No Fixed Vectors

"The energy representation has no non-zero fixed vectors," by B. Driver and Brian Hall.

We consider the ``energy representation'' $W$ of the group $\mathcal{G}$ of smooth mappings of a Riemannian manifold $M$ into a compact Lie group $G.$ Our main result is that if $W\left( g\right) f=f$ for all $g\in \mathcal{G},$ then $f=0.$ In the language of quantum field theory this says that there are no ``states.'' Our result follows from the irreducibility of the energy representation whenever the irreducibility theorems of Ismagilov, Gelfand--Graev--Ver\v{s}ik, Albeverio--Hoegh-Krohn--Testard, or Wallach apply. Our result, however, applies in general, even in cases where the energy representation is known to be reducible.

We work in the more general context of the ``Gaussian regular representation'' of the Euclidean group of a real separable Hilbert space. We show that if a function is invariant under the action of any subgroup of the Euclidean group that has unbounded orbits, then this function must be identically zero. Our result about the energy representation is a special case.

11/26/2018 10:37 AM