Kalender Department Mathematik

Mon, 26.06.2017, 14:15

Mon, 26.06.2017, 15:45
Unitary Highest Weight Representations of U_1(\ell^2) and First Order Cohomology
AG Lie-Gruppen

Referent: Manuel Herbst
Veranstalter: K.-H. Neeb
Raum: Übungsraum 4

In the work of K.--H. Neeb, one finds the result that the irreducible unitary highest weight representations of the group $U(\infty)$, which is the countable inductive limit of the compact unitary groups $U(n)$, are classified by the orbits of the weights $\lambda \in \Z^{\N}$ under the Weyl group $S_{(\N)}$ of finite permutations. A highest weight representation $\pi=\pi_\lambda$ extends to the group $U_1(\ell^2)$, which is the Banach-completion of $U(\infty)$ with respect to the metric $d(g,h)= Tr(|g-h|)$, if and only if the entries of the weight $\lambda$ form a bounded sequence in $\Z$. In the talk, I will determine for which weights $\lambda$ the first cohomology space $H^1 = H^1(\lambda)$ of the group $U_1(\ell^2)$ is non-trivial. This is the case for almost all finitely supported weights and one thus finds that the group $U_1(\ell^2)$ does not have Property (T). This work is part of my PhD-project.