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

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.