Atlas: G-invariant differential operators and the Frobenius decomposition of affine G-varieties by Ilka Agricola

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International Congress on Differential Geometry in memory of Alfred Gray (1939-1998)
September 18-23, 2000
Universidad del País Vasco
Bilbao, Spain

Organizers
M. Fernández (chairman), R. Ibáñez, M. Macho-Stadler

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G-invariant differential operators and the Frobenius decomposition of affine G-varieties
by
Ilka Agricola
Humboldt-Universitaet zu Berlin, Germany

Oral Communication

Consider a reductive connected complex algebraic group G acting on an smooth irreducible affine variety M, and denote by D^G(M) the algebra of G-invariant algebraic differential operators on M. We show that in the decomposition of the affine ring C[M] of M under the G action the multiplicity spaces are irreducible pairwise inequivalent D^G(M)-modules, admit a central character and are uniquely determined by it. This generalizes a former result of N. Wallach for linear G actions on vector spaces. I will present examples of singular varieties for which this theorem badly fails, thus showing that the smoothness assumption is crucial.

As applications, we get the two following characterisations of multiplicity free actions (the first one known previously, of course, in many special cases):

The action of G on M is multiplicity free if and only if the algebra of G-invariant differential operators D^G(M) is commutative;

The action of G on M is multiplicity free if and only if the (algebraic) quotient of the moment map

\psi_G: T^*M//G --> \mathfrakg^*//G

is a finite map, i.e. if C[T^*M]^G is a finitely generated C[\mathfrakg^*]^G-module.

The final part of my talk will be about global aspects of the Frobenius decomposition of C[M] if M is a symmetric space G/K. In their classical paper, Kostant and Rallis studied the K-action on the tangent space \mathfrakp=T_eKG/K and showed (among other) that there exist K-invariant polynomial coefficient differential operators inducing the K-isotypic decomposition of C[\mathfrakp]. We shall review what is known about K-invariant differential operators on G/K, and show the existence of enough such operators to get the desired decomposition in a few examples. Thus, the concept of K-harmonical functions makes sense on a symmetric space from a differential operator point of view (deep results by Richardson had shown their algebraic existence much earlier). The analogy to the local and global invariant theory of reflection groups will also be discussed.

http://www-irm.mathematik.hu-berlin.de/~agricola

Date received: May 29, 2000