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An explicit construction of the metaplectic representation over a finite field. (English) Zbl 1026.22018

The paper gives an explicit elementary construction of the Weil representation of a symplectic group over a finite field \(K\).
The idea of the proof explicitly uses the finiteness of the field. The Weil representation \(\mu\), as a projective representation of \(\text{SP}(2n,K)\), is given in terms of the following system of generating elements for the symplectic group \(\text{SP}(2n,K)\): \[ u(b)=\begin{pmatrix}\text{I} &0\cr b&I\cr\end{pmatrix}, \] with \(b\in\text{M}(n,K)\) such that \(b^*=b\) (where \(^*\) denotes transposition), \[ s(a)=\begin{pmatrix} a&0\cr 0&a^{*-1}\cr\end{pmatrix}, \] with \(a\in\text{GL}(n,K)\) and \[ J=\begin{pmatrix} 0&\text{I}\cr\end{pmatrix}. \] Then the author shows that \(\omega=\kappa\mu\) is an ordinary representation of \(\text{SP}(2n,K)\), where \[ \kappa\colon\text{SP}(2n,K)\to\mathbb C \] is the function defined by \(\kappa(g)=\det\mu(g) \det\mu^+(g)^{-2}\), with \(\mu^+\) being the restriction of \(\mu\) to the invariant subspace \(V^+=\{f\in\mathbb C^{K^n} : f(-x)=f(x),\forall x\in K^n\}\). The character of \(\omega\) is computed on the above generating elements of \(\text{SP}(2n,K)\).
The results obtained are motivated by connection with classical quantum kinematics, whose underlying configuration spaces are vector spaces over finite fields.

MSC:

22E50 Representations of Lie and linear algebraic groups over local fields
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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