Related papers: The equivariant analytic index for proper groupoid…

We introduce $C^*$-algebras associated with directed graphs, along with two generalizations of this concept, namely Exel-Pardo $C^*$-algebras associated with a self-similar action of a group on a directed graph, and the $C^*$-algebras…

In this paper we define twisted equivariant K-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid, this defines a periodic cohomology theory on the category of finite CW-complexes with equivariant stable projective…

We construct an equivariant version of Ray-Singer analytic torsion for proper, isometric actions by locally compact groups on Riemannian manifolds, with compact quotients. We obtain results on convergence, metric independence, vanishing for…

Let $\GR \to B$ be a bundle of compact Lie groups acting on a fiber bundle $Y \to B$. In this paper we introduce and study gauge-equivariant $K$-theory groups $K_\GR^i(Y)$. These groups satisfy the usual properties of the equivariant…

In this paper, we study invariants of linear differential operators with respect to algebraic Lie pseudogroups. Then we use these invariants and the principle of n-invariants to get normal forms (or models) of the differential operators and…

In this expository article, we consider first order elliptic differential operators acting on smooth vector bundles over compact manifolds, and certain invariants derived from the analysis of these operators, namely the eta invariant} and…

This paper presents algebraic methods for the study of polynomial relative invariants, when the group G formed by the symmetries and relative symmetries is a compact Lie group. We deal with the case when the subgroup H of symmetries is…

The paper is devoted to the index theory of orbital and transverse elliptic operators on manifolds with a proper Lie group action. It corrects errors of my previous paper (published in JNCG in 2016) on transverse operators and contains new…

In the first part of this series, we defined an equivariant index without assuming the group acting or the orbit space of the action to be compact. This allowed us to generalise an index of deformed Dirac operators, defined for compact…

We construct a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group. We show that the index is preserved by a certain class of non-compact cobordisms and…

Let $G$ be a compact Lie group acting smoothly on a smooth, compact manifold $M$, let $P \in \psi^m(M; E_0, E_1)$ be a $G$--invariant, classical pseudodifferential operator acting between sections of two vector bundles $E_i \to M$, $i =…

We prove a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of…

In this article we review the question of constructing geometric quotients of actions of linear algebraic groups on irreducible varieties over algebraically closed fields of characteristic zero, in the spirit of Mumford's geometric…

Boundary groupoids were introduced by the second author, which can be used to model many analysis problems on singular spaces. In order to investigate index theory on boundary groupoids, we introduce the notion of {\em a deformation from…

If $\mathfrak{g} \subseteq \mathfrak{h}$ is an extension of Lie algebras over a field $k$ such that ${\rm dim}_k (\mathfrak{g}) = n$ and ${\rm dim}_k (\mathfrak{h}) = n + m$, then the Galois group ${\rm Gal} \, (\mathfrak{h}/\mathfrak{g})$…

Let $X$ be a smooth complex projective variety equipped with an action of a linear algebraic group $G$ over $\mathbb{C}$. Let $D$ be a reduced effective divisor on $X$ that is invariant under the $G$--action on $X$. Let $s_D$ be the…

Let X be an algebraic variety with an action of an algebraic group G. Suppose X has a full exceptional collection of sheaves, and these sheaves are invariant under the action of the group. We construct a semiorthogonal decomposition of…

Action Lie groupoids are used to model spaces of orbits of actions of Lie groups on manifolds. For each such action groupoid $M\rtimes H$ we construct a locally convex bialgebroid $\mathord{\mathrm{Dirac}}(M\rtimes H)$ with an antipode over…

We construct of a family of fundamental solutions for elliptic partial differential operators with real constant coefficients. The elements of such a family are expressed by means of jointly real analytic functions of the coefficients of…

In this paper we study a natural decomposition of $G$-equivariant $K$-theory of a proper $G$-space, when $G$ is a Lie group with a compact normal subgroup $A$ acting trivially. Our decomposition could be understood as a generalization of…