Non-semi-stable loci in Hecke stacks and Fargues' conjecture
Ildar Gaisin, Naoki Imai
TL;DR
The paper proves the Harris–Viehmann conjecture for a general reductive group under a Hodge–Newton reducibility condition, using a generalized diamond that covers the non-semi-stable locus of the Hecke stack. It shows that the cohomology of the non-semi-stable locus with cuspidal Langlands parameter coefficients vanishes, and derives a parabolic-induction description for the cohomology in the non-basic, reducible case. As an application, it establishes the Hecke eigensheaf property in Fargues’ conjecture for cuspidal parameters in the GL_2 case, with non-abelian Lubin–Tate theory providing an explicit realization in GL_n. Together, these results clarify the geometric side of the local Langlands correspondence in the Fargues–Fontaine setting and connect non-semi-stable geometry to Levi-subgroup filtrations and automorphic representations.
Abstract
We show the Harris--Viehmann conjecture under some Hodge--Newton reducibility condition for a generalization of the diamond of a non-basic Rapoport--Zink space at infinite level, which appears as a cover of the non-semi-stable locus in the Hecke stack. We show also that the cohomology of the non-semi-stable locus with coefficient coming from a cuspidal Langlands parameter vanishes. As an application, we show the Hecke eigensheaf property in Fargues' conjecture for cuspidal Langlands parameters in the GL(2)-case.