Distinguished representations, base change, and reducibility for unitary groups

This paper describes our method of pairing automorphic distributions. This represents a third technique for obtaining the analytic properties of automorphic L-functions, in addition to the existing methods of integral representations (Rankin-Selberg) and Fourier coefficients of Eisenstein series (Langlands-Shahidi). We recently used this technique to establish new cases of the full analytic continuation of the exterior square L-functions. The paper here gives an exposition of...

We propose a geometric interpretation of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program. We show that the geometric Langlands conjecture for an irreducible unramified local system $E$ of rank $n$ on a curve implies the existence of automorphic sheaves corresponding to the universal deformation of $E$. Then we calculate the `scalar product' of two automorphic sheaves attached to this universal deformation.

Let $E/F$ be a quadratic extension of number fields and let $\pi$ be an $\mathrm{SL}_n(\mathbb{A}_F)$-distinguished cuspidal automorphic representation of $\mathrm{SL}_n(\mathbb{A}_E)$. Using an unfolding argument, we prove that an element of the $\mathrm{L}$-packet of $\pi$ is distinguished if and only if it is $\psi$-generic for a non-degenerate character $\psi$ of $N_n(\mathbb{A}_E)$ trivial on $N_n(E+\mathbb{A}_F)$, where $N_n$ is the group of unipotent upper triangular m...

Let $F$ be a totally real number field and $E/F$ a totally imaginary quadratic extension of $F$. Let $\Pi$ be a cohomological, conjugate self-dual cuspidal automorphic representation of $GL_n(\mathbb A_E)$. Under a certain non-vanishing condition we relate the residue and the value of the Asai $L$-functions at $s=1$ with rational structures obtained from the cohomologies in top and bottom degrees via the Whittaker coefficient map. This generalizes a result in Eric Urban's the...

Let F be a nonarchimedean local field of characterisitic 0 and odd residual characteristic. We describe explicitly the two base change lifts of supercuspidal representations of U(1,1)(F). This represents a step towards the goal of describing base change of endoscopic supercuspidal L-packets of U(2,1)(F).

Let $F$ be a non-Archimedean local field and $G$ be the general linear group $\mathrm{GL}_n$ over $F$. Based on the previous results of the author, we can describe the Langlands parameter of an essentially tame supercuspidal representation of $G(F)$ by those admissible embeddings of L-groups constructed by Langlands and Shelstad. We therefore provide a different interpretation on Bushnell and Henniart's essentially tame local Langlands correspondence.

Let $F$ be a $p$-adic field, and $K$ a quadratic extension of $F$. Using Tadic's classification of the unitary dual of $GL(n,K)$, we give the list of irreducible unitary representations of this group distinguished by $GL(n,F)$, in terms of distinguished discrete series. As it is known that a generalised Steinberg representation $St(k,\rho)$ is distinguished if and only if the cuspidal representation $\rho$ is $\eta^{k-1}$-distinguished, for $\eta$ the character of $F^*$ with ...

We prove a prime number theorem first for the classical Rankin-Selberg L-function $L(s,\pi\times\pi')$ over any Galois extension with $\pi$ and $\pi'$ unitary automorphic cuspidal representations of $GL_n$ and $GL_m$ respectively with at least one of the representations subject to a self-contragredient assumption. We then extend these results to two representations $\pi$ defined on $GL_n/E$ and $\pi'$ defined on $GL_m/F$ with $E$ and $F$ cylic algebraic number fields of copri...

Let $F/F_{0}$ be a quadratic extension of non-archimedean locally compact fields of residue characteristic $p\neq 2$. Let $R$ be an algebraically closed field of characteristic different from $p$. For $\pi$ a supercuspidal representation of $G=\mathrm{GL}_{n}(F)$ over $R$ and $G^{\tau}$ a unitary group in $n$ variables contained in $G$, we prove that $\pi$ is distinguished by $G^{\tau}$ if and only if $\pi$ is Galois invariant. When $R=\mathbb{C}$ and $F$ is a $p$-adic field,...

In this article, we study Prasad's conjecture for regular supercuspidal representations based on the machinery developed by Hakim and Murnaghan to study distinguished representations, and the fundamental work of Kaletha on parameterization of regular supercuspidal representations. For regular supercuspidal representations, we give some new interpretations of the numerical quantities appearing in Prasad's formula, and reduce the proof to the case of tori. The proof of Prasad's...