Let \(G\) denote a double cover of \(\text{Sp}(n,\mathbb R)\), \(\text{SO}^\star (2n)\), \(\text{SU}(p,q)\). Fix \(K\) a maximal compact subgroup. \(\mathfrak g=\mathfrak p +\mathfrak k= \mathfrak k + \mathfrak p^+ +\mathfrak p^+\) the Cartan decomposition. \(\mathfrak t\subset\mathfrak k\) denotes a Cartan subalgebra of \(K\). \(\Delta^+(\mathfrak g):=\Delta^+(\mathfrak k) \cup \Delta(\mathfrak p^+)\) is the system of positive roots obtained from a system of positive roots in \(\mathfrak k\) and the roots of \(\mathfrak t \) in \(\mathfrak p^+\). Let \(X\) denote an irreducible Harish-Chandra module of infinitesimal character \(\Lambda\) so that \(\Lambda\) is dominant with respect to \(\Delta^+(\mathfrak g)\) and set \(H_D(X)\) for the Dirac cohomology for \(X\). The main purpose of this note is to compute \(H_D(X)\) if moreover \(X\) is a unitarizable lowest weight module whose lowest \(K\)-type is one-dimensional. For this, let \(w_1 \Lambda, \dots, w_n \Lambda\) denote the \(W(\mathfrak g, \mathfrak t)\) translates of \(\Lambda\) which are dominant and regular for \(\Delta^+(\mathfrak k),\) \(w_j\) are chosen to be of shortest length. The authors show:
\[ H_D(X)=\oplus _{j=1}^{n} E_{w_j\Lambda -\rho_k}. \]
Moreover, for each \(w_i\) there is a unique \(K\)-type \(E_i\) of \(X,\) appearing in \(X\) with multiplicity one, and a unique \(K\)-type \( E_{s(w_i)}\) of the spin representation \(S\) of \(\text{SO}(\mathfrak p)\) followed by the adjoint representation of \(K\) restricted to \(\mathfrak p\) such that \(E_{w_i\Lambda -\rho_k}\) is the irreducible component of \(E_i \otimes E_{s(w_i)}\) whose highest weight is conjugated to \(s(w_i)\) plus the lowest weight of \(E_i\). The paper contains a nice review of Dirac cohomology and Wallach representations.
Complementary work to the note can be found in: [J.-S. Huang, Y.-F. Kang and P. Pandžić, Transform. Groups 14, No. 1, 163–173 (2009; Zbl 1179.22013)] and [J.-S. Huang and P. Pandžić, Dirac operators in representation theory. Mathematics: Theory & Applications. Basel: Birkhäuser (2006; Zbl 1103.22008)]