Let G be a simple algebraic group over an algebraically closed field of characteristic p, and assume that p is a separably good prime for G. Let P be a parabolic subgroup whose unipotent radical U P has nilpotence class less than p. We show that there exists a particularly nice Springer isomorphism for G which restricts to a certain canonical isomorphism Lie(U P ) ∼ − → U P defined by J.-P. Serre. This answers a question raised both by G. McNinch in [M2], and by J. Carlson et. al in [CLN]. For the groups SLn, SOn, and Sp 2n , viewed in the usual way as subgroups of GLn or GL 2n , such a Springer isomorphism can be given explicitly by the Artin-Hasse exponential series.
In this paper we construct an "abstract Fock space" for general Lie types that serves as a generalisation of the infinite wedge q-Fock space familiar in type A. Specifically, for each positive integer , we define a Z[q, q −1 ]-module F with bar involution by specifying generators and "straightening relations" adapted from those appearing in the Kashiwara-Miwa-Stern formulation of the q-Fock space. By relating F to the corresponding affine Hecke algebra we show that the abstract Fock space has standard and canonical bases for which the transition matrix produces parabolic affine Kazhdan-Lusztig polynomials. This property and the convenient combinatorial labeling of bases of F by dominant integral weights makes F a useful combinatorial tool for determining decomposition numbers of Weyl modules for quantum groups at roots of unity.
Let H be a linear algebraic group over an algebraically closed field of characteristic p > 0. We prove that any "exponential map" for H induces a bijection between the variety of r-tuples of commuting [p]-nilpotent elements in Lie(H) and the variety of height r infinitesimal one-parameter subgroups of H. In particular, we show that for a connected reductive group G in pretty good characteristic, there is a canonical exponential map for G and hence a canonical bijection between the aforementioned varieties, answering in this case questions raised both implicitly and explicitly by Suslin, Friedlander, and Bendel.
In this paper the authors produce a projective indecomposable module for the Frobenius kernel of a simple algebraic group in characteristic p that is not the restriction of an indecomposable tilting module. This yields a counterexample to Donkin’s longstanding Tilting Module Conjecture. The authors also produce a Weyl module that does not admit a p-Weyl filtration. This answers an old question of Jantzen, and also provides a counterexample to the
{(p,r)}
-Filtration Conjecture.
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