Reduced \(C ^{*}\)-algebra of the \(p\)-adic group \(GL(n)\). II
Let \(F\) be a non-Archimedean local field, let \(G=GL(n;F),\) and let \(C^*_{\text{red}}(G)\) denote the reduced \(C^*\)-algebra of \(G.\) Let \({\mathcal H}(G)\) be the Hecke algebra of \(G,\) and let \(\Omega\) be a component in the Bernstein variety \(\Omega (G).\) The Bernstein decomposition \({\mathcal H}(G)=\oplus {\mathcal H}(\Omega)\) is a canonical decomposition of \({\mathcal H}(G)\) into two-sided ideals \({\mathcal H}(\Omega).\) A minimal refinement of this decomposition is given. Let \(C^*_{\text{red}}(G//I)\) be the reduced Iwahori-Hecke \(C^*\)-algebra of \(G.\) Structure theorems for this unital \(C^*\)-algebra and the reduced spherical \(C^*\)-algebra are proved using standard techniques of non-commutative Fourier analysis. These results lead to a very explicit description of the tempered dual of \(G=GL(n;F)\) in terms of Bernstein parameters and extended quotients. It it proved that the Plancherel measure on the tempered dual of a reductive \(p\)-adic group is rotation-invariant.
scientific article