Invariant distributions and Kirillov conjecture

ESTHER GALINA; YVES LAURENT,

Buenos Aires

Conferencia; Conference in Algebraic Geometry, D-modules and Foliations; 2008

UBA

The study of  restricted representations is a vast area inRepresentation Theory of Lie  groups. In general, irreducibilityof unitary representations  is a property not preserved byrestriction to a subgroup. Kirillov´s conjecture says that this isnot the case for the group $G=GL(n,mathbb R)$ and the subgroup$P$ that less invariant the vector $(0,dots,1)$; that is everyrestricted unitary representation is irreducible in this case.Barush proved it some years ago [Annals of Math. 158 (2003),207-252]. The proof is based on a stronger analogue of theRegularity Theorem of Harish-Chandra for this case combined with adetailed study of nilpotent orbits. In a joint work with YvesLaurent, we can obtain another proof of Kirillov´s conjectureusing $mathcal D$-modules techniques and our previous results oninvariant eigendistributions to prove this stronger regularitytheorem.