Abstract
In this paper we study higher Deligne–Lusztig representations of reductive groups over finite quotients of discrete valuation rings. At even levels, we show that these geometrically constructed representations, defined by Lusztig, coincide with certain explicit induced representations defined by Gérardin, thus giving a solution to a problem raised by Lusztig. In particular, we determine the dimensions of these representations. As an immediate application we verify a conjecture of Letellier for \(\mathrm {GL}_2\) and \(\mathrm {GL}_3\).
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Acknowledgements
We are grateful to A.-M. Aubert and E. Letellier for helpful discussions, and are thankful to G. Lusztig for his encouragement. During the preparation of this work, ZC was partially supported by CSC/201308060137, and AS partially supported by EPSRC Grant EP/K024779/1.
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Chen, Z., Stasinski, A. The algebraisation of higher Deligne–Lusztig representations. Sel. Math. New Ser. 23, 2907–2926 (2017). https://doi.org/10.1007/s00029-017-0349-z
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DOI: https://doi.org/10.1007/s00029-017-0349-z