A generalization of Pillen's theorem for principal series modules II

Abstract

Let G be a connected, semisimple and simply connected algebraic group and G(q) the corresponding finite Chevalley group over the finite field of order q = p r . In a recent paper the author determined a direct sum decomposition of the kG(q)-submodule generated by a highest weight vector of a certain Weyl module when q is not too small, which is a generalization of Pillen's result in 1997. In this article, we claim that the result does not need the assumption on q.

Irreducible Modules of Reductive Groups with Borel-stable Line

“…Following [11] and [29], for each θ ∈ T, a ∈ N, and w ∈ W I(θ) , define T w,a ∈ E θ,a := End kGa (kG a 1 θ ) by T w,a (1 θ ) = U w,a w −1 1 θ . For each J ⊂ I(θ), set e J,a = w∈WJ T w,a , o J,a = (−1) ℓ(wJ ) T wJ ,a .…”

“…[29, Proposition 4.5]). Let θ ∈ T, J ⊂ I(θ), and a ∈ N. Let {π J |J ⊂ I(θ)} be a set of orthogonal primitive idempotents in E θ,a satisfying 1 = J⊂I(θ) π J and π J ∈ e J,a o J ,a E θ,a , whereJ = I(θ)\J.…”

The decomposition of permutation module for infinite Chevalley groups

“…For k =F q , it is known that Ind G B 1 B decomposes into a direct sum of indecomposable modules, each with simple socle, and there is a bijection between the direct summands and the subsets of I (cf. [YY,Proposition 4.5]). However, we have End kG (M(tr)) ≃ k for any field k, since it is clear that f (1 tr ) ∈ M(tr) U = k1 tr .…”

Abstract induced modules for reductive algebraic groups with Frobenius maps

“…The induced modules from a one dimensional module of a Borel subgroup of a finite reductive group have been investigated in great detail (cf. [17], [21], and [32]). For example, in [17] Jantzen constructed a filtration for such induced modules and gave the sum formulas of these filtrations correspond to those of the well known Jantzen filtrations of generic Weyl modules.…”