AG Lie-Gruppen: Milan Niestijl, TU Delft

Positive Energy Representations of Gauge Groups With Support at a Fixed
Point

Vortragender: Milan Niestijl, TU Delft
Veranstalter: Karl-Hermann Neeb

Abstract:
Let $\\mathcal{K} \\to M$ be a locally trivial smooth bundle of Lie groups
equipped with an action of some Lie group $P$ by bundle automorphisms.
Complementing recent progress B. Janssens and K.H. Neeb on the case where the
$P$-action on $M$ has no fixed-points, projective unitary representations
$\\overline{\\rho}$ of the locally-convex Lie group $\\mathcal{G} :=
\\Gamma_c(\\mathcal{K})$ are studied which are of positive energy and
factor
entirely through the germs at some fixed point $a \\in M$ of the $P$-action.
Under suitable assumptions, it is shown that the kernel of a particular
quadratic form on $\\R[[x_1,\\cdots, x_d]]$ generates an ideal in
$\\mathfrak{G} :=
\\Lie(\\mathcal{G})$ on which the derived representation $d\\rho$ must
vanish. This
leads in particular to sufficient conditions for $d\\rho$ to factor through a
finite jet space $J^k_x(\\mathcal{K})$ or through
$J^\\infty_a(N,\\mathcal{K})$ for
some usually lower-dimensional submanifold $N$. Some examples are considered.
If
time permits, we will have a closer look at the special case where $P=S^1$.