  We prove orthogonality and compute explicitly the (quadratic) norms for
super-Jack polynomials $SP_\lambda((z_1,\ldots,z_n),(w_1,\ldots,w_m);\theta)$
with respect to a natural positive semi-definite, but degenerate, Hermitian
product $\langle\cdot,\cdot\rangle_{n,m}^\prime$. In case $m=0$ (or $n=0$), our
product reduces to Macdonald's well-known inner product
$\langle\cdot,\cdot\rangle_n^\prime$, and we recover his corresponding
orthogonality results for the Jack polynomials
$P_\lambda((z_1,\ldots,z_n);\theta)$. From our main results, we readily infer
that the kernel of $\langle\cdot,\cdot\rangle_{n,m}^\prime$ is spanned by the
super-Jack polynomials indexed by a partition $\lambda$ not containing the
$m\times n$ rectangle $(m^n)$. As an application, we provide a Hilbert space
interpretation of the deformed trigonometric Calogero-Moser-Sutherland
operators of type $A(n-1,m-1)$.
