# Weyl modules and reduction modulo $p$.

Representation theory of Lie groups and Lie algebras are quite close and in a suitable sense they become equivalent. However, some subjects are typically found on a Lie group theory language. For example, Weyl modules almost always start from a (Chevalley) group context and not from a Lie algebra context. Another examples are about hyperalgebras.

Let's just focus on Weyl modules. Starting with a finite-dimensional complex Lie algebra $g$ one constructs Weyl modules as a module given by generators and relations, i.e as a quotient of the traditional $U(g)$, and then one can proceed with an algebraic approach.

QUESTION: What is necessary to start from a purely algebraic setting the following ideas?

Let $G$ be a simply connected Chevalley group over an algebraically closed field of characteristic $p$. Let $G_\mathbb{C}$ be the Chevalley group over the complex numbers which corresponds to $G$.

A Weyl module is the reduction modulo $p$ of an irreducible module in characteristic $0$. The Weyl module $\overline{W}(\lambda)$, the reduction modulo p of the $G_\mathbb{C}$-module of highest weight $\lambda$, has a unique maximal submodule $M(\lambda)$ and $V(\lambda) = \overline{W}(\lambda)/M(\lambda)$ is the irreducible $G$-module of highest weight $\lambda$.

It was found in http://www.jstor.org/pss/2374222.

Particularly, what is the reference for the affirmation "A Weyl module is the reduction modulo $p$ of an irreducible module in characteristic $0$" ?

-
The notion of reduction modulo $p$ requires a lift of $G$ to a ring that admits ring homomorphisms both to $\mathbb{C}$ and to your characteristic $p$ field. The standard choice is a finite extension of $\mathbb{Z}_p$. You can construct highest-weight modules over such rings and then reduce. – S. Carnahan Jun 22 '11 at 5:43
IIRC the characteristic 0 irreducible modules have several non-isomorphic reductions modulo $p$. After all, the reduction depends on the chosen $\mathbf{Z}$-form. For example the induced module will have the highest weight irreducible module at the socle after reduction mod $p$, but the Weyl module will have it at the top – Jyrki Lahtonen Jun 19 '12 at 16:05

It's not difficult to answer this question, but for this it's useful to sketch briefly the origins of the term Weyl module. As usual in mathematics, the history and attributions are somewhat convoluted, but the main developments can be reconstructed in outline form:

1) Following decades of classical development in Lie theory, a major breakthrough came with Chevalley's 1955 Tohoku paper in which he showed uniformly how to construct a $\mathbb{Z}$-basis (unique up to sign choices) for each simple Lie algebra over $\mathbb{C}$ which permits a good "reduction mod $p$ for arbitary primes. In effect he incorporated factorial denominators into his basis construction, so certain "exponentials" can be constructed as polyunomials even in characteristic $p$. After change of basis to an arbitrary field $k$, these elements generate a Chevalley group, essentially a $k$-form of an associated simple algebraic group of adjoint type. Over finite fields he gets related simple groups including some never constructed before.

2) In his 1956-58 Paris seminar on classification of semisimple algebraic groups, Chevalley observed that groups of all possible types exist based on his $\mathbb{Z}$-form approach. His Bourbaki seminar talk (exp. 219, 1960-61) explained how to convert this into something like scheme language in terms of a $\mathbb{Z}$-form of the function algebra. But not all details are given (and some are out of focus). A more definitive version was given by Lusztig in J. Amer. Math. Soc. 2009.

3) Steinberg's Yale lectures on Chevalley groups (1967-68) streamlined the construction of all types of groups, including the various twisted groups, and began the study of representations in characteristic $p$ in terms of reduction mod $p$ of simple finite dimensional modules with dominant integral highest weights. A crucial role is played by Kostant's construction of a suitable integral form in the universal enveloping algebra, which he presented at the 1965 Boulder summer conference (and where Steinberg also lectured, while Verma and I were among the graduate students attending). The basic constructions here were included in my 1972 Springer graduate text. Meanwhile Warren Wong had extended some of the ideas in terms of "contravariant" forms.

4) In my 1971 J. Algebra paper, I didn't yet give a name to the modules obtained from a minimal "admissible" $\mathbb{Z}$-form, but I was able to extend the theory of these highest weight modules and their possible composition factors. This work got taken up by Verma in his lecture notes from the 1971 Budapest summer school organized by Gelfand (published some years later); he emphasized the hidden but essential role of an affine Weyl group. In a footnote he also called attention to the 1974 paper by Carter and Lusztig (Proc. London Math. Soc.), where the interplay of representations for special linear and symmetric groups was treated in the Chevalley group setting mod $p$. They made my "linkage" result more precise and dubbed the unnamed modules of highest weight Weyl modules (in part at least because their formal characters are still given by Weyl's formula even though they tend to have complicated composition series in characteristic $p$).

5) Independently, Jantzen was making rapid progress from a different angle on the same problems; he quickly adopted the label "Weyl module" for arbitrary semisimple groups, using as I did the module obtained from a minimal admissible lattice. For the special linear case, he was also able to show by ad hoc methods that these Weyl modules have the natural universal property among finite dimensional highest weight modules. Taking advantage of Kempf's 1976 vanishing theorem for dominant line bundles in characteristic $p$, I observed that this would easily prove the universal property of Weyl modules in general (written down by Jantzen in Satz 1 of his 1980 Crelle paper). In effect Weyl modules are dual to global section modules of line bundles (with an obvious dualization of the highest weight). This became in fact the preferred definition of "Weyl module" in Jantzen's book Representations of Algebraic Groups.

6) The 1979 Invent. Math. paper by Kazhdan and Lusztig led not only to their conjecture on composition factor multiplicities for Verma modules in characteristic 0 but also to Lusztig's analogous conjecture for Weyl modules. This is still not quite proved in the correct generality (and fails to cover primes smaller than the Coxeter number). The story is not complete, but the literature by now is extensive.

-

It does not make sense to "reduce a complex representation mod $p$". For a finite group $G$ and a complex representation you first have to find a lattice invariant under $\mathbb{Z}G$. Then you can reduce mod $p$. This is explained clearly in the book by Serre.

Once you are ready to move on to your setting of Lie algebras and Lie groups you will find that the question is to make sense of the analogue of "invariant under $\mathbb{Z}G$".

-
@Chris: "characteristic $0$" does not imply "complex." – Qiaochu Yuan Jun 19 '12 at 23:21
@Qiaochu: Of course it does not imply and I never said it. You have to look at the whole phrase! Once we have the claim that we have "reduction modulo $p$ of an irreducible module in characteristic $0$", then we have the claim "reduction modulo $p$ of a complex irreducible representation". Anyway, this comment doesn't help in anything here! – Chris Jun 20 '12 at 4:38