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On tilting modules for algebraic groups. (English) Zbl 0798.20035
Let \(G\) be a reductive algebraic group over an algebraically closed field \(K\) of characteristic \(p > 0\). A rational \(G\)-module \(M\) is called a (partial) tilting module if both \(M\) and its dual \(M^*\) may be filtered by modules \(\nabla(\lambda) = \text{Ind}^ G_ B K_ \lambda\) (\(\lambda\) a dominant weight), induced from one dimensional modules for a Borel subgroup \(B\) of \(G\). This paper is an investigation of tilting modules for \(G\). There is an indecomposable tilting module \(M(\lambda)\) for each dominant weight \(\lambda\). Various results on tilting modules are established. In particular it is shown that the indecomposable tilting modules for \(G\) behave well on truncation to a Levi subgroup of \(G\). It is conjectured that one gets all the (restricted) principal indecomposable modules for the Lie algebra \(\mathfrak g\) of \(G\) from certain specified tilting modules by restricting the action from \(G\) to \(\mathfrak g\); and it is observed that this is true if the characteristic \(p\) is at least \(2h - 2\) (by results of Jantzen), where \(h\) is the Coxeter number of \(G\).
It is shown that the Ringel conjugate \(S(n,r)'\) (defined by means of tilting modules) of the Schur algebra \(S(n,r)\) is a “generalized Schur algebra” and in fact \(S(n,r)\) is self conjugate when \(r \leq n\). Two applications of this are given: a reciprocity formula stating that the filtration multiplicity of the induced module \(\nabla(\mu)\) in the tilting module \(M(\lambda)\) (in case \(G = GL(n)\)) is equal to a certain decomposition number; and a new proof of a result of Akin and Buchsbaum on Ext between induced modules (again with \(G = GL(n)\)). In a related paper by the author [Invent. Math. 110, No. 2, 389-401 (1992)] the theory of tilting modules as developed here is used to determine generators of the algebra of polynomial invariants for the action of the general linear group \(GL(n)\), by simultaneous conjugation, on \(m\)-tuples of \(n \times n\) matrices.
Reviewer: S.Donkin (London)

MSC:
20G05 Representation theory for linear algebraic groups
20G15 Linear algebraic groups over arbitrary fields
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)
16W20 Automorphisms and endomorphisms
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