Restrictions of irreducible unitary representations.

Given a reductive Lie group G and P a closed subgroup an
important problem is to know if the restriction to P of an irreducible unitary representation of G is irreducible. If G=GL(n,K), K=R or C, and P is the isotropy group of the vector
(0,...,0,1), this is exactly the Kirillov's conjecture. Here we present the ideas of Barush's proof of it and how they could be solved using D-modules arguments. This point of view will permit solve the problem for other pairs (G,P) with similar properties.