Quantum Satake in Type A: The General and Generic Case
Abstract
In work of the first author, the geometric Satake equivalence was (non-rigorously) reinterpreted as an equivalence between two algebraically-defined $2$-categories, one built from representations of a Lie algebra, and one built using singular Soergel bimodules. It was then explained how to q-deform this equivalence in type A, replacing the special linear Lie algebra with its quantum group, and using singular Soergel bimodules for a deformed reflection representation. Both the algebraic reformulation and its q-deformation were only proven in types A1 and A2. In this paper, we prove the result in type An−1 for n ≥ 4, while working generically: more precisely, we work over a field of characteristic zero, where q is not a root of unity, and having adjoined an n-th root of q. Along the way we generalize certain results in the literature (e.g. the Soergel-Williamson categorification theorem, the Soergel conjecture for spherical elements, various symmetries) to the deformed reflection representation.
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BibTeX
@misc{elias2026quantum,
title = {{Quantum Satake in Type A: The General and Generic Case}},
author = {Ben Elias and Koppara Philip Thomas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23253},
eprint = {2609.23253},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23253},
}
RIS
TY - UNPB
TI - Quantum Satake in Type A: The General and Generic Case
AU - Ben Elias
AU - Koppara Philip Thomas
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.23253
UR - https://arxiv.org/abs/2609.23253
AB - In work of the first author, the geometric Satake equivalence was (non-rigorously) reinterpreted as an equivalence between two algebraically-defined $2$-categories, one built from representations of a Lie algebra, and one built using singular Soergel bimodules. It was then explained how to q-deform this equivalence in type A, replacing the special linear Lie algebra with its quantum group, and using singular Soergel bimodules for a deformed reflection representation. Both the algebraic reformulation and its q-deformation were only proven in types A_1 and A_2. In this paper, we prove the result in type A_n−1 for n ≥ 4, while working generically: more precisely, we work over a field of characteristic zero, where q is not a root of unity, and having adjoined an n-th root of q. Along the way we generalize certain results in the literature (e.g. the Soergel-Williamson categorification theorem, the Soergel conjecture for spherical elements, various symmetries) to the deformed reflection representation.
ER -