Differential operators and the surjectivity theorem
Thursday 11th December, 11:05-11:55pm,
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.