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.
Read more about Harish-Chandra's Regularity Theorem: Statement, Proof
Famous quotes containing the words regularity and/or theorem:
“The question of whether it’s God’s green earth is not at center stage, except in the sense that if so, one is reminded with some regularity that He may be dying.”
—Edward Hoagland (b. 1932)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)