Harmonic oscillator on Weyl chambers

Let C+ denote a Weyl chamber distinguished in the framework of a finite reflection group W acting on Rd. We consider the harmonic oscillator −Δ+|x|2 as an operator on L2(C+) with approprietely chosen domains and construct families of corresponding self-adjoint extensions. These extensions are parametrized by the homomorphisms η∈Hom(W,Zˆ2) and are based on an η-symmetrization procedure of functions on the whole Rd. The analysis of the associated η-heat semigroups is included and positivity on C+ of the corresponding η-heat kernels is established.