Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces

TL;DR

This paper simplifies the computation of Lubin-Tate space cohomology by using classical representation theory of the mirabolic group and geometric arguments, building on previous work involving vanishing cycles and Shimura varieties.

Contribution

It introduces a new approach to analyze the cohomology of Lubin-Tate spaces, avoiding complex spectral sequence arguments through classical representation theory techniques.

Findings

Simplified the computation of Lubin-Tate space cohomology.

Connected the cohomology analysis with classical mirabolic group representations.

Provided a geometric argument to replace spectral sequence control.

Abstract

In my 2009 paper at Inventiones, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle $\Psi$ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of $\Psi$. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.