Infinite Dimensional Multiplicity Free Spaces III: Matrix Coefficients and Regular Functions

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 abelian algebra under convolution. In this paper, we consider the case when $K=O(n)$. In [1], we know the only possible Galfand pairs for $(K,F(n))$ is $(O(n),F(n))$, $(SO(n),F(n))$. So we just consider the case $(O(n),F(n))$, the other case ca...

Preface (A.Vershik) - about these texts (3.); I.Interpolation between inductive and projective limits of finite groups with applicatons to linear groups over finite fields; II.The characters of the groups of almost triangle matrices over finite filed; III.A Law of Large Numbers for the characters of GL_n(k) over finite field k; IV.An outline of construction of factor representations of the group GLB(F_q).

Let $G$ be a locally compact group and $1\leq p<\infty$. A continuous unitary representation $\pi\!: G\to B(\mathcal{H})$ of $G$ is an $L^p$-representation if the matrix coefficient functions $s\mapsto \langle \pi(s)x,x\rangle$ lie in $L^p(G)$ for sufficiently many $x\in \mathcal{H}$. Brannan and Ruan defined the $L^p$-Fourier algebra $A_{L^p}(G)$ to be the set of matrix coefficient functions of $L^p$-representations. Similarly, the $L^p$-Fourier-Stieltjes algebra $B_{L^p}(G)...

Let $G$ be a complex reductive algebraic group with Lie algebra $\mathfrak{g}$ and let $G_{\mathbb{R}}$ be a real form of $G$ with maximal compact subgroup $K_{\mathbb{R}}$. Associated to $G_{\mathbb{R}}$ is a $K \times \mathbb{C}^{\times}$-invariant subvariety $\mathcal{N}_{\theta}$ of the (usual) nilpotent cone $\mathcal{N} \subset \mathfrak{g}^*$. In this article, we will derive a formula for the ring of regular functions $\mathbb{C}[\mathcal{N}_{\theta}]$ as a representat...

Let $G$ be a connected semisimple Lie group with finite centre and $K$ be a maximal compact subgroup thereof. Given a function $u$ on $G$, we define $\mathcal{A} u$ to be the root mean square average over $K$, acting both on the left and the right, of $u$. We show that for all unitary representations $\pi$ of $G$, there exists a unique minimal positive-real-valued spherical function $\phi_{\lambda}$ on $G$ such that $\mathcal{A} \langle \pi(\cdot) \xi, \eta \rangle \leq \Vert...

Let $N$ be a simply connected, connected nilpotent Lie group which admits a uniform subgroup $\Gamma.$ Let $\alpha$ be an automorphism of $N$ defined by $\alpha\left( \exp X\right) =\exp AX.$ We assume that the linear action of $A$ is diagonalizable and we do not assume that $N$ is commutative. Let $W$ be a unitary wavelet representation of the semi-direct product group $\left\langle \cup_{j\in\mathbb{Z}}\alpha^{j}\left( \Gamma\right) \right\rangle \rtimes\left\langle \alpha\...

The Segal algebra $\mathbf{S}_{0}(G)$ is well defined for arbitrary locally compact Abelian Hausdorff (LCA) groups $G$. It is a Banach space that exhibits a kernel theorem similar to the well-known Schwartz kernel theorem. Specifically, we call this characterization of the continuous linear operators from $\mathbf{S}_{0}(G_{1})$ to $\mathbf{S}'_{0}(G_{2})$ by generalized functions in $\mathbf{S}'_{0}(G_{1} \times G_{2})$ the "outer kernel theorem". The main subject of this pa...

We study the Segal-Bargmann transform, or the heat transform, $H_t$ for a compact symmetric space $M=U/K$. We prove that $H_t$ is a unitary isomorphism $H_t : L^2(M) \to \cH_t (M_\C)$ using representation theory and the restriction principle. We then show that the Segal-Bargmann transform behaves nicely under propagation of symmetric spaces. If $\{M_n=U_n/K_n,\iota_{n,m}\}_n$ is a direct family of compact symmetric spaces such that $M_m$ propagates $M_n$, $m\ge n$, then this ...

The spectrum of a Gelfand pair of the form (K lx N, K), where N is a nilpotent group, can be embedded in a Euclidean space Rd . The identification of the spherical transforms of K-invariant Schwartz functions on N with the restrictions to the spectrum of Schwartz functions on Rd has been proved already when N is a Heisenberg group and in the case where N = N3,2 is the free two-step nilpotent Lie group with three generators, with K = SO3 [2, 3, 11]. We prove that the same id...

In this paper, we study the group Fourier transform and the Kohn-Nirenberg quantization for homogeneous Lie groups as mappings between certain Gelfand triples. For this, we restrict our considerations to the case, where the homogeneous Lie group $G$ admits irreducible unitary representations, that are square integrable modulo the center $Z(G)$ of $G$, and where $\dim Z(G)=1$. Replacing the Schwartz space by a certain subspace $\mathcal S_*(G) \hookrightarrow \mathcal S(G)$, w...