Closed product formulas for extensions of generalized Verma modules

Abstract

We give explicit combinatorial product formulas for the polynomials encoding the dimensions of the spaces of extensions of ( g , p ) (g,p) -generalized Verma modules, in the cases when ( g , p ) (g,p) corresponds to an indecomposable classic Hermitian symmetric pair. The formulas imply that these dimensions are combinatorial invariants. We also discuss how these polynomials, defined by Shelton, are related to the parabolic R R -polynomials introduced by Deodhar.