In a Betti/topological setting, applying categorical and 2-categorical traces to a Hecke action yields universal shtukas, excursion operators, and an S=T identity.
Finiteness of cohomology groups of stacks of shtukas as modules over Hecke algebras, and applications
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abstract
In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on the space of cuspidal automorphic forms, to the space of automorphic forms with compact support. This gives the Langlands parametrization for some quotient spaces of the latter, which is compatible with the constant term morphism.
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A toy model for the Drinfeld-Lafforgue shtuka construction
In a Betti/topological setting, applying categorical and 2-categorical traces to a Hecke action yields universal shtukas, excursion operators, and an S=T identity.