Related papers: Fixed point formula and loop group actions

We use the formalism of traces in higher categories to prove a common generalization of the holomorphic Atiyah-Bott fixed point formula and the Grothendieck-Riemann-Roch theorem. The proof is quite different from the original one proposed…

We give a systematic treatment of the stability theory for action of a real reductive Lie group G on a topological space. More precisely, we introduce an abstract setting for actions of non-compact real reductive Lie groups on topological…

The involution fixity ${\rm ifix}(G)$ of a permutation group $G$ of degree $n$ is the maximum number of fixed points of an involution. In this paper we study the involution fixity of primitive almost simple exceptional groups of Lie type.…

We prove a multiplicity formula for Riemann-Roch numbers of reductions of Hamiltonian actions of loop groups. This includes as a special case the factorization formula for the quantum dimension of the moduli space of flat connections over a…

An action of a complex reductive group $\mathrm G$ on a smooth projective variety $X$ is regular when all regular unipotent elements in $\mathrm G$ act with finitely many fixed points. Then the complex $\mathrm G$-equivariant cohomology…

Let $ \; G \; $ be a group acting on a compact Riemann surface $ \; {\mathcal X} \; $ and $ \; D \; $ be a $ \; G$-invariant divisor on $\; {\mathcal X}. \; $ The action of $ \; G \; $ on $ \; {\mathcal X} \; $ induces a linear…

We give a general formula for the equivariant complex $K$-theory $K_G^*(V)$ of a finite dimensional real linear space $V$ equipped with a linear action of a compact group $G$ in terms of the representation theory of a certain double cover…

We introduce a practical construction of group-equivariant and permutation-invariant functions of $N$ variables given a finite-dimensional space stable with respect to the group action. The construction applies to any connected linear Lie…

Let $S$ be an oriented surface of finite type, $\mathcal{MCG}(S)$ its mapping class group, and $\mathcal{T}(S)$ its Teichm\"uller space with the Teichm\"uller metric. Let $H \leq \mathcal{MCG}(S)$ be a finite subgroup and consider the…

Let $G$ be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to 1, e.g. $G= \SL_2 (\C) \times \SL_2 (\C)$ or $\SL_3 (\C)$. Then the fundamental rank of $G$ is $2,$ and according to the conjecture…

When a torus acts on a compact oriented manifold with isolated fixed points, the equivariant localization formula of Atiyah--Bott and Berline--Vergne converts the integral of an equivariantly closed form into a finite sum over the fixed…

We show that an effective action of the one-dimensional torus $\mathbb{G}_m$ on a normal affine algebraic variety $X$ can be extended to an effective action of a semi-direct product $\mathbb{G}_m\rightthreetimes\mathbb{G}_a$ with the same…

On a smooth closed oriented $4$-manifold $M$ with a smooth action by a compact Lie group $G$, we define a $G$-monopole class as an element of $H^2(M;\Bbb Z)$ which is the first Chern class of a $G$-equivariant Spin$^c$ structure which has a…

In this second part we prove that, if $G$ is one of the groups $\mathrm{PSL}_2(q)$ with $q>5$ and $q\equiv 5\pmod {24}$ or $q\equiv 13 \pmod{24}$, then the fundamental group of every acyclic $2$-dimensional, fixed point free and finite…

We study isometric actions of Steinberg groups on Hadamard manifolds. We prove some rigidity properties related to these actions. In Particular we show that every isometric action of $St_n(F_p\langle t_1,\ldots ,t_k \rangle)$ on Hadamard…

Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact Lie group…

The goal of this paper is to prove a version of the non-abelian localization theorem for the rational equivariant K-theory of a smooth variety $X$ with the action of a linear algebraic group $G$. We then use this to prove a Riemann-Roch…

A well-known result of W. Ray asserts that if $C$ is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping $T$: $C\to C$ that has no fixed point. In this paper we establish some common fixed point properties…

We prove that for each integer k of at least 2, there is an open neigborhood \nu_k of the identity map of the 2-sphere S^2, in C^1-topology such that: if G is a nilpotent subgroup of Diff^1(S^2) with length k of nilpotency, generated by…

We consider compact homogeneous spaces G/H, where G is a compact connected Lie group and H is its closed connected subgroup of maximal rank. The aim of this paper is to provide an effective computation of the universal toric genus for the…