Loop group actions on categories and Whittaker invariants

Journal article

The present paper is divided in three parts. In the first one, we develop the theory of D-modules on ind-schemes of pro-finite type. This allows to define D-modules on (algebraic) loop groups and, consequently, the notion of strong loop group action on a DG category. In the second part, we construct the functors of Whittaker invariants and Whittaker coinvariants, which take as input a DG category acted on by G((t)), the loop group of a reductive group G. Roughly speaking, the Whittaker invariant category of C is the full subcategory C N((t)),χ ⊆C consisting of objects that are N((t))-invariant against a fixed non-degenerate character χ:N((t))→G a of conductor zero. (Here N is the maximal unipotent subgroup of G.) The Whittaker coinvariant category C N((t)),χ is defined by a dual construction. In the third part, we construct a functor Θ:C N((t)),χ →C N((t)),χ , which depends on a choice of dimension theory for G((t)). We conjecture this functor to be an equivalence. After developing the Fourier–Deligne transform for Tate vector spaces, we prove this conjecture for G=GL n . We show that both Whittaker categories can be obtained by taking invariants of C with respect to a very explicit pro-unipotent group subscheme (not indscheme!) of G((t)).

Authors: Beraldo, D

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Advances in Mathematics
• Volume: 322
• Pages: 565-636
• Publication date: 2017-11-13
• Acceptance date: 2017-10-18
• DOI: 10.1016/j.aim.2017.10.024
• ISSN: 0001-8708
• Keywords: indschemes of pro-finite type; loop groups; Whittaker invariants; geometric Langlands; Fourier–Deligne transform; group actions on categories