Harish-Chandra's Regularity Theorem - Proof

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.