Analytic parameter dependence of Harish-Chandra modules for real reductive Lie groups - a family affair

Abstract

This thesis is written in the subfield of mathematics known as representation theory of real reductive Lie groups. Let G be a Lie group in the Harish-Chandra class with maximal compact subgroup K and Lie algebra g. Let Omega be a connected complex manifold. By a family of G-representations parametrized by Omega we understand an admissible K-representation on a Frėchet space V together with a continuous map pi from Omega times G to GL(V), holomorphic in the first variable such that for each parameter value zeta in Omega the corresponding map pi(zeta, .) equips V with the structure of an admissible G-representation. The typical example are family principal series representations, where Omega is the dual of the complexified Lie algebra of A. By a family of (g, K)-modules for G, parametrized by Omega we understand an admissible, countable dimensional K-representation V together with a map pi from Omega times U(g) to End(V), holomorphic in the first variable and linear in the second such that for each fixed parameter value zeta in Omega the corresponding map pi(zeta, .) equips V with the structure of a (g, K) module. These families of (g, K)-modules are the object of study in this thesis. Our main result include: 1. Suppose that for one parameter value the corresponding (g, K) module is finitely generated (i.e. a Harish-Chandra module) then every (g, K)-module in the family is finitely generated. In such case we will speak of a family of Harish-Chandra modules. The generating subspace can be chosen uniformly over compact subsets of the parameter space Omega. 2. The set of parameter values at which the family is reducible is a closed analytic subvariety of Omega. In case the parameter dependence is moreover polynomial this set is a locally finite union of countably many closed algebraic submanifolds. When this subset has positive codimension we speak of a generically irreducible family. 3. Suppose that G has real rank one. The K-finite matrixcoefficients of a family of Harish-Chandra modules are real analytic as function on Omega times G and in addition holomorphic in the first variable. 4. (Subrepresentation theorem for families). Assume that G has real rank 1 and that Omega is one dimensional. Let (V, pi) be a family of Harish-Chandra modules. Then each zeta in Omega has a neighborhood U such that the restriction of the family to U embeds holomorphically into a family of induced representations, induced from a family finite dimensional representations of the minimal parabolic subgroup. 5. Under the assumptions of 4, let (V, pi) be generically irreducible. Then every zeta in Omega has a neighborhood which has a finite cover U such that the pull back to U of the family (V, pi) is isomorphic to the family of Harish-Chandra modules of a family of G-representations, parametrized by U. Moreover if the infinitesimal character of the family depends holomorphic on the parameter, then there is no need to pass to a cover. We first discuss the main ideas in the more insightful case of the group SL(2, R).