Holomorphic discrete series vs. discrete series

Question

(I apologize in advance if this question is too naive for experts.) Let $G$ be a real semisimple Lie group. I know that holomorphic discrete series representations are only a part of all the discrete series of $G$, and the definition and construction depends on the Hermitian (complex) structure of the related symmetric spaces (domains).

I want to ask: can we characterize the holomorphic ones among all discrete series following Harish-Chandra's parametrization of all discrete series?

To be more concrete: we fix a maximal compact subgroup $K$ of $G$, and a Cartan subgroup $T$ of $K$. Let $\Delta=\Delta(G,K)$ be the root system and $\Delta_c=\Delta(K,T)=\Delta_c^+\cap \Delta_c^-$ the set of compact roots. Let $P(\Delta):=\{\lambda\in \mathfrak{t}_{\mathbb{C}}^*: \langle \lambda ,\alpha^\vee\rangle \in\mathbb{Z}\}$ be the weight lattice of $\Delta$. Then Harish-Chandra's parametrization is that the discrete series are in bijective correspondence with the set $$\{\lambda\in P(\Delta): \langle \lambda,\alpha^\vee\rangle \neq 0~(\forall \alpha\in \Delta),\text{ and }\langle \lambda, \alpha^\vee\rangle >0~(\forall \alpha\in \Delta_c^+)\}.$$ Of course, there're some conditions related to the representations (such as $K$-types and infinitesimal characters) that characterize the corresponding discrete series representation.

Then my question is: given $G$ and the above data, suppose we know the existence of the Hermitian structure on $G/K$, can we characterize parameters of holomorphic discrete series in the above parameter set?

(Thanks a lot in advance for any suggestions!)

Answer 1

The holomorphic discrete series are precisely parametrized by the $\lambda$ as indicated, which satisfy the following additional condition. Let $\Delta^+=\{\alpha\in\Delta\mid \langle\lambda,\alpha^\vee\rangle >0\}$. This is a set of positive roots of $\Delta(G,T)$. Then every simple root of $\Delta^+$ but one is compact. See Hecht/Schmid, A Proof of Blattner's Conjecture, Section 4.

Answer 2

If I remember correctly, then holomorphic discrete series are a subset of the unitarizable highest (or lowest) weight modules which are Verma modules. There are several parametrizations of these modules, so take your pick.

For example in Enright-Howe-Wallach one picks a normalization of the generator $\zeta$ of $\Delta_c^\perp$ and then the holomorphic discrete series are exhausted by $\{\lambda_0 + z\zeta | z < 0, \lambda_0 \in \Delta_c \text{-integral, dominant} \}$.