# Tensor products of Weyl modules in positive characteristic

Let $G$ be a simple algebraic group over a field $k$, and let $U$ be the unipotent radical of a Borel subgroup $B$. Because $B$ normalises $U$, the group $H = B/U$ acts on the coordinate ring $\mathcal{O} = k[X]$ of the basic affine space $X = G/U$ via $(h.f)(x) = f(xh)$. We get a decomposition of $\mathcal{O}$ into a direct sum

$\mathcal{O} = \oplus_{\lambda \in \Lambda^+} \mathcal{O}^\lambda$

where the Weyl module $\mathcal{O}^\lambda$ is the set of all $f \in \mathcal{O}$ such that $h.f = \lambda(h)f$ for all $h \in H$. Because the action of $H$ commutes with the action of $G$ on $k[X]$ given by $g.f(x) = f(g^{-1}x)$, each $\mathcal{O}^\lambda$ is a $G$-submodule of $\mathcal{O}$. We can also identify $\mathcal{O}^\lambda$ with the space of global sections $H^0(G/B, \lambda)$.

Next, multiplication in $\mathcal{O}$ induces a $G$-module map $\mathcal{O}^\lambda \otimes \mathcal{O}^\mu \to \mathcal{O}^{\lambda + \mu}$ for any $\lambda, \mu \in \Lambda^+$. Since $\mathcal{O}$ has no zero-divisors, this map is non-zero.

Now if the base field $k$ has characteristic zero, it is well-known that the $G$-modules $\mathcal{O}^\lambda$ for $\lambda \in \Lambda^+$ are irreducible, so the multiplication map above must be surjective. Does this remain true when the characteristic of $k$ is positive, when the Weyl modules $\mathcal{O}^\lambda$ are no longer irreducible in general?

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You should be careful here -- your question is confusing as stated; the modules $\cal O^\lambda$ are not Weyl modules. They are dual Weyl modules. –  Chuck Hague Jun 15 '10 at 18:44
A small point: $X=G/U$ isn't affine, although it is open in its affine hull $Spec\ k[X]$. E.g., when $G=SL_2$, $X$ is the punctured affine plane $\mathbb A^2\setminus (0,0)$. –  inkspot Aug 27 '10 at 12:38
@Konstantin: In 2014 I got around to writing up some informal notes on "Weyl modules" (and their duals), which might clarify the transition in terminology which Chuck commented on. See people.math.umass.edu/~jeh/pub/weyl.pdf –  Jim Humphreys Jan 26 at 18:19

The question has an affirmative answer and a fairly long history as well, but the proof uses some nontrivial ideas. The notation used here is nonstandard relative to that found in Jantzen's book Representations of Algebraic Groups (second edition, AMS, 2003). Also, a "Weyl module" (in the usual sense) of a given highest weight is the dual of the module of global sections for a related line bundle on the flag variety, using Kempf's vanishing theorem (1976). The term "Weyl module" was coined by Carter and Lusztig in their paper on special linear groups, partly because the formal character is given by Weyl's formula. A Weyl module has a unique simple quotient, while the corresponding module has this module as its unique simple submodule.

There was a series of papers by Lakshmibai-Musili-Seshadri on the geometry of flag varieties in prime characteristic, in which they stated along the way that the tensor product of these dual Weyl modules maps onto the one specified by the sum of highest weights. (Their proof may not be rigorous. In any case, Kempf's theorem comes into play here.) A focused reference is the paper in J. Algebra 27 (1982) by Jian-pan Wang, "Sheaf cohomology on $G/B$ and tensor product of Weyl modules". That paper followed up a suggestion of mine that such a tensor product should have a filtration with appropriate Weyl modules as subquotients. The paper by Olivier Matthieu in Duke Math. J. 59 (1989) used Frobenius splitting techniques to prove this in full generality after the partial results by Wang and then by Steve Donkin in Springer Lecture Notes 1140 (1985). Eventually all of this gets folded into the general theory of "tilting modules" for reductive algebraic groups (Chapter G in Jantzen).

[ADDED] As Ekedahl just pointed out, a treatment is given in the more recent and more extensive book by Brion and Kumar along with history.

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It is probably worth saying that for simple G, Donkin's proof worked except when p=2 and G has type E7 or E8. Also, van der Kallen's "Lectures on Frobenius Splittings and B-modules" (Tata Institute Research Lectures 84, 1993) should be mentioned; I think (?) the treatment of Mathieu's result given there is not identical to either that in Jantzen or in Brion-Kumar. –  George McNinch Jun 15 '10 at 21:53

Yes, it is true in general. I found it as Thm 3.1.2 of Brion, Michel(F-GREN-IF); Kumar, Shrawan(1-NC) Frobenius splitting methods in geometry and representation theory. Progress in Mathematics, 231. Birkhäuser Boston, Inc., Boston, MA, 2005. x+250 pp. ISBN: 0-8176-4191-2. The result itself is earlier (see historical remarks at the end of Chapter 3).

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A belated thank you. I've chosen to accept Jim's answer since he gives a very nice account of the related history... –  Konstantin Ardakov Dec 30 '10 at 17:15

Yes.

A map to a Weyl module is surjective if and only if the composition with the projection to its irreducible quotient is surjective (since any submodule of the Weyl module has trivial image there), so you're back to working with irreducibles, where the result is obvious.

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Unless I am mistaken these types of Weyl modules have the irreducible as unique minimal submodule rather than quotient. Consider the case of $S^pV$ which has $V^{(p)}$ as submodule not quotient module. For this reason they seem to be known as dual Weyl modules. –  Torsten Ekedahl Jun 15 '10 at 18:28
Yes, there are two lines of development, coming from algebraic geometry and from modular representation theory. An important point of contact is the characterization of one type of module as the dual of the other type (with "dual" highest weights involved), which involves an easy application of Kempf's vanishing theorem as found in Jantzen's 1980 Crelle paper. In both geometry and representation theory there are many interesting outgrowths, so it's difficult to credit everyone involved. But another important contributor has been Henning Andersen. –  Jim Humphreys Jun 15 '10 at 22:48
The modules the OP (presumably) wants to mention are indeed the "dual Weyl mods" AKA "standard mods". These standrd moduless are the representations with arguably the more "natural" defn: they are given by global sections of G-linearized line bundles on G/B. Stndrd modules have simple socle (soc=max'l semisimple sub) while Weyl mods have unique max'l sub. And indeed, the Weyl mods are precisely the contragredients of the stndrd mods (see next comment for a more intrinsic defn of Weyl mods). –  George McNinch Jun 15 '10 at 22:48
For an a priori defn from the point of view of Kostant's $\mathbf{Z}$-form $U_\mathbf{Z}$ of the env. alg of the corresponding complex Lie algebra $\mathfrak{g}_\mathbf{C}$, the Weyl modules arise as the reduction mod $p$ of minimal $U_\mathbf{Z}$-stable lattices in simple modules for $\mathfrak{g}_\mathbf{C}$. –  George McNinch Jun 15 '10 at 22:48