Kirillov's conjecture and $\CaD$-modules

TL;DR

This paper proves Kirillov's conjecture for certain Lie groups by analyzing the associated regular holonomic D-module and calculating the roots of its b-functions, linking representation theory with D-module theory.

Contribution

It provides an explicit proof of Kirillov's conjecture using D-module theory and b-function root calculations, connecting representation theory and algebraic analysis.

Findings

Kirillov's conjecture holds for Gl(n,R) and Gl(n,C).

The proof involves explicit calculation of b-function roots.

The approach links irreducibility preservation to D-module regularity.

Abstract

In the theory of Lie groups, the irreducibility of a unitary representation is not preserved in general by restriction to a subgroup. Kirillov's conjecture says that it is preserved for the groups Gl(n,R) or Gl(n,C) when the subgroup is the subgroup of matrices leaving invariant a non zero vector. This conjecture was proved by Barush using a detailed study of nilpotent orbits. In fact, it is not difficult to see that the conjecture is equivalent to the fact that some system of partial differential equations has no singular distributions as solutions. This system of equations is a regular holonomic D-module and we give a proof of the result by an explicit calculation of the roots of the b-functions associated to this D-module.