Proving Admissibility of Irreducible Unitaries

• Higher commutators • Godement’s principle: bounding dimensions of irreducibles • The von Neumann (strong) density theorem • Lie’s Theorem on irreducible finite-dimensional representations of connected solvable groups • Separating families of irreducible finite-dimensional representations • Subrepresentation Theorem for finite-dimensional representations of reductive linear real Lie groups • Theorem on admissibility for reductive real linear Lie groups • Background for the p-adic case • HarishChandra’s reduction to the supercuspidal case • The Bernstein estimate and a corollary • Kazhdan’s estimate on commutative subalgebras of Mn(C) • Proof of Bernstein’s estimate