Representations of solvable Lie groups and differential operators

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So-called "Polynomial conjecture" of Corwin-Greenleaf is a well known difficult conjecture for monomial representations of a connected and simply connected nilpotent Lie group. It has been a central aim of this research project. As there exists a strong parallelism for inducing and restricting representations, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A.Baklouti, G.Lion, J.Ludwig and B.Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.1.Let χ be a unitary character of an analytic subgroup H of G. We consider the monomial representation τ induced by χ up to G. The algebra of invariant differential operators on the line bundle over G/H associated to these data is algebraic over a system of generators of the set of central elements of Corwin-Greenleaf if and only if τ is of finite multiplicities.2.(Polynomial conjecture of Corwin - Greenleaf) Suppose that the monomial representation τ is of finite multiplicities. Then, the algebra of invariant differential operators on the line bundle over G/H associated to these data is isomorphic to the algebra of H-invariant polynomial functions on a certain affine subspace of the linear dual of the Lie algebra of G.3. The above result 1 and the Frobenius reciprocity in distribution sense obtained in the previous research program have their counterpart for the restrictions. We formulated them for the restriction π|K of an irreducible unitary representation π of G to an analytic subgroup K. Then, we proved them in certain particular cases.