Mogens Flensted-Jensen

Discrete series for semisimple symmetric spaces

We give a sufficient condition for the existence of minimal closed G-invariant subspaces of L2(G/H) for a semisimple symmetric space G/H. As a semisimple Lie group with finite center may always be considered as a symmetric space, this provides, in particular, a new and elementary proof of Harish-Chandras result that G has a discrete series if rank (G) = rank (K), where K is a maximal compact subgroup. Let G be a connected noncompact semisimple Lie group, let z be an involution on G, and let H be the connected component of the fixed-point group Gr containing the identity. Then G/H is a semisimple symmetric space, and the group G acts by left translation on C*(G/H) and L2(G/H). In the introduction we will, for simplicity, assume that G has a finite center. By the discrete series for G/H we shall mean the set of equivalence classes of the representations of G on minimal closed invariant subspaces of L2(G/H). Let a be a Cartan involution commuting with z. The fixed-point group K for a is a maximal compact subgroup. Our main result is THEOREM 1.1. The discrete series for G/H is nonempty and infinite if

Boundedness of Certain Unitarizable Harish-Chandra Modules

This chapter discusses the boundedness of certain unitarizable Harish-Chandra modules. It presents an assumption where f is an element of C ∞ G) or a column vector of elements of C ∞ (G). It presents a problem where V f is a unitarizable Harish-Chandra module. It explains that that f satisfies some conditions, such as, f corresponds to a section of the G -homogeneous vector bundle associated to a representation of a certain subgroup of G and/or f satisfies certain differential equations etc. For example, if f is a zonal spherical function, then one can conclude that f is bounded because f coincides with the matrix coefficient with respect to a normalized K -fixed vector of the corresponding irreducible unitary representation of G .

Basic harmonic analysis for pseudo-Riemannian symmetric spaces

We give a survey of the present knowledge regarding basic questions in harmonic analysis on pseudo-Riemannian symmetric spaces G /H, where G is a semisimple Lie group: The definition of the Fourier transform, the Plancherel formula, the inversion formula and the Paley-Wiener theorem.