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# 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.

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 (Harish-Chandra 1964a, 1964b, 1964c, 1965a, 1965b). 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.

Most proofs can be broken up into several steps as follows.

Step 1. If Θ is an invariant eigendistribution then it is analytic on the regular elements of G. This follows from elliptic regularity, by showing that the center of the universal enveloping algebra has an element that is "elliptic transverse to an orbit of G" for any regular orbit.

Step 2. If Θ is an invariant eigendistribution then its restriction to the regular elements of G is locally integrable on G. (This makes sense as the non-regular elements of G have measure zero.) This follows by showing that ΔΘ on each Cartan subalgebra is a finite sum of exponentials, where Δ is essentially the denominator of the Weyl denominator formula, with 1/Δ locally integrable.

Step 3. By steps 1 and 2, the invariant eigendistribution Θ is a sum S+F where F is a locally integrable function and S has support on the singular elements of G. The problem is to show that S vanishes. This is done by stratifying the set of singular elements of G as a union of locally closed submanifolds of G and using induction on the codimension of the strata. While it is possible for an eigenfunction of a differential equation to be of the form S+F with F locally integrable and S having singular support on a submanifold, this is only possible if the differential operator satisfies some restrictive conditions. One can then check that the Casimir operator of G does not satisfy these conditions on the strata of the singular set, which forces S to vanish.

References

Atiyah, Michael (1988), "Characters of semi-simple Lie groups", Collected works. Vol. 4, Oxford Science Publications, The Clarendon Press Oxford University Press, pp. 491–557, ISBN 978-0-19-853278-1, MR951895

Harish-Chandra (1955), "On the characters of a semisimple Lie group", Bulletin of the American Mathematical Society 61 (5): 389–396, doi:10.1090/S0002-9904-1955-09935-X, ISSN 0002-9904, MR0071715

Harish-Chandra (1956), "The characters of semisimple Lie groups", Transactions of the American Mathematical Society 83: 98–163, ISSN 0002-9947, JSTOR 1992907, MR0080875

Harish-Chandra (1963), "Invariant eigendistributions on semisimple Lie groups", Bulletin of the American Mathematical Society 69: 117–123, doi:10.1090/S0002-9904-1963-10889-7, ISSN 0002-9904, MR0145006

Harish-Chandra (1964a), "Invariant distributions on Lie algebras", American Journal of Mathematics 86: 271–309, ISSN 0002-9327, JSTOR 2373165, MR0161940

Harish-Chandra (1964b), "Invariant differential operators and distributions on a semisimple Lie algebra", American Journal of Mathematics 86: 534–564, ISSN 0002-9327, JSTOR 2373023, MR0180628

Harish-Chandra (1964c), "Some results on an invariant integral on a semisimple Lie algebra", Annals of Mathematics. Second Series 80: 551–593, ISSN 0003-486X, JSTOR 1970664, MR0180629

Harish-Chandra (1965a), "Invariant eigendistributions on a semisimple Lie algebra", Publications Mathématiques de l'IHÉS (27): 5–54, ISSN 1618-1913, MR0180630

Harish-Chandra (1965b), "Invariant eigendistributions on a semisimple Lie group", Transactions of the American Mathematical Society 119: 457–508, ISSN 0002-9947, JSTOR 1994080, MR0180631

Harish-Chandra (1978), "Admissible invariant distributions on reductive p-adic groups", in Rossmann, Wulf, Lie theories and their applications (proceedings of the 1977 annual seminar of the Canadian mathematical congress, Queen's University in Kingston, Ontario, 1977), Queen's Papers in Pure Appl. Math., 48, Kingston, Ont.: Queen's Univ., pp. 281–347, MR0579175, Reprinted in volume 4 of his collected works.

Harish-Chandra (1999), DeBacker, Stephen; Sally, Paul J. Jr., eds., Admissible invariant distributions on reductive p-adic groups, University Lecture Series, 16, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2025-4, MR1702257

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