Notes on Sobolev spaces on compact classical groups and Stein--Sahi representations

TL;DR

This paper analyzes kernels on compact classical groups, deriving explicit Plancherel formulas, positive-definiteness intervals, and models for Sahi's unipotent representations, advancing understanding of harmonic analysis on these groups.

Contribution

It provides explicit Plancherel formulas and models for unipotent representations on classical groups, extending harmonic analysis and representation theory.

Findings

Explicit Plancherel formulas for kernels on classical groups

Intervals of positive-definiteness identified

Models for Sahi's unipotent representations constructed

Abstract

We discuss kernels on complact classical groups $G=\U(n)$, $\OO(2n)$, $\Sp(n)$ defined by the formula $K(z,u)=|\det(1-zu^*)|^s$. We obtain the explicit Plancherel formula for these kernels and the interval of positive-definiteness. We also obtain explicit models for Sahi's 'unipotent' representations.