Related papers: Theta distinguished representations, inflation and…

Howe and Tan (1993) investigated a degenerate principal series representation of indefinite orthogonal groups $\mathrm{O}(V)$ and explicitly described its composition series. They showed that there exists a unique unitarizable irreducible…

For Brylinski-Deligne covering groups of an arbitrary split reductive group, we consider theta representations attached to certain exceptional genuine characters. The goal of the paper is to determine when a theta representation has exactly…

Let $D$ be a quaternion division algebra over a non-archimedean local field $F$ of characteristic zero. Let $E/F$ be a quadratic extension and $\rm{SL}_{n}^{*}(E) = {\rm{GL}}_{n}(E) \cap \rm{SL}_{n}(D)$. We study distinguished…

In this article, we consider a dual pair $(G, G')$ in the symplectic group $Sp(W)$ with $G$ compact and let $(\tilde{G}, \tilde{G}')$ be the preimages of $G$ and $G'$ in the metaplectic group $\widetilde{Sp(W)}$. For every irreducible…

Given irreducible representations $\Pi$ and $\pi$ of the rank one special orthogonal groups $G=SO(n+1,1)$ and $G'=SO(n,1)$ with nonsingular integral infinitesimal character, we state in terms of $\theta$-stable parameter necessary and…

For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${\rm GL}_2(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is…

We show that the theta representations on certain covers of general linear groups support certain types of unique functionals. The proof involves two types of Fourier coefficients. The first are semi-Whittaker coefficients, which generalize…

Let $K/F$ be a quadratic extension of $p$-adic fields, and $n$ a positive integer. A smooth irreducible representation of the group $GL(n,K)$ is said to be distinguished, if it admits on its space a nonzero $GL(n,F)$-invariant linear form.…

Let $K$ be a non-archimedean local field of residual characteristic $p\neq 2$. Let $G$ be a connected reductive group over $K$, let $\theta$ be an involution of $G$ over $K$, and let $H$ be the connected component of $\theta$-fixed subgroup…

Based on recent work of Kaletha, we apply Hakim--Murnaghan's result to study distinguished regular supercuspidal representations of tamely ramified reductive $p$-adic groups. Assuming $p$ is sufficiently large, we obtain a necessary and…

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields of residual characteristic $p\neq2$ with Galois automorphism $\sigma$, and let $R$ be an algebraically closed field of characteristic $\ell\notin\{0,p\}$. We…

Let $G=\mathrm{GL}_{2n}(\mathbb{R})$ or $G=\mathrm{GL}_n(\mathbb{H})$ and $H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. Here, $\mathbb{H}$ is the quaternion division algebra over $\mathbb{R}$. For a character $\chi$ on…

Let (G,d) be a first order differential *-calculus on a *-algebra A. We say that a pair (\pi,F) of a *-representation \pi of A on a dense domain D of a Hilbert space and a symmetric operator F on D gives a commutator representation of G if…

Let E/F be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the smooth irreducible representations of GL(n,E) that are distinguished by its…

Let $F/F_0$ be a quadratic extension of non-Archimedean locally compact fields with residual characteristic $p\neq2$, and $\ell$ be a prime number different from $p$. We classify those $\ell$-modular cuspidal irreducible representations of…

Let $\pi$ be an unitary irreducible representation of a Lie group $G$. $\pi$ defines a moment set $I_\pi$, subset of the dual $\mathfrak g^*$ of the Lie algebra of $G$. Unfortunately, $I_\pi$ does not characterize $\pi$. However, we…

Let $F$ be a field which is, either local non archimedean, or finite, of residual charcateristic $p$ but of characteristic different from $2$. Let $W$ be a symplectic space of finite dimension over $F$. Suppose $R$ is a field of…

As a consequence of his numerical local Langlands correspondence for $GL(n)$, Henniart deduced the following theorem: If $F$ is a nonarchimedean local field and if $\pi$ is an irreducible admissible representation of $GL(n,F)$, then, after…

Suppose G is a real reductive Lie group in Harish-Chandra's class. We propose here a structure for the set \Pi_u(G) of equivalence classes of irreducible unitary representations of G. (The subscript u will be used throughout to indicate…

An irreducible smooth representation of a $p$-adic group $G$ is said to be distinguished with respect to a subgroup $H$ if it admits a non-trivial $H$-invariant linear form. When $H$ is the fixed group of an involution on $G$ it is…