Loop group actions on categories and Whittaker invariants

Abstract

We develop some aspects of the theory of $D$-modules on ind-schemes of pro-finite type. These notions are used to define $D$-modules on (algebraic) loop groups and, consequently, actions of loop groups on DG categories. Let $N$ be the maximal unipotent subgroup of a reductive group $G$. For a non-degenerate character $\chi :N((t))\to {\mathbb{G}}_a$ and a category $\mathcal{C}$ acted upon by $N((t))$, we define the category ${\mathcal{C}}^{N((t)),\chi }$ of $(N((t)),\chi )$-invariant objects, along with the coinvariant category ${\mathcal{C}}_{N((t)),\chi }$. These are the Whittaker categories of $\mathcal{C}$, which are in general not equivalent. However, there is always a family of functors ${\Theta }_k:{\mathcal{C}}_{N((t)),\chi }\to {\mathcal{C}}^{N((t)),\chi }$, parametrized by $k\in \mathbb{Z}$. We conjecture that each ${\Theta }_k$ is an equivalence, provided that the $N((t))$-action on $\mathcal{C}$ extends to a $G((t))$-action. Using the Fourier-Deligne transform (adapted to Tate vector spaces), we prove this conjecture for $G=G{L}_n$ and show that the Whittaker categories can be obtained by taking invariants of $\mathcal{C}$ with respect to a very explicit pro-unipotent group subscheme (not ind-scheme) of $G((t))$.

References in corpus (6)

• Higher Topos Theory
• Noncommutative Counterparts of the Springer Resolution
• Quantum Langlands Correspondence
• On the notion of spectral decomposition in local geometric Langlands
• On the extended Whittaker category
• W-algebras and Whittaker categories

Cited by in corpus (21)

• On the notion of spectral decomposition in local geometric Langlands
• Symmetries of categorical representations and the quantum Ngô action
• On the extended Whittaker category
• W-algebras and Whittaker categories
• Fundamental local equivalences in quantum geometric Langlands
• Homological methods in semi-infinite contexts
• The topological chiral homology of the spherical category
• The center of the categorified ring of differential operators
• Affine Beilinson-Bernstein localization at the critical level for $G{L}_2$
• Sheaves of categories with local actions of Hochschild cochains
• Semi-infinite cohomology and the linkage principle for $W$-algebras
• Tempered D-modules and Borel-Moore homology vanishing
• The affine Hecke category is a monoidal colimit
• Localization for affine $W$-algebras
• Twisted Whittaker category on affine flags and the category of representations of the mixed quantum group
• Categorical Moy-Prasad theory
• A colimit presentation of $\mathcal{D}(G(K))$ via the Bott-Samelson hypercover
• Nearby cycles on Drinfeld-Gaitsgory-Vinberg Interpolation Grassmannian and long intertwining functor
• On the geometric Ramanujan conjecture
• The Drinfeld--Sokolov reduction of admissible representations of affine Lie algebras
• Affine Harish-Chandra bimodules and Steinberg--Whittaker localization