Affine Harish-Chandra center in positive characteristic
Abstract
Let G be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish--Chandra centers of the associated Kac--Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group checkG. Namely, given a noncritical level κ for G, consider the associated Kac--Moody vertex algebra Vκ(g) defined on a formal disc 𝒟, and its Harish--Chandra center of arc group invariants Vκ(g)𝒥G ⊂ Vκ(g). Write checkκ for the dual level for checkG, checkκp − checkκ for its image under the Artin--Schreier map, and OpcheckG(𝒟(1))checkκp − checkκ for the moduli space of (checkκp − checkκ)-opers on the Frobenius twisted disc 𝒟(1). We establish a canonical isomorphism Spec Vκ(g)𝒥 G ≃ OpcheckG(𝒟(1))checkκp − checkκ. There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level κ, where instead of 𝒟 we need to consider the punctured disc 𝒟×. The presence of these Harish--Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod p of the characteristic zero Harish--Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.
Related
- Simple Harish-Chandra bimodules in positive characteristics
- Representation theory of the real Gelfand order and real Harish-Chandra modules for SL2(ℝ)
- The Harish-Chandra isomorphism for supersymmetric spaces and ghost distributions
- Classification of Simple Harish-Chandra Modules over the Loop Mirror HeisenbergVirasoro Algebra
- Equivalued affine springer fibers in mixed characteristic
BibTeX
@misc{dhillon2026affine,
title = {{Affine Harish-Chandra center in positive characteristic}},
author = {Gurbir Dhillon and Ivan Losev},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23560},
eprint = {2609.23560},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23560},
}
RIS
TY - UNPB
TI - Affine Harish-Chandra center in positive characteristic
AU - Gurbir Dhillon
AU - Ivan Losev
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23560
UR - https://arxiv.org/abs/2609.23560
AB - Let G be a split reductive group defined over a field of characteristic bigger than the Coxeter numbers of its simple factors. We identify the Harish--Chandra centers of the associated Kac--Moody vertex algebras and enveloping algebras at noncritical levels with algebras of functions on moduli spaces of connections for the Langlands dual group checkG. Namely, given a noncritical level κ for G, consider the associated Kac--Moody vertex algebra V_κ(g) defined on a formal disc 𝒟, and its Harish--Chandra center of arc group invariants V_κ(g)^𝒥G ⊂ V_κ(g). Write checkκ for the dual level for checkG, checkκ^p − checkκ for its image under the Artin--Schreier map, and Op_checkG(𝒟^(1))_checkκ^p − checkκ for the moduli space of (checkκ^p − checkκ)-opers on the Frobenius twisted disc 𝒟^(1). We establish a canonical isomorphism Spec V_κ(g)^𝒥 G ≃ Op_checkG(𝒟^(1))_checkκ^p − checkκ. There is a similar identification for the Harish--Chandra center of the filtered complete enveloping algebra at level κ, where instead of 𝒟 we need to consider the punctured disc 𝒟^×. The presence of these Harish--Chandra centers is a genuinely new phenomenon for loop groups at noncritical level in positive characteristic: these centers are nontrivial, unlike for loop groups at noncritical level in characteristic zero, and in particular are not the reductions mod p of the characteristic zero Harish--Chandra centers, unlike for finite dimensional reductive groups or loop groups at critical level.
ER -