Deligne's topological central extension is universal.
Let \(F\) be a local field different from the field of complex numbers. Let \(G\) be an absolute (almost) simple, simple connected algebraic group over the field \(F\). \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 84, 35-89 (1996; Zbl 0930.20043)] has constructed a topological central extension \[ (*) 1 \rightarrow \mu(F) \rightarrow G(F)\tilde \rightarrow G(F) \rightarrow 1. \] of \(G(F),\) where \(\mu(F)\) is a finite cyclic group of roots of identity contained in \(F\). \textit{G. Prasad} and \textit{A.S. Rapinchuk} [Publ. Math., Inst. Hautes Étud. Sci. 84, 91-187 (1996; Zbl 0941.22019)] proved that, if \(G\) is \(F\)-isotropic, then extension (*) is a universal topological central extension of \(G(F)\). This proof used global arguments. In the present paper the author gives a purely ``local'' proof of this result.
scientific article; zbMATH DE number 2032762