Monomial representation of solvable Lie groups

众所周知，诱导和限制表征之间存在着很强的平行关系。在这项研究中，我在著名的轨道方法框架下研究了幂零李群的对偶性。在与A.Baklti、G.Lion和B.Magneron的合作下，我获得了以下主要结果。设G是连通的，单连通的幂零李群。(Duflo的交换性猜想)设χ是G的一个解析子群H的么正特征标，则由τ诱导到G的单项表示χ具有有限重性当且仅当G/H上的线丛上与这些数据相关的不变微分算子的代数是可交换的。(Frobenius互易)设π是G的一个不可约么正表示，π在τ的典范中心分解中的重数等于(H，χ)半不变广义向量空间的维度。上述结果1与这些限制具有对应关系。也就是说，让我们把G的一个不可约么正表示限制在一个解析子群K上，那么，这个限制是有限多的当且仅当K不变微分算子的相关代数是可交换的。

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.