 
Summary: NonLinear 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 2fold 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
