Algebraic results on representations of semisimple Lie groups

Abstract:

Let G be a noncompact connected real semisimple Lie group with finite center, and let K be a maximal compact subgroup of G. Let $\mathfrak {g}$ and $\mathfrak {k}$ denote the respective complexified Lie algebras. Then every irreducible representation $\pi$ of $\mathfrak {g}$ which is semisimple under $\mathfrak {k}$ and whose irreducible $\mathfrak {k}$-components integrate to finite-dimensional irreducible representations of K is shown to be equivalent to a subquotient of a representation of $\mathfrak {g}$ belonging to the infinitesimal nonunitary principal series. It follows that $\pi$ integrates to a continuous irreducible Hilbert space representation of G, and the best possible estimate for the multiplicity of any finite-dimensional irreducible representation of $\mathfrak {k}$ in $\pi$ is determined. These results generalize similar results of Harish-Chandra, R. Godement and J. Dixmier. The representations of $\mathfrak {g}$ in the infinitesimal nonunitary principal series, as well as certain more general representations of $\mathfrak {g}$ on which the center of the universal enveloping algebra of $\mathfrak {g}$ acts as scalars, are shown to have (finite) composition series. A general module-theoretic result is used to prove that the distribution character of an admissible Hilbert space representation of G determines the existence and equivalence class of an infinitesimal composition series for the representation, generalizing a theorem of N. Wallach. The composition series of Weylgroup-related members of the infinitesimal nonunitary principal series are shown to be equivalent. An expression is given for the infinitesimal spherical functions associated with the nonunitary principal series. In several instances, the proofs of the above results and related results yield simplifications as well as generalizations of certain results of Harish-Chandra.