The Laplacian with mixed Dirichlet-Neumann boundary conditions on Weyl chambers

Let W be a finite reflection group associated with a root system R in R-d. Let C+ denote a positive Weyl chamber distinguished by a choice of R+, a set of positive roots. We investigate realizations in L-2(C+) of the Laplacian on C+, subject to mixed Dirichlet-Neumann boundary conditions imposed on the facets of C. These conditions are determined by a homomorphism eta is an element of Hom(W, (Z) over cap (2)), where (Z) over cap (2) = {1, -1} with multiplication. The essential part of the paper contains thorough analysis of the corresponding eta-heat kernels together with proof of their positivity on C+. (C) 2022 Elsevier Inc. All rights reserved.