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We prove a generalization of the quantum de Finetti theorem when the localspace is an infinite-dimensional Fock space. In particular, instead ofconsidering the action of the permutation group on $n$ copies of that space, weconsider the action of the unitary group $U(n)$ on the creation operators ofthe $n$ modes and define a natural generalization of the symmetric subspace asthe space of states invariant under unitaries in $U(n)$. Our first result is acomplete characterization of this subspace, which turns out to be spanned by afamily of generalized coherent states related to the special unitary group$SU(p,q)$ of signature $(p,q)$. More precisely, this construction yields aunitary representation of the noncompact simple real Lie group $SU(p,q)$. Wetherefore find a dual unitary representation of the pair of groups $U(n)$ and$SU(p,q)$ on an $n(p+q)$-mode Fock space.  The (Gaussian) $SU(p,q)$ coherent states resolve the identity on thesymmetric subspace, which implies a Gaussian de Finetti theorem stating thattracing over a few modes of a unitary-invariant state yields a state close to amixture of Gaussian states. As an application of this de Finetti theorem, weshow that the $n\times n$ upper-left submatrix of an $n\times n$ Haar-invariantunitary matrix is close in total variation distance to a matrix of independentnormal variables if $n^3 =O(m)$.

$SU(p,q)$ coherent states and a Gaussian de Finetti theorem