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The cohomology of semi-infinite Deligne–Lusztig varieties

Charlotte Chan

Abstract

We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h . Boyarchenko’s two conjectures are on the maximality of X h and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n in the case h = 2 (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h attains its Weil–Deligne bound, so that the cohomology of X h is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ( X h ) are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ H c i ( X h ) [ θ ] agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p-adic groups in general.

Funding statement: This work was partially supported by NSF grants DMS-0943832 and DMS-1160720.

Acknowledgements

I would like to thank Bhargav Bhatt for several helpful conversations and Alex Ivanov for helpful comments on an earlier draft. I would also like to thank the anonymous referee for numerous observations and suggestions that have improved this paper.

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Received: 2018-09-05
Revised: 2019-09-10
Published Online: 2020-01-17
Published in Print: 2020-11-01

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