Invariant sectors of the regular representation of loop groups
Let \(LG\) be the loop group of a compact semisimple Lie group \(G\). The standard Brownian measure \(\nu\) on the continuous loop space \(L^cG\) is quasi-invariant with respect to the left and right actions of \(LG\). This yields the left (and right) regular representation \(U^L\) (and \(U^R)\) of \(LG\) on \(L^2(L^cG,\nu)\). Let \(\Omega_0\) denote the vacuum: \(\Omega_0 (x)=1\) for all \(x\in L^cG\). It is shown in the paper under review that \(\Omega_0\) is not a cyclic vector for \(U^L\). Namely, the space \(L^2 (L^cG,\nu)\) decomposes into the direct sum of infinitely many \(LG\)-invariant subspaces (weight subspaces with respect to the right action of the maximal torus \(T\) of \(G\), being labelled by characters of \(T)\); the subspace corresponding to the trivial character of \(T\) contains the cyclic component of \(\Omega_0\). At the moment, the author does not know whether the latter subspace admits \(\Omega_0\) as a cyclic vector.
scientific article; zbMATH DE number 922378