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This must be an easy question but I don't have a good argument for it and have not found a counterexample: Let $G$ be a connected semisimple algebraic group over $\mathbb{Q}$ such that the center of $G$, $Z(G)$ is trivial, i.e. $Z(G)={1}$ is it true that the center of $G_{\mathbb{R}}$, denoted $Z(G_{\mathbb{R}})$ is also trivial?

edited Dec 22 at 17:03

asked Dec 22 at 16:17

Jack
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Dear Jack, The $\mathbb R$ points will be Zariski dense in $G$, and so any real-valued element of the centre will be in the centre of $G$. Thus if $G$ has trivial centre, the real points of the group will also have trivial centre. Regards, – Emerton Dec 22 at 16:48

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If $G$ is not assumed connected, some argument is maybe needed to justify that $G_\mathbf{R}$ is Zariski dense. – Yves Cornulier Dec 22 at 17:00

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Actually Emerton's argument used implicitly a result of Rosenlicht of density. But what you ask is easier. The center is defined in a way that is independent of the base field. When you ask if a certain variety is trivial, you don't have to refer to a base field. In particular the answer for your question is true for an arbitrary algebraic group over any field of characteristic zero (in characteristic $p$ it holds if you refer to the schematic center, otherwise there may be a reduceness issue). – Yves Cornulier Dec 22 at 17:11

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Dear Jack: Your question is about the definition of $Z(G)$ and its compatibility with ground field extension. If $X$ is geometrically reduced lft over a field $k$ then $X(k_s)$ is relatively schematically dense in $X_{k_s}$: for a $k_s$-scheme $S$, any $S$-map from $X_S$ to a separated $S$-scheme is uniquely determined by its restriction to $X(k_s)$. This is an instructive exercise. It implies that if $G$ is a smooth $k$-group and $A$ is a $k$-algebra then any $g \in G(A)$ centralizing $G(k_s)$ inside $G(A_{k_s})$ centralizes the scheme $G_A$! This underlies good behavior of $Z(G)$. – kreck Dec 22 at 17:19

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Dear Jack: I think what you are seeking is to know that there is a good notion of "scheme-theoretic center" (as Yves indicates): if $G$ is a smooth $k$-group then the functor that assigns to any $k$-algebra $A$ the set of $g \in G(A)$ whose conjugation action on $G_A$ is trivial is represented by a closed $k$-subgroup scheme $Z(G)$ of $G$. Once this is known, the formation of $Z(G)$ trivially commutes with ground field extension. Build $Z(G)$ using Galois descent and my previous comment: descend the schematic intersection inside $G_{k_s}$ of schematic centralizers of all $g \in G(k_s)$! – kreck Dec 22 at 17:23

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anon · Accepted Answer · 2012-12-22 19:15:49Z UTC

The short answer is that the center Z(G) of a semisimple algebraic group is a well-defined (finite) algebraic subgroup which commutes with extension of the base field, so if it's trivial over $\mathbb{Q}$, then its trivial over every field. But presumably, that is not what you meant to ask. Perhaps you mean: does $Z(\mathbb{Q})$ trivial imply $Z(\mathbb{R})$ trivial? The answer is no. You have a finite group $Z(\bar{\mathbb{Q}})$ with an action of the absolute Galois group of $\mathbb{Q}$ and you are asking: if only 1 is fixed by the full Galois group is only 1 fixed by complex conjugation? Obviously, not necessarily. It's easy to makes lots of different Galois modules as centers of semisimple groups over $\mathbb{Q}$. Or perhaps you mean: if $Z(G)$ is trivial (as an algebraic subgroup), is the center of $G(\mathbb{R})$ trivial? The answer is yes, because $G(\mathbb{R})$ is dense in $G$ for the Zariski topology.

Jim Humphreys · Answer 2 · 2012-12-23 21:39:56Z UTC

Since the question and its answer have both gone through a number of edits, and are accompanied by numerous comments, it may be helpful (or not) to sort the question out further.

1) The interface between semisimple Lie groups (real or complex) and semisimple algebraic groups over $\mathbb{C}$ is somewhat tricky and has been approached historically from different directions. (Borel for instance got heavily involved in correlating these theories.) There is no single reference which takes all viewpoints into account, but it may be helpful to look at Chapter 5 of Onishchik-Vinberg Lie Groups and Algebraic Groups. See for example Section 1 and the first few problems there, with hints and cross-references given later. How you think about the question here is partly determined by your upbringing.

2) It's worth noting that "defined over $\mathbb{Q}$ " in this context is not really needed, since a connected semisimple algebraic group in characteristic 0 always arises (by Chevalley's work) from a $\mathbb{Z}$ -form and is automatically defined over $\mathbb{Q}$ . Aside from this, the essential problem is to relate the complex group $G$ to its subgroup of real points. Here is where the interaction with Lie groups comes in, adding a layer of complication in terminology and notation.

3) To amplify Emerton's comment in matrix terms, think of each center as the kernel of an adjoint representation. Say $G$ is a complex algebraic matrix group, so its Lie algebra (in the sense of either algebraic groups or Lie groups) contains the Lie algebra of the Lie group $G(\mathbb{R})$ . Under the adjoint (conjugation) action any real matrix in the center of $G(\mathbb{R})$ then centralizes the complex Lie algebra of $G$ and thus lies in the (trivial) center of $G$ .

4) This kind of relationship between real and complex groups goes back to E. Cartan and Chevalley, but the interaction with algebraic group language owes most to Borel. In any case, the theory involved was in place over half a century ago. It doesn't require the explicit classificaiton of semsiimple groups or Lie algebras, nor does it require any scheme language. (In the characteristic 0 Borel-Chevalley theory, it's usually safe to identify an algebraic group with its points over $\mathbb{C}$ .)

Center of the algebraic group G_{\mathbb{R}} for a centerless G

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