Related papers: On the direct integral decomposition in branching …

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak{n}$ of dimension $n.$ Let $H$ be a subgroup of the automorphism group of $N.$ Assume that $H$ is a commutative, simply connected,…

Let $G_1,G_2$ be real reductive groups and $(\pi,V)$ a smooth, irreducible, admissible representation of $G_1 \times G_2$. We prove that $(\pi,V)$ is the completed tensor product of $(\pi_i,V_i)$, $i=1,2$, where $(\pi_i,V_i)$ is a…

Let $E/F$ be a unramified quadratic extension of non-archimedean local fields of odd characteristic $p$, and $G$ be the unramified unitary group $U(2, 1)(E/F)$. For an irreducible smooth representation $\pi$ of $G$ over…

Let $G$ be a discrete countable group, and let $\Gamma$ be an almost normal subgroup. In this paper we investigate the classification of (projective) unitary representations $\pi$ of $G$ into the unitary group of the Hilbert space…

In this paper we study branching laws for certain unitary representations. This is done on the smooth vectors via a version of the {\it period integrals}, studied in number theory, and also closely connected to the {\it symmetry-breaking…

In this note we collect several characterizations of unitary representations $(\pi, \mathcal{H})$ of a finite dimensional Lie group $G$ which are trace class, i.e., for each compactly supported smooth function $f$ on $G$, the operator…

Let $G$ be a reductive group in the Harish-Chandra class e.g. a connected semisimple Lie group with finite center, or the group of real points of a connected reductive algebraic group defined over $\R$. Let $\sigma$ be an involution of the…

Consider a unitary representation $\pi$ of a discrete group $G$, which, when restricted to an almost normal subgroup $\Gamma\subseteq G$, is of type II. We analyze the associated unitary representation $\overline{\pi}^{\rm{p}}$ of $G$ on…

Let $\Gamma=\langle T_{k},M_{l}:k\in\mathbb{Z}^{d},l\in B\mathbb{Z}% ^{d}\rangle $ be a group of unitary operators where $T_{k}$ is a translation operator and $M_{l}$ is a modulation operator acting on $L^{2}\left( \mathbb{R}^{d}\right) .$…

For an essentially tame supercuspidal representation $\pi$ of a connected reductive $p$-adic group $G$, we establish two distinct and complementary sufficient conditions for the irreducible components of its restriction to a maximal compact…

Given a connected semisimple Lie group $G$ and an arithmetic subgroup $\Gamma$, it is well-known that each irreducible representation $\pi$ of $G$ occurs in the discrete spectrum $L^2_{\text{disc}}(\Gamma\backslash G)$ of…

Let $G=\operatorname{O}(1,n+1)$ with maximal compact subgroup $K$ and let $\Pi$ be a unitary irreducible representation of $G$ with non-trivial $(\mathfrak{g},K)$-cohomology. Then $\Pi$ occurs inside a principal series representation of…

In this paper, we study the extended Hamilton-Jacobi Theory in the context of dynamical systems with symmetries. Given an action of a Lie group $G$ on a manifold $M$ and a $G$-invariant vector field $X$ on $M$, we construct complete…

We present a finite algorithm for computing the set of irreducible unitary representations of a real reductive group G. The Langlands classification, as formulated by Knapp and Zuckerman, exhibits any representation with an invariant…

Let $G(\mathbb{R})$ be a real reductive group. Suppose $\pi$ is an irreducible representation of $G(\mathbb{R})$ having a Whittaker model, and consider three invariants of $\pi$ related to nilpotents elements of the Lie algebra of $G$ (or…

In this paper we develop two types of tools to deal with differentiability properties of vectors in continuous representations $\pi \: G \to \GL(V)$ of an infinite dimensional Lie group $G$ on a locally convex space $V$. The first class of…

A host algebra of a (possibly infinite dimensional) Lie group $G$ is a $C^*$-algebra whose representations are in one-to-one correspondence with certain continuous unitary representations $\pi \colon G \to \U(\cH)$. In this paper we present…

We consider branching laws for the restriction of some irreducible unitary representations $\Pi$ of $G=O(p,q)$ to its subgroup $H=O(p-1,q)$. In Kobayashi (arXiv:1907.07994), the irreducible subrepresentations of $O(p-1,q)$ in the…

Let $G$ be a real reductive connected Lie group and $\sigma$ an involution of $G$. Let $H$ denote the identity component of the group of fixed points of $\sigma$, $\mathfrak g$ the Lie algebra of $G$ and $\mathfrak q$ the -1 eigenspace of…

Let $H \leq S_n$ be an intransitive group with orbits $\Omega_1, \Omega_2, \ldots ,\Omega_k$. Then certainly $H$ is a subdirect product of the direct product of its projections on each orbit, $H|_{\Omega_1} \times H|_{\Omega_2} \times…