Gaussian maps and tensor products of irreducible representations

Abstract

Let G be a simply-connected complex simple Lie group, P a parabolic subgroup, L an ample line bundle on G/P. Then Γ(L) is an irreducible G-module, and every such arises in this way. In this paper, we initiate the study the G-filtration of the tensor product Γ(L)⊗Γ(M) by order of vanishing along the diagonal of G/P×G/P. Each subquotient of the filtration is contained in some Γ(L⊗M⊗SiΩ G/P 1 , so these (reducible) G-modules are examined. The first non-trivial piece of the filtration is given by a Gaussian map of [W3]; we conjecture the surjectivity in case L, M are ample, and prove this in a number of cases. One is led immediately to the study of a natural P-filtration of Γ(L) by order of vanishing of hyperplanes at a fixed point of G/P; tensoring this filtration with Γ(M) and inducing up to G gives the diagonal filtration, from which many familiar results are deduced.