Some remarks on the uniqueness of liftings of principal indecomposable modules for infinitesimal subgroups of semisimple groups
Let \(G\) denote a semisimple, simply connected algebraic group over an algebraically closed field of positive characteristic \(p\) and \(G_1\) denote the first Frobenius kernel of \(G\). Let \(L(\lambda)\) denote the simple \(G\)-module of highest weight \(\lambda\). If \(\lambda\) is a \(p\)-restricted weight, then \(L(\lambda)\) remains simple upon restriction to \(G_1\). Let \(Q_1(\lambda)\) denote the invective envelope of \(L(\lambda)\) as a \(G_1\)-module. In the category of \(G_1\)-modules, \(Q_1(\lambda)\) is also the projective cover of \(L(\lambda)\). A long-standing conjecture of Humphreys and Verma asserts that \(Q_1(\lambda)\) should admit the structure of a rational \(G\)-module. The conjecture is known to hold for \(p \geq 2h - 2\) (where \(h\) denotes the Coxeter number of the associated root system) and for \(\mathrm{SL}_3\) for all primes. A related question is whether such a lifting is unique if it exists. In the generic \(p \geq 2h - 2\) setting, it is known that the lifting is in fact unique (and may be identified with a certain indecomposable tilting module). In this paper, it is shown that the uniqueness of the lifting also holds for \(\mathrm{SL}_3\). The approach makes use of some potentially useful general criteria for uniqueness. Considering \(L(\lambda)\) as a simple \(G\)-module, let \(I(\lambda)\) denote its injective envelope. The author considers a finite-dimensional \(G\)-submodule \(Y(\lambda)\) of \(I(\lambda)\) which is defined to be the largest submodule whose composition factors have highest weights in a certain finite set determined by \(\lambda\). The author shows that \(Y(\lambda)\) necessarily contains any \(G\)-module lift of \(Q_1(\lambda)\). Further, two different conditions are given that would allow one to conclude that \(Y(\lambda)\) is the unique \(G\)-lift of \(Q_1(\lambda)\) if such a lift exists, and \(Y(\lambda)\) would be an indecomposable tilting module.
scientific article; zbMATH DE number 6690404