Transfer of Unitary Representations

Introduction. This paper has as its primary purpose a clear exposition of the idea in [21] which we describe as transfer between real forms of a semi-simple Lie group over C. The basic point is that there are representations of one real form that can be more easily understood in the context of another real from. An interesting example is a minimal representation (that is annihilated by the Joseph ideal) of a split group over R. In the case when the complexification admits a Hermitian symmetric real form, minimal representations of that real form are part of the “analytic continuation of the holomorphic discrete series” (cf. [5]). For example, in the case of SO(4, 4) or split E7 one can “transfer” the holomorphic minimal representations of SO(6, 2) and Hermitian symmetric E7 respectively. If there is no Hermitian symmetric real form then since there is always a quaternionic real form one can do a similar transfer. We feel that this exposition is necssary since the original discussion had many misprints which could be confusing and also lacked, in some cases, proper reference to earlier related work. The secondary purpose is to give some new examples of its applicability. These examples give more evidence of a possible deep connection between the notion of transfer and Howe’s theory of dual pairs. We will now give a description of the paper. Let GC be a connected, simply connected semi-simple Lie group over C. LetG and G be two real forms of GC with respective maximal compact subgroups K and K . Let θ and θ be corresponding Cartan involutions looked upon as automorphisms of GC. We assume that θθ ′ = θθ. Set M = K∩K . Let g, g, gC denote the Lie algebras of G,G, GC respectively. We shall use similar notation throughout the article thus the Lie groups will be denoted by capital letters and the corresponding Lie algebra by the corresponding lower case fraktur letter. Let (π, V ) be an irreducible admissible (g,K )-module. Then we may look upon it as a (gC,M)-module. We can apply the Zuckerman functors (Γ K M ) i to V and get (g,K)-modules