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's regularity theorem states that any invariant eigendistribution on a semisimple group or Lie algebra is a locally integrable function.
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.
Atiyah (1988) gave an exposition of the proof of Harish-Chandra's regularity theorem for the case of SL2(R), and sketched its generalization to higher rank groups.