Dirac cohomology of Wallach representations

Abstract

Let G be either the metaplectic double cover of Sp(2n, ‫,)ޒ‬ or SO * (2n), or SU( p, q). Let g be the complexified Lie algebra of G and let K be a maximal compact subgroup of G. Let X be one of the Wallach modules for the pair (g, K ). In other words, X corresponds to a discrete point in the classification of unitary lowest weight modules with scalar lowest Ktype. The purpose of this paper is to calculate the Dirac cohomology of X. Our approach is based on the explicit knowledge of the K-types of X. We establish a bijection between certain K-types E i of X and certainKtypes F i of the spin module, whereK is the spin double cover of K . The Dirac cohomology is then realized as the set of Parthasarathy-Ranga-RaoVaradarajan components of E i ⊗ F i .

Unitary representations with non-zero Dirac cohomology for complex E 6

“…. , ̟ 5 , 1 4 ζ} satisfy (14). Since E µ is assumed to be a K(R)-type, it follows that the coordinates [a, b, c, d, e, f ] of {µ − ρ (j) n } + ρ c meet the requirement (14).…”

“…After the work [12], Dirac cohomology became a refined invariant for Lie group representations, and a natural problem arose: could we classify G(R) The current paper aims to achieve the classification for E 6(− 14) , by which we actually mean the group E6 h in atlas. Note that atlas [28] is a software which computes many aspects of questions pertaining to Lie group representations.…”

Dirac cohomology and the bottom layer K-types

“…Our assertion follows from this last characterization and Parthasarathy's Dirac inequality D 2 ≥ 0. For a more detailed explanation, see [HPP,Proposition 3.2].…”

Computing the associated cycles of certain Harish-Chandra modules

“…it is a module for the spin double coverK of K R (finite-dimensional if M is admissible). This invariant was introduced in [V2], it turned out to be very interesting and also quite computable; see for example [HP1,HP2,HKP,HPR,HPP,HPZ,BP1,BP2,BPT,MP,MZ,DH].…”