The (cyclic) enhanced nilpotent cone via quiver representations. (English) Zbl 1477.16021
The main result of the paper under review is a new parametrization of orbits in the cyclic enhanced nilpotent cone \(\mathcal N_\infty(\ell,n)\), where \(\ell\) and \(n\) denote natural numbers. The authors use this description to their study of admissible \(\mathcal D\)-modules in [G. Bellamy and M. Boos, Kyoto J. Math. 61, No. 1, 115–170 (2021; Zbl 1481.14036)].
We first recall the definition of \(\mathcal N_\infty(\ell,n)\). Let \(\mathcal Q_\infty(\ell)\), the cyclic quiver with \(\ell\)-vertices together with the framing \(\infty \to 0\) at the vertex \(0\). The cyclic enhanced nilpotent cone \(\mathcal N_\infty(\ell,n)\) is defined as the space of quiver representations of \(\mathcal Q_\infty(\ell)\) with dimension one at the framing vertex and such that the endomorphism at vertex \(0\) obtained by going once around the cycle has nilpotency \(\leq n\).
For example, note that if we fix \(\ell=1\) and consider only representation with dimension vector \((1,n)\), then we recover the enhanced nilpotent cone \(V \times \mathcal N\), where \(\mathcal N\) denotes the space of nilpotent \(n\times n\)-matrices.
Returning to the general setting, let \(G = \prod_{i =0}^{\ell-1} \mathrm{GL}_i\). Then \(G\) naturally acts on \(\mathcal N_\infty(\ell,n)\) and the action has finitely many orbits. In the case \(\ell=1\), the \(G\)-orbits were parametrized by P. N. Achar and A. Henderson [Adv. Math. 219, No. 1, 27–62 (2008; Zbl 1205.14061)] and R. Travkin [Sel. Math., New Ser. 14, No. 3–4, 727–758 (2009; Zbl 1230.20047)] independently and for general \(\ell\) by C.P. Johnson in his thesis [Enhanced nilpotent representations of a cyclic quiver. Salt Lake City: University of Utah (PhD Thesis) (2010)].
The authors give a different parametrization. Namely they show that the \(G\)-orbits in \(\mathcal N_\infty(\ell,n)\) are in natural bijection with the set: \[ \mathcal Q(n,\ell) := \{ (\lambda;\nu) \in \mathcal P \times \mathcal P_\ell \mid \mathrm{res}_\ell(\lambda) + \mathrm{sres}_\ell(\nu) = n\delta \}, \] where \(\mathcal P\) (resp. \(\mathcal P_\ell\)) denotes the set of partitions (resp. \(\ell\)-multipartions), \(\mathrm{res}(\lambda)\) and \(\mathrm{sres}(\nu)\) are the (shifted) \(\ell\)-residues of the corresponding partitions and \(\delta\) is the minimal imaginary root for the cyclic quiver. While this algebra does not have finite representation type in general, by restricting to the representations to have dimension one at the framing vertex, the authors are able to overcome this complication.
The authors also give an explicit translation between the different parametrizations.
We first recall the definition of \(\mathcal N_\infty(\ell,n)\). Let \(\mathcal Q_\infty(\ell)\), the cyclic quiver with \(\ell\)-vertices together with the framing \(\infty \to 0\) at the vertex \(0\). The cyclic enhanced nilpotent cone \(\mathcal N_\infty(\ell,n)\) is defined as the space of quiver representations of \(\mathcal Q_\infty(\ell)\) with dimension one at the framing vertex and such that the endomorphism at vertex \(0\) obtained by going once around the cycle has nilpotency \(\leq n\).
For example, note that if we fix \(\ell=1\) and consider only representation with dimension vector \((1,n)\), then we recover the enhanced nilpotent cone \(V \times \mathcal N\), where \(\mathcal N\) denotes the space of nilpotent \(n\times n\)-matrices.
Returning to the general setting, let \(G = \prod_{i =0}^{\ell-1} \mathrm{GL}_i\). Then \(G\) naturally acts on \(\mathcal N_\infty(\ell,n)\) and the action has finitely many orbits. In the case \(\ell=1\), the \(G\)-orbits were parametrized by P. N. Achar and A. Henderson [Adv. Math. 219, No. 1, 27–62 (2008; Zbl 1205.14061)] and R. Travkin [Sel. Math., New Ser. 14, No. 3–4, 727–758 (2009; Zbl 1230.20047)] independently and for general \(\ell\) by C.P. Johnson in his thesis [Enhanced nilpotent representations of a cyclic quiver. Salt Lake City: University of Utah (PhD Thesis) (2010)].
The authors give a different parametrization. Namely they show that the \(G\)-orbits in \(\mathcal N_\infty(\ell,n)\) are in natural bijection with the set: \[ \mathcal Q(n,\ell) := \{ (\lambda;\nu) \in \mathcal P \times \mathcal P_\ell \mid \mathrm{res}_\ell(\lambda) + \mathrm{sres}_\ell(\nu) = n\delta \}, \] where \(\mathcal P\) (resp. \(\mathcal P_\ell\)) denotes the set of partitions (resp. \(\ell\)-multipartions), \(\mathrm{res}(\lambda)\) and \(\mathrm{sres}(\nu)\) are the (shifted) \(\ell\)-residues of the corresponding partitions and \(\delta\) is the minimal imaginary root for the cyclic quiver. While this algebra does not have finite representation type in general, by restricting to the representations to have dimension one at the framing vertex, the authors are able to overcome this complication.
The authors also give an explicit translation between the different parametrizations.
Reviewer: Carl Mautner (Bonn)
MSC:
| 16G20 | Representations of quivers and partially ordered sets |
| 16G60 | Representation type (finite, tame, wild, etc.) of associative algebras |
| 17B08 | Coadjoint orbits; nilpotent varieties |
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