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 $H_c^i\left(X_h\right)$ 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 $\theta \mapsto H_c^i\left(X_h\right)\left[\theta \right]$ 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.

中文翻译：

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半无限Deligne–Lusztig变种的同调

我们证明了1979年Lusztig关于半无限Deligne–Lusztig变种与局部场上的代数代数的同调性的猜想。我们还证明了博雅琴科关于这些变体的两个猜想。众所周知，在这种情况下，半无限Deligne–Lusztig变种是由某些有限类型方案的极限组成的ind方案$X_H$。博亚琴科的两个猜想是关于$X_H$以及它们的同调环面特征空间的行为。这两个猜想仅在具有Hasse不变性的除法代数中才广为人知$\mathrm{1个}/ñ$ 在这种情况下 $H=2$（“最低水平”）由博亚琴科克-温斯坦的作品研究了鲁宾-泰特塔中特殊仿射类的同调性。我们证明了有理点的数量$X_H$ 达到其Weil–Deligne界，因此 $X_H$在很强的意义上是纯净的。我们证明了同调群的圆环本征空间$H_C^{\mathrm{一世}}（X_H）$是不可约的表示，并且在一个同调程度内得到支持。最后，我们给出了与任何除法代数相连的半无限Deligne-Lusztig变体的同源群的完整描述，从而给出了这些群的一大类超尖峰表示的几何实现。而且，对应$\theta \mapsto H_C^{\mathrm{一世}}（X_H）\left[\theta \right]$同意当地的Langlands和Jacquet-Langlands的信件。一般而言，本文开发的技术应可用于研究p- adic组的这些构造。