arXiv subjects

Linda Saal

Publications and source records attributed to Linda Saal.

5 recordsLinked to original sources

arXiv2025

A novel approach through spherical functions in the characterization of invariant functions

Given a compact subgroup K of the orthogonal group acting on the Euclidean space Rn, Gerald Schwarz proved that every smooth K-invariant function on Rn can be expressed as a smooth function of a generating set of $K$-invariant polynomials on n variables. The goal of this work is to provide an alternative and more straightforward proof of this result, based on Gelfand theory, with a particular focus on spherical functions.

math.FA↗

arXiv2020

On commutative homogeneous vector bundles attached to nilmanifolds

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 commutative homogeneous vector bundles. Being a Gelfand pair is a necessary condition of being a commutative homogeneous vector bundle. For the case in which G/K is a nilmanifold having square-integrable representations, in a previous article we determined a big family of commutative homogeneous vector bundles. In this paper, we complete that classification.

math.RT↗

arXiv2018

Matrix spherical analysis on nilmanifolds

Given a nilpotent Lie group $N$, a compact subgroup $K$ of automorphisms of $N$ and an irreducible unitary representation $(τ,W_τ)$ of $K$, we study conditions on $τ$ for the commutativity of the algebra of $\mathrm{End}(W_τ)$-valued integrable functions on $N$, with an additional property that generalizes the notion of $K$-invariance. A necessary condition, proved by F. Ricci and A. Samanta, is that $(K\ltimes N,K)$ must be a Gelfand pair. In this article we determine all the commutative algebras from a particular class of Gelfand pairs constructed by J. Lauret.

math.RT↗

arXiv2012

Explicit fundamental solutions of some second order differential operators on Heisenberg groups

Let $p,q,n$ be natural numbers such that $p+q=n$. Let $\FF$ be either $\CC$, the complex numbers field, or $\HH$, the quaternionic division algebra. We consider the Heisenberg group $N(p,q,\FF)$ defined as $N(p,q,\FF)=\FF^{n}\times \mathfrak{Im}\FF$, with group law given by $$(v,ζ)(v',ζ')=(v+v', ζ+ζ'-{1/2} \mathfrak{Im} B(v,v')),$$ where $B(v,w)=\sum_{j=1}^{p} v_{j}\bar{w_{j}} - \sum_{j=p+1}^{n} v_{j}\bar{w_{j}}$. Let $U(p,q,\FF)$ be the group of $n\times n$ matrices with coefficients in $\FF$ that leave invariant the form $B$. In this work we compute explicit fundamental solutions of some second order differential operators on $N(p,q,\FF)$ which are canonically associated to the action of $U(p,q,\FF)$.

math.RT↗

arXiv2003

Reality of non-Fock Spinors

The infinite dimensional Clifford Algebra has a maze of irreducible unitary representations. Here we determine their type -real, complex or quaternionic. Some, related to the Fermi-Fock representations, have no real or quetrnionic structures. But there are many on L(2) of the circle that do and which seem to have analytic meaning.

math.RT↗