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Algebra Seminar
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Anthony Joseph
Weizmann Institute

Differential operators and the surjectivity theorem

Thursday 11th December, 11:05-11:55pm, Carslaw 173.

The "Orbit Method" leads to the following question. Let g be a finite dimensional Lie algebra, O a coadjoint orbit and L a Lagrangian subvariety of O. Can one give the algebra R[L] of regular functions on L the structure of a g-module? This is straightforward if L is a polarization of O; but this circumstance is rare. Another approach is to consider the algebra D[L] of differential operators on L. The hope is then that D[L] contains an image of the enveloping algebra U(g) of g. Then the natural action of D[L] on R[L] restricts to an action of U(g) on R[L].

The above approach works remarkably well for an orbital variety L contained in the nilradical m of a parabolic of a semisimple Lie algebra, when m is commutative. Indeed when L is strictly contained in m one has the remarkable fact that D[L] actually coincides with an image of U(g). This result was reported by Goncharov when L is a minimal orbital variety. It was generalized by Levasseur-Stafford for all classical g through case-by-case analysis. Here we present a general proof.

Quite remarkably, as a U(g) module R[L] is unitarizable, and indeed this construction gives all unitarizable highest-weight modules with a spherical vector which occur beyond the first reduction point.

We conclude with some further recent results on the "quantization" of so-called hypersurface orbital varieties.