Spectral decomposition of Dunkl Laplacian and application to a radial integral representation for the Dunkl kernel

For the Dunkl Laplacian associated with a Coxeter group, the Von Neumann spectral decomposition is given. As a consequence, for the Dunkl kernel an integral representation with respect to the Lebesgue measure is given (DIR). Also, this integral representation (DIR) is used to write the Dunkl kernel as a radial integral representation with respect to a radial probability measure (DIR)(rad). Moreover, we compute the Schwartz kernel associated with the spectral density of the Dunkl Laplacian and we prove that it is nothing but the kernel of the generalized spectral projector as given in Ben Said and Mejjaoli (J Funct Spaces, 2020). Furthermore, we use this Schwartz kernel to build a more general functional calculus for the Dunkl Laplacian. Finally, we apply this functional calculus to give an integral representation for the wave kernel of the Dunkl Laplacian.