# Timeline for Definitions of real reductive groups

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Feb 24 '21 at 11:27 history edited
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S Jun 25 '13 at 21:11
Jun 25 '13 at 19:19 comment 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.
Jun 25 '13 at 18:51 answer timeline score: 14
Jun 21 '10 at 20:51 comment I'll leave this to specialists, but keep in mind that Elie Cartan would not have recognized the term real reductive group. The notion of "reductive" for Lie or algebraic groups is more recent and also somewhat confusing. The connotation of "completely reducible" inspired some papers by Hochschild and Mostow, where your definition 1 plays a role. I'd tend to rely mostly on Vogan, Wallach, Knapp.
Jun 21 '10 at 1:36 comment Andrea, the connectedness tripped me up for relating #1 and #2. The condition in #1 should be satisfied by identity component in #2, and so your condition #5 should indeed be equivalent to #1, but #2 with algebraically connected $G$ is probably the "best" situation in which to find oneself. As you know, connectedness is a persistent nuisance on the real-analytic side, whereas algebraically one has very nice connectedness results valid for any ground field (such as $\mathbf{R}$ even though it isn't algebraically closed). That's a reason why #2 with algebraically connected $G$ is "best".
Jun 20 '10 at 23:34 comment 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.
Jun 20 '10 at 21:00 comment I have also seen the definition: 5. Open subgroup of a group of the form #2. Maybe this is equivalent to #1
Jun 20 '10 at 19:06 comment 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.
Jun 20 '10 at 16:46 comment Andrea: sorry, I misread #3 near the end as saying "is of the form". Since connectedness is a main issue for you, I highly recommend focusing on condition #2 if possible. It is a wonderful thing to promote Lie algebra facts to global group facts for a connected real algebraic group (connected in the Zariski topology) even when the real points are disconnected. I once heard from Vogan that away from #2 (perhaps with connectedness in the algebraic sense), the main issue is to decide in what ways you want things to begin breaking down.
Jun 20 '10 at 16:15 comment The exact definition is in the link. Essentially the torus can be decomposed in a compact part and a vector part too (this depends on the group, not only on the algebra) A connected torus does not violate 3, because both Ad(G) and Int(**g**^C) are trivial
Jun 20 '10 at 16:10 comment 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")
Jun 20 '10 at 16:09 comment 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).
Jun 20 '10 at 16:00 comment I corrected 2 and added a description of 4.
Jun 20 '10 at 15:59 history edited