A right gspace is a manifold y with a smooth right gaction e e y. By an equivariant version of the bgg correspondence, m defines an object f in the bounded derived category of g equivariant coherent sheaves on projective space. Consider the trivial g vector bundle b x s s over b where s is a g module. Is analgebraic g vector bundleoverarepresentation bofgtriviali. M such that e is a g space and the action of g preserves bers, i. G x, the category of g equivariant vector bundles on a topological space x. This makes the collection of all principal g bundles over b into a category. Pdf geometric quantization and equivariant cohomology. Given a representation w of g, a gequivariant vector bundle of. Examples are constructed for the well controlled case of equivariant vector bundles over compact coadjoint orbits. Everyg equivariant complex vector bundle carries a natural spincstructure because there is a unique group homomorphism uk spinc2k that lifts the map uk. There is a chern character map from equivariant ktheory to equivariant ordinary cohomology. Consider arbitrary equivariant mappings of total spaces of principal gbundles with arbitrary bases. We show that every g equivariant vector bundle on an af.
Equivariant ktheory ningchuan zhang august 16, 2019 1 classical equivariant ktheory 1. Gequivariant vector bundles on x, and from these bundles construct the tan gent bundle of x, the bundle of exterior iforms, and the canonical line bundle of x. A gvector bundle over x is a gmap e nx which is a vector bundle such that g acts linearly on the bers, i. Quantization of equivariant vector bundles springerlink. The set of isomorphism classes of principal g bundles over b will be denote p gb. If m is a gmanifold and z c m is a compact invariant submanifold then 7. Pdf equivariant vector bundles on certain affine gvarieties. So, analagously to ktheory, we may dene a gequivariant cohomology theory which in degree 0 is given by the grothendieck ring k gx of gequivariant vector bundles on x.
Namely, there is a notion of a g equivariant complex vector bundle over a g space x. Gvarieties u and i, moru, vg denotes the equivariant morphisms algebraic gmaps from u to i. Constructing equivariant vector bundles via the bgg. Let g be a semisimple algebraic group, and hc a character on a maximal torus h. Equivalently, a frame can be viewed as a linear isomorphism. Now, gequivariant vector bundles over any space with a transitive gaction correspond to representations of the isotropy group, so the vector bundle to which al corresponds must coincide with that obtained by regarding v as a gerepresentation. Equivariant vector bundles on drinfelds upper half space 3 is the dual of the steinberg representation. A coadjoint orbit is a symplectic manifold with a transitive, semisimple symmetry group. It can be constructed by a method entirely analogous to that of the ordinary frame bundle. Introduction for an arbitrary finite group g, the paper 5 contains an existence theorem for equivariant embeddings of principal g bundles into g vector bundles with trivial.
Integral closure of invariant ideals, toroidal resolution. Thetools discussed beloware applied to the following twoproblems. A projective vector bundle e associated to an azumaya bundle aover x lifts to a vector bundle eb over pwith an action of sun which is equivariant with respect to the pun action on pand in which the center zn acts as the nth roots of unity. In preparation for the main result, the quantization of coadjoint orbits is discussed in detail. In this note, we will study the problem of concretely describing the category vecgx of g equivariant vector bundles on prehomogeneous g varieties x. This is called algebraic ktheory if we wish to generalize this last example to the equivariant setting, we have to be. The set of isomorphims classes of complex real g vector bundles over xwill be denoted by bdlgx obdlgx. Equivariant semialgebraic vector bundles request pdf. Kvect g x is known as equivariant topological ktheory. Equivariant algebraic vector bundles over a product of affine varieties kayo masuda received april 7, 1996 0. The vector bundle e on y 0 corresponds to a graded g m equivariant locally free module over b. The tangent bundle of p sm is a g equivariant vector bundle on p sm, and can therefore be written the pullback of a smooth vector bundle e on x. Throughout this paper, the base field is the field c of complex numbers.
This module corresponds to a vector bundle eon projb which is y, whose restriction to y 0 is e. Torus equivariant vector bundles on projective spaces. We show that every gequivariant vector bundle on an af. In this case, note that the projectivized vector bundle pe projsyme also. Di erential operators on equivariant vector bundles over. Twistor theory of manifolds with grassmannian structures. A right g space is a manifold y with a smooth right g action e e y. We describe a strategy for the construction of finitely generated g equivariant zgraded modules m over the exterior algebra for a finite group g. A g spinc vector bundle is a smooth real g vector bundle with a given g spincstructure.
Equivariant vector bundles on the lubintate moduli space. Equivariant algebraic ktheory northeastern university. Let v be the underlying vector space of the representation the ring homomorphism. Moreover, htt uses this formula to claim a number of facts, e. Notes on principal bundles and classifying spaces mit math. Therefore vector principal g bundles are always trivial. The orthonormal frame bundle of e, denoted f o e, is the set of all orthonormal frames at each point x in the base space x. G which is linear on each fiber and makes the following. This yields the g homotopy category of left g spaces.
G, where g is the pullback by left multiplication by g and ad is the adjoint representation. A ghomotopy or gequivariant homotopy, or equivariant homotopy between gmaps is a homotopy f. Equivariant algebraic vector bundles over representations. M such that e is a gspace and the action of g preserves bers, i. For general g, an equivariant characteristic class is the image of a particular element of h g pt. If we say a property holds for f up to translation and twist, e. Gvarieties u and i, moru, vg denotes the equivariant morphisms algebraic g maps from u to i. A g variety x is called prehomogeneous if g acts on x with a zariski dense orbit. Constructing equivariant vector bundles via the bgg correspondence 2.
As constants act trivially and the quotient is singular, we think of the homotopy quotient of css by the group g g u1. We establish a connection between equivariant integrally closed ideal sheaves on a g fibration y over a g spherical variety x with an affine fiber v and equivariant vector bundles on the universal toroidal resolution of x. From the endomorphism ring r of the g vector bundle s a construction of g vector bundles over b is given. Equivariant vector bundles on certain affine gvarieties article pdf available in pure and applied mathematics quarterly 24 may 2006 with 70 reads how we measure reads.
For g g a compact lie group or more generally a compact topological group the concept of g g spectrum or g g equivariant spectrum is the generalization of that of spectrum as one passes from stable homotopy theory to equivariant stable homotopy theory, or more generally, as g g is allow to vary, to global equivariant stable homotopy theory. V will denote the tfold direct sum of v with itself. Everything is defined over the field c of complex numbers. Equivariant algebraic vector bundles over representations of. Gequivariantvector bundles on x, and from these bundles construct the tan gent bundle ofx, the bundle of exterior iforms,and the canonical line bundle of x. Every gequivariant vector bundle over eg gives rise to a compatible collection of representations of the finite subgroups of g. A g homotopy or g equivariant homotopy, or equivariant homotopy between g maps is a homotopy f. The bundles constructed this way have the property that. The natural question that arises is whether it is possible to give a concrete description of the category of g equivariant vector bundles on an arbitrary normal prehomogeneous g variety. This includes the traditional notion of a characteristic class for a. Equivariant vector bundles on drinfelds upper half space. A vector bundle w of rank r on m is called equivariant if w corresponds to an equivariant principal gl r, cbundle.
This makes intuitive sense, for the deformation bundle is the restriction to m of the tangent bundle to m g. In many cases it is possible to associate to a gequivariant hermitian vector bundle. It extends to a borel subgroup bc, giving a one dimensional representation w. An introduction to equivariant cohomology and the equivariant. Y in the usual sense, with the added condition that f be gequivariant here g acts trivially on the i coordinate. Kgx k0 g x is the grothendieck group of the monoid of isomorphism classes of g. Let x be a gscheme and let e be a gvector bundle over x. Ageometricdescriptionofequivariantkhomologyforproper. In part iv, we restrict equivariant bundles to the general fibre x k, which is a rigid analytic space over the quotient field k of a, isomorphic to the open. A prequantum bundle with connection is a principal circle bundle p m for which c 1p. A riemannian gvector bundle v over a point is an orthogonal representation. G equivariant vector bundle is a vector bundle e on x with an action of g. Standard ideal sheaves let v be a g equivariant vector bundle on x.
Introduction let g be a reductive complex affine algebraic group and z a complex affine g variety with a g fixed base point z 0 ez. We refer the reader to 12, section 1 and 20, section i. Pdf we give a concrete description of the category of g equivariant vector bundles on certain affine g varieties where g is a reductive linear. Taking the g m nite vectors, we get a graded locally free module over b. If x is a smooth gvariety, where g is a connected linear algebraic group, and l is an arbitrary line bundle over x, then some positive power l. We develop a necessary condition for f being isomorphic to a vector bundle that can be. Pdf equivariant generalized homotopy of fibre bundles. Let us determine the gvector bundles on the space of cosets gh, when h is a closed subgroup of g. Xis equivariant and t acts linearly on the bers of. Let e x be a real vector bundle of rank k over a topological space x. This is called algebraic ktheory if we wish to generalize this last example to the equivariant. Gross equivariant vector bundles 25 the condition on the action ofuf on lief is simply that theset hoffirf, f ofall homomorphismsover r is anamodule,with addition and amultiplicationdefined using the operations of f. An equivariant principal circle bundle p m is a principal circle bundle with the action of g on the total space p covering the action of m.
Equivariant vector bundles, their derived category and k. E is the character of a representation of gin the vector space hix. So, analagously to ktheory, we may dene a g equivariant cohomology theory which in degree 0 is given by the grothendieck ring k g x of g equivariant vector bundles on x. The projective bundle pe has a natural structure of a g scheme so that the natural morphism p. Equivalently, an equivariant vector bundle is a pair w. C cohx, the category of coherent sheaves on an algebraic variety x. We will write this a module simply as homf, f if the base ring r is fixed. An equivariant or toric vector bundle eon xis a locally free sheaf of nite rank on xwith a taction on the corresponding geometric vector bundle ve specsyme such that the projection. Equivariant vector bundles, their derived category and ktheory on af.
On equivariant vector bundles on an almost homogeneous. Let g be a reductive algebraic group and let b be an affine variety with an algebraic action of g. Vector bundles and homogeneous spaces,differential geometry, proc. Pdf a construction of equivariant bundles on the space. The simplest instance of this problem is the case when x is a homogeneous g variety. Gequivariant vector bundles on prehomogeneous gvarieties x. Now, if e is a super projective vector bundle over x, it.
By assumption we have a vector bundle e on x 0 with. If x and y are gspaces, a gequivariant map is a map. In this paper we discuss about equivariant generalized homotopy of fibre bundles and sheaves and g topological spaces. G equivariant vector bundles on a class of spherical varieties consisting of certain smooth g.
An isomorphism of g vector bundles over xis just an isomorphism of vector bundles that is g equivariant. Spec x n 0 s n x vwe denotetheprojection xv xby pv. The set of gvector bundle morphisms from f to s is denoted. An equivariant vector bundle eover xis a smooth vector bundle e. This definition is independent of the value of t chosen. Y x be a torsor under the vector bundle space e considered as a group scheme over x and g acts on y so that f and the action map e. The only available examples of vector g bundles are thus of the form m x g, where g is both a vector space and a lie group. The presentation matrix of these bundles attains westwicks upper bound. Integrally closed ideals and vector bundles in this section, we assume that xis g spherical.
Y in the usual sense, with the added condition that f be g equivariant here g acts trivially on the i coordinate. The representation ring of g g over the complex numbers is the g g equivariant ktheory of the point, or equivalently by the greenjulg theorem, if g g is a compact lie group, the operator ktheory of the group algebra the groupoid convolution algebra of the delooping groupoid of g g. A ghomotopy or gequivariant homotopy, or equivariant homotopy between gmaps. A morphism of principal bundles over b is an equivariant map p. We prove that any semialgebraic g vector bundle over a semialgebraic g set has a semialgebraic. Namely, there is a notion of a gequivariant complex vector bundle over a gspace x. Equivariant vector bundles over the upper half plane. If the space x is a point, a g vector bundle over x is just a linear representation of the group g. This yields the ghomotopy category of left gspaces. The homotopy classification theorem for vector bundles then. Now, just as a vector bundle over the trivial space is just a vector space. Request pdf equivariant semialgebraic vector bundles let g be a compact semialgebraic group.