Related papers: Geometric Weil representation: local field case

We construct a Langlands parameterization of supercuspidal representations of $G_2$ over a $p$-adic field. More precisely, for any finite extension $K / \QQ_p$ we will construct a bijection \[ \CL_g : \CA^0_g(G_2,K) \rightarrow \CG^0(G_2,K)…

Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such…

It is well known(cf. Weil, G\'erardin's works) that there are two different Weil representations of a symplectic group over an odd finite field. By a twisted action, we show that one can reorganize them as a representation of a related…

Let F be a non-archimedean local field and let G be a connected reductive group defined over F. We assume that G splits over a tame extension of F and that the residual characteristic p does not divide the order of the Weyl group. To each…

We prove that the reduction mod \ell of the local Langlands correspondence between supercuspidal representations of GL_n(F), where F is a finite extension of Q_p, and representations of the Galois group of F is well-defined. The results and…

Let $R$ be a commutative $\mathbb{Z}[1/p]$-algebra, let $m \leq n$ be positive integers, and let $G_n=\text{GL}_n(F)$ and $G_m=\text{GL}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\times G_m]$-module…

In this paper we provide, under some mild explicit assumptions, a geometric description of the category of representations of the centralizer of a regular unipotent element in a reductive algebraic group in terms of perverse sheaves on the…

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil…

We extend the Weil representation of infinite-dimensional symplectic group to a representation a certain category of linear relations.

We introduce and motivate -- based on ongoing joint work with Germ\'an Stefanich -- the notion of potent categorical representations of a complex reductive group $G$, specifically a conjectural Langlands correspondence identifying potent…

We investigate the representations of the hyperalgebras associated to the map algebras $\mathfrak g\otimes \mathcal A$, where $\mathfrak g$ is any finite-dimensional complex simple Lie algebra and $\mathcal A$ is any associative commutative…

Let $\mathrm{Mp}(2n)$ be the metaplectic group over a local field $F \supset \mathbb{Q}_p$ defined by an additive character of $F$ of conductor $4\mathfrak{o}_F$. Gan-Savin ($p \neq 2$) and Takeda-Wood ($p=2$) obtained an equivalence…

Let $F$ be a p-adic local field and $G=GL_2(F)$. Let $\mathcal{H}^{(1)}$ be the pro-p Iwahori-Hecke algebra of $G$ with coefficients in an algebraic closure of $\mathbb{F}_p$. We show that the supersingular irreducible…

The geometric Satake isomorphism is an equivalence between the categories of spherical perverse sheaves on affine Grassmanian and the category of representations of the Langlands dual group. We provide a similar description for derived…

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell\neq p$ be a prime number, and $\mathrm{W}_F$ the Weil group of $F$. We classify the indecomposable $\mathrm{W}_F$-semisimple Deligne…

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k) \to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the…

We extend the lifting methods of our previous paper to lift reducible odd representations $\bar{\rho}:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when…

We introduce support varieties for rational representations of a linear algebraic group $G$ of exponential type over an algebraically closed field $k$ of characteristic $p > 0$. These varieties are closed subspaces of the space $V(G)$ of…

The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on…

An intrinsic construction of the tensor category of finite dimensional representations of the Langlands dual group of G in terms of a tensor category of perverse sheaves on the loop group, LG, is given. The construction is applied to the…