  Let V be a symplectic vector space and let $\mu$ be the oscillator
representation of Sp(V). It is natural to ask how the tensor power
representation $\mu^{\otimes t}$ decomposes. If V is a real vector space, then
Howe-Kashiwara-Vergne (HKV) duality asserts that there is a one-one
correspondence between the irreducible subrepresentations of Sp(V) and the
irreps of an orthogonal group O(t). It is well-known that this duality fails
over finite fields. Addressing this situation, Gurevich and Howe have recently
assigned a notion of rank to each Sp(V) representation. They show that a
variant of HKV duality continues to hold over finite fields, if one restricts
attention to subrepresentations of maximal rank. The nature of the
rank-deficient components was left open. Here, we show that all rank-deficient
Sp(V)-subrepresentations arise from embeddings of lower-order tensor products
of $\mu$ and $\bar\mu$ into $\mu^{\otimes t}$. The embeddings live on spaces
that have been studied in quantum information theory as tensor powers of
self-orthogonal Calderbank-Shor-Steane (CSS) quantum codes. We then find that
the irreducible Sp(V) subrepresentations of $\mu^{\otimes t}$ are labelled by
the irreps of orthogonal groups O(r) acting on certain r-dimensional spaces for
r <= t. The results hold in odd charachteristic and the "stable range" t <= 1/2
dim V. Our work has implications for the representation theory of the Clifford
group. It can be thought of as a generalization of the known characterization
of the invariants of the Clifford group in terms of self-dual codes.
