Tilting modules and cellular categories

In this paper we study categories of tilting modules. Our starting point is the tilting modules for a reductive algebraic group G in positive characteristic. Here we extend the main result in [8] by proving that these tilting modules form a (strictly object-adapted) cellular category. We use this result to specify a subset of cellular basis elements, which generates all morphisms in this category. In a different direction we generalize the earlier results to the case where G is replaced by the infinitesimal thickenings G_rT of a maximal torus T in G by the Frobenius subgroup schemes G_r. Here our procedure leads to a special set of generators for the morphisms in the category of projective G_rT- modules. Our methods are rather general (applying to "quasi hereditary like" categories). In particular, there are completely analogous results for tilting modules of quantum groups at roots of unity. As examples we treat the tilting modules in the ordinary BGG category O, and in the modular case we examine G = SL_2 in some details.