Notes on Sobolev spaces on compact classical groups and Stein--Sahi representations
Abstract
We discuss kernels on complact classical groups $G=\U(n)$, $\OO(2n)$, $\Sp(n)$ defined by the formula
K(z,u)=|det(1−z
u
∗
)
|
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.