Monomial representation of solvable Lie groups

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It is well known that there exists a strong parallelism for inducing and restricting representations. In this research, I studied this duality for nilpotent Lie groups in the framework of celebrated orbit method. In collaboration with A. Baklouti, G. Lion and B. Magneron, I obtained the following main results. Let G be a connected, simply connected nilpotent Lie group.1. (Commutativity conjecture of Duflo, Corwin Greenleaf) Let χ be a unitary character of an analytic subgroup H of G. Then, the monomial representation τ induced by χ up to G is of finite multiplicities if and only if the algebra of invariant differential operators on the line bundle over G/H associated to these data is commutative.2. (Frobenius reciprocity) Let π be an irreducible unitary representation of G. The multiplicity of π in the canonical central decomposition of τ is equal to the dimension of the space of (H, χ ) semi-invariant generalized vectors.3. The above result 1 has its counterpart for the restrictions. Namely, let's restrict an irreducible unitary representation of G to an analytic subgroup K. Then, this restriction is of finite multiplicities if and only if the associated algebra of K invariant differential operators is commutative.