Realizations and pinnings (épinglages) of reductive groups

Let $G$ be a reductive group over an (say, algebraically closed) field $k$. Springer (in his book on algebraic groups) calls for a chosen maximal torus $T$ in $G$ a family $(u_\alpha) _{\alpha \in \Phi(G,T)}$ of immersions $u _\alpha:\mathbf{G}_a \rightarrow G$ such that

(i) $t u_\alpha(c) t^{-1} = u_\alpha( \alpha(t) c)$ for all $c \in \mathbf{G}_a$ and $t \in T$,

(ii) $n_\alpha := u_\alpha(1) u_{-\alpha}(-1) u_\alpha(1)$ lies in $\mathrm{N}_G(T) \setminus T$,

(iii) $u_\alpha(x) u_{-\alpha}(-x^{-1}) u_\alpha(x) = \alpha^\vee(x) n_\alpha$ for all $x \in k^\times$,

a realization of (G,T) (or $\Phi(G,T)$) in $G$. We then have $\mathrm{Im}(u_\alpha) = U_\alpha$.

In the book of Conrad-Gabber-Prasad on pseudo-reductive groups a pinning of $G$ is defined as a tuple $(T,\Phi^+,(\varphi_\alpha)_ {\alpha \in \Delta})$ where $T$ is a maximal torus, $\Phi^+$ is a positive system for $\Phi(G,T)$, $\Delta$ is the corresponding basis and $\varphi _{\alpha}: (\mathrm{SL} _2, \mathrm{SL} _2 \cap \mathrm{D} _2) \rightarrow (G _\alpha, G _\alpha \cap T)$ are central isogenies such that $\varphi _\alpha( \mathrm{diag}(x,x^{-1}) ) = \alpha^\vee(x)$ for all $x \in k^\times$, where $G _\alpha = \langle U _\alpha,U _{-\alpha} \rangle$.

My question is: are these two notions somehow equivalent? If a pinning is given, then by defining

$u_\alpha(x) = \varphi_\alpha\begin{pmatrix} 1 & x \\\\ 1 & 0 \end{pmatrix}$, $u_{-\alpha}(x) = \varphi_\alpha\begin{pmatrix} 1 & 0 \\\\ x & 1 \end{pmatrix}$

I get closed immersions satisfying the properties above, but unfortunately, as I have $\varphi_\alpha$ only for $\alpha \in \Delta$ this does not yet define a realization. How can I define the $u_\alpha$ for $\alpha \notin \Delta \cup -\Delta$? What about the other direction?

Moreover (as C-G-P also mentions) in SGA3, exposé XXIII, there is defined the notion of épinglages and Conrad mentions that these carry the same information as the pinnings above. Can somebody make this precise? Moreover in SGA, it is mentioned that an épinglage induces monomorphisms $p_\alpha: \mathbf{G}_a \rightarrow G$ for $\alpha \in \Delta \cup -\Delta$. I suspect that these are the morphisms I defined above, but again, can I get a realization from this?

A further problem is the following: For a given realization and a total order on $\Phi(G,T)$ Springer defines structure constants which appear in the expression of the commutator $\lbrack u_\alpha(x), u_\beta(y) \rbrack$ in terms of $u_\gamma$ for linearly independent $\alpha, \beta \in \Phi$. Springer shows that for root systems NOT of type $G_2$ a realization with integral structure constants exist. Demazure also calculates these commutators in SGA3, exposé XXII, for the $p_\alpha$ mentioned above in case of rank 2 root systems. Here, I was surprised that the structure constants seem to be independent of the pinning chosen. Is this now a rank 2 phenomenon that is also true for realizations or does this mean that pinnings/épinglages are more restrictive than realizations?

I hope, somebody can help me here.

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The obvious exercise to do here is surely to let $G$ be $GL_3$, define $\phi_\alpha$ to be the obvious thing for $\alpha$ a simple root, and then attempt to define $\phi_\alpha$ for the other positive root and to see if there is any flexibility. What happens if you do this exercise? Alternatively you can just wait for Brian to wake up. –  Kevin Buzzard Mar 9 '10 at 11:12
I could figure out that one can extend the "partial realization" on the base to a realization by conjugating with elements of the Weyl group and introducing additional coefficients (this is how the $p_{ir + js}$ are defined in SGA and this is probably what happens in 9.2.1 in Springer). But my understanding is far from complete yet, so any help is still welcome! :) –  user717 Mar 9 '10 at 16:05
I'll send an answer later. This question raises a very good point which should be addressed in the p-red book (the issue of defining $\phi_a$ for all roots $a$). It is certainly something I thought about during the writing of Appendix A.4 of the book. I will discuss with my co-authors about the cleanest way to handle it and get back to you. –  BCnrd Mar 9 '10 at 17:06
Morning Brian . –  Kevin Buzzard Mar 9 '10 at 19:03
Poor, beautiful, butterflies, pinned to the page. –  Scott Morrison Mar 10 '10 at 2:16

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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I should also add that the theory of split groups over the prime field explains why the structure constants should be able to be chosen to depend only on the characteristic of the field, and the theory of split groups over $\mathbf{Z}$ explains why the structure constants can be chosen in a universal manner depending only on the root datum. (Of course, one has to then prove the Isom/Existence/Isogeny Theorems over $\mathbf{Z}$, which is done in SGA3 using a lot of preliminaries and Weil's birational group laws over Dedekind domains.) –  BCnrd Mar 10 '10 at 0:48
Great! Thanks a lot for your detailed answer. –  user717 Mar 11 '10 at 17:11