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The Steinberg quotient of a tilting character

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Abstract

Let G be a simple algebraic group over an algebraically closed field of prime characteristic. If M is a finite dimensional G-module that is projective over the Frobenius kernel of G, then its character is divisible by the character of the Steinberg module. In this paper we study such quotients, showing that if M is an indecomposable tilting module, then the multiplicities of the orbit sums appearing in its “Steinberg quotient” are well behaved.

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Notes

  1. If \(p \ge 2h-2\), then \(T((p-1)\rho +\lambda ) \cong \widehat{Q}_1((p-1)\rho +w_0\lambda )\), so these characters are the same in this case. It is currently not known exactly when this isomorphism holds, only that it does not hold in general [5].

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Acknowledgements

This paper was in a preliminary stage when I began a two-week visit with Stephen Donkin at the University of York. Time spent working with Stephen has greatly impacted the present form of this paper, and I am grateful for his generosity, for communicating shorter proofs of some of these results, and for directing me to the paper by Ye. I thank the London Mathematical Society, and the University of York, for financial support during this visit. I would also like to thank Henning Haahr Andersen for sending a number of detailed comments and thought-provoking questions after reading an earlier draft of this paper. This final version was also improved thanks to many detailed comments from the anonymous referee.

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  1. Department of Mathematics, Georgia Southern University, Statesboro, USA

    Paul Sobaje

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Sobaje, P. The Steinberg quotient of a tilting character. Math. Z. 297, 1733–1747 (2021). https://doi.org/10.1007/s00209-020-02576-8

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  • DOI: https://doi.org/10.1007/s00209-020-02576-8

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