Counting irreducible representations of general linear groups and unitary groups

Let G G be a general linear group over R R , C C , or H H , or a real unitary group. In this paper, we precisely describe the number of isomorphism classes of irreducible Casselman–Wallach representations of G G with a given infinitesimal character and a given complex associated variety, expressed in terms of certain combinatorial data called painted Young diagrams and assigned Young diagrams.