On the notion of spectral decomposition in local geometric Langlands

Abstract

The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with non-abelian reductive groups and with ramification. The present work is meant to clarify the situation, showing that the two obvious candidates for such a notion coincide. More broadly, we study the moduli space of (possibly irregular) de Rham local systems from the perspective of homological algebra. We show that, in spite of its infinite-dimensional nature, this moduli space shares many of the nice features of an Artin stack. Less broadly, these results support the belief in the existence of a version of geometric Langlands allowing arbitrary ramification. Along the way, we give some apparently new, if unsurprising, results about the algebraic geometry of the moduli space of connections, using Babbitt-Varadarajan's reduction theory for differential equations.

References in corpus (4)

• Nilpotence and descent in equivariant stable homotopy theory
• Loop group actions on categories and Whittaker invariants
• Formal loops, Tate objects and tangent Lie algebras
• Acyclic complexes and 1-affineness

Cited by in corpus (7)

• Loop group actions on categories and Whittaker invariants
• Structures symplectiques et de Poisson sur les champs en catégories
• W-algebras and Whittaker categories
• Homological methods in semi-infinite contexts
• Recent progress in geometric Langlands theory
• Acyclic complexes and 1-affineness
• Jordan Decomposition for Formal G-Connections