Filtrations on \(G_1T\)-modules (Q2766402)

Let \(G\) be an almost simple and simply connected algebraic group defined and split over the prime field \(\mathbb{F}_p\) with a maximal split torus \(T\). In previous work of \textit{H. H. Andersen} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 30, No. 3, 353-366 (1997; Zbl 0885.17008)], tilting modules for algebraic groups over fields of prime characteristic were studied by constructing a Jantzen type filtration on the space of group homomorphisms from a Weyl module to a tilting module. The representation theory of \(G\) is closely related to that of \(G_1T\) where \(G_1\) denotes the first Frobenius kernel of \(G\). The main goal of this work is to construct an analogous filtration for \(G_1T\)-modules using the deformation theory developed by \textit{H. H. Andersen, J. C. Jantzen}, and \textit{W. Soergel} [Astérisque 220, 321 p. (1994; Zbl 0802.17009)].NEWLINENEWLINENEWLINEFor \(G_1T\)-modules, a tilting module is the same as a projective module. The authors construct a filtration of the vector space \(\Hom_{G_1T}(Z(\lambda)^\tau,Q)\) for a projective \(G_1T\)-module \(Q\) and weight \(\lambda\) for \(T\). Here \(Z(\lambda)^\tau\) denotes the (contravariant) dual of the standard \(G_1T\)-module (or baby Verma module) \(Z(\lambda)\). They prove a sum formula for the dimensions of the factors of the filtration in terms of the Jantzen filtration of a baby Verma module. Following the above mentioned work of Andersen, the authors further consider the effect of translation functors on the filtration and show that certain behavior under ``wall-crossing'' would imply Lusztig's conjecture on the characters of an irreducible \(G_1T\)-module. Finally, the authors expand upon the original motivating work of Andersen and consider an analogous theory for certain quantum groups and the group \(G\) itself.