\(L^2\)-singular dichotomy for orbital measures of classical compact Lie groups
Let \(G\) be a compact Lie group. For any \(h\in G\) let \(C_h\) denote the conjugacy class containing \(h\). The orbital measure \(\mu_h\) supported on \(C_h\) is the \(G\)-invariant measure defined by \(\int_G f\, d\mu_h= \int_G f(ghg^{-1})\, dm_G(g)\) for \(f\) continuous, where \(m_G\) is the Haar measure on \(G\). In the paper under review the authors prove that for any classical, compact, simple, connected Lie group \(G\), the \(G\)-invariant orbital measures supported on non-trivial conjugacy classes satisfy a surprising ``\(L^2\)-singular dichotomy'': Either \(\mu_h^k\in L^2(G)\) or \(\mu_h^k\) is singular to the Haar measure on \(G\), where \(\mu_h^k=\mu_h*\dots*\mu_h\) is the convolution power. The minimum exponent \(k\) for which \(\mu_h^k\in L^2(G)\) is specified; it depends on the Lie properties of the element \(h\in G\). As a corollary, the authors complete the solution to a classical problem -- to determine the minimum exponent \(k\) such that \(\mu^k\in L^1(G)\) for all central, continuous measures \(\mu\) on \(G\).
scientific article; zbMATH DE number 5614813
scientific article