Equivariant formality and representation theory
Abstract
Let G be a compact connected Lie group and K its connected Lie subgroup. Using the K-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over G/K, we find two characterizations of equivariant formality of the isotropy action of K on G/K. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of G to that of K. The second characterization asserts that equivariant formality amounts to the equivariant K-theoretic index pairing of two certain special K-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where K is a torus of dimension one less than the rank of G, culminating in a complete classification of equivariant formality where G is further assumed to be simple.
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BibTeX
@misc{fok2026equivariant,
title = {{Equivariant formality and representation theory}},
author = {Chi-Kwong Fok},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.00448},
eprint = {2609.00448},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00448},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Equivariant formality and representation theory
AU - Chi-Kwong Fok
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2609.00448
UR - https://arxiv.org/abs/2609.00448
AB - Let G be a compact connected Lie group and K its connected Lie subgroup. Using the K-theoretic version of equivariant formality developed in [F2] which involves analysis of vector bundles over G/K, we find two characterizations of equivariant formality of the isotropy action of K on G/K. The first one equates equivariant formality with a "smoothness" condition of the restriction map of the representation ring of G to that of K. The second characterization asserts that equivariant formality amounts to the equivariant K-theoretic index pairing of two certain special K-theory classes being nontrivial in some sense. By applying these characterizations, we are able to give a representation theoretic criterion for equivariant formality of the isotropy action by a circle subgroup, as well as an invariant theory criterion for the case where K is a torus of dimension one less than the rank of G, culminating in a complete classification of equivariant formality where G is further assumed to be simple.
ER -