Tensor products of maximal degenerate series representations of the group SL( n, R)

Abstract

This paper has two goals: firstly to study the maximal degenerate series representations π ± μ, gn, μ Π ± u,v , u∈C ν = 0, 1 of SL( n, ℝ) and, secondly, to decompose the tensor products π + μ, gn ⊗ π − μ, gn μ ℝ, ν =0, 1 (or canonical representations) into irreducible representations. These tensor products have an invariant Hermitian form, which is closely connected with the Berezin form, considered in an earlier paper. We determine the decomposition of the tensor products with respect to this Hermitian form. Generally speaking, this form is not positive-definite. Though one can consider all concepts in the framework of para-Hermitian symmetric spaces, explicit results are only known so far for the rank one spaces, considered in this paper.