Abstract
In 1979, Lusztig proposed a cohomological construction of supercuspidal representations of reductive p-adic groups, analogous to Deligne–Lusztig theory for finite reductive groups. In this paper we establish a new instance of Lusztig’s program. Precisely, let X be the Deligne–Lusztig (ind-pro-)scheme associated to a division algebra D over a non-Archimedean local field K of positive characteristic. We study the \(D^\times \)-representations \(H_\bullet (X)\) by establishing a Deligne–Lusztig theory for families of finite unipotent groups that arise as subquotients of \(D^\times \). There is a natural correspondence between quasi-characters of the (multiplicative group of the) unramified degree-n extension of K and representations of \(D^{\times }\) given by \(\theta \mapsto H_\bullet (X)[\theta ]\). For a broad class of characters \(\theta ,\) we show that the representation \(H_\bullet (X)[\theta ]\) is irreducible and concentrated in a single degree. After explicitly constructing a Weil representation from \(\theta \) using \(\chi \)-data, we show that the resulting correspondence matches the bijection given by local Langlands and therefore gives a geometric realization of the Jacquet–Langlands transfer between representations of division algebras.
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Acknowledgements
I am deeply grateful to Mitya Boyarchenko for introducing me to this area of research. I’d also like to thank Stephen DeBacker, Tasho Kaletha, Jake Levinson, David Speyer, Kam-Fai Tam, and Jared Weinstein for helpful conversations. Finally, I’d like to thank the referees for numerous helpful comments on both the exposition and the mathematics. This work was partially supported by NSF Grants DMS-0943832 and DMS-1160720.
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Chan, C. Deligne–Lusztig constructions for division algebras and the local Langlands correspondence, II. Sel. Math. New Ser. 24, 3175–3216 (2018). https://doi.org/10.1007/s00029-018-0410-6
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DOI: https://doi.org/10.1007/s00029-018-0410-6