Tilting Modules for Algebraic Groups

Abstract

The author surveys what is known and what is conjectured abouttilting modules for a reductive algebraic group in characteristicp>0, based on his papers [Ann. Sci. Ecole Norm. Sup. (4) 30(1997), no. 3, 353 366; MR 98h:20076; ``A sum formula for tiltingfiltrations'', to appear]. He also reformulates his earlierconjectures about the connections with tilting modules for arelated quantum group at a pth root of unity. The hope is thatthe formal characters of the modules in these two settings willagree when p is at least the Coxeter number and the highestweight is in the lowest p\sp 2 alcove for the affine Weylgroup.\par {For the entire collection see MR 99i:20002.}