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There are several definitions of real reductive groups, sometimes subtly inequivalent. The following come to my mind:

A closed subgroup of GL(n,C) closed under conjugate transpose.
The set of real points G(R) of a real algebraic group such that G(C) is reductive.
Knapp's definition i.e. a Lie group G with a reductive Lie algebra g, a Cartan decomposition g = k + p, a maximal compact subgroup K such that G = K.exp(p) and such that every automorphism of g of the form Ad(g), g in G, is in Int(g^C).
Harish-Chandra definition i.e. 3. plus the condition that the connected component of the identity of a semisimple factor of G has finite center.

Maybe there are others too. What are the relations between these different definitions?

edited Jun 20 2010 at 15:59

asked Jun 20 2010 at 15:06

Andrea Altomani
427●1●6

It's more a question of the scope of the theory: for example, Harish-Chandra considered non-linear groups with compact center, which doesn't fall under 1 or 2. Knapp used a variant of 1 in "Beyond the introduction". Is there a particular reason that you need to use different definitions? – Victor Protsak Jun 20 2010 at 15:54

I corrected 2 and added a description of 4. – Andrea Altomani Jun 20 2010 at 16:00

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For #3, please define "Cartan decomposition" when the Lie algebra is not semisimple. Note also that a torus violates #3 since its adjoint representation is trivial. These two points lead me to ask if Harish-Chandra considered $G$ whose Lie algebra is not semisimple. I thought he always worked with groups having semisimple Lie algebra (and then in inductive arguments with centralizers of tori would handle extra central stuff directly). – Boyarsky Jun 20 2010 at 16:09

Boyarsky, connectedness is one of the main issues for me. Victor, it's mainly for efficient usage of the literature. Often the definition used in a paper is not explicitly stated.Personally, I use definition 3 because I like Knapp's book (actually 3 is from "Beyond an Introduction") – Andrea Altomani Jun 20 2010 at 16:10

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Aspects of the relationship between #2 and #4 are addressed in section 24.C of Borel's book on linear algebraic groups. I have a vague memory (but cannot recall from where) that a Cartan involution can always be identified with conjugate transpose as in #1 (maybe this is due to Mostow?), and so I think #1 and #2 may be equivalent, at least modulo connectedness issues on which I'm less confident. – BCnrd Jun 20 2010 at 19:06

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Definitions of real reductive groups

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