Some remarks on the ring of twisted tilting modules for algebraic groups

In a recent paper [S. Doty, A. Henke, Decomposition of tensor products of modular irreducibles for SL2, Q. J. Math. 56 (2005) 189–207], Doty and Henke give a decomposition of the tensor product of two rational simple modules for the special linear group of degree 2 over an algebraically closed field of characteristic p>0. In performing this calculation it proved useful to know that the simple modules are twisted tensor products of tilting modules. It seems natural therefore to consider the ring of twisted tilting modules for a semisimple group G (a subring of the representation ring of G). However, we quickly specialize to the case in which G is the special linear group of degree 2. We show that (in this case) the ring is reduced and describe associated varieties. We give formulas from which one may determine the multiplicities of the indecomposable module summands of the tensor product of twisted tilting modules.