Weil representations via abstract data and Heisenberg groups: a\n comparison

2019/06/08 by James Cruickshank, Cruickshank, James, Luis Gutiérrez Frez +3
Chemistry · Mathematics · #11T24 #15B33 #20C15 #20F05 #20H25 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Molecular spectroscopy and chirality #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.1906.03468

openalex publication_date 2019/06/08 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Let B be a ring, not necessarily commutative, having an involution * and\nlet mathrm U_2m(B) be the unitary group of rank 2m associated to a\nhermitian or skew hermitian form relative to *. When B is finite, we\nconstruct a Weil representation of mathrm U_2m(B) via Heisenberg groups\nand find its explicit matrix form on the Bruhat elements. As a consequence, we\nderive information on generalized Gauss sums. On the other hand, there is an\naxiomatic method to define a Weil representation of mathrm U_2m(B), and\nwe compare the two Weil representations thus obtained under fairly general\nhypotheses. When B is local, not necessarily finite, we compute the index of\nthe subgroup of mathrm U_2m(B) generated by its Bruhat elements. Besides\nthe independent interest, this subgroup and index are involved in the foregoing\ncomparison of Weil representations.\n

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