arXiv subjects

Peter Crooks

Publications and source records attributed to Peter Crooks.

30 records · Page 2Linked to original sources

arXiv2019

An application of spherical geometry to hyperkähler slices

This work is concerned with Bielawski's hyperkähler slices in the cotangent bundles of homogeneous affine varieties. One can associate such a slice to the data of a complex semisimple Lie group $G$, a reductive subgroup $H\subseteq G$, and a Slodowy slice $S\subseteq\mathfrak{g}:=\mathrm{Lie}(G)$, defining it to be the hyperkähler quotient of $T^*(G/H)\times (G\times S)$ by a maximal compact subgroup of $G$. This hyperkähler slice is empty in some of the most elementary cases (e.g. when $S$ is regular and $(G,H)=(\operatorname{SL}_{n+1},\operatorname{GL}_{n})$, $n\geq 3$), prompting us to seek necessary and sufficient conditions for non-emptiness. We give a spherical-geometric characterization of the non-empty hyperkähler slices that arise when $S=S_{\text{reg}}$ is a regular Slodowy slice, proving that non-emptiness is equivalent to the so-called $\mathfrak{a}$-regularity of $(G,H)$. This $\mathfrak{a}$-regularity condition is formulated in several equivalent ways, one being a concrete condition on the rank and complexity of $G/H$. We also provide a classification of the $\mathfrak{a}$-regular pairs $(G,H)$ in which $H$ is a reductive spherical subgroup. Our arguments make essential use of Knop's results on moment map images and Losev's algorithm for computing Cartan spaces.

math.SG↗

arXiv2018

Hessenberg varieties, Slodowy slices, and integrable systems

This work is intended to contextualize and enhance certain well-studied relationships between Hessenberg varieties and the Toda lattice, thereby building on the results of Kostant, Peterson, and others. One such relationship is the fact that every Lagrangian leaf in the Toda lattice is compactified by a suitable choice of Hessenberg variety. It is then natural to imagine the Toda lattice as extending to an appropriate union of Hessenberg varieties. We fix a simply-connected complex semisimple linear algebraic group $G$ and restrict our attention to a particular family of Hessenberg varieties, a family that includes the Peterson variety and all Toda leaf compactifications. The total space of this family, $X(H_0)$, is shown to be a Poisson variety with a completely integrable system defined in terms of Mishchenko--Fomenko polynomials. This leads to a natural embedding of completely integrable systems from the Toda lattice to $X(H_0)$. We also show $X(H_0)$ to have an open dense symplectic leaf isomorphic to $G/Z \times S_{\text{reg}}$, where $Z$ is the centre of $G$ and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. This allows us to invoke results about integrable systems on $G\times S_{\text{reg}}$, as developed by Rayan and the second author. Lastly, we witness some implications of our work for the geometry of regular Hessenberg varieties.

math.AG↗

arXiv2018

Slodowy slices and the complete integrability of Mishchenko-Fomenko subalgebras on regular adjoint orbits

This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve upon this result, showing that each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits (i.e. orbits of maximal dimension). Our approach incorporates the theory of regular $\mathfrak{sl}_2$-triples and associated Slodowy slices, as developed by Kostant.

math.SG↗

arXiv2017

An equivariant description of certain holomorphic symplectic varieties

This short note considers varieties of the form $G\times S_{\text{reg}}$, where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in hyperkähler geometry, theoretical physics, and in the theory of abstract integrable systems developed by Fernandes, Laurent-Gengoux, and Vanhaecke. In particular, previous work of the author and Rayan uses a Hamiltonian $G$-action to endow $G\times S_{\text{reg}}$ with a canonical abstract integrable system. One might therefore wish to understand, in some sense, all examples of abstract integrable systems arising from Hamiltonian $G$-actions. Accordingly, we consider a holomorphic symplectic variety $X$ carrying an abstract integrable system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show that there must exist a $G$-equivariant variety isomorphism $X\cong G\times S_{\text{reg}}$.

math.SG↗

arXiv2017

Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices

We study holomorphic integrable systems on the hyperkähler manifold $G\times S_{\text{reg}}$, where $G$ is a complex semisimple Lie group and $S_{\text{reg}}$ is the Slodowy slice determined by a regular $\mathfrak{sl}_2(\mathbb{C})$-triple. Our main result is that this manifold carries a canonical \textit{abstract integrable system}, a foliation-theoretic notion recently introduced by Fernandes, Laurent-Gengoux, and Vanhaecke. We also construct traditional integrable systems on $G\times S_{\text{reg}}$, some of which are completely integrable and fundamentally based on Mishchenko and Fomenko's argument shift approach.

math.SG↗

arXiv2017

Complex adjoint orbits in Lie theory and geometry

This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau-Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author's knowledge, have not yet appeared in the research literature.

math.AG↗

arXiv2016

Hessenberg varieties for the minimal nilpotent orbit

For a connected, simply-connected complex simple algebraic group $G$, we examine a class of Hessenberg varieties associated with the minimal nilpotent orbit. In particular, we compute the Poincaré polynomials and irreducible components of these varieties in Lie type $A$. Furthermore, we show these Hessenberg varieties to be GKM with respect to the action of a maximal torus $T\subseteq G$. The corresponding GKM graphs are then explicitly determined. Finally, we present the ordinary and $T$-equivariant cohomology rings of our varieties as quotients of those of the flag variety.

math.AG↗

arXiv2015

Properties of nilpotent orbit complexification

We consider aspects of the relationship between nilpotent orbits in a semisimple real Lie algebra $\mathfrak{g}$ and those in its complexification $\mathfrak{g}_{\mathbb{C}}$. In particular, we prove that two distinct real nilpotent orbits lying in the same complex orbit are incomparable in the closure order. Secondly, we characterize those $\mathfrak{g}$ having non-empty intersections with all nilpotent orbits in $\mathfrak{g}_{\mathbb{C}}$. Finally, for $\mathfrak{g}$ quasi-split, we characterize those complex nilpotent orbits containing real ones.

math.AG↗

arXiv2015

Some results on equivariant contact geometry for partial flag varieties

We study equivariant contact structures on complex projective varieties arising as partial flag varieties $G/P$, where $G$ is a connected, simply-connected complex simple group of type $ADE$ and $P$ is a parabolic subgroup. We prove a special case of the LeBrun-Salamon conjecture for partial flag varieties of these types. The result can be deduced from Boothby's classification of compact simply-connected complex contact manifolds with transitive action by contact automorphisms, but our proof is completely independent and relies on properties of $G$-equivariant vector bundles on $G/P$. A byproduct of our argument is a canonical, global description of the unique $SO_{2n}(\mathbb C)$-invariant contact structure on the isotropic Grassmannian of $2$-planes in $\mathbb C^{2n}$.

math.RT↗

arXiv2014

Generalized Equivariant Cohomology and Stratifications

For $T$ a compact torus and $E_T^*$ a generalized $T$-equivariant cohomology theory, we provide a systematic framework for computing $E_T^*$ in the context of equivariantly stratified smooth complex projective varieties. This allows us to explicitly compute $E_T^*(X)$ as an $E_T^*(\text{pt})$-module when $X$ is a direct limit of smooth complex projective $T_{\mathbb{C}}$-varieties with finitely many $T$-fixed points and $E_T^*$ is one of $H_T^*(\cdot;\mathbb{Z})$, $K_T^*$, and $MU_T^*$. We perform this computation on the affine Grassmannian of a complex semisimple group.

math.AT↗

arXiv2013

The Torus-Equivariant Cohomology of Nilpotent Orbits

We consider aspects of the geometry and topology of nilpotent orbits in finite-dimensional complex simple Lie algebras. In particular, we give the equivariant cohomologies of the regular and minimal nilpotent orbits with respect to the action of a maximal compact torus of the overall group in question.

math.AG↗

arXiv2009

On Projective Equivalence of Univariate Polynomial Subspaces

We pose and solve the equivalence problem for subspaces of ${\mathcal P}_n$, the $(n+1)$ dimensional vector space of univariate polynomials of degree $\leq n$. The group of interest is ${\rm SL}_2$ acting by projective transformations on the Grassmannian variety ${\mathcal G}_k{\mathcal P}_n$ of $k$-dimensional subspaces. We establish the equivariance of the Wronski map and use this map to reduce the subspace equivalence problem to the equivalence problem for binary forms.

math.QA↗