Realizations and pinnings (épinglages) of reductive groups

OK, here's the deal.

I. First, the setup for the benefit of those who don't have books lying at their side. Let $(G,T)$ be a split connected reductive group over a field $k$, and choose $a in Phi(G,T)$ (e.g., a simple positive root relative to a choice of positive system of roots). Let $G_a$ be the $k$-subgroup generated by the root groups $U_a$ and $U_{-a}$. (Recall that $U_a$ is uniquely characterized as being a nontrivial smooth connected unipotent $k$-subgroup normalized by $T$ and on which $T$ acts through the character $a$.) This is abstractly $k$-isomorphic to ${rm{SL}}_ 2$ or ${rm{PGL}}_ 2$, and $G_a cap T$ is a split maximal torus.
So there is a central isogeny $phi:{rm{SL}}_ 2 rightarrow G_a$ (either isomorphism or with kernel $mu_2$), and since ${rm{PGL}}_ 2(k)$ is the automorphism group of ${rm{SL}}_ 2$ and of ${rm{PGL}}_ 2$ over $k$ there is precisely this ambiguity in $phi$ (via precomposition with its action on ${rm{SL}}_ 2$). The burning question is: to what extent can we use $T$ and $a$ to nail down $phi$ uniquely?

The action of $G_a cap T$ on $U_a$ is via the nontrivial character $a$, and among the two $k$-isomorphisms $mathbf{G}_ m simeq G_a cap T$ the composition with this character is $t mapsto t^{pm 2}$ in the ${rm{SL}}_ 2$-case and $t mapsto t^{pm 1}$ in the ${rm{PGL}}_ 2$-case. Fix the unique such isomorphism making the exponent positive.

Now back to the central isogeny $phi:{rm{SL}}_ 2 rightarrow G_a$. By conjugacy of split maximal tori (when they exist!) we can compose with a $G_a(k)$-conjugation if necessary so that $phi$ carries the diagonal torus $D$ onto $G_a cap T$. Recall that we used $a$ to make a preferred isomorphism of $G_a cap T$ with $mathbf{G}_ m$. The diagonal torus $D$ also has a preferred identification with $mathbf{G}_ m$, namely $t mapsto {rm{diag}}(t, 1/t)$. Thus, $phi: D rightarrow G_a cap T$ is an endomorphism of $mathbf{G}_ m$ with degree 1 or 2, so it is $t mapsto t^{pm 2}$ (${rm{PGL}}_ 2$-case) or $t mapsto t^{pm 1}$ (${rm{SL}}_ 2$-case). Since the standard Weyl element of ${rm{SL}}_ 2$ induces inversion on the diagonal torus, by composing with it if necessary we can arrange that $phi$ between these tori uses the positive exponent. That is exactly the condition that $phi$ carries the standard upper triangular unipotent subgroup $U^+$ onto $U_a$ (rather than onto $U_{-a}$).

II. So far, so good: we have used just $T$ and the choice of $a in Phi(G,T)$ to construct a central isogeny $phi_a:{rm{SL}}_ 2 rightarrow G_a$ carrying $U^+$ onto $U_a$ and $D$ onto $G_a cap T$, with the latter described uniquely in terms of canonically determined identifications with $mathbf{G}_ m$ (as $t mapsto t$ or $t mapsto t^2$). The remaining ambiguity is precomposition with the action of elements of ${rm{PGL}}_ 2(k)$ that restrict to the identity on $D$, which is to say the action of $k$-points of the diagonal torus $overline{D}$ of ${rm{PGL}}_ 2$. Such action restrict to one on $U^+$ that identifies $overline{D}(k)$ with the $k$-automorphism group of $U^+$ (namely, $k^{times}$ with its natural identification with ${rm{Aut}}_ k(mathbf{G}_ a)$).

Summary: to nail down $phi$ uniquely it is equivalent to specify an isomorphism of $U^+$ with $mathbf{G}_ a$. But $phi$ carries $U^+$ isomorphically onto $U_a$. So it is the same to choose an isomorphism of $U_a$ with $mathbf{G}_ a$. Finally, the Lie functor clearly defines a bijection

$${rm{Isom}}_ k(mathbf{G}_ a, U_ a) simeq {rm{Isom}}(k, {rm{Lie}}(U_ a)).$$
So we have shown that to specify a pinning in the sense of Definition A.4.12 of C-G-P is precisely the same as to specify a pinning in the sense of SGA3 Exp. XXIII, 1.1.

III. Can we improve a pinning to provide unambiguous $phi_c$'s for all roots $c$? No, there is a discrepancy of units which cannot be eliminated (or at least not without a tremendous amount of work, the value of which is unclear, especially over $mathbf{Z}$), and if we insist on working over $k$ and not $mathbf{Z}$ then there are further problems caused by degeneration of commutation relations among positive root groups in special cases in small nonzero characteristics (e.g., ${rm{G}}_ 2$ in characteristic 3 and ${rm{F}}_ 4$ in characteristic 2).
As we saw above, to nail down each $phi_c$ it is equivalent to do any of the following 3 things: fix an isomorphism $mathbf{G}_ a simeq U_c$, fix a basis of ${rm{Lie}}(U_ c)$, or fix a central isogeny ${rm{SL}}_ 2 rightarrow G_c$ carrying $D$ onto $G_c cap T$ via $t mapsto t$ or $t mapsto t^2$ according to the canonical isomorphisms of $mathbf{G}_ m$ with these two tori (the case of $G_c cap T$ being determined by $c$). This latter viewpoint provides $phi_{-c}$ for free once $phi_c$ has been defined (compose with conjugation by the standard Weyl element), so the problem is to really define $phi_c$ for $c in Phi^+$.

Consider the unipotent radical $U$ of the Borel corresponding to $Phi^+$, so $U$ is directly spanned (in any order) by the $U_c$'s for positive $c$. If we choose an enumeration of $Phi^+$ to get an isomorphism $prod_c U_c simeq U$ of varieties via multiplication, then for simple $c$ we have a preferred isomorphism of $U_c$ with $mathbf{G}_ a$ and one can ask if the isomorphism $mathbf{G}_ a simeq U_c$ can be determined for the other positive $c$ so that the group law on $U$ takes on an especially simple form. This amounts to working out the commutation relations among the $U_a$'s for $a in Delta$ (when projected in $U_c$ for various $c$), and by $T$-equivariance such relations will involve monomials in the coordinates of the $U_a$'s along with some coefficients in $k^{times}$ (and some coefficients of 0). These are the confusing "structure constants". Chevalley developed a procedure over $mathbf{Z}$ to make the choices so that the structure constants come out to be positive integers (when nonzero), but there remained some ambiguity of signs since

$${rm{Aut}}_ {mathbf{Z}}(mathbf{G}_ a) = mathbf{Z}^{times} = {pm 1}.$$

Working entirely over $k$, there are likewise $k^{times}$-scalings that cannot quite be removed. I am told that Tits developed a way to eliminate all sign confusion, but honestly I don't know a reason why it is worth the heavy effort to do that. For most purposes the pinning as above is entirely sufficient, and this is "justified" by the fact that all ambiguity from $(T/Z_G)(k)$-action is eliminated in the Isomorphism Theorem when improved to include pinnings (see Theorem A.4.13 in C-G-P).

IV. What about the realizations in the sense of Springer's book? All he's doing is making use of the concrete description of the group law on ${rm{SL}}_ 2$ to describe a central isogeny $phi_c$ for every positive root $c$. (His conditions relate $c$ with $-c$, which amounts to the link between $phi_c$ and $phi_{-c}$ in the preceding discussion.) As long as he restricts to $alpha in pm Delta$ then he's just defined a pinning in the above sense. But he goes further to basically do what is described in II but without saying so explicitly. He then has to confront the puzzle of the structure constants. (It is a real puzzle, since in the theory over $mathbf{Z}$, which logically lies beyond the scope of his book, the structure constants are not always $0$ and $pm 1$; in some rare cases one gets coefficients of $pm 2$ or $pm 3$, which implies that if one insists on working over fields and not over $mathbf{Z}$ then life in characteristic 2 and 3 will look a bit funny in some cases.) The only conceptual way I know of to overcome the puzzle of the structure constants is to work over $mathbf{Z}$ and to follow either SGA3 or Chevalley in this respect. For the former, one has to really do the whole theory of reductive groups over a base that is not necessarily a field. For the latter, perhaps the (unpublished?) Yale notes of Steinberg are the best reference.

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