Sphericity and multiplication of double cosets for infinite-dimensional classical groups

TL;DR

This paper constructs spherical subgroups in infinite-dimensional classical groups, explores the semigroup structure of double cosets, and extends operator colligation concepts to these groups, revealing new algebraic and representation-theoretic insights.

Contribution

It introduces new spherical subgroups in infinite-dimensional classical groups and develops a semigroup structure on double cosets, extending operator colligation frameworks.

Findings

Semigroup structure on double cosets in infinite-dimensional groups

Construction of spherical subgroups not necessarily symmetric

Semigroup actions on spaces of fixed vectors in unitary representations

Abstract

We construct spherical subgroups in infinite-dimensional classical groups $G$ (usually they are not symmetric and their finite-dimensional analogs are not spherical). We present a structure of a semigroup on double cosets $L\setminus G/L$ for various subgroups $L$ in $G$, moreover these semigroups act in spaces of $L$-fixed vectors in unitary representations of $G$. We also obtain semigroup envelops of groups $G$ generalizing constructions of operator colligations.