Harish-Chandra's Regularity Theorem

In mathematics, Harish-Chandra's regularity theorem, introduced by Harish-Chandra (1963), 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. Harish-Chandra (1978, 1999) proved a similar theorem for semisimple p-adic groups.

Harish-Chandra (1955, 1956) 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.