Convolution algebras associated to representations
Abstract
Given a complex reductive group G, a representation V of G and a Borel-stable subspace M ⊂ V, we consider the associated Steinberg-type variety Z. We prove that, under a certain condition on (V,M), called gluability, the equivariant Borel-Moore homology or K-theory of Z, equipped with the convolution product, is obtained as the intersection of two copies of the nil-Hecke algebra inside its localization. We also provide a description of these new algebras in terms of poles and residues. Similar results are obtained when G is replaced by its loop group. This generalizes results of Ginzburg, Kapranov and Vasserot describing the affine Hecke algebra and DAHA, as well as a result of Teleman and Gannon--Webster that realizes certain Coulomb branches by gluing two copies of the universal centralizer.
Related
- Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem
- Middle convolution for Lie algebra representations
- Ramified convolution and monodromy bases for symmetrizable Kac--Moody algebras
- Kleisli convolution representations of power monoids
- Dunkl theory, convolution algebras, and related Markov processes
BibTeX
@misc{crisan2026convolution,
title = {{Convolution algebras associated to representations}},
author = {Dragos Crisan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.19783},
eprint = {2606.19783},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.19783},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convolution algebras associated to representations
AU - Dragos Crisan
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2606.19783
UR - https://arxiv.org/abs/2606.19783
AB - Given a complex reductive group G, a representation V of G and a Borel-stable subspace M ⊂ V, we consider the associated Steinberg-type variety Z. We prove that, under a certain condition on (V,M), called gluability, the equivariant Borel-Moore homology or K-theory of Z, equipped with the convolution product, is obtained as the intersection of two copies of the nil-Hecke algebra inside its localization. We also provide a description of these new algebras in terms of poles and residues. Similar results are obtained when G is replaced by its loop group. This generalizes results of Ginzburg, Kapranov and Vasserot describing the affine Hecke algebra and DAHA, as well as a result of Teleman and Gannon--Webster that realizes certain Coulomb branches by gluing two copies of the universal centralizer.
ER -