Semibounded Unitary Representations of Double Extensions of Hilbert--Loop Groups
Semibounded Unitary Representations of Double Extensions of Hilbert--Loop Groups
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希尔伯特二重扩张的半有界酉表示--环群
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发表时间：
2012
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通讯作者：
K. Neeb
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作者：
K. Neeb
一个可能是无限维的李群G的酉表示称为半有界的，如果来自导出表示的相应算子i\dd\pi（x）$在G的李代数g的某个非空开子集上自上而下一致有界。我们对扭圈群$\cL_\phi（K）$的双重扩张群$\hat\cL_\phi（K）$的所有不可约半有界表示进行分类，其中$K$是一个简单的希尔伯特-李群（在这个意义上，它的李代数上的标积是不变的），$\phi$是$K$的有限阶自同构这导致了7个不可约局部仿射根系统与他们的典型$\Z$-分次之一。为了实现这一目标，我们将全纯归纳法推广到某些Fr 'echet-Lie群上，并证明了Fr' echet-BCH-Lie群上解析算子值正定函数的无穷小刻画.
A unitary representation of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i\dd\pi(x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $\g$ of $G$. We classify all irreducible semibounded representations of the groups $\hat\cL_\phi(K)$ which are double extensions of the twisted loop group $\cL_\phi(K)$, where $K$ is a simple Hilbert--Lie group (in the sense that the scalar product on its Lie algebra is invariant) and $\phi$ is a finite order automorphism of $K$ which leads to one of the 7 irreducible locally affine root systems with their canonical $\Z$-grading. To achieve this goal, we extend the method of holomorphic induction to certain classes of Fr\'echet-Lie groups and prove an infinitesimal characterization of analytic operator-valued positive definite functions on Fr\'echet--BCH--Lie groups.