Notes on Weyl modules for semisimple algebraic groups

Over many decades of development, there has been some evolution in the language and notation used for Weyl modules in the theory of semisimple algebraic groups in prime characteristic. Here we survey this evolution briefly, in the hope of clarifying what goes on in the literature. We also discuss briefly the related notion of tilting module. Since our emphasis is on rational representations, it is convenient to fix an arbitrary algebraically closed field K of characteristic p > 0 and take G to be a connected semisimple algebraic group over K which is moreover simply connected. (Similar ideas apply to a connected reductive group whose derived group is simply connected, but with added bookkeeping needed for characters of the central torus.) We fix a pair of opposite Borel subgroups B+ and B− intersecting in a maximal torus T and corresponding to respective choices of positive and negative roots, while W = NG(T )/T is the Weyl group. Though notation for root systems and related concepts differs widely in the literature, we take Φ to be the root system and X := X(T ) the character group of T , identifiable in our simply connected case with the full lattice of integral weights. Relative to the positive roots Φ+ we get a cone of dominant integral weights in X, denoted X+. Since G is simply connected, the half-sum of positive roots ρ (= sum of fundamental dominant weights) lies in X+.