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On the inductive Alperin–McKay conditions in the maximally split case

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Abstract

The Alperin–McKay conjecture relates height zero characters of an \(\ell \)-block with the ones of its Brauer correspondent. This conjecture has been reduced to the so-called inductive Alperin–McKay conditions about quasi-simple groups by the third author. The validity of those conditions is still open for groups of Lie type. The present paper describes characters of height zero in \(\ell \)-blocks of groups of Lie type over a field with q elements when \(\ell \) divides \(q-1\). We also give information about \(\ell \)-blocks and Brauer correspondents. As an application we show that quasi-simple groups of type \(\mathsf C\) over \(\mathbb {F}_q\) satisfy the inductive Alperin–McKay conditions for primes \(\ell \ge 5\) and dividing \(q-1\). Some methods to that end are adapted from Malle and Späth (Ann. Math. (2) 184:869–908, 2016).

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Acknowledgements

Part of this work was completed while the authors were in residence at the MSRI in Berkeley, California during the Spring 2018 program on Group Representation Theory and Applications, supported by the NSF Grant DMS-1440140. Part was also completed at the Isaac Newton Institute for Mathematical Sciences during the Spring 2020 program Groups, Representations, and Applications: New Perspectives, supported by EPSRC grant EP/R014604/1. The authors thank both institutes and the organizers of the programs for making their stays possible and providing a collaborative and productive work environment. The second-named author was also supported in part by grants from the Simons Foundation (Award #351233) and the NSF (Award # DMS-1801156). She would also like to thank the Graduate School at Universität Wuppertal for its hospitality during her visits in August 2018 and April 2019 in the framework of the research training group GRK 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology, which is funded by the DFG. We thank Gunter Malle and the referee for numerous remarks on the manuscript.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu-PRG, CNRS, 8 Place Aurélie Nemours, 75205, Paris Cedex 13, France

    Marc Cabanes

  2. Department of Mathematics and Statistics, Metropolitan State University of Denver, Denver, CO, 80217, USA

    A. A. Schaeffer Fry

  3. School of Mathematics and Natural Sciences, Universität Wuppertal, Gaußstr. 20, 42119, Wuppertal, Germany

    Britta Späth

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  1. Marc Cabanes
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  2. A. A. Schaeffer Fry
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  3. Britta Späth
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Correspondence to A. A. Schaeffer Fry.

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Cabanes, M., Schaeffer Fry, A.A. & Späth, B. On the inductive Alperin–McKay conditions in the maximally split case. Math. Z. 299, 2419–2441 (2021). https://doi.org/10.1007/s00209-021-02764-0

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