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Let $G_1, G_2$ be connected semisimple Lie groups, let us assume for simplicity that both groups are complex (even though, I am interested in the real Lie groups as well). Let $f: G_1\to G_2$ be a monomorphism which sends regular semisimple elements to regular semisimple elements. Does it follow that $f$ also sends regular unipotent elements to regular ones?

I suspect that the answer is well-known, but I could not find it. (Actually, I do not even know if there is a standard name for homomorphisms preserving regularity.) This question is motivated by study of discrete subgroups of higher rank Lie groups, but explaining the precise motivation will take us a bit too far.

Edit: I should have thought a bit more before asking, since there is an obvious counter-example: The reducible (faithful) representation $SL(2)\to SL(3)$ preserves regularity of semisimple elements but does not preserve regularity of unipotent elements. However, the reducible representation $SL(n)\to SL(n+1), n\ge 3$, fails to preserve regularity of semisimple elements, so maybe there is a hope to classify all counter-examples.

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@Misha: I got distracted by the Lie group language and skipped over the key word "monomorphism". So my comment on centralizers isn't relevant except in the image of the map. –  Jim Humphreys May 8 '13 at 20:59
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2 Answers 2

Probably it's more natural to talk about the Jordan decomposition and regularity when the groups are interpreted as semisimple algebraic groups (or real forms thereof). I don't think there is a special name for the homomorphisms you describe. But your question should have an affirmative answer in the algebraic group setting as an application of ideas in the paper by A. Borel and J. Tits, Homomorphismes "abstraits" de groupes algebriques simples, Ann. of Math. 97 (1973), 499-571. This paper has the most comprehensive treatment of what is possible for abstract homomorphisms relative to various fields of definition, etc.

An important feature of regular semisimple elements in such an algebraic group is their density, whereas Borel-Tits show in effect that abstract homomorphisms are fairly close to being algebraic group morphisms that would respect semisimple and unipotent elements along with their centralizers. I'll take another look at the paper to see how close it comes to answering your question directly, but anyway it's available online through JSTOR.

ADDED:

1) Working with these groups over $\mathbb{C}$ simplifies matters a lot. For example, a connected semisimple Lie group is the same as a connected semisimple algebraic group (Chevalley), where the Jordan-Chevalley decomposition exists and is preserved under rational homomorphisms. In this algebraically closed characteristic 0 setting, the Borel-Tits study of abstract homomorphisms also simplifies and overlaps earlier papers on Chevalley groups, etc. Since Borel-Tits aim for maximum generality, their hypotheses get fairly technical and are not always needed over $\mathbb{C}$. Indeed, I'm not quite convinced that you need your hypothesis on the behavior of regular semisimple elements.

2) However, working with real Lie groups is appreciably more complicated. For example, some of these are not linear algebraic groups (making the notion of semisimple or unipotent element less obvious). Borel-Tits mostly avoid considering anisotropic algebraic groups over a field which is not algebraically closed. For semisimple Lie groups, anisotropic = compact. Fortunately however, all elements of a compact Lie group are "semisimple" (while a compact Lie group itself is algebraic over $\mathbb{R}$); so your question doesn't arise here.

3) Though it's probably not directly relevant to what you are looking at, there is a fairly long history involving real Lie groups (for instance continuity of their abstract homomorphisms), going back to work of Freudenthal and others. Following the Borel-Tits paper, Tits himself focused more directly on Lie groups in a concisely written conference article. The promised sequel with more details apparently never appeared: Homorphismes “abstraits” de groupes de Lie. Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), pp. 479–499. Academic Press, London, 1974.

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Thank you, Jim, I will take a look. –  Misha May 8 '13 at 5:20
    
Jim: In general (even over C), algebraic monomorphisms need not preserve regularity of semisimple elements (and of unipotent ones). For instance, reducible representations $SL(n)\to SL(n+1)$ will not preserve regularity (for $n\ge 3$), while all irreducible representations $SL(2)\to SL(n)$ preserve regularity of unipotents. –  Misha May 8 '13 at 18:14
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If memory serves, results similar to those you are interest in are proved in

Seitz, Gary M. "Abstract homomorphisms of algebraic groups." Journal of the London Mathematical Society 56.1 (1997): 104-124.

More recent references include the work by Caprace and his students, the starting point being

Caprace, Pierre-Emmanuel. "Abstract" Homomorphisms of Split Kac-Moody Groups. Amer Mathematical Society, 2009.

The main trick, already noted in Borel--Tits, is the following: by Jacobson-Morozov (on the group level), the nilpotent element you are interested in is included in a group of type $A_1$, i.e. $\text{SL}_2(\mathbf C)$ or $\text{PSL}_2(\mathbf C)$. Now either use representation theory (Caprace) or the fact that your nilpotent element is included in the derived group of a solvable group (Borel--Tits).

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Yes, this is what I tried to do before finding a trivial counter example! –  Misha May 8 '13 at 18:19
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