  In this paper, Hermite polynomials related to quantum systems with orthogonal
O(m)-symmetry, finite reflection group symmetry G < O(m), symplectic symmetry
Sp(2n) and superspace symmetry O(m) x Sp(2n) are considered. After an overview
of the results for O(m) and G, the orthogonality of the Hermite polynomials
related to Sp(2n) is obtained with respect to the Berezin integral. As a
consequence, an extension of the Mehler formula for the classical Hermite
polynomials to Grassmann algebras is proven. Next, Hermite polynomials in a
full superspace with O(m) x Sp(2n) symmetry are considered. It is shown that
they are not orthogonal with respect to the canonically defined inner product.
However, a new inner product is introduced which behaves correctly with respect
to the structure of harmonic polynomials on superspace. This inner product
allows to restore the orthogonality of the Hermite polynomials and also
restores the hermiticity of a class of Schroedinger operators in superspace.
Subsequently, a Mehler formula for the full superspace is obtained, thus
yielding an eigenfunction decomposition of the super Fourier transform.
Finally, an extensive comparison is made of the results in the different types
of symmetry.
