Weil representations of the symplectic group (Q1273401)

Let \(q\) be an odd prime power and \(\text{Sp}:=\text{Sp}_{2n}(q)\) be the finite symplectic group over the field of \(q\) elements. A Weil representation of \(\text{Sp}\) is a complex representation \(R\) of degree \(q^n\), which decomposes as the sum of two irreducible constituents, \(R_0\) of degree \((q^n+1)/2\) and \(R_1\) of degree \((q^n-1)/2\). Weil representations have been the subject of intensive research since the 1960's. The purpose of the paper is to give explicit constructions of Weil representations. \(\text{Sp}\) acts on the Heisenberg group \(H\) of order \(q^{2n+1}\) and acts trivially on the centre of \(H\). Hence any irreducible complex representation of \(H\) of degree \(q^n\) is stabilized by \(\text{Sp}\) and thus gives rise to a projective representation \(R'\) of \(\text{Sp}\). Having constructed an explicit model of \(R'\), the author shows that \(R'\) is equivalent to an ordinary representation \(R\), without referring to known results about the Schur multiplier of \(\text{Sp}\). Next he shows that \(R_0\) and \(R_1\) can be realized over the smallest possible field, namely its character value field \(\mathbb{K}\), except for the case of \(R_1\) with \(\mathbb{K}\subseteq\mathbb{R}\), where \(R_1\) can be realized over a quadratic extension of \(\mathbb{K}\). Again, his construction is explicit and does not require Schur index calculations.