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 . Boyarchenko’s two conjectures are on the maximality of 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 in the case (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 attains its Weil–Deligne bound, so that the cohomology of is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group 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 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.
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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