Block-compatible metaplectic cocycles

Abstract

has cardinality r ≥ 1. Let G be the F-rational points of a simple Chevalley group defined over F. In his thesis, Matsumoto [5] gave a beautiful construction for the metaplectic cover G of G, a central extension of G by μr(F) whose existence is intimately connected with the deep properties of the r-th order Hilbert symbol (·, ·)F : F × × F → μr(F). Metaplectic groups figure prominently in the study of number theory, representation theory, and physics, arising naturally in the theory of theta functions, dual pair correspondences, Weil representations, and spin geometry. In this paper we study the class of central extensions of a simple Chevalley group over an arbitrary infinite field, of which the metaplectic groups form an important subclass. Metaplectic groups were constructed quite explicitly in Weil’s memoir [10] in the case that G is symplectic. In [3] and [4], Kubota gave the construction of the r-fold metaplectic cover of GL2(F). Moreover, he described an explicit 2-cocycle σK on GL2(F) that represents the second cohomology class of the extension (cf. §3 Corollary 8), which makes it possible to deal quite rapidly with many concrete problems in this setting. Steinberg [9] and Moore [7] considered the algebraic problem of determining the central extensions of a simple Chevelley group over an arbitrary field; they were also led to the metaplectic groups. This line of investigation was completed by Matsumoto [5], whose work forms the foundation of the present paper. To summarize our results, let F be an infinite field, G the F-rational points of a simple Chevalley group defined over F, A an abelian group, and c : F × F → A a Steinberg symbol that is bilinear if G is not symplectic (cf. §1). In this paper we describe an explicit 2-cocycle σG in Z (G;A) that represents the cohomology class in H(G;A) of the central extension G of G by A constructed by Matsumoto [5]