  Based on the theory of Dunkl operators, this paper presents a general concept
of multivariable Hermite polynomials and Hermite functions which are associated
with finite reflection groups on $\b R^N$. The definition and properties of
these generalized Hermite systems extend naturally those of their classical
counterparts; partial derivatives and the usual exponential kernel are here
replaced by Dunkl operators and the generalized exponential kernel K of the
Dunkl transform. In case of the symmetric group $S_N$, our setting includes the
polynomial eigenfunctions of certain Calogero-Sutherland type operators. The
second part of this paper is devoted to the heat equation associated with
Dunkl's Laplacian. As in the classical case, the corresponding Cauchy problem
is governed by a positive one-parameter semigroup; this is assured by a maximum
principle for the generalized Laplacian. The explicit solution to the Cauchy
problem involves again the kernel K, which is, on the way, proven to be
nonnegative for real arguments.
