arXiv subjects
Publications and source records attributed to Augustin-Liviu Mare.
The main result of the paper is a Borel type description of the $Sp(1)^n$-equivariant cohomology ring of the manifold $Fl_n(\mathbb{H})$ of all complete flags in $\mathbb{H}^n$. To prove this, we obtain a Goresky-Kottwitz-MacPherson type description of that ring.
We show that the small quantum product of the generalized flag manifold $G/B$ is a product operation on $H^*(G/B)\otimes \bR[q_1,..., q_l]$ uniquely determined by the fact that it is a deformation of the cup product on $H^*(G/B)$, it is commutative, associative, graded with respect to $°(q_i)=4$, it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the "quantum Giambelli problem" for $G/B$. This result gives new proofs of the quantum Chevalley formula (proved by Peterson and Fulton-Woodward), and of Fomin, Gelfand and Postnikov's description of the quantization map for $Fl_n$.
Let $P=G/K$ be a semisimple non-compact Riemannian symmetric space, where $G=I_0(P)$ and $K=G_p$ is the stabilizer of $p\in P$. Let $X$ be an orbit of the (isotropy) representation of $K$ on $T_p(P)$ ($X$ is called a real flag manifold). Let $K_0\subset K$ be the stabilizer of a maximal flat, totally geodesic submanifold of $P$ which contains $p$. We show that if all the simple root multiplicities of $G/K$ are at least 2 then $K_0$ is connected and the action of $K_0$ on $X$ is equivariantly formal. In the case when the multiplicities are equal and at least 2, we will give a purely geometric proof of a formula of Hsiang, Palais and Terng concerning $H^*(X)$. In particular, this gives a conceptually new proof of Borel's formula for the cohomology ring of an adjoint orbit of a compact Lie group.
Atiyah's 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.
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 $κ_K$. We start with the case when K is a torus T: we determine the kernel of the equivariant Kirwan map (defined by Goldin in [Go]) corresponding to a generic circle S in T, and show how to recover from this the kernel of $κ_T$, as described by Tolman and Weitsman. (In the situation when the fixed point set of the torus action is finite, similar results have been obtained in our previous papers [Je], [Je-Ma]). For a compact nonabelian Lie group K we will use the ``non-abelian localization formula'' of [Je-Ki1] and [Je-Ki2] to establish relationships -- some of them obtained by Tolman and Weitsman in [To-We] -- between $\ker(κ_K)$ and $\ker(κ_T)$, where T is a maximal torus in K. An Appendix generalizes Theorem 1.8 to the case of singular values of $κ_T$.
A purely combinatorial construction of the quantum cohomology ring of the flag manifold $G/B$ is presented. We show that the ring we construct is commutative, associative and satisfies the usual grading condition. By using results of two of our previous papers, we obtain a presentation of this ring in terms of generators and relations, as well as formulas for quantum Giambelli polynomials. We show that these polynomials satisfy a certain orthogonality property, which - for G=SL_n(C) - was proved previously by Fomin, Gelfand and Postnikov.
Consider the infinite dimensional flag manifold $LK/T$ corresponding to the simple Lie group $K$ of rank $l$ and with maximal torus $T$. We show that, for $K$ of type $A$, $B$ or $C$, if we endow the space $H^*(LK/T)\otimes \bR[q_1,...,q_{l+1}]$ (where $q_1,...,q_{l+1}$ are multiplicative variables) with an $\bR[\{q_j\}]$-bilinear product satisfying some simple properties analogous to the quantum product on $QH^*(K/T)$, then the isomorphism type of the resulting ring is determined by the integrals of motion of a certain periodic Toda lattice system, in exactly the same way as the isomorphism type of $QH^*(K/T)$ is determined by the integrals of motion of the non-periodic Toda lattice (see the theorem of Kim). This is a generalization of a theorem of Guest and Otofuji.
We obtain a complete description of collections of n conjugacy classes in SU(2) with the property that the multiplication map from the product of these n conjugacy classes to SU(2) is surjective. The basic instrument is a characterization of collections of n+1 conjugacy classes in SU(2) such that the image of the multiplication map from these conjugacy classes to SU(2) contains the identity matrix, which generalizes a result of Jeffrey and Weitsman.
Using the notion of equivariant Kirwan map, as defined by Goldin, we prove that -- in the case of Hamiltonian torus actions with isolated fixed points -- Tolman and Weitsman's description of the kernel of the Kirwan map can be deduced directly from the residue theorem of Jeffrey and Kirwan. A characterization of the kernel of the Kirwan map in terms of residues of one variable (i.e. associated to Hamiltonian circle actions) is obtained.
We show how the quantum Chevalley formula for G/B, as stated by Peterson and proved rigorously by Fulton and Woodward, combined with ideas of Fomin, S. Gelfand and Postnikov, leads to a formula which describes polynomial representatives of the Schubert cohomology classes in the canonical presentation of QH^*(G/B) in terms of generators and relations.