Questions tagged [equivariant]

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3 votes
0 answers
123 views

group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
2 votes
0 answers
230 views

Moment map of equivariant line bundles

I'm reading Szabo's `Equivariant Cohomology and Localization of Path Integrals'. I've stumbled upon an equation I can't make sense of, in the discussion about $G$-equivariant line bundles on ...
5 votes
1 answer
524 views

Do equivariant morphisms induce representable maps of quotient stacks?

Let $f: X \to Y$ be a $G$-equivariant map between schemes $X$, $Y$ with action of a flat group scheme $G$. Then why is the induced map of algebraic stacks $[X/G] \to [Y/G]$ representable?
3 votes
0 answers
629 views

Equivariant Riemann-Roch on DM stacks?

Does an equivariant version of (Toen)-Riemann-Roch theorem hold say over a smooth Deligne-Mumford stack $X$ over the complex numbers? Any references that state this explicitely? Are there formulas ...
6 votes
0 answers
113 views

2-functoriality of equivariant derived categories

I am wondering about the 2-functoriality in equivariant derived categories, and I hope that someone can clarify... (apologies if this is a stupid question) For the more precise formulation, recall ...
5 votes
2 answers
1k views

Atiyah-Guillemin-Sternberg convexity theorem

I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory. So ...
4 votes
0 answers
210 views

Milnor's model of $EG$ and Kac-Moody groups

I am working with non-compact Kac-Moody groups $\mathcal{K}$. We can use Milnor's join model for $E\mathcal{K}=\varinjlim \mathcal{K}^{*n}$, where $\mathcal{K}^{*n}$ is the iterated join (see page 20 ...
3 votes
0 answers
408 views

Equivariant form of Nagata's compactification theorem?

Given a finite group $G$ acting on an algebraic variety $X$ (let's say over $\mathbb C$, if that helps), is there always a proper variety $\bar X$ with a $G$ action such that $X \to \bar X$ is a $G$-...
3 votes
0 answers
98 views

good choice of extension of equivariant map

Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that $...
2 votes
1 answer
242 views

How to extend an equivariant map from a compact Lie group

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$...
3 votes
1 answer
141 views

What's the topology on the mapping space $Map_H(G, Y)$ when $G$ is not finite

When $G$ is a finite group and $H$ a closed subgroup of it, the sets of right cosets $H\backslash G$ has the discrete topology on it. Let $Y$ be a $H-$space. We have the $G-$homeomorphism \begin{...
1 vote
1 answer
648 views

Beilinson-Bernstein localization: $\mathfrak{g}$ action on $G$-equivariant sheaf

I have a few elementary questions related to Beilinson-Bernstein localization. Let $G$ be a semisimple algebraic group over $\mathbb{C}$ with Lie algebra $\mathfrak{g}$. Consider the setup of ...
2 votes
1 answer
672 views

Definition of Givental $J$-function of cotangent bundle of flag variety

I would like to know the definition of Givental $J$-function of cotangent bundle of flag variety. To state my question more precisely, let us briefly recall the definition of the Givental $J$-function ...
9 votes
3 answers
2k views

Is this a definition of equivariant derived category?

Let $X$ be a topological space and $G$ be a topological group acting on $X$, both locally compact Hausdorff. Denote by $D^b(X)$ the derived category of sheaves (say of abelian groups) on $X$. We ...
16 votes
2 answers
2k views

Equivariant version of Morse theory

Is there some variant on Morse or Morse-Bott theory yielding equivariant (co)homology instead of singular homology? Any reference/idea would be greatly appreciated. Crossposted on StackExchange.
1 vote
0 answers
151 views

some intuition about the degree of a map

Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some ...
6 votes
1 answer
500 views

Cocycle condition for equivariant sheaves

Let $G$ be an affine group that acts on a variety $X$. Equivariant sheaves on $X$ could be defined in the following way. Consider the simplicial space $X_\bullet$ : $X_n := G^n \times X$, $s_0:X_0 \...
8 votes
1 answer
720 views

Equivariant algebraic K-theory of affine space

Unlike algebraic K-theory, equivariant K-theory of affine space (over a field $k$) can be quite nontrivial, depending on the action of the group in question. For example, if one takes the standard ...
1 vote
1 answer
408 views

Do there exist equivariant sheafs that are not equivariant vector bundles?

For $F \subset G$ two algebraic groups, consider a homogeneous space $H$ of the form $G/F$. Now every vector bundle over $H$ is a coherent sheaf, but the converse is not true. What happens in the ...
3 votes
0 answers
157 views

Monodromy along strata of a pushforward

Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived. Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
3 votes
1 answer
423 views

Vector bundles on a weighted projective stack

Put $X := \mathbb A^{n+1}\!-\lbrace0\rbrace$. Let $G=\mathbb C^*$ act on $X$ with (positive) weights $w_0,\dots,w_n$. The quotient stack $[X/G]$ is called the weighted projective stack. Each vector ...
0 votes
1 answer
430 views

Non-Symmetric Equivariant Riemannian Metrics on Homogeneous Spaces

For a homogeneous space $M = G/H$, the number of $H$-equivariant Riemannian metrics on $M$ is usually much smaller than the space of Riemannian metrics. I am wondering what happens when the symmetric ...
3 votes
1 answer
242 views

Classifying all Equivariant Bilinear Forms on a Finite-Dimensional Module

Given a finite dimensional (real) vector space $V$, and two non-degenerate bilinear forms $(\cdot,\cdot)_1$ and $(\cdot,\cdot)_2$, one can use a basic linear algebra argument to show that there exists ...
6 votes
2 answers
777 views

Weights on equivariant cohomology?

Let $X$ be a quasi-projective variety over the complex numbers, equipped with an action of a linear algebraic group $G$. Is there a natural mixed Hodge structure on its equivariant cohomology? Is ...
4 votes
1 answer
181 views

Equivariant versus retractive spaces: a reference request

Let $T$ be the category of compactly generated weak Hausdorff spaces with model structure given by Serre fibrations, Serre cofibrations and weak homotopy equivalences. Let $G = |G.|$ be the (geometric)...
5 votes
0 answers
606 views

What is the right notion of equivariant Cech cohomology?

What is the right definition of equivariant Cech cohomology is so that given a $G$-space $X$, $H^1_G(X;H)$ classifies $G$-equivariant principal $H$-bundles on $X$?
2 votes
1 answer
583 views

How to calculate the equivariant cohomology ring of $P^2$?

It is well known that Kirwan's injection theorem gives an ring injection from $H^{\ast}_T(M)$ to $H^{\ast}_T(M^T)$ which is induced by the inclusion $M^T \to M$, where $T$ is a torus acting on ...
5 votes
2 answers
719 views

Jets of Equivariant Vector Bundles

Let $M$ be a (compact) $G$-homogeneous space with fibre group $H$, and let ${\cal E}$ be a $G$-equivariant $k$-dimensional vector bundle over $M$ with corresponding representation $\pi:H \to $R$^k$. ...
21 votes
0 answers
1k views

Can we describe equivariant vector bundles of free group action in terms of descent theory (Barr-Beck theorem)?

It is well known that for a compact topological group $G$ acts (say, from the right) freely on a compact space $X$. Then the category of equivariant complex vector bundles on $X$, $\text{Vect}_G(X)$, ...
5 votes
0 answers
324 views

T-Equivariant trivialization of a principal G-bundle

Let $k$ be a field, let $G$ be an algebraic group scheme over $k$ and let $T = \textrm{Spec } k[x, x^{-1}]$ be a one-dimensional torus. Does there exist a scheme $X$ over $k$, an algebraic $T$-action ...
4 votes
0 answers
290 views

What is the equivariant derived category good for?

Given a topological group acting on a topological space, Bernstein and Lunts construct the equivariant derived category, which looks like the derived category of the quotient would, if action was free ...
18 votes
8 answers
3k views

Reference request: Equivariant Topology

I am teaching a graduate seminar in equivariant topology. The format of the course is that I will give 2-3 weeks of background lectures, then each week a student will present a topic. The students ...
2 votes
1 answer
360 views

When is restriction an equivalence of categories of equivariant vector bundles?

Suppose a (linear algebraic) group $G$ acts on a variety $X$ and that $U$ is a $G$-invariant open subvariety. My question is: under what conditions is the restriction functor $i^*: Vect^G(X) \...
18 votes
2 answers
1k views

What is the role of equivariance in the Atiyah-Singer index theorem?

I'm trying to read through "The index of elliptic operators I," and here's what I understand of the structure of the proof: Define (using purely K-theoretic means) a homomorphism $K_G(TX) \to R(G)$...
0 votes
0 answers
436 views

Lifting of torus action to line bundle

Let $\mathbb{P}(V) = \mathbb{P}(\mathbb C \oplus \mathbb C)$ be with a $\mathbb C^*$ action : $\lambda (u,v) = (u,\lambda v)$. There are two fixed points of this action, say $0$ and $\infty$. What ...
3 votes
1 answer
284 views

For a G-variety, what could one say about the motif of the corresponding simplicial variety

Let G be an algerbraic group, and X be a G-variety (that I will assume to be smooth). Then one can consider a simlicial variety whose terms are $G^i\times X$. This simplicial variety yields a 'complex ...
4 votes
0 answers
398 views

Equivariant sheaves and simplicial varieties

I would like to proof the following theorem: Let $\pi:X\rightarrow X/G$ be a principal $G$-bundle (say of varieties, Zariski locally trivial), then $\pi^*$ induces an equivalence between modules on ...
22 votes
0 answers
2k views

Is the equivariant cohomology an equivariant cohomology?

Suppose a finite group $G$ acts piecewise linearly on a polyhedron $X$. Then there are two kinds of equivariant cohomology (or homology). $\bullet$ With coefficients in a $\Bbb Z G$-module $M$. A ...
6 votes
6 answers
597 views

Equivariant homology of $\Omega X$\/-space (references needed)?

Let $(X, *)$ be pointed a (1-connected) space, and let $\Omega X$ denote its based loops space. Then, as one knows very well, $\Omega X$ is a group up to homotopy (this includes all the necessary ...
10 votes
0 answers
2k views

Why is the Nil-Hecke Algebra appearing?

The Nil-Hecke algebra is defined to be the subalgebra of the endomorphism ring of $\mathbb{C}[x_1,\ldots,x_n]$ generated by the operators of multiplication by $x_i$ and the divided difference ...
4 votes
0 answers
231 views

Equivariant sheaves basics reference

I am looking for a reference for basic facts about actions of linear algebraic groups and their Lie-algebras on $\mathcal O_X$-modules. For example I could not find a reference the following: Let $...
9 votes
1 answer
763 views

Are these notions of strongly equivariant D-modules equivalent?

It seems that there are two notions of strongly equivaraint $D_X$- Modules and I would like to know if they are equivalent, or at least how they are related. Let $\rho: G\times X \rightarrow X$ be an ...
11 votes
1 answer
885 views

What does this naive attempt at $S^1$-equivariant homology describe?

After reading Cohen and Voronov's notes on string topology, one can find the following construction: Suppose we have a topological space $X$ with continuous action of $S^1$. This means we have a map $\...
1 vote
1 answer
617 views

The fiber of the sheaf of invariants

Let us suppose the the group $G:=\mathbb{Z}/2\mathbb{Z}=(1,i)$ freely act on a smooth projective variety/$k$ $X$ and denote by $Y$ the G.I.T. quotient $X/G$. Let $\pi:X\longrightarrow Y$ the quotinet ...
1 vote
0 answers
182 views

G-Modules on X=G/H modules on X/H ?

I think it is true that $G$-equivariant sheaves on $X$ are equal to $G/H$ equivariant sheaves on $X/H$. More precisely I'm interested in the following statement: Given an algebraic group $G$ with ...
17 votes
1 answer
2k views

Geometric interpretation of filtered rings and modules

Let $A$ be a commutative algebra, say over $\mathbb{C}$. Giving a grading on $A$ corresponds at least morally to giving a $\mathbb{C}^*$ action on spec(A): $A_i$ can be thought of as those ...

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