Let G be a connected semisimple Lie group with a finite center. There are two general methods to construct representations of G, namely, ordinary parabolic induction and cohomological parabolic induction. We define Eisenstein integrals relative to cohomological inductions which generalize Flensted-Jensen's fundamental functions for discrete series. They are analogous to Harish-Chandra's Eisenstein integrals related to ordinary inductions. Fix a maximal compact subgroup K of G. Given an admissible continuous representation of G on a Hilbert space M, and a K type α occurring in M, we introduce the notion of Li positivity of α which is extremely useful in the study of branching laws. Roughly speaking, it says that the α component matrix coefficient of M takes positive definite values on th...[ Read more ]

Let G be a connected semisimple Lie group with a finite center. There are two general methods to construct representations of G, namely, ordinary parabolic induction and cohomological parabolic induction. We define Eisenstein integrals relative to cohomological inductions which generalize Flensted-Jensen's fundamental functions for discrete series. They are analogous to Harish-Chandra's Eisenstein integrals related to ordinary inductions. Fix a maximal compact subgroup K of G. Given an admissible continuous representation of G on a Hilbert space M, and a K type α occurring in M, we introduce the notion of Li positivity of α which is extremely useful in the study of branching laws. Roughly speaking, it says that the α component matrix coefficient of M takes positive definite values on the split part of G. Use our integral, we show that the minimal K types of many interesting representations are Li positive. These include all irreducible unitary representations with nonzero cohomology. In the first chapter, we study the algebraic structure on the space of all matrix coefficients on G. In the second chapter, we relate functions on G to functions on a Riemannian symmetric space. In chapter 3 and chapter 4, we define the analogous Eisenstein integral and relate it to cohomological induction. We define and study Li positivity in the last chapter.