Remarks on Gelfand pairs associated to non-type-I solvable Lie groups (Q1279644)

Suppose that \(S\) is a connected and simply connected compact Lie group acting on \(S\) as automorphisms. \((K;S)\) is called a Gelfand pair if the Banach \(*\)-algebra \(L^1_K(S)\) of all \(K\)-invariant integrable functions on \(S\) is a commutative algebra. The main result is: \((K;S)\) is a Gelfand pair if and only if there exists a sufficiently large family \(F\subset\widehat S\) of irreducible unitary representations of \(S\) such that for each \(\pi\in F\): (1) the stabilizer \(K_\pi\) in \(K\) is compact; (2) each irreducible projective representation of \(K_\pi\) appears at most once in the decomposition of \(W_\pi\), where \(\widehat S\) is the unitary dual.