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Non-Linear Covers of Real Groups Jeffrey Adams
 

Summary: Non-Linear Covers of Real Groups
Jeffrey Adams
September 2, 2004
Abstract
Let G be a semisimple, simply connected, algebraic group defined
over R. The set of real points G of G is not necessarily topologi-
cally simply connected, in which case G admits a non­trivial covering
group. We give simple uniform proofs of several basic properties of
real non­linear groups, in particular a simple criterion for when such
a cover exists. Some of these properties were previously known from a
case­by­case check based on the classification of real groups and their
covers.
1 Introduction
Let G be a semisimple, simply connected, algebraic group defined over R.
The set of real points G of G is not necessarily topologically simply con-
nected, in which case G admits a non­trivial covering group. An example
is the metaplectic group, the 2-fold cover of Sp(2n, R). Such a group is not
realizable as a linear (or algebraic, or matrix) group, and is said to be a
non­linear cover of G.
Such groups are very common. In the p­adic case every isotropic group

  

Source: Adams, Jeffrey - Department of Mathematics, University of Maryland at College Park

 

Collections: Mathematics