A combinatorial interpretation of the Bernstein degree of modules of covariants

We introduce sets R () consisting of semistandard tableaux of shape, subject to a certain restriction on the initial column. These R () are generalizations of the rational, symplectic, or orthogonal tableaux (which furnish a weight basis for the irreducible finite-dimensional representations U_ of the classical groups H = GLₖ, Sp₂₊, or Oₖ, respectively). As our main result, we show that the cardinality of R () gives the Bernstein degree (i. e. , multiplicity) of the modules of covariants (of type U_) 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 Gₑ. Notably, our result using R () is valid regardless of the rank of H, whereas the previous result of Nishiyama-Ochiai-Taniguchi (expressing the Bernstein degree in terms of U_) holds only when k is at most the real rank of Gₑ. In effect, as k increases beyond this range, |R () | interpolates between U_ and the dimension of a certain "limiting" K-module.