On some explicit formulas in the theory of Weil representation
Let \(W\) be the finite subgroup of the symplectic group \(Sp(X)\) consisting of all \(\sigma\) such that \(\{e_ i, e_ i^*\}\subseteq \{\pm e_ i, \pm e_ i^*\}\) for each \(i\), where \(e_ i\), \(e_ j^*\) form a symplectic basis of \(X= V+V^*\). Then using the Bruhat decomposition \(Sp(X)= PWP\) (where \(P\) is the stabilizer of \(V^*\)) the author shows the existence of the Haar measures \(\mu_ \sigma\) such that for the operators \(\xi(\cdot)\); \(\xi(p_ 1 \sigma p_ 2)= \xi(p_ 1) \xi(\sigma) \xi(p_ 2)\) and \(\xi(\sigma_ 1 \sigma_ 2)= \xi(\sigma_ 1) \xi(\sigma_ 2)\), \(\sigma_ 1,\sigma_ 2\in W\), \(p_ 1, p_ 2\in P\). For the standard model of \(\mu\) he describes the 2-cocycle \(c(\sigma_ 1, \sigma_ 2)\) (the Weil index of the Leray invariant of the Lagrangian subspaces \(V^*\), \(V^* \sigma_ 2^{-1}\), \(V^* \sigma_{1\cdot}\)) in terms of the Leray invariant, and generalizes the Weil formula for triplets of elements belonging to the big Bruhat cell. The author finds the normalizing constant such that the standard model is metaplectic and gives the explicit formula for the corresponding mutiplier \(c(\sigma_ 1,\sigma_ 2)\).
scientific article; zbMATH DE number 508867