Factoring Tilting Modules for Algebraic Groups

Let G be a semisimple, simply-connected algebraic group over an algebraically closed field of characteristic p > 0. We observe that the tensor product of the Steinberg module with a minuscule module is always indecomposable tilting. Although quite easy to prove, this fact does not seem to have been observed before. It has the following consequence: If p > 2h − 2 and a given tilting module has highest weight p-adically close to the rth Steinberg weight, then the tilting module is isomorphic to a tensor product of two simple modules, usually in many ways. Let G be a semisimple, simply-connected algebraic group over an algebraically closed field k of characteristic p > 0. For convenience we assume the underlying root system is indecomposable. Tensor products are over k unless otherwise specified. Fix a maximal torus T in G and write X(T ) for the character group of T . Note that X(T ) ≃ Z for some n. By “G-module” we mean “rational G-module”. Fix a Borel subgroup B containing T and let the negative roots be determined by B. Let X(T ) = {λ ∈ X(T ) : (α, λ) > 0, all simple roots α} be the set of dominant weights and Xr(T ) = {λ ∈ X(T ) + : (α, λ) < p, all simple roots α}. The set X1(T ) is known as the restricted region and its elements are often called restricted weights. For any λ ∈ X(T ) let ∆(λ) = the Weyl module of highest weight λ; ∇(λ) = the dual Weyl module of highest weight λ; L(λ) = the simple G-module of highest weight λ. The main properties of these families of modules are summarized in [8], to which the reader should also refer for any unexplained notation or terminology. Date: 29 June 2009. Research supported by the Mercator Programme, DFG.