A Hesselink-type formula for the nilpotent cone of Lie algebra representations
Abstract
A well-known result of Hesselink gives a formula for the q-character of the nilpotent cone of a semisimple Lie algebra in terms of a q-analog of the Kostant partition function. For a reductive Lie algebra g, we define a class of \emph{Hesselink-type representations}, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain q-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of sl2 we call an \emph{extended quiver representation of trivial type}. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.
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BibTeX
@misc{krylov2026hesselinktype,
title = {{A Hesselink-type formula for the nilpotent cone of Lie algebra representations}},
author = {Vasily Krylov and Frank Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.03314},
eprint = {2608.03314},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.03314},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Hesselink-type formula for the nilpotent cone of Lie algebra representations
AU - Vasily Krylov
AU - Frank Wang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2608.03314
UR - https://arxiv.org/abs/2608.03314
AB - A well-known result of Hesselink gives a formula for the q-character of the nilpotent cone of a semisimple Lie algebra in terms of a q-analog of the Kostant partition function. For a reductive Lie algebra g, we define a class of \emph{Hesselink-type representations}, for which we prove an analog of Hesselink's formula under certain additional freeness assumptions. Moreover, we prove that the formula still holds for certain examples of Hesselink-type representations without the freeness assumptions. Using these methods, we obtain q-character formulas for the nilpotent cone of a representation of a cyclic quiver with equal dimensions, a representation of a cyclic quiver with two vertices, and a representation of a product of copies of sl_2 we call an \emph{extended quiver representation of trivial type}. In the process, we give a general framework for proving formulas of this type for representations that are not necessarily Hesselink-type. We also provide some counterexamples to natural questions regarding Hesselink-type representations.
ER -