指数李群的单项表示的可容许性
Admissibility For Monomial Representations of Exponential Lie Groups
摘要：
设$G$是单连通指数可解李群，$H$是闭连通子群，$τ$是由$H$的酉特征标$χ_f$导出的$G$的表示.通过轨道方法，$τ$的谱对应于满足谱变量$A_τ= f + \h^\perp$的共伴轨道的集合$G\cdot A_τ/ G$。证明了$τ$的谱测度关于Plancherel测度是绝对连续的当且仅当$H$自由作用在$A_τ$的某点上.作为推论，我们证明了如果$G$是非幺模的，则$τ$有容许向量当且仅当前面的轨道条件成立.
英文摘要：
Let $G$ be a simply connected exponential solvable Lie group, $H$ a closed connected subgroup, and let $τ$ be a representation of $G$ induced from a unitary character $χ_f$ of $H$. The spectrum of $τ$ corresponds via the orbit method to the set $G\cdot A_τ/ G$ of coadjoint orbits that meet the spectral variety $A_τ= f + \h^\perp$. We prove that the spectral measure of $τ$ is absolutely continuous with respect to the Plancherel measure if and only if $H$ acts freely on some point of $A_τ$. As a corollary we show that if $G$ is nonunimodular, then $τ$ has admissible vectors if and only if the preceding orbital condition holds.
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