Infinite-Dimensional Representations

Abstract

This third chapter presents a brief and somewhat sketchy introduction to the theory of unitary representations of reductive and semisimple Lie groups G. The basic fact for an irreducible unitary representation π of G on a Hilbert space ℋ, is that every irreducible representation к of a maximal compact subgroup K ⊂ G has multiplicity m(к, π| к) ≤ dim к. This yields up the infinitesimal character χπ: L(g)→ ℂ and the distribution character θπ: C c ∞ (G) → ℂ, and consequently the differential equations $$z({\theta _\pi }) = \,{\chi _\pi }(z){\theta _\pi }$$ for $$z \in L{\text{(g)}}$$ which are the starting point for serious harmonic analysis on G.