Augustin-Liviu Mare

Cohomology of symplectic reductions of generic coadjoint orbits

Let O λ be a generic coadjoint orbit of a compact semi-simple Lie group K. Weight varieties are the symplectic reductions of O λ by the maximal torus T in K. We use a theorem of Tolman and Weitsman to compute the cohomology ring of these varieties. Our formula relies on a Schubert basis of the equivariant cohomology of O λ , and it makes explicit the dependence on A and a parameter in Lie(T)* =: t*.

REAL LOCI OF BASED LOOP GROUPS

Let (G, K) be a Riemannian symmetric pair of maximal rank, where G is a compact simply connected Lie group and K is 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 × S1 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 × S1 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

A Characterization of the Quantum Cohomology Ring of

{\mathbb{Z}_2}

G/B

of Ω(G) and Ω(G/K). Namely, the two cohomology rings are isomorphic, by a degree-halving isomorphism (Bott and Samelson [BS] 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 [HHP].

and Applications

We observe that the small quantum product of the generalized flag manifold G/B is a product operation ⋆ on H∗(G/B) ⊗ R[q1, . . . , ql] uniquely determined by the facts that it is a deformation of the cup product on H∗(G/B); it is commutative, associative, and graded with respect to deg(qi) = 4; it satisfies a certain relation (of degree two); and the corresponding Dubrovin connection is flat. Previously, we proved that these properties alone imply the presentation of the ring (H(G/B)⊗ R[q1, . . . , ql], ⋆) in terms of generators and relations. In this paper we use the above observations to give conceptually new proofs of other fundamental results of the quantum Schubert calculus forG/B: the quantumChevalley formula of D. Peterson (see also Fulton andWoodward) and the “quantization by standard monomials” formula of Fomin, Gelfand, and Postnikov for G = SL(n, C). The main idea of the proofs is the same as in Amarzaya–Guest: from the quantum D-module of G/B one can decode all information about the quantum cohomology of this space. Received by the editors September 6, 2005; revised April 25, 2006. AMS subject classification: Primary: 14M15; secondary: 14N35. c ©Canadian Mathematical Society 2008. 875

Connectivity properties of moment maps on based loop groups

For a compact, connected, simply-connected Lie group G, the loop group LG is the infinite-dimensional Hilbert Lie group consisting of H^1-Sobolev maps S^1-->G. The geometry of LG and its homogeneous spaces is related to representation theory and has been extensively studied. The space of based loops Omega(G) is an example of a homogeneous space of

Connectivity and Kirwan surjectivity for isoparametric submanifolds

LG

The Kirwan map, equivariant Kirwan maps, and their kernels

and has a natural Hamiltonian T x S^1 action, where T is the maximal torus of G. We study the moment map mu for this action, and in particular prove that its regular level sets are connected. This result is as an infinite-dimensional analogue of a theorem of Atiyah that states that the preimage of a moment map for a Hamiltonian torus action on a compact symplectic manifold is connected. In the finite-dimensional case, this connectivity result is used to prove that the image of the moment map for a compact Hamiltonian T-space is convex. Thus our theorem can also be viewed as a companion result to a theorem of Atiyah and Pressley, which states that the image mu(Omega(G)) is convex. We also show that for the energy functional E, which is the moment map for the S^1 rotation action, each non-empty preimage is connected.

Products of conjugacy classes in SU(2)

Atiyahs formulation of what is nowadays called the convexity theorem of Atiyah-Guillemin-Sternberg has two parts: (a) the image of the moment map arising from a Hamiltonian action of a torus on a symplectic manifold is a convex polytope, and (b) all preimages of the moment map are connected. Part (a) was generalized by Terng to the wider context of isoparametric submanifolds in euclidean space. In this paper we prove a generalization of part (b) for a certain class of isoparametric submanifolds (more precisely, for those with all multiplicities strictly greater than 1). For generalized real flag manifolds, which are an important class of isoparametric submanifolds, we give a surjectivity criterium of a certain Kirwan map (involving equivariant cohomology with rational coefficients) which arises naturally in this context. Examples are also discussed.

On the complete integrability of the periodic quantum Toda lattice

Abstract Consider a Hamiltonian action of a compact Lie group K  on a compact symplectic manifold. We find descriptions of the kernel of the Kirwan map corresponding to a regular value of the moment map κK . We start with the case when K  is a torus T : we determine the kernel of the equivariant Kirwan map (defined by Goldin in [R. F. Goldin , An effective algorithm for the cohomology ring of symplectic reductions, Geom. Func. Anal. 12 (2002), 567–583]) corresponding to a generic circle S  ⊂ T, and show how to recover from this the kernel of κT , as described by Tolman and Weitsman in [S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [L. C. Jeffrey, The residue formula and the Tolman-Weitsman theorem, J. reine angew. Math. 562 (2003), 51–58], [L. C. Jeffrey and A.-L. Mare, The kernel of the equivariant Kirwan map and the residue formula, Quart. J. Math. Oxford 54 (2004), 435–444].) For a compact nonabelian Lie group K  we will use the ‘‘non-abelian localization formula’’ of [L. C. Jeffrey and F. C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995), 291–327] and [L. C. Jeffrey and F. C. Kirwan, Localization and the quantization conjecture, Topology 36 (1995), 647–693] to establish relationships—some of them obtained by Tolman and Weitsman in [S. Tolman and J. Weitsman, The cohomology rings of symplectic quotients, Comm. Anal. Geom. 11 No. 4 (2003), 751–773]—between Ker(κK ) and Ker(κT ), where T  ⊂ K  is a maximal torus. In the appendix we prove that the same relationships remain true in the case when 0 is no longer a regular value of μT .