Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a re... more Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H * T (M ) → H * (M 0 ). For an equivariant cohomology class η ∈ H * T (M ) we present new localization formulas which express M 0 κ 0 (η) as sums of certain integrals over the connected components of the fixed point set M T . To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T ) we give a new proof of the Jeffrey-Kirwan localization formula [JK1]. This paper is dedicated to Alan Weinstein on the occasion of his 60th birthday.
gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-spa... more gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-space in terms of iterated residues. The definition of quasi-Hamiltonian spaces was introduced in [1]. In [2] a localization formula for equivariant de Rham cohomology of a compact quasi-Hamiltonian G-space was proved. In this paper we prove a residue formula for intersection pairings of reduced spaces of quasi-Hamiltonian Gspaces, by constructing a corresponding Hamiltonian G-space. Our formula is a close analogue of the result in . In this article we rely heavily on the methods of ; for the general class of compact Lie groups G treated in [2], we rely on results of Szenes and Brion-Vergne concerning diagonal bases.
Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in ... more Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in rational equivariant cohomology from M to a level set Φ -1 (p) is a surjection, and we denote the kernel by I p . When T has isolated fixed points, we show that I p distinguishes the chambers of the moment polytope for M . In particular, counting the number of distinct ideals I p as p varies over different chambers is equivalent to counting the number of chambers.
This paper studies intersection theory on the compactified moduli space M(n, d) of holomorphic bu... more This paper studies intersection theory on the compactified moduli space M(n, d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface Σ of genus g ≥ 2 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups IH * (M(n, d)) defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities M(n, d) of M(n, d). Based on our earlier work , we give a precise formula for the intersection cohomology pairings and provide a method to calculate pairings on M(n, d). The case when n = 2 is discussed in detail. Finally Witten's integral is considered for this singular case.
gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-spa... more gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-space in terms of iterated residues. The definition of quasi-Hamiltonian spaces was introduced in [?]. In [?] a localization formula for equivariant de Rham cohomology of a compact quasi-Hamiltonian G-space was proved. In this paper we prove a residue formula for intersection pairings of reduced spaces of quasi-Hamiltonian Gspaces, by constructing a corresponding Hamiltonian G-space. Our formula is a close analogue of the result in [?]. In this article we rely heavily on the methods of [?]; for the general class of compact Lie groups G treated in [?], we rely on results of Szenes and Brion-Vergne concerning diagonal bases.
In this article we investigate the Duistermaat-Heckman theorem using the theory of hyperfunctions... more In this article we investigate the Duistermaat-Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite dimensional manifolds, this more general theory seems to be necessary in order to accomodate the existence of the infinite order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. We will quickly review of the theory of hyperfunctions and their Fourier transforms. We will then apply this theory to construct a hyperfunction analogue of the Duistermaat-Heckman distribution. Our main goal will be to study the Duistermaat-Heckman hyperfunction of ΩSU ( ), but in getting to this goal we will also characterize the singular locus of the moment map for the Hamiltonian action of T × S 1 on ΩG. The main goal of this paper is to present a Duistermaat-Heckman hyperfunction arising from a Hamiltonian action on an infinite dimensional manifold.
Let G = SU (2) and let ΩG denote the space of continuous based loops in G, equipped with the poin... more Let G = SU (2) and let ΩG denote the space of continuous based loops in G, equipped with the pointwise conjugation action of G. It is a classical fact in topology that the ordinary cohomology H * (ΩG) is a divided polynomial algebra Γ[x]. The algebra Γ[x] can be described as an inverse limit as k → ∞ of the symmetric subalgebra in Λ(x 1 , . . . , x k ) where Λ(x 1 , . . . , x k ) is the usual exterior algebra in the variables x 1 , . . . , x k . We compute the R(G)-algebra structure of the G-equivariant K-theory K * G (ΩG) which naturally generalizes the classical computation of H * (ΩG) as Γ[x]. Specifically, we prove that K * G (ΩG) is an inverse limit of the symmetric (S 2r -invariant) subalgebra (K * G ((P 1 ) 2r )) S 2r of K * G ((P 1 ) 2r ), where the symmetric group S 2r acts in the natural way on the factors of the product (P 1 ) 2r and G acts diagonally via the standard action on each factor.
Let G = SU (2) and let ΩG denote the space of based loops in SU (2). We explicitly compute the R(... more Let G = SU (2) and let ΩG denote the space of based loops in SU (2). We explicitly compute the R(G)-module structure of the topological equivariant K-theory K * G (ΩG) and in particular show that it is a direct product of copies of K * G (pt) ∼ = R(G). (We describe in detail the R(G)-algebra (i.e. product) structure of K * G (ΩG) in a companion paper.) Our proof uses the geometric methods for analyzing loop spaces introduced by Pressley and Segal (and further developed by Mitchell). However, Pressley and Segal do not explicitly compute equivariant K-theory and we also need further analysis of the spaces involved since we work in the equivariant setting. With this in mind, we have taken this opportunity to expand on the original exposition of Pressley-Segal in the hope that in doing so, both our results and theirs would be made accessible to a wider audience. CONTENTS 1. Introduction 1 2. Classical Results 4 3. The Grassmannian Gr z (K) and its subspaces 7 4. Description of filtration quotients as Thom spaces 12 5. The G-homotopy equivalence SGr ′ z bdd,r (K) → S Gr z α(psm),r (K) 14 6. Proof of the main theorem 23 References 24
We compute the symplectic volume of the symplectic reduced space of the product of N coadjoint or... more We compute the symplectic volume of the symplectic reduced space of the product of N coadjoint orbits of a compact connected Lie group G. We compare our result with the result of Suzuki and Takakura , who study this in the case G = SU(3) starting from geometric quantization.
Jeffrey and Kirwan [19] gave expressions for intersection pairings on the reduced space μ(0)/G of... more Jeffrey and Kirwan [19] gave expressions for intersection pairings on the reduced space μ(0)/G of a particular Hamiltonian G-space M in terms of iterated residues. The definition of quasi-Hamiltonian spaces was introduced in [1]. In [3] a localization formula for equivariant de Rham cohomology of a compact q-Hamiltonian G-space was proved. In this paper we prove a residue formula for intersection pairings of reduced spaces of certain quasi-Hamiltonian G-spaces, by constructing the corresponding Hamiltonian G-space. We show that in an important class of special cases our result agrees with that in [3]. In this article we rely heavily on the methods of [19]; for the more general class of compact Lie groups G treated in [3], we rely on results of Szenes and Brion-Vergne concerning diagonal bases. Date: April 16, 2008. 1 2 LISA JEFFREY AND JOON-HYEOK SONG
Let G be a semisimple compact connected Lie group. An N -fold reduced product of G is the symplec... more Let G be a semisimple compact connected Lie group. An N -fold reduced product of G is the symplectic quotient of the Hamiltonian system of the Cartesian product of N coadjoint orbits of G under diagonal coadjoint action of G. Under appropriate assumptions, it is a symplectic orbifold. Using the technique of nonabelian localization and the residue formula of Jeffrey and Kirwan, we investigate the symplectic volume of an N -fold reduced product of G. Suzuki and Takakura gave a volume formula for the N -fold reduced product of SU(3) in [25] by using geometric quantization and the Riemann-Roch formula. We compare our volume formula with theirs and prove that our volume formula agrees with theirs in the case of triple reduced products of SU(3).
This chapter investigates the Duistermaat–Heckman theorem using the theory of hyperfunctions. In ... more This chapter investigates the Duistermaat–Heckman theorem using the theory of hyperfunctions. In applications involving Hamiltonian torus actions on infinite-dimensional manifolds, the more general theory seems to be necessary in order to accommodate the existence of the infinite-order differential operators which arise from the isotropy representations on the tangent spaces to fixed points. The chapter quickly reviews the theory of hyperfunctions and their Fourier transforms. It then applies this theory to construct a hyperfunction analogue of the Duistermaat–Heckman distribution. The main goal will be to study the Duistermaat–Heckman distribution of the loop space of SU(2) but it will also characterize the singular locus of the moment map for the Hamiltonian action of T×S1 on the loop space of G. The main goal of this chapter is to present a Duistermaat–Heckman hyperfunction arising from a Hamiltonian group action on an infinite-dimensional manifold.
Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a re... more Given a compact symplectic manifold M with the Hamiltonian action of a torus T , let zero be a regular value of the moment map, and M 0 the symplectic reduction at zero. Denote by κ 0 the Kirwan map H * T (M ) → H * (M 0 ). For an equivariant cohomology class η ∈ H * T (M ) we present new localization formulas which express M 0 κ 0 (η) as sums of certain integrals over the connected components of the fixed point set M T . To produce such a formula we apply a residue operation to the Atiyah-Bott-Berline-Vergne localization formula for an equivariant form on the symplectic cut of M with respect to a certain cone, and then, if necessary, iterate this process using other cones. When all cones used to produce the formula are one-dimensional we recover, as a special case, the localization formula of Guillemin and Kalkman [GK]. Using similar ideas, for a special choice of the cone (whose dimension is equal to that of T ) we give a new proof of the Jeffrey-Kirwan localization formula [JK1]. This paper is dedicated to Alan Weinstein on the occasion of his 60th birthday.
Let K be a compact Lie group. We introduce the process of symplectic implosion, which associates ... more Let K be a compact Lie group. We introduce the process of symplectic implosion, which associates to every Hamiltonian K-manifold a stratified space called the imploded cross-section. It bears a resemblance to symplectic reduction, but instead of quotienting by the entire group, it cuts the symmetries down to a maximal torus of K. We examine the nature of the singularities and describe in detail the imploded cross-section of the cotangent bundle of K, which turns out to be identical to an affine variety studied by Gelfand, Vinberg, Popov, and others. Finally we show that "quantization commutes with implosion". Contents 1. Introduction 1 2. The construction 2 3. Abelianization 5 4. The universal imploded cross-section 7 5. The stratification 12 6. Kähler structures 16 7. Quantization and implosion 24 8. Notation index 28 References 29
Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected ... more Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K the fixed point set of an involutive automorphism σ. This induces an involutive automorphism τ of the based loop space Ω(G). There exists a maximal torus T ⊂ G such that the canonical action of T × S 1 on Ω(G) is compatible with τ (in the sense of Duistermaat). This allows us to formulate and prove a version of Duistermaat's convexity theorem. Namely, the images of Ω(G) and Ω(G) τ (fixed point set of τ ) under the T × S 1 moment map on Ω(G) are equal. The space Ω(G) τ is homotopy equivalent to the loop space Ω(G/K) of the Riemannian symmetric space G/K. We prove a stronger form of a result of Bott and Samelson which relates the cohomology rings with coefficients in Z 2 of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [Bo-Sa] had proved that the Betti numbers are equal). A version of this theorem involving equivariant cohomology is also proved. The proof uses the notion of conjugation space in the sense of Hausmann, Holm, and Puppe [Ha-Ho-Pu].
gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-spa... more gave expressions for intersection pairings on the reduced space of a particular Hamiltonian G-space in terms of iterated residues. The definition of quasi-Hamiltonian spaces was introduced in [?]. In [?] a localization formula for equivariant de Rham cohomology of a compact quasi-Hamiltonian G-space was proved. In this paper we prove a residue formula for intersection pairings of reduced spaces of quasi-Hamiltonian Gspaces, by constructing a corresponding Hamiltonian G-space. Our formula is a close analogue of the result in [?]. In this article we rely heavily on the methods of [?]; for the general class of compact Lie groups G treated in [?], we rely on results of Szenes and Brion-Vergne concerning diagonal bases.
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