Harish-Chandra's regularity theorem


In mathematics, Harish-Chandra's regularity theorem, introduced by, states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function. proved a similar theorem for semisimple p-adic groups.
had previously shown that any invariant eigendistribution is analytic on the regular elements of the group, by showing that on these elements it is a solution of an elliptic differential equation. The problem is that it may have singularities on the singular elements of the group; the regularity theorem implies that these singularities are not too severe.

Statement

A distribution on a group G or its Lie algebra is called invariant if it is invariant under conjugation by G.
A distribution on a group G or its Lie algebra is called an eigendistribution if it is an eigenvector of the center of the universal enveloping algebra of G (identified with the left and right invariant differential operators of G.
Harish-Chandra's regularity theorem states that any invariant eigendistribution on a semisimple group or Lie algebra is a locally integrable function.
The condition that it is an eigendistribution can be relaxed slightly to the condition that its image under the center of the universal enveloping algebra is finite-dimensional. The regularity theorem also implies that on each Cartan subalgebra the distribution can be written as a finite sum of exponentials divided by a function Δ that closely resembles the denominator of the Weyl character formula.

Proof

Harish-Chandra's original proof of the regularity theorem is given in a sequence of five papers.
gave an exposition of the proof of Harish-Chandra's regularity theorem for the case of SL2, and sketched its generalization to higher rank groups.
Most proofs can be broken up into several steps as follows.