  We study generalized Hermite polynomials with rectangular matrix arguments
arising in multivariate statistical analysis and the theory of zonal
polynomials. We show that these are well-suited for expressing the Wiener-Ito
chaos expansion of functionals of the spectral measure associated with Gaussian
matrices. In particular, we obtain the Wiener chaos expansion of Gaussian
determinants of the form $\det(XX^T)^{1/2}$ and prove that, in the setting
where the rows of $X$ are i.i.d. centred Gaussian vectors with a given
covariance matrix, its projection coefficients admit a geometric interpretation
in terms of intrinsic volumes of ellipsoids, thus extending the content of
Kabluchko and Zaporozhets (2012) to arbitrary chaotic projection coefficients.
Our proofs are based on a crucial relation between generalized Hermite
polynomials and generalized Laguerre polynomials. In a second part, we
introduce the matrix analog of the classical Mehler's formula for the
Ornstein-Uhlenbeck semigroup and prove that matrix-variate Hermite polynomials
are eigenfunctions of these operators. As a byproduct, we derive an
orthogonality relation for Hermite polynomials evaluated at correlated Gaussian
matrices. We apply our results to vectors of independent arithmetic random
waves on the three-torus, proving in particular a CLT in the high-energy regime
for a generalized notion of total variation on the full torus.
