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

Rank-deficient representations in the Theta correspondence over finite fields arise from quantum codes