# The Shale-Weil representation associated to a quadratic form

## Abstract

Let (V, B) be a symplectic space. The representation \( \mathop{R}\limits^{\sim } \) of \(\mathop{{Sp}}\limits^{\sim } \) is “almost” irreducible. It is a sum of two irreducible representations corresponding to even and odd functions. We will similarly study the decomposition into irreducible components of the k-tensor products of the representation \(\mathop{R}\limits^{\sim } \). The principle is the following: Let (E, S) be an orthogonal vector space of dimension k, with a quadratic form S. Then the space (V ⊗ E, B ⊗ S) with the bilinear form B ⊗ S is a symplectic space. The representation R of the group Sp(B × S) restrict to Sp(B) × O(S) in R_{S}. When S is positive definite, R_{S}, as a representation of Sp(B), is the k-tensor product of R and the compact group O(k) plays an analogous role for the de-k composition of \(\mathop{ \otimes }\limits^{k} \) that the symmetric group σ_{k} for the representation of GL(n, ℂ) \(\mathop{ \otimes }\limits^{k} {{\rm{C}}^{n}} \) let us consider an irreducible representation X of O(k) occurring in R_{S}, then the restriction of the representation R_{S} to the space of vectors of type X under O(k) is jointly irreducible under Sp(B) x O(k). Thus the unitary representation of Sp(B) occurring in R_{S} are in this way naturally parametrized by representations of the compact group O(k).

## Keywords

Irreducible Representation Unitary Representation Harmonic Polynomial Symplectic Space Lagrangian Subspace## Preview

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