Cohomology of the Bruhat-Tits strata in the supersingular locus of the $\mathrm{GU}(1,n-1)$ Shimura variety at a ramified prime

Abstract

The supersingular locus of the $\mathrm{GU}(1,n-1)$ Shimura variety at a ramified prime $p$ is stratified by Coxeter varieties attached to finite symplectic groups. In this paper, we compute the $\ell$-adic cohomology of the Zariski closure of any such stratum. These are known as closed Bruhat-Tits strata. We prove that the cohomology groups of odd degree vanish, and those of even degree are explicitely determined as representations of the symplectic group with a Frobenius action. Each closed Bruhat-Tits stratum is linearly stratified by Coxeter varieties attached to smaller symplectic groups. Thanks to results of Lusztig who computed the cohomology of Coxeter varieties for classical groups, we make use of the spectral sequence associated to this stratification and describe explicitely all the terms at infinity. We point out that the closed Bruhat-Tits strata have isolated singularities when the dimension is greater than 1. Our analysis requires discussing the smoothness of the blow-up at the singular points, as well as comparing the ordinary $\ell$-adic cohomology with intersection cohomology. A by-product of our computations is that these two cohomologies actually coincide, so that surprisingly the presence of singularities does not interfere with the cohomology.