Related papers: Realizing the local Weil representation over a num…

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…

Let F be a local field. The action of GL(n,F) on the Grassmann variety Gr(m,n,F) induces a continuous representation of the maximal compact subgroup of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible constituents of this…

Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual…

Let k be an algebraically closed field of characteristic two. Let R be the ring of Witt vectors of length two over k. We construct a group stack \hat G over k, the metaplectic extension of the Greenberg realization of Sp_{2n}(R). We also…

We prove that when $q$ is a power of $2$, every complex irreducible representation of $\mathrm{Sp}(2n, \mathbb{F}_q)$ may be defined over the real numbers, that is, all Frobenius-Schur indicators are 1. We also obtain a generating function…

A method to construct irreducible unitary representations of a hyperspecial compact subgroup of a reductive group over p-adic field with odd p is presented. Our method is based upon Cliffods theory and Weil representations over finite…

In analogy with the study of representations of $GL_{2n}(F)$ distinguished by $Sp_{2n}(F)$, where $F$ is a local field, in this paper we study representations of $U_{2n}(F)$ distinguished by $Sp_{2n}(F)$. (Only quasi-split unitary groups…

By a global approach, we prove the arithmetic fundamental lemma conjecture for unitary groups in $n$ variables over $\mathbb{Q}_p$ when $p\geq n$.

Let $A$ be a ring with $1\neq 0$, not necessarily finite, endowed with an involution~$*$, that is, an anti-automorphism of order $\leq 2$. Let $H_n(A)$ be the additive group of all $n\times n$ hermitian matrices over $A$ relative to $*$.…

The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \oplus ... \oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume…

The classical construction of the Weil representation, with complex coefficients, has long been expected to work for more general coefficient rings. This paper exhibits the minimal ring $\mathcal{A}$ for which this is possible, the integral…

Over a p-adic field of odd residual characteristic, Gan and Savin proved a correspondence between the Bernstein components of the even and odd Weil representations of the metaplectic group and the components of the trivial representation of…

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 any reductive group G over a non-Archimedean local field has a cuspidal complex representation.

Weil's representation is a basic object in representation theory which plays a crucial role in many places: construction of unitary irreducible representations in the frame of the orbit method, Howe correspondence, Theta series,... The…

Let C be the centralizer in a finite Weyl group of an elementary abelian 2-subgroup. We show that every complex representation of C can be realized over the field of rational numbers. The same holds for a Sylow 2-subgroup of C.

Consider a quartic $q$-Weil polynomial $f$. Motivated by equidistribution considerations we define, for each prime $\ell$, a local factor which measures the relative frequency with which $f\bmod \ell$ occurs as the characteristic polynomial…

We compare the dimensions of the irreducible Sp(2g,K)-modules over a field K of characteristic p constructed by Gow with the dimensions of the irreducible Sp(2g,F_p)-modules that appear in the first approximation to representations of…

We prove several multiplicity one theorems in this paper. For k a local field not of characteristic two, and V a symplectic space over k, any irreducible admissible representation of the symplectic similitude group GSp(V) decomposes with…

The finite symplectic group Sp(2g) over the field of two elements has a natural representation on the vector space of Siegel modular forms of given weight for the principal congruence subgroup of level two. In this paper we decompose this…