  We investigate vector-valued Gabor frames (sometimes called Gabor
superframes) based on Hermite functions $H_n$. Let $h= (H_0, H_1, ..., H_n)$ be
the vector of the first $n+1$ Hermite functions. We give a complete
characterization of all lattices $\Lambda \subseteq \bR ^2$ such that the Gabor
system
  $\{e^{2\pi i \lambda_2 t} \boh (t-\lambda_1): \lambda = (\lambda_1,
\lambda_2) \in \Lambda \}$ is a frame for $L^2 (\bR, \bC ^{n+1})$. As a
corollary we obtain sufficient conditions for a single Hermite function to
generate a Gabor frame and a new estimate for the lower frame bound.
  The main tools are growth estimates for the Weierstrass $\sigma $-function, a
new type of interpolation problem for entire functions on the Bargmann-Fock
space, and structural results about vector-valued Gabor frames.
