Tony Feng: Modular Functoriality in Local Langlands

Background

Let $F$ be a local field with residue characteristic $p$. Let $G$ be a reductive group over $F$. To simplify the situation, we “pretend” that $G$ is split.

Let $\mathbb{k}$ be a coefficient field with $\text{char}\neq p$.

Let $W_ {F}\subseteq \operatorname{Gal}(F^{s}/F)$ be the Weil group. Let $\check{G}$ be the dual group of $G$.

Local Langlands correspondence relates $\mathrm{IrrRep}_ {\mathbb{k}}(G(F)) \leftrightarrow \{W_ {F}\rightarrow \check{G}(k)\}/\sim$.

Fargues-Scholze constructed $\pi\mapsto \rho_ {\pi}$, which should be semi-simplification of the conjectured correspondence.

Example 1.

  • $\operatorname{GL}_ {n}\leftrightarrow\operatorname{GL}_ {n}$
  • $\operatorname{SO}_ {2n+1}\leftrightarrow\operatorname{Sp}_ {2n}$
  • $\operatorname{SO}_ {2n}\leftrightarrow\operatorname{SO}_ {2n}$

This construction is not functorial. $\operatorname{Sp}_ {2a}\times\operatorname{Sp}_ {2b}\subseteq \operatorname{Sp}_ {2a+2b}$, but there is no nontrivial map $\operatorname{SO}_ {2a+1}\times\operatorname{SO}_ {2b+1}\rightarrow \operatorname{SO}_ {2a+2b+1}$

Langlands functoriality relates $\mathrm{Rep}_ {\mathbb{k}}(H(F))\rightarrow \operatorname{Rep}_ {\mathbb{k}}(G(F))$ for $\check{H}\rightarrow \check{G}$.

Example 2. For $P=MN\subseteq G$ a parabolic, $\check{H}:=\check{M}\subseteq \check{G}$. Then $\operatorname{Rep}_ {\mathbb{k}}(M(F))\rightarrow \operatorname{Rep}_ {\mathbb{k}}(G(F))$ is the parabolic induction.

Adjoint is given by Jacquet module $\operatorname{Rep}_ {\mathbb{k}}(G(F))\rightarrow \operatorname{Rep}_ {\mathbb{k}}(M(F))$.

Modular functoriality

Goal: analogue of Jacquet module for more situations. Modular functoriality (Trevmann-Venkatesh):

  • $\text{char}(\mathbb{k})=\ell\neq p$,
  • $\sigma\in \operatorname{Aut}(G)$ order $\ell$,
  • $H=G^{\sigma}$ connected reductive.

Example 3 (cyclic base change). Start with $H$ and $\operatorname{Gal}(E/F)=\mathbb{Z}/\ell$, $G=\operatorname{Res}_ {E/F}(H_ {E})$. Let $\sigma\in \operatorname{Gal}(E/F)$ be a generator. Then $G^{\sigma} = H$.

Example 4 (Triality). Consider $G=\operatorname{Spin}(8)$ and $H=G_ {2}$.

Feng-tikzcd1

Claim: there exists a “natural” $\check{H}_ {\mathbb{k}}\rightarrow \check{G}_ {\mathbb{k}}$ in this situation.

Example 5. Let $G=\operatorname{Sp}_ {2a+2b}$ and $\text{char}(F)\neq 2$, $\ell=2$. Let $\sigma$ be conjugation by $\begin{pmatrix} \operatorname{Id}_ {2a} & 0 \\ 0 & -\operatorname{Id}_ {2b} \end{pmatrix}.$

Looking for $\operatorname{SO}_ {2a+1}\times \operatorname{SO}_ {2b+1}\rightarrow \operatorname{SO}_ {2a+2b+1}$, In char 2, it does exist.

Note that $\operatorname{SO}_ {2a+1}$ preserves $x_ {0}^{2}+x_ {1}x_ {2}+\cdots +x_ {2a-1}x_ {2a}$ and $\operatorname{SO}_ {2b+1}$ preserves $y_ {0}^{2}+y_ {1}y_ {2}+\cdots + y_ {2b-1}y_ {2b}$. Then in char 2, $x_ {0}^{2}+y_ {0}^{2}=z_ {0}^{2}$ for $z_ {0}=x_ {0}+y_ {0}$. This gives the map.

Let $[G]:=G(F)/G(\mathcal{O}_ {F})$ acted by $G(F)$. Then by satake isomorphism $$\mathscr{O}(\check{G}_ {\mathbb{k}}//\check{G}_ {\mathbb{k}})\xrightarrow{\cong}\mathscr{H}(G,\mathbb{k}):=\operatorname{Fun}_ {G(F)}^{c}([G]^{2},\mathbb{k}).$$

Trevmann-Venkatesh: there is an “exceptional” $\operatorname{br}:\mathscr{H}(G,\mathbb{k})\rightarrow \mathscr{H}(H,\mathbb{k})$ called Braver homomorphism.

Key point: assume that $[G]^{\sigma} = [H]$. Then $\mathscr{H}(G,\mathbb{k})^{\sigma}\xrightarrow{\text{restrict}}\mathscr{H}(H,\mathbb{k})$ is an algebra homomorphism.

Proof.

Additivity is straightforward. By definition $f_ {1}\star_ {G}f_ {2}(x,z)=\sum_ {y\in [G]}f_ {1}(x,y)f_ {2}(y,z)$. Then the restriction $f_ {1}|_ {H}\star f_ {2}|_ {H}(x,z) = \sum_ {y\in [H]} f_ {1}(x,y)f_ {2}(y,z)$. Therefore, the difference is $$\sum_ {y\in [G]\backslash [H]} f_ {1}(x,y)f_ {2}(y,z).$$ In general this term does not vanish. But in our situation, $\sigma$ acts freely on $[G]\backslash [H]$. Therefore, the difference is equal to $$\sum_ {y\in [G]\backslash [H]} f_ {1}(x,\sigma y)f_ {2}(\sigma y,z).$$

However, $f_ {i}(x,\sigma y) = f_ {i}(\sigma_ {x}, \sigma y) = f_ {i}(x,y)$, where the first identity uses $x\in H=G^{\sigma}$ and second identity uses $f$ is stable under $\sigma$.

Therefore, everything cancels in orbits.

After more work, one can show that there is an exceptional

Question: does this come from $\check{H}_ {\mathbb{k}}\rightarrow \check{G}_ {\mathbb{k}}$?

Theorem 6 (TV, omitting $E_ {6}$). Yes if $G$ is simply connected and $H$ is semi-simple.

Proof is via classification and case exhaustion.

Theorem 7 (Feng). Yes if $\ell$ is not too small (i.e. good for $\check{G}$) via uniform proofs.

Example 8. $\sigma$ = conjugation by strongly regular $s\in G(F)$. Then $H=$(not necessarily split) torus $T$. Then ${}^{L}T\rightarrow {}^{L}G=\check{G}\rtimes W_ {F}$.

Idea of Proof.

We upgrade $\mathscr{H}(G,K)$ to $\operatorname{Sat}(\operatorname{Gr}_ {G},\mathbb{k})$. We have geometric satake $$\operatorname{Sat}(\operatorname{Gr}_ {G},\mathbb{k})\xrightarrow{\cong}\operatorname{Rep}(\check{G}).$$

The hard work is to construct a Braver functor $\operatorname{Sat}(\operatorname{Gr}_ {G},\mathbb{k})\rightarrow \operatorname{Sat}(\operatorname{Gr}_ {H},\mathbb{k}).$

Applications to LLC

Let $G,H,\sigma$ be as before.

We have $$\operatorname{FS}: \{\operatorname{IrrRep}_ {\mathbb{k}}(G(F))\}\rightarrow \{\rho_ {\pi}:W_ {F}\rightarrow \check{G}(\mathbb{k})\},$$

  • surjectivity?
  • finite?
  • nontriviality

Kaletha explicit parametrization over $\mathbb{C}$ $$\operatorname{IrrRep}^{\text{r.s.c}}_ {\mathbb{k}}(G(F))\leftarrow \{T \text{ elliptic , regular }\chi: T(F)\rightarrow \mathbb{C}^{\times} \}$$

and

$$\text{L-parameters}^{\text{s.c.}}\leftarrow \{W_ {F}\xrightarrow{\iota_ {\chi}\cdot \epsilon} {}^{L}T \hookrightarrow {}^{L}G\}.$$

Theorem 9 (Feng). Given ${}^{L}\Psi: \check{H}\rightarrow \check{G}$, if $\pi\in H(F)$ has parameter $\rho_ {\pi}$, then ${}^{L}\Psi\circ \rho_ {\pi}$ arises from some $\Pi\in \operatorname{Rep}_ {\mathbb{k}}(G(F)).$

Let $\Pi\in \operatorname{IrrRep}_ {\mathbb{k}}(G(F))$ acted by $\sigma$. Assume that $[\Pi]$ is fixed, then $T^{0}(\Pi) = \frac{\Pi^{\sigma}}{(1+\sigma+\cdots+\sigma^{\ell-1})\Pi}\in \operatorname{Rep}_ {\mathbb{k}}(H(F)).$

Theorem 10. Irreducible subquotient $\pi$ of $T^{\circ}\Pi$ transfers to $\Pi\otimes_ {\operatorname{Fr}}\mathbb{k}=:\Pi^{(1)}$.

Theorem 11 (Work in progress). If $\mathbb{k}=\mathbb{C}$, then $(T,\chi)\sim \Pi_ {(T,\chi)}$. If $T$ is unramified, then $\rho_ {\Pi_ {(T,\chi)}}$ agrees with Kaletha’s parametrization.