Homological methods in semi-infinite contexts

Abstract

Actions of algebraic groups on DG categories provide a convenient, unifying framework in some parts of geometric representation theory, especially the representation theory of reductive Lie algebras. We extend this theory to loop groups and affine Lie algebras, extending previous work of Beraldo, Gaitsgory and the author. Along the way, we introduce some subjects of independent interest: ind-coherent sheaves of infinite type indschemes, topological DG algebras, and weak actions of group indschemes on categories. We also present a new construction of semi-infinite cohomology for affine Lie algebras, based on a modular character for loop groups. As an application of our methods, we establish an important technical result for Kac-Moody representations at the critical level, showing that the appropriate ("renormalized") derived category of representations admits a large class of symmetries coming from the adjoint action of the loop group and from the center of the enveloping algebra.

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Cited by in corpus (6)

• Affine Beilinson-Bernstein localization at the critical level for $G{L}_2$
• Equivariant localization in factorization homology and applications in mathematical physics I: Foundations
• Equivariant localization in factorization homology and applications in mathematical physics II: Gauge theory applications
• An Extension of the Kazhdan-Lusztig Equivalence
• The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras
• Affine Harish-Chandra bimodules and Steinberg--Whittaker localization