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arXiv 1208.4137math.RT
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infinite dimensional李群的主列表示，II：诱导表示的构造

Principal series representations of infinite dimensional Lie groups, II: Construction of induced representations

Joseph A. Wolf

备注：
Second in a series of three papers
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摘要：

研究了由极小抛物子群$P$的因子表示导出的经典无穷维实单李群$G$的表示.这充分利用了最近发展起来的抛物子群和子代数的结构理论。一般来说，无穷维经典李群中的抛物线比有限维情形中的抛物线要复杂一些，并且不是有限维抛物线的直接极限。我们扩展了他们的结构理论，并将其用于经典主列表示的无穷维模拟。为了做到这一点，我们检查两种类型的PDMATHAMN条件：标志关闭条件和极小。本文利用黎曼对称空间理论证明了：若$P$是旗闭的，则$G$的任意极大lim-compact子群$K$在$G /P$ \，.当$P$是最小的，我们证明了它是顺从的，我们使用顺从群的性质，以诱导酉表示$P$ $τ$的连续表示$\Ind_P^G(τ)$的$G$上完整的局部凸拓扑向量空间。当$P$是最小的和标志封闭的，我们有一个分解$P = MAN$类似的有限维的情况下，我们展示了如何给$K$ -频谱信息$\Ind_P^G(τ)|_K = \Ind_M^K(τ|_M)$。

英文摘要：

We study representations of the classical infinite dimensional real simple Lie groups $G$ induced from factor representations of minimal parabolic subgroups $P$. This makes strong use of the recently developed structure theory for those parabolic subgroups and subalgebras. In general parabolics in the infinite dimensional classical Lie groups are are somewhat more complicated than in the finite dimensional case, and are not direct limits of finite dimensional parabolics. We extend their structure theory and use it for the infinite dimensional analog of the classical principal series representations. In order to do this we examine two types of conditions on $P$: the flag-closed condition and minimality. We use some riemannian symmetric space theory to prove that if $P$ is flag-closed then any maximal lim-compact subgroup $K$ of $G$ is transitive on $G /P$\,. When $P$ is minimal we prove that it is amenable, and we use properties of amenable groups to induce unitary representations $τ$ of $P$ up to continuous representations $\Ind_P^G(τ)$ of $G$ on complete locally convex topological vector spaces. When $P$ is both minimal and flag-closed we have a decomposition $P = MAN$ similar to that of the finite dimensional case, and we show how this gives $K$--spectrum information $\Ind_P^G(τ)|_K = \Ind_M^K(τ|_M)$.

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