### 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.