The geometric Weil representation (Q2481738)

Let \((V, \omega)\) be a \(2N\)-dimensional symplectic vector space over the finite field \(k\). Let \(H\) be the associated Heisenberg group. According to the Stone-von Neumann theorem, there exists a unique irreducible representation \((\pi, H,\mathcal{H})\) with central character \(\psi\). There exists a Schrödinger model for the realization of this representation, which relies on a Lagrangian splitting of \(V\). However, representation theory suggests a more ``correct'' realization, \(S(V)\) the function space on \(V\), which is called Weyl transform. Let \(G=Sp(V,\omega)\) be the symplectic group of \((V,\omega)\). It gives an action on \(H\). Every element \(g\) in \(G\) gives a new Heisenberg representation \(\pi^g\) on the same space of \(\pi\). It turns out that \(\pi^g\) is isomorphic to \(\pi\). It gives the so-called Weil representation. Based on the idea of invariant presentation, i.e. the Weyl transform, the Weil representation can be realized as a single function on \(G\times V\), satisfying some convolution property. From this point of view, the authors use the idea of function-sheaf correspondence to geometrize the above function. More precisely, they construct perverse sheaves on \(G\times V\) which are called geometric Weil representations, satisfying a convolution property and they induce the above functions after taking the trace of the Frobenius morphism.