On some explicit formulas in the theory of Weil representation
type: Item
instance of: scholarly article
title: On some explicit formulas in the theory of Weil representation
zbMATH Open document ID: 0794.58017
author: R. Ranga Rao
published in: Pacific Journal of Mathematics
publication date: 1994-09-05T00:00:00Z
review text: 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 \(c(\sigma_ 1,\sigma_ 2)\).
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 508867
zbMATH Keywords: Lagrangian space
zbMATH Keywords: symplectic group
zbMATH Keywords: Leray invariant
zbMATH Keywords: Bruhat cell
reviewed by: Stanisław Janeczko
MaRDI profile type: Publication
full work available at URL: pjm.1993.157.335
OpenAlex ID: W2038031528
DOI: 10.2140/PJM.1993.157.335
description: scientific article; zbMATH DE number 508867