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We study integrality of the Frobenius-Perron eigenvalues of the Cartan matrices for the principal blocks of some finite groups of Lie type with noncyclic abelian Sylow p-subgroups. [ABSTRACT FROM AUTHOR]

We determine the finite simple groups of Lie type of characteristic p, for which the Steinberg character lies in the principal ℓ-block for every prime ℓ ≠ p dividing the order of the group. [ABSTRACT FROM AUTHOR]

M. Cabanes and J. Rickard showed in [3] that the Alvis–Curtis character duality of a finite group of Lie type is induced in non defining characteristic ℓ by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-ℓ-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group. [ABSTRACT FROM AUTHOR]