In mathematics, a holomorphic discrete series representation is a discrete series representation

Discrete series representation

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²...

 of a semisimple Lie group that can be represented in a natural way as a Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

 of holomorphic functions. The simple Lie groups with holomorphic discrete series are those whose symmetric space is Hermitian

Hermitian symmetric space

In mathematics, a Hermitian symmetric space is a Kähler manifold M which, as a Riemannian manifold, is a Riemannian symmetric space. Equivalently, M is a Riemannian symmetric space with a parallel complex structure with respect to which the Riemannian metric is Hermitian...

. Holomorphic discrete series representations are the easiest discrete series representations to study because they have highest or lowest weights, which makes their behavior similar to that of finite dimensional representations of compact Lie groups.

found the first examples of holomorphic discrete series representations, and classified them for all semisimple Lie groups.

and described the characters of holomorphic discrete series representations.