A combinatorial interpretation of the Bernstein degree of modules of covariants

We introduce sets R(σ) consisting of semistandard tableaux of shape σ, subject to a certain restriction on the initial column. These R(σ) are generalizations of the rational, symplectic, or orthogonal tableaux (which furnish a weight basis for the irreducible finite-dimensional representations U_σ of the classical groups H = GLₖ, Sp₂ₖ, or Oₖ, respectively). As our main result, we show that the cardinality of R(σ) gives the Bernstein degree (i.e., multiplicity) of the modules of covariants (of type U_σ) of each classical group H. Via Howe duality, these modules can also be viewed as (g,K)-modules of unitary highest weight representations of a real reductive group GR. Notably, our result using R(σ) is valid regardless of the rank of H, whereas the previous result of Nishiyama-Ochiai-Taniguchi (expressing the Bernstein degree in terms of U_σ) holds only when k is at most the real rank of GR. In effect, as k increases beyond this range, |R(σ)| interpolates between U_σ and the dimension of a certain "limiting" K-module.