Component groups
Last updated: October 8, 2005
Let G be a connected complex reductive algebraic group, with a fixed inner
class, and fix a real form in this inner class, represented by an involution
θ. Let G(R) be the group of real points of G for the real form, and
denote K the fixed point group of θ. As explained
here, K is (isomorphic to) the complexification
of a maximal compact subgroup of G(R); in particular, the component group of
K is equal to the component group of G(R).
The following facts are known:
-
when G is semisimple and simply connected, the group K is connected (cf. [1],
thm. 8.1);
-
if T is a maximally split θ-stable torus in G, then T(R) meets all
components of G(R).
From these facts, it follows immediately that the component group of G(R) is
always an elementary abelian 2-group (because this is true for tori), and in
fact it is possible to deduce an algorithm to compute the group from the
root datum of G and the involution θ. The algorithm is explained in
the 2004 notes by Fokko du Cloux
here. This information is available in the program through the
"components" command.
[1] |
R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math.
Soc. 80 (1968), pp. 1-108.
|
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