Some remarks on measures on non-compact semi-simple Lie groups
Let G be an arbitrary connected, non-compact, semi-simple Lie group with finite centre, K a fixed maximal compact subgroup of G, and m a fixed Haar measure on G. The main results are the following three theorems. Theorem A: Let E be a right K-invariant Borel set in G such that \(0<m(E)<\infty\). If \(\mu\) is a complex (finite) right K-invariant measure on G such that \(\mu(gE)=0\) for all \(g\in G\), then \(\mu =0.\) Theorem B: Let E be a K-bi-invariant Borel set in G with \(\bar E\) compact and \(m(E)>0\). Then there exists an elementary spherical function \(\phi\) such that \(\int_{gE}\phi(x)dm(x)=0\) for all \(g\in G.\) Theorem C: Let G be the group SL(2,R) and E a K-bi-invariant Borel set in G with \(0<m(E)<\infty\). Then there exists a non-trivial function \(f\in L^ 1(G)\) such that \(\int_{gE}f(x)dm(x)=\int_{Eg}f(x)dm(x)=0\) for all \(g\in G\).
measures on non-compact semi-simple Lie groups
scientific article; zbMATH DE number 3844348