Kenta Suzuki: Fargues Categorical Conjecture for Elliptic Parameters for SL(n)

When do classical constrcution and construction of Fargues-Scholze match?

It is known in many cases, e.g.

  • $\mathrm{GL}_ {n}$ (Fargues-Scholze),
  • $\operatorname{GSp}_ {4}$ (Hamman),
  • $U_ {n}$ (Hamman, Peng)
  • etc.

Theorem 1. Fargues-Scholze construction and Gelbert-Knopp’s construction match for $G=\operatorname{SL}_ {n}$ and elliptic parameters.

Classical Picture

There is a surjection $\operatorname{Irr}(\operatorname{GL}_ {n}(F))\rightarrow \{n\text{-dim representations of }W_ {F}\}/\sim$ which is a bijection over irreducible representations of Weil group, by Hennirat, Harris-Taylor, and Scholze.

When $G=\operatorname{SL}_ {n}$ and $\pi\in \operatorname{IrrRep}(\operatorname{SL}_ {n}(F)),$ one can find $\Pi\in \operatorname{Rep}(\operatorname{GL}_ {n}(F))$, such that $\pi\xhookrightarrow{\oplus}\Pi|_ {\operatorname{SL}_ {n}(F)}.$

Expected compatibility of LLX with isognenies imply that there exists $\varphi_ {\Pi}: W_ {F}\rightarrow \operatorname{GL}_ {n}$, and define $\overline{\varphi_ {\Pi}}=W_ {F}\xrightarrow{\varphi_ {\Pi}}\operatorname{GL}_ {n}\twoheadrightarrow\operatorname{PGL}_ {n}$, and then $$\varphi_ {\pi}=\overline{\varphi_ {\Pi}}.$$

This is not a bijection. If $\Pi|_ {\operatorname{SL}_ {n}}=\pi_ {1}\oplus\cdots\oplus\pi_ {r},$ then $\varphi_ {\pi_ {1}}=\cdots=\varphi_ {\pi_ {r}}.$ The adjoint action $\operatorname{GL}_ {n}$ on $\operatorname{SL}_ {n}$ is outer automorphism, and $\operatorname{GL}_ {n}/\operatorname{SL}_ {n}\times F^{\times}$ acts on $\{\pi_ {1},\dots,\pi_ {r}\}.$

We know that $\operatorname{GL}_ {n}/\operatorname{SL}_ {n}\times F^{\times}\xrightarrow[\text{det}]{\cong}F^{\times}/(F^{\times})^{n}.$ The action is transitive. Then there is a bijection $$(F^{\times}/(F^{\times})^{n}) /\operatorname{Stab}_ {\pi_ {1}}\xrightarrow{\cong} \{\pi_ {1},\dots,\pi_ {r}\}.$$

This translates to $L$-parameters. Let $S_ {\varphi}=Z_ {\operatorname{PGL}_ {n}}(\overline{\varphi}).$ Then there is a map $F^{\times}/(F^{\times})^{n}\twoheadrightarrow \operatorname{Hom}(S_ {\varphi},\underline{\mathbb{Q}}_ {\ell}^{\times}),$ and the kernel is exactly $\operatorname{Stab}(\pi_ {1})$ by Gelbert-Knopp.

Theorem 2 (GK). There is a surjection $\operatorname{Irr}(\operatorname{SL}_ {n}(F))\rightarrow \{\varphi:W_ {F}\rightarrow \operatorname{PGL}_ {n}(F), \chi\in \operatorname{IrrRep}(S_ {\varphi})\}/\text{conj}$, bijection over elliptic parameters.

Categorical picture

Fargues-Scholze constructed an action $\operatorname{Perf}(\operatorname{Par}_ {\check{G}})$ on $D_ {\text{lis}}(\operatorname{Bun}_ {G}).$ Then it predicts that \begin{align} D_ {\operatorname{Coh}}(\operatorname{Par}_ {\check{G}}) &\rightarrow \operatorname{D}_ {\text{lis}}(\operatorname{Bun}_ {G}) \\ M&\mapsto M\star W_ {\psi}
\end{align} is an equivalence over elliptic parameters.

Conjecture 3 (Fargues). Let $G$ be split and $\varphi$ is an elliptic parameter. There exists a unique generic supercuspidal representation $\pi$ of $G(F)$, such that $\varphi_ {\pi}$ is given by the Fargues-Scholze map, and the functor $$\operatorname{Perf}(\ast/S_ {\varphi})\rightarrow D_ {\text{lis}}^{C_ {\varphi}}(\operatorname{Bun}_ {G})$$ is $t$-exact equivalence.

Theorem 4. Yes for $\operatorname{SL}_ {n}$.