It appears to me that there are at least two working definitions of the term "Chevalley group" operative in the literature. For example, one can consider Steinberg's notes on the subject. Starting from a complex Lie algebra $\frak{g}$ and a finite dimensional representation $V$ of it, he constructs a lattice $M$ in $V$ and a family of operators $x_\alpha(a) \in GL(M \otimes_\mathbb{Z} A)$ for any root $\alpha$ and element $a$ of a ring $A.$ These operators generate a subgroup $G_{\mathfrak{g}, V, St}(A)$ of $GL(M \otimes_\mathbb{Z} A)$ which Steinberg calls a Chevalley group. The choice of $M$ also defines a $\mathbb{Z}$-structure on $V.$ That is, we say a polynomial on $V$ has integer coordinates if this is so when the basis for $V$ used to write it in coordinates is actually a $\mathbb{Z}$-basis for $M$. Taking the integer coordinate polynomials on $V$ that vanish on $G_{\mathfrak{g}, V, St}(\mathbb{Q})$ we get a group scheme $G_{\mathfrak{g}, V, StCh}.$ My impression is that this group-scheme is also sometimes called a "Chevalley group." Is this correct? In Bourbaki Expose 219, Chevalley gives a slightly different construction of a family of group schemes $\{G_{\mathfrak{g}, V, Ch}\}$ where now $\mathfrak{g}$ ranges over "anticompact" Lie algebras over $\mathbb{Q}.$ Can anyone confirm that $G_{\mathfrak{g} \otimes_\mathbb{Q} \mathbb{C} , V \otimes_\mathbb{Q} \mathbb{C} , StCh}$ and $G_{\mathfrak{g}, V, Ch}$ are the same group-scheme for each $\mathfrak{g}$ and $V$ as in Chevalley? Is defining a Chevalley group as ``one of the group schemes $G_{\mathfrak{g}, V, Ch}$ equivalent to the definition given in these notes by Brian Conrad? And, finally, are there any other inequivalent definitions of the term "Chevalley group" operative in the literature? Thanks for your time!
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1$\begingroup$ Prop. 1.3 and Thm 1.4 of the link near the end shows there is exactly one notion in the group scheme setting: for a root datum $\Phi$, up to isomorphism there is a unique smooth affine $\mathbf{Z}$-group $G=G_{\Phi}$ with connected reductive fibers and split $\mathbf{Q}$-fiber of root datum $\Phi$ (and $G$ is then split over $\mathbf{Z}$ in the sense of SGA3). Steinberg's notion is the subgroup of $G(A)$ generated by $A$-points of root subgroups for a pinning over $\mathbf{Z}$ for simply connected semisimple root data $\Phi$ (Lie algebra over $\mathbf{Q}$ is insensitive to isogenies). $\endgroup$– nfdc23Commented Oct 9, 2016 at 21:57
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1$\begingroup$ Hmm, maybe Steinberg's groups are for adjoint-type semisimple root data instead. Anyway, it doesn't seem necessary to check the compatibility of SGA3 with the constructions by Chevalley or Steinberg, since everything one wants can be deduced from the known properties of Chevalley groups in the sense of SGA3 (though it is satisfying to have the unique characterization over $\mathbf{Z}$ mentioned above so that one can think about these things independently of any particular construction); e.g., have you see Thm. 5.3.5 and Example 5.3.9 in math.stanford.edu/~conrad/papers/luminysga3.pdf? $\endgroup$– nfdc23Commented Oct 9, 2016 at 22:11
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1$\begingroup$ What seems clear is that "Chevalley groups" in the broadened usage of Steinberg are exactly the split groups (over $\mathbb{Z}$) of all possible isogeny types. It's true that Chevalley's 1955 paper considered only the adjoint groups, starting with a Chevalley basis of the Lie algebra, but Steinberg's viewpoint has largely prevailed by now (reinforced by work of Lusztig). There are of course quite a few non-split groups over various fields, which were studied by Borel-Tits and codified by Conrad-Gabber-Prasad following SGA3. $\endgroup$– Jim HumphreysCommented Oct 9, 2016 at 22:34
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$\begingroup$ P.S. Chevalley's Bourbaki talk is not entirely rigorous, so a more reliable recent source for working over $\mathbb{Z}$ is the 2009 paper by Lusztig ams.org/mathscinet-getitem?mr=2505299 $\endgroup$– Jim HumphreysCommented Oct 9, 2016 at 22:42
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$\begingroup$ @nfdc23: Steinberg allows any lattice between the root lattice and the full weight lattice in his construction of "Chevalley groups". $\endgroup$– Jim HumphreysCommented Oct 9, 2016 at 22:43
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