Universality of Holomorphic Discrete Series

The goal here is to recover an apocryphal result on the structure of holomorphic discrete series representations of symplectic groups Spn(R) and unitary groups U(p, q) for sufficiently high highest weight of the lowest K-type. The same sort of argument applies to other groups of hermitian type, for example the classical groups O(n, 2) and O∗(2n). For Sp(n), the maximal compact is isomorphic to U(n). For ρ with highest weight (m1, . . . ,mn) it is sufficient to assume that m1 ≥ . . . ≥ mn ≥ n to reach the conclusions below. For U(p, q), the maximal compact is U(p) × U(q), and for ρ with highest weight (m1, . . . ,mp)× (m1, . . . ,mq) it is sufficient to assume that m1 ≥ . . . ≥ mp ≥ p+ q − 1 2 and m1 ≥ . . . ≥ mq ≥ p+ q − 1 2