Finite Dimensional Representations of Invariant Differential Operators
Abstract: Let G be a complex reductive group and V a rational G-module. If G is finite, then the algebra of invariant polynomial differential operators D(V)G is simple. If G is positive dimensional, this is no longer the case, and little is known about the irreducible or finite dimensional representations of D(V)G, except when G is a torus. This remains true even for the adjoint representation of G on g
The first case to consider is G=SL2. Here, remarkably, the algebra of invariant differential operators D(sl2)SL2 is isomorphic to the universal enveloping algebra U(sl2). The representation theory of U(sl2) is well understood. We consider what happens for the simple Lie algebras of rank 2. We have rather complete results for SL3. The finite dimensional representations V(m,i) are parametrized by pairs of positive integers (m,i) such that
(1) 2 2i m +2, and
(2) m + i is not congruent to zero modulo 3
and V(m,i) has dimension (m + 3)i(m + 3 - i)/6.