David Yang：On the Classification of $G(\!(t)\!)$-categories
Philosophy/Motivations:
Use $\mathrm{DGCat}^{\text{alg-geom}}$ to encode Drinfeld’s lemma and overcome the failure of Kunneth formula of $\ell$-adic sheaves. In this talk, let’s pretend there is a Kunneth formula for $\ell$-adic sheaves so that $\text{Tr}(\operatorname{Fr}^{\ast},\mathrm{Shv}(X_ {\overline{\mathbb{F}_ {q}}}))\cong \operatorname{Fun}(X(\mathbb{F}_ {q}), \overline{\mathbb{Q}_ {\ell}}).$
Take $F=\mathbb{F}_ {q}(\!(t)\!)$. Let $F^{\text{unr}}:= \overline{\mathbb{F}_ {q}}(\!(t)\!).$ Let $G$ be a group over $F$, and $G^{\text{unr}}$ is an ind-scheme over $\overline{\mathbb{F}_ {q}}$. Then $\text{Tr}(\mathrm{Fr}, \mathrm{Shv}(G^{\text{unr}}))\cong \mathscr{H}$ convolution algebra of compactly supported locally constant functions on $G(F)$ (not literally true!).
In short: from a category with an action of $\mathrm{Shv}(G^{\text{unr}})$ and a suitable endomorphism, one gets a representation of $G(F)$.
Warning 1. There are technicality and subtle issues involved and the above assertions should be understood with caution (to my understanding).
Goal: classify $G(\!(t)\!)$-categories.
On the arithmetic setting, there is
- exhaustion theorem for representations,
- equalities of Hecke algebras ([Adler-Fintzen-Mishra-Ohara]).
Goal is to see them geometrically.
Slogan: Local geometric Langlands holds “up to Five Lemma”. Joint work with G. Dhillon and Y. Varshavsky. By five lemma, he means five lemma for categories do not hold as naive modules over a naive algebra as in an elementary homological algebra class.
A filtration on category of $G(\!(t)\!)$-categories indexed by a poset $I$: assign to each object $X\in G(\!(t)\!)$-Cat, a sequence of full subcategories $X_ {i}\subseteq X$ stable under $G(\!(t)\!)$-action, and $X_ {i}\subseteq X_ {j}$ for $i<j$.
For $i\in I$, one can take $G(\!(t)\!)\text{-Cat}^{i}$ consisting of objects $X$ with $X_ {j}=0$ if $j$ is not $\geq i$ and $X_ {j}=X$ for $j\geq i$.
Local geometric Langlands: the statement is $$G(\!(t)\!)\text{-Cat}\cong \operatorname{QCoh}(\operatorname{LocSys})\text{-Cat}$$ and there exists filtrations on both sides such that associated grades are isomorphic.
Warning 2. The statement may not be quiet correct for de Rham setting on the nose. Need verification.
For every $G(\!(t)\!)$-category $\mathscr{C}$. For each rational number $r$, one defines full subcategory $\mathscr{C}^{\leq r}$ to be the $G(\!(t)\!)$-category generated by all $\mathscr{C}^{K_ {x},r+}$ where $K_ {x,r+}$ is the Moy-Prasad group and $x$ runs over $\mathscr{B}(G,F)$.
Enrich this slightly: Define a filtration indexed by pairs $(r,G_ {0})$, where $G_ {0}\rightarrow G$ is a twisted Levi, i.e. becomes a levi over $\overline{\mathbb{k}(\!(t)\!)}$.
Theorem 3 (Dhillon-Varshavsky-Yang, $1_ {r}$). $G(\!(t)\!)\text{-Cat}^{=(r,G_ {0})} \cong (G_ {0}(\!(t)\!)\text{-Cat}^{=r,G_ {0}},\text{gen})^{W_ {\text{rel}}}$, where $W_ {\text{rel}} = N_ {G}(G_ {0})/G_ {0}$.
This allows one to reduce understanding pieces of $G(\!(t)\!)$-Cat to understanding pieces with $G=G_ {0}$. If $G=G_ {0}$, $G(\!(t)\!)\text{-Cat}^{=(r,G)}\cong G(\!(t)\!)\text{-Cat}^{<r}\otimes_ {Z\text{-Cat}^{?}}Z\text{-Cat}^{??}$.
Applying these two reductions together, one can reduce to depth 0 case. The depth 0 case is understood by Dhillon-Li-Yun-Zhu.
