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$\pm 1$-equivariant perverse sheaves on the affine line

Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
Sergey Guminov's user avatar
2 votes
0 answers
65 views

Equivariant $K$-theory and proper actions of discrete groups

The work of Lück and Oliver describes the generalization of equivariant $K$-theory to infinite discrete groups. When $X$ is a finite proper $G$-CW complex, there exist Bott isomorphisms $K^n_G(X)\cong ...
user519810's user avatar
5 votes
0 answers
148 views

Equivalent descriptions of equivariant K-theory

I am looking at references for computing $$K_{T}(G/H)$$ where $G$ is a compact connected Lie group with maximal torus $T$, and $H\subset G$ is a corank one Lie subgroup such that $G/H\cong S^{2k-1}$ ...
Yun Liu's user avatar
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0 votes
1 answer
168 views

Equivariant sheaves on $\mathbb P^1$

Let $K(\mathbb P^1)$ be the Grothendieck group of sheaves on $\mathbb P^1$. I want to show that the map $K^{{\rm PGL}(2)\times \{\pm 1\}}(\mathbb P^1) \to K(\mathbb P^1)$ is not onto. I read somewhere ...
Yellow Pig's user avatar
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2 votes
0 answers
88 views

$\mathrm{GL}(n, \mathbb{Z})$-equivariant maps on $\mathrm{GL}(n, \mathbb{R})$

$\DeclareMathOperator\GL{GL}$Can you describe the maps from $\GL(n, \mathbb{R})$ to $\GL(n, \mathbb{R})$ that are equivariant w.r.t. right multiplication by $\GL(n, \mathbb{Z})$? I'm interested even ...
gm01's user avatar
  • 327
3 votes
0 answers
49 views

Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...
Malkoun's user avatar
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1 vote
0 answers
89 views

Invariant category and coinvariant category under group action

Let $\mathcal{C}$ be a category with a finite group action $G$, There is a notion called G-equivariant category, denoted by $\mathcal{C}^G$. In the paper Kuznetsov's Fano threefold conjecture via K3 ...
user41650's user avatar
  • 1,952
1 vote
1 answer
298 views

Short exact sequence of equivariant line bundles on $\mathbb P^1$

I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^...
IntegrableSystemsEnthusiast's user avatar
2 votes
0 answers
48 views

A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors)

Let $G$ be a compact semisimple Lie group and let $T$ be a maximal torus in $G$. On the Lie algebra level, we have a real Lie algebra $\mathfrak{g}$ and a (particular) real slice, say $\mathfrak{t}$, ...
Malkoun's user avatar
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6 votes
0 answers
188 views

G-sheaves on spaces with a free G-action

Let $X$ be a topological space equipped with an action of a group $G$. In the Tohoku paper, Grothendieck defined "$G$-sheaves" on $X$ as sheaves equipped with $G$-action on the etale space, ...
Misha Verbitsky's user avatar
5 votes
0 answers
155 views

equivariant Steenrod algebra

From Hu-Kriz "Real-oriented homotopy theory and an analogue of the Adams-Novikov spectral sequence" we know about calculation of $C_2$ - equivariant Steenrod algebra. Where can I find (if it ...
Dr.Martens's user avatar
5 votes
3 answers
665 views

Deequivariantisation of indecomposable sheaves

Let G be a connected group acting on a space X. All spaces should be reasonable, so e.g. G is a complex affine algebraic group acting on an algebraic variety X, with everything done using the usual ...
Peter McNamara's user avatar
2 votes
0 answers
40 views

On the higher-dimensional Berry-Robbins problem

Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...
Malkoun's user avatar
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3 votes
1 answer
295 views

What is the pointed Borel construction of the $0$-sphere?

From what I understand, the Borel construction takes a $G$-space $X$ and produces a topological space $X\times_{G}\mathbf{E}G$―the homotopy quotient $X/\!\!/G$ of $X$ by $G$ in the $\infty$-category ...
Emily's user avatar
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4 votes
0 answers
126 views

Pushforward of invariant measures (equivariant Moser theorem)

There is a well-known theorem that between any two absolutely continuous Borel probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$ there is an increasing triangular transformation $T : \mathbb{R}^n ...
ivan's user avatar
  • 41
5 votes
0 answers
121 views

Which operations commute with fractional translation?

Let $\mathbf{v}$ be a real signal (i.e., an infinitely long vector). A translation operation $T_{s}\mathbf{v}$ with integer $s$ is trivially defined by $\forall i,s:\left(T_{s}\mathbf{v}\right)_{i}=v_{...
Daniel Soudry's user avatar
2 votes
1 answer
154 views

Approximation of $C^1$-smooth equivariant maps by infinitely smooth ones

Let $M,N$ be smooth closed manifolds acted by a finite group $G$. Let $f\colon M\to N$ be a $C^1$-smooth $G$-equivariant map. Is it true that for any $\varepsilon>0$ there exists a $C^\infty$-...
asv's user avatar
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4 votes
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184 views

Are there mathematical/physical applications of these Weyl equivariant maps?

Let $G$ be a compact Lie group and $T$ a choice of maximal torus. Denote the corresponding Lie algebras by $\mathfrak{g}$ and $\mathfrak{t}$. Elements of $\mathfrak{t} \otimes \mathbb{R}^3$ are called ...
Malkoun's user avatar
  • 5,031
7 votes
0 answers
239 views

Ample divisors on $T$-varieties

Question: how does one use a torus action to help decide whether a divisor or line bundle is ample? In more detail: I have a (normal, but usually rather singular) $T$-variety $X$, where $T$ is an ...
Geordie Williamson's user avatar
3 votes
0 answers
67 views

Equivariant smooth approximation

Suppose we have a compact manifold $M$ with the action of a compact group $G$. Consider the space of $C^l$ $G$-equivariant diffeomorphisms $\text{Diff}_G^{l}(M)$ with the $C^l$ topology and the space ...
cr1t1cal's user avatar
  • 755
6 votes
1 answer
364 views

What is equivariant chains on a representation sphere?

For a finite group $G$ and a finite-dimensional real representation $V$ of $G$, denote by $S^V$ the one-point compactification of $V$, with basepoint at infinity. What is the reduced chain complex $...
John Pardon's user avatar
  • 18.4k
4 votes
1 answer
164 views

Explicit example of an equivariant embedding of $U(n)/( U(k) \times U(n-k))$ into a finite dimensional $U(n)$-representation

We know that if $H$ is a closed subgroup of a compact Lie group $G$ one can find a finite dimensional $G$-representation $V$ and an element $v_0 \in V$ such that $\textrm{Stab}(v_0)= H$. This gives a $...
MLV's user avatar
  • 73
6 votes
0 answers
170 views

Equivariant Morse theory for non-compact Lie groups

Let $G$ be a Lie group acting properly on a smooth manifold $M$. The (non-equivariant) definition of a Morse function does not carry over to equivariant functions $M \rightarrow \mathbb{R}$ (where $\...
Lukas's user avatar
  • 198
5 votes
0 answers
109 views

Is there a smooth $W_{G_2}$-equivariant map from the flag manifold of $U(4)$ to that of $G_2$?

The Weyl group $W$ of $G_2$, is a group of order $12$ which is generated by the subgroup of permutations of $e_1$, $e_2$ and $e_3$ and by by the element $\tau$ which maps $(e_1,e_2,e_3)$ to $(-e_1,-...
Malkoun's user avatar
  • 5,031
6 votes
0 answers
225 views

A group action on another group action quotient: how to best describe the resulting structure and does it have a name?

Suppose I have an action $\alpha:G\times X\to X$ of a group $G$ on a set $X$ and, on top of that, an action $\beta:H\times(X/G)\to(X/G)$ of another group $H$ on the set of $G$-orbits. Is there a nice ...
მამუკა ჯიბლაძე's user avatar
23 votes
1 answer
809 views

Vector bundles on $\mathbb{A}^n / G$

Let $G$ be a finite group acting linearly on $\mathbb{A}^n$. Do we expect algebraic vector bundles on $X := \mathbb{A}^n/G$ to be trivial? Here by the quotient I mean the singular scheme, not the ...
Evgeny Shinder's user avatar
1 vote
0 answers
76 views

Singular chain complex of balanced products

Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring) $$f:C_*(V) \...
FKranhold's user avatar
  • 1,623
1 vote
0 answers
307 views

Definition of an equivariant connection and equivariant curvature

Can anyone give me a reference which precisely stated the definition of an equivariant connection and equivariant curvature? Precisely, If G be a compact lie group acting linearly on a smooth ...
Anantadulal paul's user avatar
8 votes
1 answer
568 views

Terminology about G- simplicial complexes

For a simplicial complex $X$ with an action of a discrete group $G$, we can impose the following condition, namely that if $g\in G$ stabilizes a given simplex $\sigma\subseteq X$, then $g:\sigma\to\...
John Pardon's user avatar
  • 18.4k
5 votes
1 answer
261 views

Graded commutativity of the $n$th Browder bracket

Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...
FKranhold's user avatar
  • 1,623
1 vote
0 answers
223 views

Equivariant vector bundles whose quotient map preserves the stabilizer

Let $G$ be a compact Lie group which act on a manifold $M$. We fix this action throughout our question. Assume that $E\to M$ is a vector bundle which has the potential of admitting ...
Ali Taghavi's user avatar
2 votes
1 answer
309 views

Resolution by locally free $G$-equivariant sheaves on varieties

I have been reading the section in the beginning of Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics by Bartocci et. al., and stumbled across the following sentence (page 26). ...
DKS's user avatar
  • 411
2 votes
0 answers
91 views

Does intermediate extension functor commutes with forgetful functor in equivariant derived category?

The forgetful functor from $D^b_G(X)$ to $D^b(X)$ carries $Perv_G(X)$ to $Perv(X)$ by definition $5.1$ in the book of Bernstein and Lunts. My question is do the following functors, intermediate ...
userabc's user avatar
  • 667
3 votes
0 answers
83 views

Could we have the simplicial definition of equivariant derived category of sheaves with arrow direction inversed?

Let $X$ be a topological space and $G$ be a topological group acting on $X$ from the left. We consider the simplicial space $[G\backslash X]_{\cdot}$ where $$ [G\backslash X]_n=\underbrace{G\times \...
Zhaoting Wei's user avatar
  • 8,727
9 votes
0 answers
196 views

Does the category of $G$-equivariant sheaves have enough injectives?

The question is related to this one. Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$. Let $G$ be a topological group which ...
Zhaoting Wei's user avatar
  • 8,727
1 vote
0 answers
125 views

The space of Riemannian structures as an orbifold.

Consider a smooth closed manifold $M$. The space of Riemannian metrics is an open cone in the space of sections of some vector bundle. On this space the group of diffeomorphisms of $M$ acts by ...
Thomas Rot's user avatar
  • 7,383
6 votes
0 answers
220 views

Fundamental class in equivariant K-theory

I'm looking for an accessible reference for the definition of the fundamental class in equivariant K-theory. The setup I'm interested in is the following: suppose $V$ is a vector space equipped with ...
clementine's user avatar
4 votes
0 answers
133 views

Spin equivariance of the Dirac operator-flat case

This question was posed on Math.SE but no one has answered it; it may be suitable for MathOverflow. Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial ...
truebaran's user avatar
  • 9,180
10 votes
0 answers
173 views

Baum Connes conjecture and abstract isomorphism

Baum-Connes conjecture states that for a locally compact group $G$ the so called assebly map $\mu$ between $G$-equivariant K-homology of the universal example for proper actions of $G$ and K-theory of ...
truebaran's user avatar
  • 9,180
8 votes
1 answer
662 views

Interactions (functors) between equivariant sheaves for different groups?

Let $G$ be a finite group and $k$ a field (alg. closed char 0 for simplicity). To every $G$ set $X$ we can assign the category of $G$-equivariant sheaves of $k$-vector spaces $Sh_G(X)$. It is ...
Saal Hardali's user avatar
  • 7,619
5 votes
2 answers
1k views

Classification of (complex algebraic) vector bundles on punctured affine space

The Quillen-Suslin theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$. Let's work over the complex numbers. What can be said about vector ...
Qfwfq's user avatar
  • 22.8k
2 votes
0 answers
84 views

Equivariant Formula for High Dimensional Isolated set

The Atiyah-Bott-Berline-Vergne-Witten localization formula says $S^1$ acting on compact manifold $M$ isolated fixing points. And for a closed equivariant form $\omega$, then $$(2\pi)^{-\frac{\dim(M)}...
DLIN's user avatar
  • 1,915
5 votes
0 answers
74 views

Equivariant corner straightening

Equivariant corner straightening is usually mentioned in the literature without further explanation. What would be a reference where this is done (more or less) carefully for compact Lie group actions ...
Igor Belegradek's user avatar
1 vote
0 answers
168 views

Generalizing approximate $\mathbb{Z}$-equivariance of a simple function

Let $f(x) := x^2 + (1-x^2)x$ and $F(x) := \log \frac{x}{1-x}-\frac{1}{x}$. It can be shown (cf. https://math.stackexchange.com/questions/1865370/) that $F$ is approximately equivariant w/r/t the $\...
Steve Huntsman's user avatar
3 votes
0 answers
124 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^...
QSR's user avatar
  • 2,213
2 votes
0 answers
231 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 ...
Meer Ashwinkumar's user avatar
5 votes
1 answer
526 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?
user89937's user avatar
3 votes
0 answers
634 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 ...
Qfwfq's user avatar
  • 22.8k
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 ...
Matthias Wendt's user avatar
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 ...
user56980's user avatar
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