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).