On the cohomology of moduli spaces of mixed characteristic shtukas
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Abstract
We study the cohomology of certain recently developed moduli spaces of mixed characteristic shtukas, which act as representation spaces for certain Lie groups and Galois groups. These spaces generalize Rapoport-Zink spaces, which provided a natural framework for studying moduli spaces of p-divisible groups. To construct them, one inputs the datum of a Lie group, a particular cocharacter for a maximal torus inside that group, and suitable data for a "G-isocrystal." In particular, for a judicious choice of this datum for the group GL_n, one recovers the Lubin-Tate tower at infinite level, whose cohomology is used to realize the transfer functors predicted by the Local Langlands Correspondence, which connect certain representations of the Lie group to representations of a Galois group. In this new setting, we discuss how some conjectures on the cohomology of Rapoport-Zink spaces at infinite level can be generalized to the setting of shtukas. We are particularly interested in the situation where the datum used to construct the spaces includes a cocharacter which is not minuscule, as the corresponding shtuka spaces have no classical analogue in the setting of Rapoport-Zink. In particular, when the Lie group involved is GL_n, we demonstrate how the Kottwitz and Harris-Viehmann conjectures for Rapoport-Zink spaces constructed via minuscule cocharacters implies similar conjectures about the cohomology of moduli spaces of shtukas. Our techniques rely crucially on very recent progress on the "Geometrization of the Local Langlands Program." In particular, the cohomological computations required to relate the conjectures will involve an interpretation using geometric Hecke operators, and this work will outline an intimate connection to Fargues's conjectures on the existence of Hecke eigensheaves on the stack of vector bundles on the Fargues-Fontaine curve. Our main results include a careful verification of the existence of a Hecke eigensheaf on the stack of rank 2 vector bundles on the curve associated to a supercuspidal representation of GL_2(ℚ_p), with an argument that works more generally given the validity of the classical conjectures for GL_n. Arguments for such a connection with Hecke eigensheaves go back to Fargues's original outline of his Geometrization program, but at the time lacked a sufficiently developed theory of diamonds and v-stacks to give a fully rigorous verification and model for all objects involved.