{e})$, where

such that $\hat{e}|_ {\hat{U_ \alpha}}:\hat{U_ \alpha}\xrightarrow{\cong}{\mathbb{G}_ a}$.

Recall that $\mathrm{Aut}(\hat{G},\hat{B},\hat{T},\hat{e})\cong\mathrm{Aut}(\text{root datum of }G)$, which bears an action of $\Gamma_ F=\mathrm{Gal}(\overline{F}/F)\twoheadrightarrow \Gamma_ {\widetilde{F}/F}$, where $\widetilde{F}$ is a finite extension of $F$. We denote this Galois action by $\xi$.

$\rho_ {\text{ad}}:=$ half sum of all positive coroots of $\hat{G}$, i.e. $$ \rho_ {\text{ad}}:\mathbb{G}_ m\rightarrow (\hat{G})_ {\text{ad}}. $$ Note that $$ \begin{split} \mathrm{Ad}_ {\rho_ {\text{ad}}}:\hat{B}&\xrightarrow{\cong}{\hat{B}}\ \hat{T}&\xrightarrow{\cong}{\hat{T}} \end{split} $$ fixing Borel and torus, while change the pinning by $\mathrm{Ad}_ {\rho_ {\text{ad}}(\lambda)}\hat{e}=\lambda\hat{e}$.

$\mathrm{Ad}_ {\rho_ {\text{ad}}}$ and $\Gamma_ {\widetilde{F}/F}$ action commutes with each other and we define $$ {}^cG:=\hat{G}\rtimes_ {\mathrm{Ad}_ {\rho_ {\text{ad}}}\times\xi}(\mathbb{G}_ m\times \Gamma_ {\widetilde{F}/F}). $$ Recall that $$ {}^LG:=\hat{G}\rtimes_ {\xi} \Gamma_ {\widetilde{F}/F} $$

Denote $(\text{cycl}^{-1},\text{pr})$ by $\widetilde{\text{pr}}$.

🔗Main Theorem

Recall that $$ R_ {\Gamma_ q,H}={(g,h)\in H^2: ghg^{-1}=h^q}, $$ which is claimed to be an affine scheme, dimension $\dim H$, and hence l.c.i and flat over $\Lambda:=\mathbb{Z}[\frac{1}{q}],$ and reduced.

Now we prove this claim. We have

where

and $H^{[q]}$ is defined to be the pullback.

Fix $\mathcal{O}\subseteq \text{char}^{-1}(\xi)$ an $H$-conjugacy class. Let $h\in\mathcal{O}$. Then $\text{res}^{-1}(h)$ is a torsor under $Z_ H(h)$. In particular, $R_ {\Gamma_ q,H}$ is generically reduced.

$(\text{char}\circ \text{res})^{-1}(\xi)$ has dimension $\dim H$ or ($-\infty$). By finiteness of choices of $\xi$ and $\mathcal{O}$, we see $$ \dim R_ {q,H}=\dim H. $$

Then by miraculous flatness theorem (fiberwisely correct dimension over a Cohen-Maucauly base), we know that $R_ {\Gamma_ q, H}$ is flat. We know it is generically reduced. Since it is l.c.i, it is actually reduced.

🔗Coherent Sheaves

Let $\mathcal{X}_ {{}^cG,F}$ be the stack of $L$-parameters over $\mathbb{Z}_ {\ell }$ as before. Then $$ \begin{split} \mathrm{Coh}(\mathcal{X}_ {{}^cG,F}) &= \mathrm{Coh}^{\hat{G}}(\mathcal{X}_ {{}^cG,F}^\square)\ &=\varinjlim_ {L}\mathrm{Coh}^{\hat{G}}(\mathcal{X}_ {{}^cG,L/F}^\square). \end{split} $$

🔗Stacks of Tame and Unipotent Parameters

Suppose that $G$ splits over a tamely ramified extension, i.e. $\widetilde{F}/F$ is tame. So $F^t\widetilde{F}=F^t$. Take $L=F^t$. So $W(L/F)=W_ F^t$ the tame quotient of $W_ F$, i.e. $$ 1\rightarrow I_ F^t\rightarrow W_ F^t\rightarrow \mathbb{Z}\rightarrow 1, $$ and $I_ F^t\cong\prod_ {(l,q)=1}\mathbb{Z}_ {\ell }(1)$, $\Gamma_ q\hookrightarrow W_ F^t$ open dense subgroup.

We have a restriction map $$ \mathcal{X}_ {{}^cG,F}^{\text{tame}}\xrightarrow{\text{res}}\hat{G}\overline{\tau}/\hat{G}\rightarrow \hat{G}\overline{\tau}//\hat{G} $$ induced by $(g,h)\mapsto h$, and note that $d(h)=(1,\overline{\tau})$.

Note that $\overline{\tau}=1$ if and only if $G$ splits over an unramified extension since $\overline{\tau}$ is a (topological) generator of the (tame) inertia. Let $\hat{U}\subseteq \hat{G}$ the unipotent variety (consisting of all unipotent elements).