# Definitions of real reductive groups

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

1. A closed subgroup of $GL(n,C)$ closed under conjugate transpose.

2. The set of real points $G(R)$ of a real algebraic group such that $G(C)$ is reductive.

3. Knapp's definition i.e. a Lie group $G$ with a reductive Lie algebra $\mathfrak g$, a Cartan decomposition $\mathfrak g =\mathfrak k +\mathfrak p$, a maximal compact subgroup $K$ such that $G = K.exp(\mathfrak p)$ and such that every automorphism of $\mathfrak g$ of the form $\operatorname{Ad}(g)$, $g \in G$, is in $\operatorname{Int}(\mathfrak g^C)$.

4. 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?

• 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 '10 at 15:54
• 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 '10 at 16:09
• 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 '10 at 19:06
• There is also a def in Wallach, Real reductive groups, which is a blend of your #2 and #3, I think, with an explicit map from G into R-points of a reductive linear algebraic group (I don't have it close by, so can't check the details). It's good to explicitly include Cartan decomposition, but I would take a pragmatic view: find which properties are needed for a particular application and quote the source which proves them, using its def. In practice, that means Wallach:) Older papers (and even books -- Vogan 1981?) liked to refer to Harish-Chandra's original papers, making it less accessible. – Victor Protsak Jun 20 '10 at 23:34
• I think what's most useful here is to know the examples that tend to distinguish one of these from another. I'll contribute the easy one: the universal cover of $SL_n({\mathbb R})$, $n>2$, which does not sit inside a finite-dimensional matrix group. – Allen Knutson Jun 25 '13 at 19:19

The definition he uses is in some ways the simplest and most natural, I think. In the general framework of finite dimensional Lie algebras over an arbitrary field $K$ of characteristic 0, there are elementary definitions of semisimple and reductive Lie algebras. When you take $K = \mathbb{R}$ and work in the classical framework of real Lie groups, it's then natural to define a connected Lie group $G$ to be reductive if its Lie algebra is. Of course, the Lie algebra only sees local behavior, so one could leave it at that. However, disconnected Lie groups come up immediately when Lie group theory is combined with linear algebraic groups in the study of representations, automorphic forms, etc. So Borel adds in this case the extra requirement that $G$ have only finitely many connected components in the euclidean topology.
Where does this condition come from? Starting with a connected linear algebraic group (scheme) $H$ over $\mathbb{R}$, the resulting group $G:= H(\mathbb{R})$ is a real Lie group but need not be connected. An obvious example is the multiplicative group. But a basic theorem states that this Lie group has only finitely many components in the euclidean topology. The theorem comes, for instance, from Whitney's older work on real affine varieties but is also a consequence of a finiteness theorem in Galois cohomology proved in the work of Borel-Serre. (I'm not sure what the best modern proof of the theorem is, but that's another question.)