Discrete-type restricted representations of exponential groups and differential operators

Let G = exp g be an exponential solvable Lie group with Lie algebra g, K = exp k an analytic subgroup of G with Lie algebra g and pi an irreducible unitary representation of G. The focus here is on the study of the restriction pi|(K) of discrete type, especially the algebra D-pi(G)(K) of K-invariant differential operators in the space of pi. Provided that the coadjoint orbit of G corresponding to pi is of maximal dimension, we diagonalize these operators by means of Penney's distributions so that the algebra D-pi(G)(K) turns out to be commutative when pi|(K) is of discrete type, and show that the converse may fail to hold as a developed example reveals. Furthermore, we show that D-pi(G)(K) is isomorphic to the algebra C[Omega](K) of the K-invariant polynomial functions on Omega. We also produce a process to provide some generators of the kernel ker pi in the enveloping algebra of g(C).