The Spectral Hecke Algebra

TL;DR

This paper introduces the spectral Hecke algebra, a derived enhancement of local Galois deformation rings, acting on moduli spaces of Galois representations, and verifies a basic form of local-global compatibility in the Langlands program.

Contribution

It proposes the spectral Hecke algebra as a new derived structure acting on Galois moduli spaces, connecting geometric Langlands and Galois deformation theory.

Findings

Verified local-global compatibility between spectral Hecke algebra and derived Galois deformation rings.

Established the spectral Hecke algebra's role in the Langlands correspondence.

Connected derived Hecke algebra actions with arithmetic group cohomology.

Abstract

We introduce a derived enhancement of local Galois deformation rings that we call the "spectral Hecke algebra", in analogy to a construction in the Geometric Langlands program. This is a Hecke algebra that acts on the spectral side of the Langlands correspondence, i.e. on moduli spaces of Galois representations. We verify the simplest form of derived local-global compatibility between the action of the spectral Hecke algebra on the derived Galois deformation ring of Galatius-Venkatesh, and the action of Venkatesh's derived Hecke algebra on the cohomology of arithmetic groups.