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Peter Crooks

Publications and source records attributed to Peter Crooks.

At least 19 recordsLinked to original sources
arXiv2026

Poisson blow-ups and the adjoint quotient

We leverage Polishchuk's Poisson blow-up criterion in the context of algebro-geometric integrable systems. In more detail, one may associate an integrable system $\tau:\mathfrak{X}\longrightarrow\mathfrak{B}$ to each affine Poisson scheme $\mathfrak{X}$ over $\mathbb{C}$. We prove that the blow-ups of $\mathfrak{X}$ along fibers of $\tau$ are Poisson schemes occurring in a family $\widetilde{\mathfrak{X}\times\mathfrak{B}}\longrightarrow\mathfrak{B}$, where $\widetilde{\mathfrak{X}\times\mathfrak{B}}$ is itself a Poisson scheme. This result is subsequently specialized to the adjoint quotient $\tau:\mathfrak{g}\longrightarrow\mathfrak{g}/\!/G=:\mathfrak{c}$ of a finite-dimensional complex semisimple Lie algebra $\mathfrak{g}$ with integrating algebraic group $G$. We show that the family $\widetilde{\mathfrak{g}\times\mathfrak{c}}\longrightarrow\mathfrak{c}$ is flat, conical, and equipped with a canonical Poisson Hamiltonian $G$-variety structure. We also obtain Poisson-geometric results on the fibers of this family, which are blow-ups of $\mathfrak{g}$ along regular adjoint orbit closures.

math.RT↗
arXiv2025

Slices for reductive group actions in algebraic and holomorphic symplectic geometry

Symplectic slice theorems elucidate the local structure of symplectic manifolds carrying Hamiltonian actions of compact Lie groups. We generalize these theorems in two natural settings. The first is based on the idea that complex reductive algebraic groups are the natural complex-geometric counterparts of compact Lie groups. Using new definitions of Poisson and symplectic slices, we prove analogues of the classical symplectic slice theorems for Hamiltonian actions of complex reductive algebraic groups. These analogues have versions in the complex-algebraic and holomorphic categories, and make extensive use of Slodowy slices and decomposition classes in complex reductive Lie algebras. The starting point for our second setting is the fact that Hamiltonian Lie group actions are special cases of Hamiltonian symplectic groupoid actions. We generalize the classical symplectic slice theorems to the latter case.

math.SG↗
arXiv2025

Grothendieck-Springer resolutions and TQFTs

The Moore-Tachikawa conjecture is that each connected complex semisimple group $G$ determines a two-dimensional TQFT in a category of Hamiltonian symplectic varieties. While it would be worthwhile to prove this conjecture outright, our objectives are drastically different. We instead view the Moore--Tachikawa conjecture as a first step in systematically assigning new TQFTs to purely Lie-theoretic data. At the same time, one should expect these new TQFTs to bear a close relation to those conjectured by Moore and Tachikawa. Our manuscript aims to integrate these two points of view. Let $\mathfrak{g}$ be the Lie algebra of $G$. Consider a conjugacy class $\mathcal{C}$ of parabolic subalgebras of $\mathfrak{g}$. This class determines partial Grothendieck--Springer resolutions $\mu_{\mathcal{C}}:\mathfrak{g}_{\mathcal{C}}\longrightarrow\mathfrak{g}^*=\mathfrak{g}$ and $\nu_{\mathcal{C}}:G_{\mathcal{C}}\longrightarrow G$. We construct a canonical symplectic groupoid $(T^*G)_{\mathcal{C}}\substack{\longrightarrow\\[-9pt] \longrightarrow}\mathfrak{g}_{\mathcal{C}}$ and quasi-symplectic groupoid $\mathrm{D}(G)_{\mathcal{C}}\substack{\longrightarrow\\[-9pt] \longrightarrow} G_{\mathcal{C}}$. By considering a Kostant slice $\mathrm{Kos}\subseteq\mathfrak{g}$ and Steinberg slice $\mathrm{Ste}\subseteq G$, we prove that the pairs $(((T^*G)_{\mathcal{C}})_{\text{reg}}\substack{\longrightarrow\\[-9pt] \longrightarrow}(\mathfrak{g}_{\mathcal{C}})_{\text{reg}},\mu_{\mathcal{C}}^{-1}(\mathrm{Kos}))$ and $((\mathrm{D}(G)_{\mathcal{C}})_{\text{reg}}\substack{\longrightarrow\\[-9pt] \longrightarrow}(G_{\mathcal{C}})_{\text{reg}},\nu_{\mathcal{C}}^{-1}(\mathrm{Ste}))$ determine new and explicit TQFTs in a $1$-shifted Weinstein symplectic category. We then show that certain symplectic varieties arising from our new TQFTs have canonical Lagrangian relations to the open Moore-Tachikawa varieties.

math.SG↗
arXiv2024

The Moore-Tachikawa conjecture via shifted symplectic geometry

We use shifted symplectic geometry to construct the Moore-Tachikawa topological quantum field theories (TQFTs) in a category of Hamiltonian schemes. Our new and overarching insight is an algebraic explanation for the existence of these TQFTs, i.e. that their structure comes naturally from three ingredients: Morita equivalence, as well as multiplication and identity bisections in abelian symplectic groupoids. Using this insight, we generalize the Moore-Tachikawa TQFTs in two directions. The first generalization concerns a 1-shifted version of the Weinstein symplectic category $\mathbf{WS}_1$. Each abelianizable quasi-symplectic groupoid $\mathcal{G}$ is shown to determine a canonical 2-dimensional TQFT $\eta_{\mathcal{G}}:\mathbf{Cob}_2\longrightarrow\mathbf{WS}_1$. We recover the open Moore-Tachikawa TQFT and its multiplicative counterpart as special cases. Our second generalization is an affinization process for TQFTs. We first enlarge Moore and Tachikawa's category $\mathbf{MT}$ of holomorphic symplectic varieties with Hamiltonian actions to $\mathbf{AMT}$, a category of affine Poisson schemes with Hamiltonian actions of affine symplectic groupoids. We then show that if $\mathcal{G} \rightrightarrows X$ is an affine symplectic groupoid that is abelianizable when restricted to an open subset $U \subseteq X$ statisfying Hartogs' theorem, then $\mathcal{G}$ determines a TQFT $\eta_{\mathcal{G}} : \mathbf{Cob}_2 \longrightarrow \mathbf{AMT}$. In more detail, we first devise an affinization process sending 1-shifted Lagrangian correspondences in $\mathbf{WS}_1$ to Hamiltonian Poisson schemes in $\mathbf{AMT}$. The TQFT is obtained by composing this affinization process with the TQFT $\eta_{\mathcal{G}|_U} : \mathbf{Cob}_2 \longrightarrow \mathbf{WS}_1$ of the previous paragraph. Our results are also shown to yield new TQFTs outside of the Moore-Tachikawa setting.

math.SG↗
arXiv2024

Scheme-theoretic coisotropic reduction

We develop an affine scheme-theoretic version of Hamiltonian reduction by symplectic groupoids. It works over $\Bbbk=\mathbb{R}$ or $\Bbbk=\mathbb{C}$, and is formulated for an affine symplectic groupoid $\mathcal{G}\rightrightarrows X$, an affine Hamiltonian $\mathcal{G}$-scheme $\mu:M\longrightarrow X$, a coisotropic subvariety $S\subseteq X$, and a stabilizer subgroupoid $\mathcal{H}\rightrightarrows S$. Our first main result is that the Poisson bracket on $\Bbbk[M]$ induces a Poisson bracket on the subquotient $\Bbbk[\mu^{-1}(S)]^{\mathcal{H}}$. The Poisson scheme $\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ is then declared to be a Hamiltonian reduction of $M$. Other main results include sufficient conditions for $\mathrm{Spec}(\Bbbk[\mu^{-1}(S)]^{\mathcal{H}})$ to inherit a residual Hamiltonian scheme structure. Our main results are best viewed as affine scheme-theoretic counterparts to an earlier paper, where we simultaneously generalize several Hamiltonian reduction processes. In this way, the present work yields scheme-theoretic analogues of Marsden-Ratiu reduction, Mikami-Weinstein reduction, \'{S}niatycki-Weinstein reduction, and symplectic reduction along general coisotropic submanifolds. The initial impetus for this work was its utility in formulating and proving generalizations of the Moore-Tachikawa conjecture.

math.SG↗
arXiv2023

Universal families of twisted cotangent bundles

Given a complex algebraic group $G$ and complex $G$-variety $X$, one can study the affine Hamiltonian Lagrangian (AHL) $G$-bundles over $X$. Lisiecki indexes the isomorphism classes of such bundles in the case of a homogeneous $G$-variety $X=G/H$; the indexing set is the set of $H$-fixed points $(\mathfrak{h}^*)^H\subset\mathfrak{h}^*$, where $\mathfrak{h}$ is the Lie algebra of $H$. In very rough terms, one may regard $\psi\in(\mathfrak{h}^*)^H$ as labeling the isomorphism class of a $\psi$-twisted cotangent bundle of $G/H$. These twisted cotangent bundles feature prominently in geometric representation theory and symplectic geometry. We introduce and examine the notion of a universal family of AHL $G$-bundles over a $G$-variety $X$, as part of a broader program on Lie-theoretic and incidence-theoretic constructions of regular Poisson varieties. This family is defined to be a flat family $\pi:\mathcal{U}\longrightarrow Y$, in which $\mathcal{U}$ is a Poisson variety, the fibers of $\pi$ form a complete list of representatives of the isomorphism classes of AHL $G$-bundles over $X$, and other pertinent properties are satisfied. Our first main result is the construction of a universal family of AHL $G$-bundles over a homogeneous base $X=G/H$, for connected $H$. In our second main result, we take $X$ to be a conjugacy class $\mathcal{C}$ of self-normalizing closed subgroups of $G$. We associate to $\mathcal{C}$ a regular Poisson variety $\mathcal{U}_{\mathcal{C}}$, defined in incidence-theoretic terms. Attention is paid to the case of conjugacy classes of normalizers of symmetric subgroups. In the case of a connected semisimple group $G$ and conjugacy class $\mathcal{C}$ of parabolic subgroups, our third main result relates $\mathcal{U}_{\mathcal{C}}$ to the partial Grothendieck-Springer resolution for $\mathcal{C}$.

math.AG↗
arXiv2023

The double Gelfand-Cetlin system, invariance of polarization, and the Peter-Weyl theorem

The bundle map $T^*\hspace{-2pt}\operatorname{U}(n)\longrightarrow\operatorname{U}(n)$ provides a real polarization of the cotangent bundle $T^*\hspace{-2pt}\operatorname{U}(n)$, and yields the geometric quantization $Q_1(T^*\hspace{-2pt}\operatorname{U}(n)) = L^2(\operatorname{U}(n))$. We use the Gelfand-Cetlin systems of Guillemin and Sternberg to show that $T^*\hspace{-2pt}\operatorname{U}(n)$ has a different real polarization with geometric quantization $Q_2(T^*\hspace{-2pt}\operatorname{U}(n))= \bigoplus_\alpha V_\alpha \otimes V_\alpha^*$, where the sum is over all dominant integral weights $\alpha$ of $\operatorname{U}(n)$. The Peter-Weyl theorem, which states that these two quantizations are isomorphic, may therefore be interpreted as an instance of ``invariance of polarization" in geometric quantization.

math.SG↗
arXiv2023

Abelianization and the Duistermaat-Heckman theorem

Fix a compact connected Lie group $G$ with Lie algebra $\mathfrak{g}$, as well as a strong Gelfand-Cetlin datum on $\mathfrak{g}^*$. Let us also fix a connected symplectic manifold $M$ endowed with an effective Hamiltonian $G$-action and proper moment map. We associate to such information a measure on $\mathbb{R}^{\mathrm{b}}$, where $\mathrm{b}=\frac{1}{2}(\dim G+\mathrm{rank}\hspace{2pt}G)$. We also express the Radon-Nikodym derivative of this measure in terms of the volumes of the symplectic quotients of $M$ by $G$, and thereby prove a non-abelian version of the Duistermaat-Heckman theorem.

math.SG↗
arXiv2022

Gelfand-Cetlin abelianizations of symplectic quotients

We show that generic symplectic quotients of a Hamiltonian $G$-space $M$ by the action of a compact connected Lie group $G$ are also symplectic quotients of the same manifold $M$ by a compact torus. The torus action in question arises from certain integrable systems on $\mathfrak{g}^*$, the dual of the Lie algebra of $G$. Examples of such integrable systems include the Gelfand-Cetlin systems of Guillemin-Sternberg in the case of unitary and special orthogonal groups, and certain integrable systems constructed for all compact connected Lie groups by Hoffman-Lane. Our abelianization result holds for smooth quotients, and more generally for quotients which are stratified symplectic spaces in the sense of Sjamaar-Lerman.

math.SG↗
arXiv2021

Symplectic reduction along a submanifold

We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in TQFT. A key feature of our construction is a concrete and systematic association of a Hamiltonian $G$-space $\mathfrak{M}_{G, S}$ to each pair $(G,S)$, where $G$ is any Lie group and $S\subseteq\mathrm{Lie}(G)^*$ is any submanifold satisfying certain non-degeneracy conditions. The spaces $\mathfrak{M}_{G, S}$ satisfy a universal property for symplectic reduction which generalizes that of the universal imploded cross-section. While these Hamiltonian $G$-spaces are explicit and natural from a Lie-theoretic perspective, some of them appear to be new.

math.SG↗
arXiv2021

On the singularities of Mishchenko-Fomenko systems

To each complex semisimple Lie algebra $\mathfrak{g}$ and regular element $a\in\mathfrak{g}_{\text{reg}}$, one associates a Mishchenko-Fomenko subalgebra $\mathcal{F}_a\subseteq\mathbb{C}[\mathfrak{g}]$. This subalgebra amounts to a completely integrable system on the Poisson variety $\mathfrak{g}$, and as such has a bifurcation diagram $\Sigma_a\subseteq\mathrm{Spec}(\mathcal{F}_a)$. We prove that $\Sigma_a$ has codimension one in $\mathrm{Spec}(\mathcal{F}_a)$ if $a\in\mathfrak{g}_{\text{reg}}$ is not nilpotent, and that it has codimension one or two if $a\in\mathfrak{g}_{\text{reg}}$ is nilpotent. In the nilpotent case, we show each of the possible codimensions to be achievable. Our results significantly sharpen existing estimates of the codimension of $\Sigma_a$.

math.SG↗
arXiv2020

Towards a quantization of the double via the enhanced symplectic category

This paper considers the enhanced symplectic "category" for purposes of quantizing quasi-Hamiltonian $G$-spaces, where $G$ is a compact simple Lie group. Our starting point is the well-acknowledged analogy between the cotangent bundle $T^*G$ in Hamiltonian geometry and the internally fused double $D(G)=G\times G$ in quasi-Hamiltonian geometry. Guillemin and Sternberg consider the former, studing half-densities and phase functions on its so-called character Lagrangians $\Lambda_{\mathcal{O}}\subseteq T^*G$. Our quasi-Hamiltonian counterpart replaces these character Lagrangians with the universal centralizers $\Lambda_{\mathcal{C}}\longrightarrow\mathcal{C}$ of regular, $\frac{1}{k}$-integral conjugacy classes $\mathcal{C}\subseteq G$. We show each universal centralizer to be a "quasi-Hamiltonian Lagrangian" in $D(G)$, and to come equipped with a half-density and phase function. At the same time, we consider a Dehn twist-induced automorphism $R:D(G)\longrightarrow D(G)$ that lacks a natural Hamiltonian analogue. Each quasi-Hamiltonian Lagrangian $R(\Lambda_{\mathcal{C}})$ is shown to have a clean intersection with every $\Lambda_{\mathcal{C}'}$, and to come equipped with a half-density and phase function of its own. This leads us to consider the possibility of a well-behaved, quasi-Hamiltonian notion of the BKS pairing between $R(\Lambda_{\mathcal{C}})$ and $\Lambda_{\mathcal{C}'}$. We construct such a pairing and study its properties. This is facilitated by the nice geometric fearures of $R(\Lambda_{\mathcal{C}})\cap\Lambda_{\mathcal{C}'}$ and a reformulation of the classical BKS pairing. Our work is perhaps the first step towards a level-$k$ quantization of $D(G)$ via the enhanced symplectic "category".

math.SG↗
arXiv2020

The log symplectic geometry of Poisson slices

Our paper develops a theory of Poisson slices and a uniform approach to their partial compactifications. The theory in question is loosely comparable to that of symplectic cross-sections in real symplectic geometry.

math.SG↗
arXiv2020

Hessenberg varieties and Poisson slices

This work pursues a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian $G$-variety $\mu:G\times\mathcal{S}\longrightarrow\mathfrak{g}$ to each complex semisimple Lie algebra $\mathfrak{g}$ with adjoint group $G$ and fixed Kostant section $\mathcal{S}\subseteq\mathfrak{g}$. This variety is one of Bielawski's hyperk\"ahler slices, and it is central to Moore and Tachikawa's work on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian $G$-varieties $\overline{\mu}_{\mathcal{S}}:\overline{G\times\mathcal{S}}\longrightarrow\mathfrak{g}$ and $\nu:\mathrm{Hess}\longrightarrow\mathfrak{g}$. The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification $\overline{G}$, while the latter is the standard family of Hessenberg varieties. Each of $\overline{\mu}$ and $\nu$ is known to be a fibrewise compactification of $\mu$. We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our main result is a canonical $G$-equivariant bimeromorphism $\mathrm{Hess}\cong\overline{G\times\mathcal{S}}$ of varieties over $\mathfrak{g}$. This bimeromorphism is shown to be a Hamiltonian $G$-variety isomorphism in codimension one, and to be compatible with a Poisson isomorphism obtained by B\u{a}libanu. We also show our bimeromorphism to be a biholomorphism if $\mathfrak{g}=\mathfrak{sl}_2$, and we conjecture that this is the case for arbitrary $\mathfrak{g}$. We conclude by discussing the implications of our conjecture for Hessenberg varieties.

math.SG↗
arXiv2020

Perverse sheaves and the cohomology of regular Hessenberg varieties

We use the Springer correspondence to give a partial characterization of the irreducible representations which appear in the Tymoczko dot-action of the Weyl group on the cohomology ring of a regular semisimple Hessenberg variety. In type A, we apply these techniques to prove that all irreducible summands which appear in the pushforward of the constant sheaf on the universal Hessenberg family have full support. We also observe that the recent results of Brosnan and Chow, which apply the local invariant cycle theorem to the family of regular Hessenberg varieties in type A, extend to arbitrary Lie type. We use this extension to prove that regular Hessenberg varieties, though not necessarily smooth, always have the "Kahler package."

math.RT↗
arXiv2019

On the fibres of Mishchenko-Fomenko systems

This work is concerned with Mishchenko and Fomenko's celebrated theory of completely integrable systems on a complex semisimple Lie algebra $\mathfrak{g}$. Their theory associates a maximal Poisson-commutative subalgebra of $\mathbb{C}[\mathfrak{g}]$ to each regular element $a\in\mathfrak{g}$, and one can assemble free generators of this subalgebra into a moment map $F_a:\mathfrak{g}\rightarrow\mathbb{C}^b$. We examine the structure of fibres in Mishchenko--Fomenko systems, building on the foundation laid by Bolsinov, Charbonnel--Moreau, Moreau, and others. This includes proving that the critical values of $F_a$ have codimension $1$ or $2$ in $\mathbb{C}^b$, and that each codimension is achievable in examples. Our results on singularities make use of a subalgebra $\mathfrak{b}^a\subseteq\mathfrak{g}$, defined to be the intersection of all Borel subalgebras of $\mathfrak{g}$ containing $a$. In the case of a non-nilpotent $a\in\mathfrak{g}_{\text{reg}}$ and an element $x\in\mathfrak{b}^a$, we prove the following: $x+[\mathfrak{b}^a,\mathfrak{b}^a]$ lies in the singular locus of $F_a^{-1}(F_a(x))$, and the fibres through points in $\mathfrak{b}^a$ form a $\mathrm{rank}(\mathfrak{g})$-dimensional family of singular fibres. We next consider the irreducible components of our fibres, giving a systematic way to construct many components via Mishchenko--Fomenko systems on Levi subalgebras $\mathfrak{l}\subseteq\mathfrak{g}$. In addition, we obtain concrete results on irreducible components that do not arise from the aforementioned construction. Our final main result is a recursive formula for the number of irreducible components in $F_a^{-1}(0)$, and it generalizes a result of Charbonnel--Moreau. Illustrative examples are included at the end of this paper.

math.SG↗
arXiv2019

Kostant-Toda lattices and the universal centralizer

To each complex semisimple Lie algebra $\mathfrak{g}$ decorated with appropriate data, one may associate two completely integrable systems. One is the well-studied Kostant-Toda lattice, while the second is an integrable system defined on the universal centralizer $\mathcal{Z}_{\mathfrak{g}}$ of $\mathfrak{g}$. These systems are similar in that each exploits and closely reflects the invariant theory of $\mathfrak{g}$, as developed by Chevalley, Kostant, and others. One also has Kostant's description of level sets in the Kostant-Toda lattice, which turns out to suggest deeper similarities between the two integrable systems in question. We study relationships between the two aforementioned integrable systems, partly to understand and contextualize the similarities mentioned above. Our main result is a canonical open embedding of a flow-invariant open dense subset of the Kostant-Toda lattice into $\mathcal{Z}_{\mathfrak{g}}$. Secondary results include some qualitative features of the integrable system on $\mathcal{Z}_{\mathfrak{g}}$.

math.SG↗