Related papers: Some questions on global distinction for $\mathrm{…

Selfdual representations of any group fall into two classes when they are irreducible: those which carry a symmetric bilinear form, and the others which carry an alternating bilinear form. The Langlands correspondence, which matches the…

For an unramified connected reductive group $G$ defined over a number field $F$, consider the part of the spherical automorphic spectrum with cuspidal support $[T,\mathcal{O}(\chi)]$, where $T$ is a maximal torus and $\chi$ is an unramified…

This manuscript has two goals: 1. To write an explicit description of the degenerate residual spectrum of the split, simple, simply-connected, exceptional groups of type $E_n$ (for $n=6,7,8$). 2. To set a practical guide for similar…

It is well known that the Tchebotarev density theorem implies that an irreducible $\ell$-adic representation $\rho$ of the absolute Galois group of a number field $K$ is determined (up to isomorphism) by the characteristic polynomials of…

We construct the field A generated by n algebraically independent elements, and show that the linear space of derivations over this field is faithfully represented by the linear space of the n-th fold Cartesian product of this field acting…

We exhibit a basis for the space of spherical characters of a distinguished supercuspidal representation $\pi$ of a connected reductive $p$-adic group, subject to the assumption that $\pi$ is obtained via induction from a representation of…

Let $F$ be an archimedean local field and let $E$ be $F\times F$ (resp. a quadratic extension of $F$). We prove that an irreducible generic (resp. nearly tempered) representation of $\operatorname{GL}_n(E)$ is $\operatorname{GL}_n(F)$…

We construct explicit finite-dimensional orthogonal representations $\pi_N$ of $\operatorname{SL}_{N}(\mathbb{Z})$ for $N \in \{3,4\}$ all of whose invariant vectors are trivial, and such that $H^{N -…

We prove the compatibility of local and global Langlands correspondences for GL_n, which was proved up to semisimplification by Harris-Taylor. More precisely, for the n-dimensional l-adic representation R_l(\Pi) of the Galois group of a…

Let $\tau_1^{(r)}$, $\tau_2^{(r)}$ be two genuine cuspidal automorphic representations on $r$-fold covers of the adelic points of the general linear groups $GL_{n_1}$, $GL_{n_2}$, resp., and let $E(g,s)$ be the associated Eisenstein series…

Let $G$ be the group of $F$-points of a split connected reductive $F$-group over a non-Archimedean local field $F$ of characteristic 0. Let $\pi$ be an irreducible smooth self-dual representation of $G$. The space $W$ of $\pi$ carries a…

Let $\rm E/\rm F$ be an unramified quadratic extension of local non archimedean fields of characteristic 0. Let $\underline{H}$ be an algebraic reductive group, defined and split over $\rm F$. We assume that the split connected component of…

Let $ M$ be a cusped hyperbolic $ 3$-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in $ \mathrm {PGL}(2,\mathbf {C})$ (up to conjugation) is of complex dimension the number $ \nu $…

Let $F/\mathbb{Q}$ be a totally real field and $A$ a modular $\GL_2$-type abelian variety over $F$. Let $K/F$ be a CM quadratic extension. Let $\chi$ be a class group character over $K$ such that the Rankin-Selberg convolution $L(s,A,\chi)$…

The present paper proves a $q$-identity, which arises from a representation $\pi_{N,\psi}$ of $\text{GL}_n(\mathbb{F}_q)$. This identity gives a significant simplification for the dimension of $\pi_{N,\psi}$, which allowed the second author…

Let $G$ be a classical group defined over a local field $F$ of characteristic zero. For any irreducible admissible representation $\pi$ of $G(F)$, which is of Casselman-Wallach type if $F$ is archimedean, we extend the study of spectral…

Let $E/F$ be a quadratic extension of finite fields. By a result of Gow, an irreducible representation $\pi$ of $G = {\rm GL}_n(E)$ has at most one non-zero $H$-invariant vector, up to multiplication by scalars, when $H$ is ${\rm GL}_n(F)$…

Let $F$ be a non Archimedean local field with odd residual characteristic, and let $K$ be a hyperspecial maximal compact subgroup of the $p$-adic symplectic group $G=\mathrm{Sp}_4(F)$. Let $\mathfrak{s}$ be an inertial class for $G$ in the…

We prove an integrality result for the value at s=1 of the adjoint L-function associated to a cohomological cuspidal automorphic representation on GL(n) over any number field. We then show that primes (outside an exceptional set) dividing…

Let $\Pi_0$ be a representation of a group $H$. We say that a representation $\tau$ is $(H,\Pi_0)$-distinguished, if it is a quotient of $\Pi_0$. It is natural to ask whether this notion "inflates" to larger groups, in the sense that a…