Related papers: Some regular symmetric pairs

A symmetric pair of reductive groups $(G,H,\theta)$ is called stable, if every closed double coset of $H$ in $G$ is preserved by the anti-involution $g\mapsto \theta(g^{-1})$. In this paper, we develop a method to verify the stability of…

Let V be a quadratic space with a form q over an arbitrary local field F of characteristic different from 2. Let $W=V \oplus Fe$ with the form Q extending q with Q(e)=1. Consider the standard embedding of O(V) into O(W) and the two-sided…

We first prove, for pairs consisting of a simply connected complex reductive group together with a connected subgroup, the equivalence between two different notions of Gelfand pairs. This partially answers a question posed by Gross, and…

In this paper we prove that the symmetric pair $(GL_{n+k}(F),GL_n(F) \times GL_k(F))$ is a Gelfand pair for any local field F of characteristic 0. For non-archimedean F it has been proven in [JR]. We use techniques developed in [AG2] to…

For a non-compact simple Lie algebra $\mathfrak{g}$ over $\mathbb{R}$, we denote by $\mathcal{O}^{\mathbb{C}}_{\min,\mathfrak{g}}$ the unique complex nilpotent orbit in $\mathfrak{g} \otimes_\mathbb{R} \mathbb{C}$ containing all minimal…

Extending results of Kazhdan to the relative case, we relate harmonic analysis over some spherical spaces G(F)/H(F), where F is a field of positive characteristic, to harmonic analysis over the spherical spaces G(E)/H(E), where E is a…

It is well known that the pair $(\mathcal{S}_n,\mathcal{S}_{n-1})$ is a Gelfand pair where $\mathcal{S}_n$ is the symmetric group on $n$ elements. In this paper, we prove that if $G$ is a finite group then $(G\wr \mathcal{S}_n, G\wr…

We consider symmetric Gelfand pairs $(G,K)$ where $G$ is a compact Lie group and $K$ a subgroup of fixed point of an involutive automorphism. We study the regularity of $K$-bi-invariant matrix coefficients of $G$. The results rely on the…

For a symmetric space (G,H), one is interested in understanding the vector space of H-invariant linear forms on a representation \pi of G. In particular an important question is whether or not the dimension of this space is bounded by one.…

Let G be a locally compact group and let K be a compact subgroup of Aut(G), the group of automorphisms of G. The pair $(G, K )$ is a Gelfand pair if the algebra $L^{1}_{K}(G)$ of K-invariant integrable functions on G is commutative under…

A strong Gelfand pair $(G, H)$ is a finite group $G$ and a subgroup $H$ where every irreducible character of $H$ induces to a multiplicity-free character of $G$. We determine the strong Gelfand pairs of the sporadic groups, their…

We prove that (GL_{2n}(C),Sp_{2n}(C)) is a Gelfand pair. More precisely, we show that for an irreducible smooth admissible Frechet representation (\pi,E) of GL_{2n}(C) the space of continuous functionals Hom_{Sp_{2n}(\cc)}(E,C) is at most…

Classical symmetric pairs consist of a symmetrizable Kac-Moody algebra $\mathfrak{g}$, together with its subalgebra of fixed points under an involutive automorphism of the second kind. Quantum group analogs of this construction, known as…

Normalized irreducible characters of the symmetric group S(n) can be understood as zonal spherical functions of the Gelfand pair $(S(n)\times S(n),\Diag S(n))$. They form an orthogonal basis in the space of the functions on the group S(n)…

The notion of Gelfand pair (G, K) can be generalized if we consider homogeneous vector bundles over G/K instead of the homogeneous space G/K and matrix-valued functions instead of scalar-valued functions. This gives the definition of…

Let $\mathcal{OG}(4)$ denote the family of all graph-group pairs $(\Gamma, G)$ where $\Gamma$ is finite, 4-valent, connected, and $G$-oriented ($G$-half-arc-transitive). A subfamily of $\mathcal{OG}(4)$ has recently been identified as…

Given a Gelfand pair of Lie groups, we identify the spectrum with a suitable subset of C^n and we prove the equivalence between Gelfand topology and euclidean topology.

Let $F(n)$ be a connected and simply connected free 2-step nilpotent lie group and $K$ be a compact subgroup of Aut($F(n)$). We say that $(K,F(n))$ is a Gelfand pair when the set of integrable $K$-invariant functions on $F(n)$ forms an…

Let F be an arbitrary local field. Consider the standard embedding of GL(n,F) into GL(n+1,F) and the two-sided action of GL(n,F) \times GL(n,F) on GL(n+1,F). In this paper we show that any GL(n,F) \times GL(n,F)-invariant distribution on…

A double-normal pair of a finite set $S$ of points from $R^d$ is a pair of points $\{p,q\}$ from $S$ such that $S$ lies in the closed strip bounded by the hyperplanes through $p$ and $q$ perpendicular to $pq$. A double-normal pair $pq$ is…