Equivariant map

About: Equivariant map is a research topic. Over the lifetime, 9205 publications have been published within this topic receiving 137115 citations.

Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante

Abstract: We show existence and uniqueness of the equivariant isometric immersions of Riemannian surfaces into Lorentz space-forms under conditions implying convexity, when we impose that the associated representations leave a point invariant.

On equivariant quantum cohomology of homogeneous spaces: Chevalley formulae and algorithms

Abstract: We prove a Chevalley formula for the equivariant quantum multiplication of two Schubert classes in the homogeneous variety X=G/P. Using this formula, we give an effective algorithm to compute the 3-point, genus zero, equivariant Gromov-Witten invariants on X, which are the structure constants of its equivariant quantum cohomology algebra

Quasi-exceptional sets and equivariant coherent sheaves on the nilpotent cone

Abstract: In math.AG/0005152 a certain $t$-structure on the derived category of equivariant coherent sheaves on the nil-cone of a simple complex algebraic group was introduced (the so-called perverse $t$-structure corresponding to the middle perversity). In the present note we show that the same $t$-structure can be obtained from a natural quasi-exceptional set generating this derived category. As a consequence we obtain a bijection between the sets of dominant weights and pairs consisting of a nilpotent orbit, and an irreducible representation of the centralizer of this element, conjectured by Lusztig and Vogan (and obtained by other means in math.RT/0010089).

Construction of group actions on four-manifolds

Abstract: It is shown that any cyclic group of odd prime order acts on any closed, simply connected topological 4-manifold, inducing the identity on integral homology. The action is locally linear except perhaps at one isolated fixed point. In the case of primes greater than three a more careful argument is used to show that the action can be constructed to be locally linear. Introduction. In this paper we shall show that every closed, simply connected topological 4-manifold M4 admits an action of any cyclic group Zp of odd prime order p. The action will be homologically trivial (that is, induce the identity on integral homology), be pseudofree (that is, have only isolated fixed points) except in certain cases when p = 3, and be locally linear except perhaps at one isolated fixed point. When p = 2 and the intersection form of M4 has even type, then the same conclusion holds. This is also due to S. Kwasik and P. Vogel [1985], by a somewhat different proof. If p = 2 and the intersection form of M has odd type, however, then an action of Z2 which is homologically trivial must have a fixed point set containing two-dimensional components (see ?7). Further it is harder to control the Kirby-Siebenmann triangulation obstruction, which is then not determined by the intersection form. It is interesting to ask whether these actions can be locally linear or smooth (when M4 is smooth). The existence of a locally linear Z2 action implies the vanishing of the Kirby-Siebenmann obstruction (see Kwasik and Vogel [1984]). We shall show, however, that (for p > 3 in general) these actions can be constructed to be locally linear. In a recent preprint Kwasik has shown that this is the case for the fake Cp2 when p is odd by a rather different proof. These actions are homologically trivial and have only isolated fixed points (except sometimes when p = 3). (The G-Signature Theorem shows that not every simply connected 4-manifold admits pseudofree, locally linear actions when p = 3. We shall see, for example, that neither the E8 manifold, the Kummer surface, nor a nontrivial connected sum of copies of Cp2 admits such an action for p = 3.) In broad outline our construction goes as follows. Let a closed, oriented, simply connected 4-manifold M be given. By studying equivariant framed links, we construct a compact smooth 4-manifold with boundary, having the intersection form of M, and admitting a smooth Zp action which is pseudofree and homologically trivial. The action is free on the boundary homology sphere E. In a paper primarily focused on equivariant plumbing diagrams for high dimensional, highly connected manifolds with even intersection forms, Weintraub [1975] carried out most of this. We give an independent development because we need to be able to better control Received by the editors October 16, 1985 and, in revised form, January 7, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57S17, 57S25, 57N15. Research supported in part by a grant from the National Science Foundation. (?)1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

A Finite Analog of the AGT Relation I: Finite W -Algebras and Quasimaps’ Spaces

Abstract: Recently Alday, Gaiotto and Tachikawa [2] proposed a conjecture relating 4-dimensional super-symmetric gauge theory for a gauge group G with certain 2-dimensional conformal field theory. This conjecture implies the existence of certain structures on the (equivariant) intersection cohomology of the Uhlenbeck partial compactification of the moduli space of framed G-bundles on \({\mathbb{P}^2}\) . More precisely, it predicts the existence of an action of the corresponding W-algebra on the above cohomology, satisfying certain properties.