Chiral principal series categories. I: Finite dimensional calculations
type: Item
instance of: scholarly article
title: Chiral principal series categories. I: Finite dimensional calculations
zbMATH Open document ID: 1469.14026
author: Sam Raskin
published in: Advances in Mathematics
publication date: 2021-07-26T00:00:00Z
review text: This manuscript begins a series studying \(D\)-modules on the Feigin-Frenkel semi-infinite flag variety from the perspective of the Beilinson-Drinfeld factorization, or chiral, theory. Let \(G\) be a split reductive group over a field \(k\) whose \(\text{char}(k)=0\). Let \(X\) be a smooth curve and let \(x\in X\) be a fixed \(k\)-point. Let \(O_x=kt_x\) be the ring of Taylor series based at \(x\), and \(K_x = k((t_x))\) be the ring of Laurent series based at \(x\). Zastava spaces are certain finite-dimensional subvarieties of the affine Grassmannian (Section 2). S. Raskin calculates Whittaker-twisted cohomology groups of Zastava spaces in Theorem 4.6.1. The author shows that such cohomology groups realize the nilradical of a Borel subalgebra for the Langlands dual group, following earlier work of \textit{B. Feigin} et al. [Transl., Ser. 2, Am. Math. Soc. 194, 113--148 (1999; Zbl 1076.14511)] and \textit{A. Braverman} and \textit{D. Gaitsgory} [Invent. Math. 150, No. 2, 287--384 (2002; Zbl 1046.11048); Geom. Funct. Anal. 17, No. 6, 1788--1850 (2008; Zbl 1234.11155)]. He compares this geometric realization of the Langlands dual group to the standard one provided by factorizable geometric Satake in Theorem 7.9.1.
reviewed by: Mee Seong Im
Mathematics Subject Classification ID: 14D24
Mathematics Subject Classification ID: 22E57
Mathematics Subject Classification ID: 14F10
Mathematics Subject Classification ID: 14M15
zbMATH DE Number: 7374911
zbMATH Keywords: geometric Langlands
zbMATH Keywords: Whittaker
zbMATH Keywords: Zastava space
MaRDI profile type: Publication
full work available at URL: j.aim.2021.107856
OpenAlex ID: W3178028055
cites work: Perverse sheaves on affine flags and Langlands dual group
cites work: Quantum groups, the loop Grassmannian, and the Springer resolution
cites work: Modules over the small quantum group and semi-infinite flag manifold
cites work: Complex Analytic Connections in Fibre Bundles
cites work: Q3795773
cites work: Q4830108
cites work: Q3320507
cites work: Loop group actions on categories and Whittaker invariants
cites work: Factorizable sheaves and quantum groups
cites work: Geometric Eisenstein series
cites work: Deformations of local systems and Eisenstein series
cites work: Intersection cohomology of Drinfeld's compactifications
cites work: Q3926802
cites work: On a theorem of Braden
cites work: Affine Kac-Moody algebras and semi-infinite flag manifolds
cites work: Semiinfinite Flags. II. Local and Global Intersection Cohomology of Quasimaps' Spaces
cites work: Semiinfinite Flags. I. Case of global curve $P^1$
cites work: Chiral Koszul duality
cites work: Local geometric Langlands correspondence and affine Kac-Moody algebras
cites work: D-modules on the affine flag variety and representations of affine Kac-Moody algebras
cites work: Whittaker patterns in the geometry of moduli spaces of bundles on curves
cites work: On de Jong's conjecture
cites work: Vertex algebras and the formal loop space
cites work: Higher Topos Theory (AM-170)
cites work: Q5185104
cites work: Geometric Langlands duality and representations of algebraic groups over commutative rings
cites work: Twisted geometric Satake equivalence via gerbes on the factorizable grassmannian
cites work: The Harder-Narasimhan stratification of the moduli stack of \(G\)-bundles via Drinfeld's compactifications
DOI: 10.1016/J.AIM.2021.107856
description: scientific article; zbMATH DE number 7374911