Gelfand–Cetlin abelianizations of symplectic quotients

We introduce the notion of a Gelfand–Cetlin datum $(\lambda_{\text{big}},\mathfrak{g}^{*}_{\text{s-reg}})$ in Definition 1. This amounts to $\lambda_{\text{big}}$ being a continuous map on $\mathfrak{g}^{*}$ that restricts to a Poisson moment map for a Hamiltonian action of a compact torus $\mathbb{T}_{\text{big}}$ on an open dense subset $\mathfrak{g}^{*}_{\text{s-reg}}\subset\mathfrak{g}^{*}$, along with some extra conditions that capture salient properties of the classical Gelfand–Cetlin systems. One of these conditions is that the open, symplectic submanifold $M_{\text{s-reg}}\coloneqq\mu^{-1}(\mathfrak{g}^{*}_{\text{s-reg}})\subset M$ be a Hamiltonian $\mathbb{T}_{\text{big}}$-space with moment map $\lambda_{M}\coloneqq(\lambda_{\text{big}}\circ\mu)\big{|}_{M_{\text{s-reg}}}$ for any Hamiltonian $G$-space $M$ with moment map $\mu:M\longrightarrow\mathfrak{g}^{*}$. The results of Guillemin–Sternberg [4, 5] imply that Gelfand–Cetlin data exist for all unitary and special orthogonal groups, while more recent results of Hoffman–Lane [8] imply that such data exist in all Lie types.

The following is the main result of our paper.

Part (ii) is strictly more general than (i). Part (i) is included for the sake of exposition and accessibility.

One may regard this theorem as an approach to abelianizing the generic symplectic quotients of a Hamiltonian $G$-space $M$, i.e. to presenting such quotients as symplectic quotients by a compact torus. An alternative approach to abelianization is pursued in the work of Guillemin–Jeffrey–Sjamaar [2].

Section 2 briefly establishes some of our conventions concerning Lie theory and Hamiltonian geometry. Section 3 subsequently motivates and contextualizes the notion of a Gelfand–Cetlin datum. Our main result is then proved in Section 4 for smooth quotients. A generalization to stratified symplectic spaces is formulated and proved in Section 5.

The authors would like to thank Megumi Harada and Jeremy Lane for exceedingly useful conversations. P.C. acknowledges support from a Utah State University startup grant, while J.W. acknowledges support from Simons Collaboration Grant # 579801.

The Lie algebra of the unitary group $\operatorname{U}(1)$ is the real vector space $i\mathbb{R}\subset\mathbb{C}$ of purely imaginary numbers. We will identify this vector space with $\mathbb{R}$ in the obvious way. It follows that $\mathbb{R}^{k}$ is the Lie algebra of $\operatorname{U}(1)^{k}$ for all non-negative integers $k$, and that

is the exponential map for $\operatorname{U}(1)^{k}$. In certain contexts, we will implicitly use the dot product to regard $\mathbb{R}^{k}$ as the dual of the Lie algebra of $\operatorname{U}(1)^{k}$.

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$ and rank $\ell$. One has the adjoint representations $\mathrm{Ad}:G\longrightarrow\operatorname{GL}(\mathfrak{g})$ and $\mathrm{ad}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g})$, as well as the coadjoint representations $\mathrm{Ad}^{*}:G\longrightarrow\operatorname{GL}(\mathfrak{g}^{*})$ and $\mathrm{ad}^{*}:\mathfrak{g}\longrightarrow\mathfrak{gl}(\mathfrak{g}^{*})$. The $G$-representations induce $G$-actions on $\mathfrak{g}$ and $\mathfrak{g}^{*}$, and thereby give rise to stabilizer subgroups

of $G$ for all $x\in\mathfrak{g}$ and $\xi\in\mathfrak{g}^{*}$. On the other hand, the $\mathfrak{g}$-representations allow us to define centralizers

for all $x\in\mathfrak{g}$ and $\xi\in\mathfrak{g}^{*}$. It follows that $\mathfrak{g}_{x}$ (resp. $\mathfrak{g}_{\xi}$) is the Lie algebra of $G_{x}$ (resp. $G_{\xi}$). Let us also note that $\dim\mathfrak{g}_{x}\geq\ell$ and $\dim\mathfrak{g}_{\xi}\geq\ell$ for all $x\in\mathfrak{g}$ and $\xi\in\mathfrak{g}^{*}$. The regular loci

are open, dense, $G$-invariant subsets of $\mathfrak{g}$ and $\mathfrak{g}^{*}$, respectively. An element $x\in\mathfrak{g}$ (resp. $\xi\in\mathfrak{g}^{*}$) then belongs to $\mathfrak{g}_{\text{reg}}$ (resp. $\mathfrak{g}_{\text{reg}}^{*}$) if and only if $\mathfrak{g}_{x}$ (resp. $\mathfrak{g}_{\xi}$) is a Cartan subalgebra of $\mathfrak{g}$. This is equivalent to $G_{x}$ (resp. $G_{\xi}$) being a maximal torus of $G$.

A few remarks on maximal tori and and Cartan subalgebras are warranted. Let $\mathfrak{t}\subset\mathfrak{g}$ be a Cartan subalgebra, and write $T\subset G$ for the maximal torus with Lie algebra $\mathfrak{t}$. The exponential map $\mathrm{exp}:\mathfrak{g}\longrightarrow G$ then restricts to a surjective homomorphism $\mathrm{exp}\big{|}_{\mathfrak{t}}:\mathfrak{t}\longrightarrow T$ of abelian groups. The kernel of the latter is a free $\mathbb{Z}$-submodule of $\mathfrak{t}$ with rank equal to $\ell$. It follows that the same is true of

Let $(M,\sigma)$ be a Poisson manifold, i.e. $\sigma\in H^{0}(M,\Lambda^{2}TM)$ is a Poisson bivector field on the manifold $M$. Note that $\sigma$ may be regarded as a skew-symmetric bilnear map from two copies of $T^{*}M$ to the trivial rank-$1$ vector bundle over $M$. Contracting $\sigma$ with cotangent vectors in the first argument then determines a vector bundle morphism $\sigma^{\vee}:T^{*}M\longrightarrow TM$. One calls $(M,\sigma)$ non-degenerate if $\sigma^{\vee}$ is an isomorphism. In this case, $(\sigma^{\vee})^{-1}=\omega^{\vee}$ for a unique symplectic form $\omega\in H^{0}(M,\Lambda^{2}T^{*}M)$, where $\omega^{\vee}:TM\longrightarrow T^{*}M$ is the vector bundle morphism obtained by contracting $\omega$ with tangent vectors in the first argument. This process gives rise to a bijective correspondence between symplectic structures on $M$ and non-degenerate Poisson structures on $M$. We will thereby make no distinction between symplectic manifolds and non-degenerate Poisson manifolds.

If $(M,\sigma)$ is a Poisson manifold, then $\sigma$ can be recovered from the Poisson bracket $\{\cdot,\cdot\}$ that it induces. This bracket associates to smooth functions $f_{1},f_{2}:M\longrightarrow\mathbb{R}$ the smooth function

At the same time, one defines the Hamiltonian vector field of a smooth function $f:M\longrightarrow\mathbb{R}$ by $X_{f}\coloneqq-\sigma^{\vee}(\mathrm{d}f)\in H^{0}(M,TM)$. It follows that

for all smooth functions $f_{1},f_{2}:M\longrightarrow\mathbb{R}$.

Now let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$ and exponential map $\mathrm{exp}:\mathfrak{g}\longrightarrow G$. If $G$ acts smoothly on a manifold $M$, then each $\eta\in\mathfrak{g}$ determines a generating vector field $\eta_{M}\in H^{0}(M,TM)$ by

for all $m\in M$. A Poisson manifold $(M,\sigma)$ with a smooth $G$-action will be called a Poisson Hamiltonian $G$-space if $\sigma$ is $G$-invariant and $M$ comes equipped with a moment map. This last term refers to $G$-equivariant smooth map $\mu:M\longrightarrow\mathfrak{g}^{*}$ satisfying $X_{\mu^{\eta}}=\eta_{M}$ for all $\eta\in\mathfrak{g}$, where $\mu^{\eta}:M\longrightarrow\mathbb{R}$ is the result of pairing $\mu$ with $\eta$ pointwise. We will reserve the term Hamiltonian $G$-space for a Poisson Hamiltonian $G$-space whose underlying Poisson structure is symplectic.

It will be advantageous to recall the Poisson Hamiltonian $G$-space structure on $\mathfrak{g}^{*}$. The Poisson bracket on $\mathfrak{g}^{*}$ is given by

for all smooth functions $f_{1},f_{2}:M\longrightarrow\mathbb{R}$ and points $\xi\in\mathfrak{g}^{*}$, where $\mathrm{d}_{\xi}f_{1},\mathrm{d}_{\xi}f_{2}\in(\mathfrak{g}^{*})^{*}=\mathfrak{g}$ denote the differentials of $f_{1},f_{2}$ at $\xi$, respectively. One finds that $\mathfrak{g}^{*}$ is a Poisson Hamiltonian $G$-space with respect to the coadjoint action, and with the identity $\mathfrak{g}^{*}\longrightarrow\mathfrak{g}^{*}$ serving as the Poisson moment map.

Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$ and rank $\ell$. Consider the quantities

and introduce the following tori of small, intermediate, and big ranks:

The respective Lie algebras of these tori are

It is instructive to consider this definition in relation to the theory of completely integrable systems. One is thereby led to the following result.

It is natural to wonder about the generality in which Gelfand–Cetlin data exist. The earliest constructions are due to Guillemin–Sternberg [4, 5], and apply to all unitary groups $\operatorname{U}(n)$ and special orthogonal groups $\operatorname{SO}(n)$. The underlying techniques are based on Thimm’s method, as described in [5]. Further details are outlined in Sections 3.3–3.5 of this paper. The case of symplectic groups is considerably more subtle and addressed in Harada’s paper [6].

Some recent work of Hoffman–Lane [8] implies the existence of Gelfand–Cetlin data for an arbitrary compact connected Lie group $G$; the reader is referred to [8, Section 6.2] for the relevant details. This Hoffman–Lane paper is part of a broader program aimed at generalizing the results of Harada–Kaveh [7].

We now discuss the construction of functions $\lambda_{1},\ldots,\lambda_{\ell}:\mathfrak{g}^{*}\longrightarrow\mathbb{R}$ satisfying Conditions (i) and (ii) in Definition 1. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak{g}$ and rank $\ell$. Choose a $G$-invariant inner product on $\mathfrak{g}$, a Cartan subalgebra $\mathfrak{t}\subset\mathfrak{g}$, and a closed, fundamental Weyl chamber $\mathfrak{t}_{+}\subset\mathfrak{t}$. The chamber $\mathfrak{t}_{+}$ is known to be a fundamental domain for the adjoint action of $G$ on $\mathfrak{g}$. Our inner product thereby identifies $\mathfrak{t}_{+}$ with a fundamental domain $\mathfrak{t}_{+}^{*}\subset\mathfrak{g}^{*}$ for the $G$-action on $\mathfrak{g}^{*}$. We may therefore define a continuous surjection $\pi:\mathfrak{g}^{*}\longrightarrow\mathfrak{t}_{+}^{*}$ by the property that $(G\cdot\xi)\cap\mathfrak{t}_{+}^{*}=\{\pi(\xi)\}$ for all $\xi\in\mathfrak{g}^{*}$, where $G\cdot\xi\subset\mathfrak{g}^{*}$ is the coadjoint orbit of $\xi$. One sometimes calls $\pi$ the sweeping map on $\mathfrak{g}^{*}$ with respect to $\mathfrak{t}$; its fibers are exactly the coadjoint orbits of $G$, and $\pi(\mathfrak{g}^{*}_{\text{reg}})$ is the interior $(\mathfrak{t}_{+}^{*})^{\circ}$ of $\mathfrak{t}_{+}^{*}$. One also finds that the commutative diagram

is Cartesian. The left vertical map

is a easily seen to be a smooth, surjective submersion.

Now choose a $\mathbb{Z}$-basis $\{\phi_{1},\ldots,\phi_{\ell}\}$ of the $\mathbb{Z}$-submodule $\Lambda_{\mathfrak{t}}\subset\mathfrak{t}$. Note that the pairing between $\mathfrak{t}$ and $\mathfrak{t}^{*}$ allows one to regard $\phi_{1},\ldots,\phi_{\ell}$ as functions on $\mathfrak{t}_{+}^{*}$. With this in mind, each $k\in\{1,\ldots,\ell\}$ determines a function

The previous paragraph implies that $\lambda_{1},\ldots,\lambda_{\ell}$ are smooth on $\mathfrak{g}^{*}_{\text{reg}}$, while being $G$-invariant and continuous as functions on $\mathfrak{g}^{*}$. Given any $\xi\in\mathfrak{g}^{*}_{\text{reg}}$, the differentials $\mathrm{d}_{\xi}\lambda_{1},\ldots,\mathrm{d}_{\xi}\lambda_{\ell}\in(\mathfrak{g}^{*})^{*}=\mathfrak{g}$ may be described as follows.

Retain the objects and notation discussed in Section 3.3. Let $G=G_{0}\supset G_{1}\supset\cdots\supset G_{m}$ be a descending filtration of $G$ by connected closed subgroups with respective Lie algebras $\mathfrak{g}=\mathfrak{g}_{0}\supset\mathfrak{g}_{1}\supset\cdots\supset\mathfrak{g}_{m}$. Let us also choose a Cartan subalgebra $\mathfrak{t}_{j}\subset\mathfrak{g}_{j}$ and closed, fundamental Weyl chamber $(\mathfrak{t}_{j})_{+}\subset\mathfrak{t}_{j}$ for each $j\in\{0,\ldots,m\}$. Our $G$-invariant inner product on $\mathfrak{g}$ gives rise to a $G_{j}$-module isomorphism $\mathfrak{g}_{j}\cong\mathfrak{g}_{j}^{*}$, by means of which $\mathfrak{t}_{j}$ and $(\mathfrak{t}_{j})_{+}$ correspond to subsets $\mathfrak{t}_{j}^{*}$ and $(\mathfrak{t}_{j}^{*})_{+}$ of $\mathfrak{g}_{j}^{*}$. As in Section 3.3, one may define a continuous surjection $\pi_{j}:\mathfrak{g}_{j}^{*}\longrightarrow(\mathfrak{t}_{j}^{*})_{+}$ by the condition that $(G_{j}\cdot\xi)\cap(\mathfrak{t}_{j}^{*})_{+}=\{\pi_{j}(\xi)\}$ for all $\xi\in\mathfrak{g}_{j}^{*}$.

Let $\ell=\ell_{0}\geq\ell_{1}\geq\cdots\geq\ell_{m}$ be the ranks of $G=G_{0}\supset G_{1}\supset\cdots\supset G_{m}$, respectively. Let us also choose a $\mathbb{Z}$-basis $\{\phi_{j1},\ldots,\phi_{j\ell_{j}}\}$ of the lattice $\Lambda_{\mathfrak{t}_{j}}\subset\mathfrak{t}_{j}$ for each $j\in\{0,\ldots,m\}$. As in Section 3.3, we define the functions

for $k\in\{1,\ldots,\ell_{j}\}$. The same section implies that $\nu_{j1},\ldots,\nu_{j\ell_{j}}$ are $G_{j}$-invariant and continuous on $\mathfrak{g}_{j}^{*}$, as well as smooth on $(\mathfrak{g}_{j}^{*})_{\text{reg}}$. We also have the following equivalent version of Proposition 4.

Let $\sigma_{j}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}_{j}^{*}$ denote the transpose of the inclusion $\mathfrak{g}_{j}\hookrightarrow\mathfrak{g}$ for each $j\in\{0,\ldots,m\}$. Consider the functions on $\mathfrak{g}^{*}$ defined by

for $j\in\{0,\ldots,m\}$ and $k\in\{1,\ldots,\ell_{j}\}$. It will be convenient to enumerate these functions as

where $\mathrm{c}\coloneqq\ell_{0}+\cdots+\ell_{m}$.

The discussion preceding Proposition 5 implies that $\lambda_{\text{big}}$ is smooth on the open subset

Let us also define

These last few sentences give context for the following consequence of [4, Theorem 3.4].

This result has the following immediate connection to Definition 1: the pair $(\lambda_{\text{big}},\mathfrak{g}^{*}_{\text{s-reg}})$ is a Gelfand–Cetlin datum if and only if $\mathrm{c}=\mathrm{b}$, $\mathfrak{g}^{*}_{\text{s-reg}}$ is dense in $\mathfrak{g}^{*}$, and $\lambda_{\text{big}}\big{|}_{\mathfrak{g}^{*}_{\text{s-reg}}}:\mathfrak{g}^{*}_{\text{s-reg}}\longrightarrow\lambda_{\text{big}}(\mathfrak{g}^{*}_{\text{s-reg}})$ is a principal $\mathbb{T}_{\text{int}}$-bundle. Guillemin–Sternberg [4, 5] explicitly show these conditions to be achievable for $G=\operatorname{U}(n)$ and $G=\operatorname{SO}(n)$. Our next section outlines the details of the Guillemin–Sternberg construction for $G=\operatorname{U}(n)$.

Fix a positive integer $n$. Consider the Lie group $G\coloneqq\mathrm{U}(n)$ of unitary $n\times n$ matrices, and its Lie algebra $\mathfrak{g}\coloneqq\mathfrak{u}(n)$ of skew-Hermitian $n\times n$ matrices. Let us also consider the real $\operatorname{U}(n)$-module $\mathcal{H}(n)$ of Hermitian $n\times n$ matrices. In what follows, we will freely identify $\mathfrak{u}(n)^{*}$ with $\mathcal{H}(n)$ by means of the non-degenerate, $G$-invariant bilinear form

Given an integer $j\in\{0,\ldots,n-1\}$, define the subgroup

The descending chain $\operatorname{U}(n)=G=G_{0}\supset G_{1}\supset\cdots\supset G_{n-1}$ then induces such a chain $\mathfrak{u}(n)=\mathfrak{g}=\mathfrak{g}_{0}\supset\mathfrak{g}_{1}\supset\cdots\supset\mathfrak{g}_{n-1}$ on the level of Lie algebras. We have

for all $j\in\{0,\ldots,n-1\}$, where the second equation implicitly uses (3.2). The transpose $\sigma_{j}:\mathfrak{g}^{*}\longrightarrow\mathfrak{g}_{j}^{*}$ of $\mathfrak{g}_{j}\subset\mathfrak{g}$ then sends $\xi\in\mathfrak{g}^{*}=\mathcal{H}(n)$ to the $(n-j)\times(n-j)$ submatrix in the bottom right-hand corner of $\xi$.

Now note that

is a Cartan subalgebra for each $j\in\{0,\ldots,n-1\}$. Let $\phi_{jk}\in\mathfrak{t}_{j}$ be the result of setting $a_{k}=1$ and $a_{p}=0$ for $p\neq k$, noting that $\{\phi_{j1},\ldots,\phi_{j(n-j)}\}$ is a $\mathbb{Z}$-basis of $\Lambda_{\mathfrak{t}_{j}}\subset\mathfrak{t}_{j}$. At the same time, consider the fundamental Weyl chamber

for each $j\in\{0,\ldots,n-1\}$. Under the pairing (3.2), $(\mathfrak{t}_{j})_{+}$ corresponds to the cone

We then have sweeping maps $\pi_{j}:\mathfrak{g}_{j}^{*}\longrightarrow(\mathfrak{t}^{*}_{j})_{+}$ and compositions $\nu_{jk}\coloneqq\phi_{jk}\circ\pi_{j}:\mathfrak{g}_{j}^{*}\longrightarrow\mathbb{R}$ for $j\in\{0,\ldots,n-1\}$ and $k\in\{1,\ldots,n-j\}$, as in Section 3.4. The functions

from Section 3.4 are therefore given by the following condition: if $\xi\in\mathcal{H}(n)$ and $j\in\{0,\ldots,n-1\}$, then $\lambda_{j1}(\xi)\geq\lambda_{j2}(\xi)\geq\cdots\geq\lambda_{j(n-j)}(\xi)$ are the eigenvalues of the $(n-j)\times(n-j)$ submatrix in the bottom right-hand corner of $\xi$.

Observe that the number of maps $\lambda_{jk}$ is

Our enumeration (3.1) therefore takes the form

Let us also consider the open dense subset

of $\mathfrak{g}^{*}=\mathcal{H}(n)$. By the paragraph following the proof of Proposition 6, $(\lambda_{\text{big}},\mathcal{H}(n)_{\text{s-reg}})$ is a Gelfand–Cetlin datum if and only if $\lambda_{\text{big}}\big{|}_{\mathcal{H}(n)_{\text{s-reg}}}:\mathcal{H}(n)_{\text{s-reg}}\longrightarrow\lambda_{\text{big}}(\mathcal{H}(n)_{\text{s-reg}})$ is a principal bundle for $\mathbb{T}_{\text{int}}=\operatorname{U}(1)^{\frac{n(n-1)}{2}}$. This later condition is verified in [4, Section 5].

Adopt the notation and conventions in Section 3.1, and let $(\lambda_{\text{big}},\mathfrak{g}^{*}_{\text{s-reg}})$ be a Gelfand–Cetlin datum. Given any $\xi\in\mathfrak{g}^{*}_{\text{reg}}$, Definition 1(ii) tells us that $\{\mathrm{d}_{\xi}\lambda_{1},\ldots,\mathrm{d}_{\xi}\lambda_{\ell}\}$ is a basis of $\mathfrak{g}_{\xi}$. This basis determines a vector space isomorphism

The torus $\mathbb{T}_{\text{small}}$ is then a universal maximal torus in the following sense.

We now prove two supplementary facts needed to establish the main result of this paper. We continue with the notation and conventions of Sections 3.1 and 4.1.