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J. Pevtsova and the author constructed a ``universal $p$-nilpotent operator" for an infinitesimal group scheme $G$ over a field $k$ of characteristic $p > 0$ which led to coherent sheaves on the scheme of 1-parameter subgroups of $G$…

Let $G$ be a connected, simply connected nilpotent group and $\pi$ be a square-integrable irreducible unitary representation modulo its center $Z(G)$ on $L^2(\mathbf{R}^d)$. We prove that under reasonably weak conditions on $G$ and $\pi$…

In this paper, we explicitly determine the local $2$-adic component of a non-selfdual automorphic representation $\Pi$ of $\mathrm{GL}_3$ constructed by van Geemen and Top. We prove that $\Pi_2$ is a parabolically induced representation of…

We prove the following result in relative representation theory of a reductive p-adic group $G$: Let $U$ be the unipotent radical of a minimal parabolic subgroup of $G$, and let $\psi$ be an arbitrary smooth character of $U$. Let $S \subset…

Let $F$ be a local non-archimedian field of odd residue characteristic and let $G=PGL(2)$. In this paper we study an analog of irreducible cuspidal representations of the group $G(F)$ when $F$ is replaced by the field $K=F((t))$. The story…

We study, for a locally compact group $G$, the compactifications $(\pi,G^\pi)$ associated with unitary representations $\pi$, which we call {\it $\pi$-Eberlein compactifications}. We also study the Gelfand spectra $\Phi_{\mathcal{A}}(\pi)}$…

We investigate local-global compatibility for cuspidal automorphic representations $\pi$ for GL(2) over CM fields that are regular algebraic of weight $0$. We prove that for a Dirichlet density one set of primes $l$ and any $\iota :…

We study the Howe correspondence for unipotent representations of irreducible dual pairs $(G',G)=(\text{U}_m(\mathbb{F}_q),\text{U}_n(\mathbb{F}_q))$ and $(G',G)=(\text{Sp}_{2m}(\mathbb{F}_q),\text{O}^\epsilon_{2n}(\mathbb{F}_q))$, where…

Let $G$ be a finite group. Let $U_1,U_2,\dots$ be a sequence of orthogonal representations in which any irreducible representation of $\oplus_{n \geq 1} U_n$ has infinite multiplicity. Let $V_n=\oplus_{i=1}^n U_n$ and $S(V_n)$ denote the…

The duality principle for group representations developed in \cite{DHL-JFA, HL_BLM} exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of…

For a central division algebra $D$ of dimension $d^2$ over a finite extension $F$ of $\mathbb Q_p$ or of $\mathbb F_p((t))$, a field $R$ of characteristic prime to $p$, and an irreducible smooth $R$-representation $\pi$ of $G=GL_n(D)$, we…

Let $G$ be a simple group over a global function field $K$, and let $\pi$ be a cuspidal automorphic representation of $G$. Suppose $K$ has two places $u$ and $v$ (satisfying a mild restriction on the residue field cardinality), at which the…

Let $G$ be a compact connected semisimple Lie group, let $K$ be a closed subgroup of $G$, let $\Gamma$ be a finite subgroup of $G$, and let $\tau$ be a finite-dimensional representation of $K$. For $\pi$ in the unitary dual $\widehat G$ of…

Let F be a non-Archimedean local field and let G be a connected reductive affine algebraic F-group. Let I be an Iwahori subgroup of G(F) and denote by H(G; I) the Iwahori-Hecke algebra, i.e. the convolution algebra of complex-valued…

In this paper, the 2-category $\mathfrak{Rep}_{{\bf 2Mat}_{\mathbb{C}}}(\mathbb{G})$ of (weak) representations of an arbitrary (weak) 2-group $\mathbb{G}$ on (some version of) Kapranov and Voevodsky's 2-category of (complex) 2-vector spaces…

A real representation $\pi$ of a finite group may be regarded as a homomorphism to an orthogonal group $\Or(V)$. For symmetric groups $S_n$, alternating groups $A_n$, and products $S_n \times S_{n'}$ of symmetric groups, we give criteria…

Let $F$ be a non-archimedean local field. In this paper we explore genericity of irreducible smooth representations of $GL_n(F)$ by restriction to a maximal compact subgroup $K$ of $GL_n(F)$. Let $(J, \lambda)$ be a Bushnell--Kutzko type…

Let $F$ be a $p$-adic field and $\mathbf{U}$ be a unipotent group defined over $F$, and set $U=\mathbf{U}(F)$. Let $\sigma$ be an involution of $\mathbf{U}$ defined over $F$. Adapting the arguments of Yves Benoist in the real case, we prove…

In this paper we suppose G is a finite group acting tamely on a regular projective curve X over Z and V is an orthogonal representation of G of dimension 0 and trivial determinant. Our main result determines the sign of the…

Diffeomorphism groups $G$ of manifolds $M$ on locally $\bf F$-convex spaces over non-Archimedean fields $\bf F$ are investigated. It is shown that their structure has many differences with the diffeomorphism groups of real and complex…