Finite reflection groups and symmetric extensions of Laplacian

Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $\Omega$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $\Omega_+=\Omega\cap C_+$ be the positive part of $\Omega$. We define a family $\{-\Delta_{\eta}^+\}$ of self-adjoint extensions of the Laplacian $-\Delta_{\Omega_+}$, labeled by homomorphisms $\eta\colon W\to \{1,-1\}$. In the construction of these $\eta$-Laplacians $\eta$-symmetrization of functions on $\Omega$ is involved. The Neumann Laplacian $-\Delta_{N,\Omega_+}$ is included and corresponds to $\eta\equiv1$. If $H^{1}(\Omega)=H^{1}_0(\Omega)$, then the Dirichlet Laplacian $-\Delta_{D,\Omega_+}$ is either included and corresponds to $\eta={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $\Psi(-\Delta_{N,\Omega})$ and $\Psi(-\Delta_{\eta}^+)$, or $\Psi(-\Delta_{D,\Omega})$ and $\Psi(-\Delta_{D,\Omega_+})$, where $\Psi$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.