Classification of Discrete Series by Minimal K-type

Following the proof by Hecht and Schmid of Blattner’s conjecture for K multiplicities of representations belonging to the discrete series it turned out that some results which were earlier known with some hypothesis on the Harish-Chandra parameter of the discrete series representation could be extended removing those hypotheses. For example this was so for the geometric realization problem. Occasionally a few other results followed by first proving them for Harish-Chandra parameters which are sufficiently regular and then using Zuckerman translation functors, wall crossing methods, etc. Recently, Hongyu He raised the question (private communication) of whether the characterization of a discrete series representation by its lowest K-type, which was proved by this author and R. Hotta with some hypothesis on the HarishChandra parameter of the discrete series representations, can be extended to all discrete series representations excluding none, using a combination of these powerful techniques. In this article we will answer this question using Dirac operator methods and a result of Susana Salamanca-Riba. 1. Preliminaries and statement of the main theorem Let G be a connected linear real semisimple Lie group whose complexification is simply connected. We assume that G admits discrete series (which implies that G has a compact Cartan subgroup). In Sections 1, 2 and 3 of this paper we write G instead of G; in Section 4G will not be assumed to be connected and G will denote the identity component. Let T be a compact Cartan subgroup of G. Let K be a maximal compact subgroup of G containing T . Denote by t, g, k the corresponding Lie subalgebras and t, g, k their complexifications. Let g = k + p be the Cartan decomposition. Here p is the orthogonal complement of k in g. Fix a positive system P in the set of roots Δ(g, t). The set of roots Δ(k, t) is a subset of Δ(g, t). Intersecting P with Δ(k, t) we get a positive system Pk of Δ(k , t). The complement of Pk in P is denoted Pn and is called the set of noncompact positive roots. The root spaces g for α ∈ Pn ⋃ −Pn span p. Let λ ∈ tR = Hom ( √ −1t,R) be a P -dominant integral linear form. Denote by ρ half the sum of the roots in P , by ρk half the sum of the roots in Pk and by ρn half the sum of the roots in Pn. Then λ + ρ is P -dominant regular integral and ρn is Pk-dominant Received by the editors July 29, 2014 and, in revised form, December 4, 2014 and August 14, 2015. 2010 Mathematics Subject Classification. Primary 22E46; Secondary 22D30.