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

Compact finite-rank nilspaces have become central in the nilspace approach to higher-order Fourier analysis, notably through their role in a general form of the inverse theorem for the Gowers norms. This paper studies these nilspaces per se, and in connection with further refinements of this inverse theorem that have been conjectured recently. Our first main result states that every compact finite-rank nilspace is obtained by taking a free nilspace (a nilspace based on an abe...

In [AG2] we explored the question what symmetric pairs are Gelfand pairs. We introduced the notion of regular symmetric pair and conjectured that all symmetric pairs are regular. This conjecture would imply that many symmetric pairs are Gelfand pairs, and in particular that any connected symmetric pair over C is a Gelfand pair. In this paper we show that the pairs $$(GL(V),O(V)), (GL(V),U(V)), (U(V),O(V)), (O(V \oplus W),O(V) \times O(W)), (U(V \oplus W),U(V) \times U(W))$$...

These notes are an expanded version of a talk given by the second author. Our main interest is focused on the challenging problem of computing Kronecker coefficients. We decided, at the beginning, to take a very general approach to the problem of studying multiplicity functions, and we survey the various aspects of the theory that comes into play, giving a detailed bibliography to orient the reader. Nonetheless the main general theorems involving multiplicities functions (con...

In a preprint from 1982, John Milnor formulated various fundamental questions concerning infinite-dimensional Lie groups. In this note, we describe some of the answers (and partial answers) obtained in the preceding years.

These informal notes concern some basic themes of harmonic analysis related to representations of groups.

It has been shown that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L^1(N)^K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)^K of K-invariant Schwartz functions on N and the space S({\Sigma}) of functions on the Gelfand spectrum {\Sigma} of L^1(N)^K which extend to Schwartz functions on Rd, once {\Sigma} is suitably embedded in Rd. We call ...

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak{n}$ having rational structure constants. We assume that $N=P\rtimes M,$ $M$ is commutative, and for all $\lambda\in \mathfrak{n}^{\ast}$ in general position the subalgebra $\mathfrak{p}=\log(P)$ is a polarization ideal subordinated to $\lambda$ ($\mathfrak{p}$ is a maximal ideal satisfying $[\mathfrak{p},\mathfrak{p}]\subseteq\ker\lambda$ for all $\lambda$ in general posit...

Let $G$ be a locally compact group, $A_p (G)$ be the Herz algebra of $G$ associated with $1 <p< \infty$. We show that $A_p (G)$ is Arens regular if and only if $G$ is a discrete group and for each countable subgroup $H$ of $G$, $A_p (H)$ is Arens regular. In the case $G$ is a countable discrete group we investigate the relations between Arens regularity of $A_p (G)$ and the iterated limit condition. We consider the problem of Arens regularity of $l^1 (G)$ as a subspace of $A_...

We consider the groups G which arise from real semisimple Jordan algebras via the Tits-Koecher-Kantor construction. Such a G is characterized by the fact that it admits a parabolic subgroup P=LN which is conjugate to its opposite, and for which the nilradical N is abelian. In this situation, the Levi component L has a finite number of orbits on N; and each orbit carries a measure which transforms by a character under L. By Mackey theory the space of L2-functions on each orb...

Let G be a Lie group which is the union of an ascending sequence of Lie groups G_n (all of which may be infinite-dimensional). We study the question when G is the direct limit of the G_n's in the category of Lie groups, topological groups, smooth manifolds, resp., topological spaces. Full answers are obtained for G the group Diff_c(M) of compactly supported smooth diffeomorphisms of a sigma-compact smooth manifold M, and for test function groups C^infty_c(M,H) of compactly su...