arXiv subjects

Peter Hochs

Publications and source records attributed to Peter Hochs.

At least 37 records · Page 2Linked to original sources

arXiv2020

Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.

math.DG↗

arXiv2019

Normal forms and invariant manifolds for nonlinear, non-autonomous PDEs, viewed as ODEs in infinite dimensions

We prove that a general class of nonlinear, non-autonomous ODEs in Fréchet spaces are close to ODEs in a specific normal form, where closeness means that solutions of the normal form ODE satisfy the original ODE up to a residual that vanishes up to any desired order. In this normal form, the centre, stable and unstable coordinates of the ODE are clearly separated, which allows us to define invariant manifolds of such equations in a robust way. In particular, our method empowers us to study approximate centre manifolds, given by solutions of ODEs that are central up to a desired, possibly nonzero precision. The main motivation is the case where the Fréchet space in question is a suitable function space, and the maps involved in an ODE in this space are defined in terms of derivatives of the functions, so that the infinite-dimensional ODE is a finite-dimensional PDE. We show that our methods apply to a relevant class of nonlinear, non-autonomous PDEs in this way.

math.AP↗

arXiv2019

Equivariant Callias index theory via coarse geometry

The equivariant coarse index is well-understood and widely used for actions by discrete groups. We extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over $C^*$-algebras of continuous functions to obtain a meaningful index. Inspired by work by Roe, we then develop a localised variant, with values in the $K$-theory of a group $C^*$-algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the $K$-theory of a group $C^*$-algebra.

math.KT↗

arXiv2018

A geometric formula for multiplicities of $K$-types of tempered representations

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be compact. Under a condition on $K$, which holds in particular if $K$ is maximal compact, we give a geometric expression for the multiplicities of the $K$-types of any tempered representation (in fact, any standard representation) $π$ of $G$. This expression is in the spirit of Kirillov's orbit method and the quantisation commutes with reduction principle. It is based on the geometric realisation of $π|_K$ obtained in an earlier paper. This expression was obtained for the discrete series by Paradan, and for tempered representations with regular parameters by Duflo and Vergne. We obtain consequences for the support of the multiplicity function, and a criterion for multiplicity-free restrictions that applies to general admissible representations. As examples, we show that admissible representations of $\mathrm{SU}(p,1)$, $\mathrm{SO}_0(p,1)$ and $\mathrm{SO}_0(2,2)$ restrict multiplicity-freely to maximal compact subgroups.

math.DG↗

arXiv2018

A geometric realisation of tempered representations restricted to maximal compact subgroups

Let $G$ be a connected, linear, real reductive Lie group with compact centre. Let $K<G$ be maximal compact. For a tempered representation $π$ of $G$, we realise the restriction $π|_K$ as the $K$-equivariant index of a Dirac operator on a homogeneous space of the form $G/H$, for a Cartan subgroup $H<G$. (The result in fact applies to every standard representation.) Such a space can be identified with a coadjoint orbit of $G$, so that we obtain an explicit version of Kirillov's orbit method for $π|_K$. In a companion paper, we use this realisation of $π|_K$ to give a geometric expression for the multiplicities of the $K$-types of $π$, in the spirit of the quantisation commutes with reduction principle. This generalises work by Paradan for the discrete series to arbitrary tempered representations.

math.RT↗

arXiv2018

Orbital integrals and $K$-theory classes

Let $G$ be a semisimple Lie group with discrete series. We use maps $K_0(C^*_rG)\to \mathbb{C}$ defined by orbital integrals to recover group theoretic information about $G$, including information contained in $K$-theory classes not associated to the discrete series. An important tool is a fixed point formula for equivariant indices obtained by the authors in an earlier paper. Applications include a tool to distinguish classes in $K_0(C^*_rG)$, the (known) injectivity of Dirac induction, versions of Selberg's principle in $K$-theory and for matrix coefficients of the discrete series, a Tannaka-type duality, and a way to extract characters of representations from $K$-theory. Finally, we obtain a continuity property near the identity element of $G$ of families of maps $K_0(C^*_rG)\to \mathbb{C}$, parametrised by semisimple elements of $G$, defined by stable orbital integrals. This implies a continuity property for $L$-packets of discrete series characters, which in turn can be used to deduce a (well-known) expression for formal degrees of discrete series representations from Harish-Chandra's character formula.

math.KT↗

arXiv2017

Shelstad's character identity from the point of view of index theory

Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding $L$-packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of $K$-theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program.

math.RT↗

arXiv2017

A fixed point formula and Harish-Chandra's character formula

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations.

math.DG↗

arXiv2016

An equivariant index for proper actions I

Equivariant indices have previously been defined in cases where either the group or the orbit space in question is compact. In this paper, we develop an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In parts II and III of this series, we will explore some properties and applications of this index.

math.KT↗

arXiv2016

Algebraic $K$-theory and a semi-finite Fuglede-Kadison determinant

In this paper we apply algebraic $K$-theory techniques to construct a Fuglede-Kadison type determinant for a semi-finite von Neumann algebra equipped with a fixed trace. Our construction is based on the approach to determinants for Banach algebras developed by Skandalis and de la Harpe. This approach can be extended to the semi-finite case since the first topological $K$-group of the trace ideal in a semi-finite von Neumann algebra is trivial. On our way we also improve the methods of Skandalis and de la Harpe by considering relative $K$-groups with respect to an ideal instead of the usual absolute $K$-groups. Our construction recovers the determinant homomorphism introduced by Brown, but all the relevant algebraic properties are automatic due to the algebraic $K$-theory framework.

math.OA↗

arXiv2016

Equivariant indices of Spin$^c$-Dirac operators for proper moment maps

We define an equivariant index of Spin$^c$-Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.

math.DG↗

arXiv2016

A fixed point theorem on noncompact manifolds

We generalise the Atiyah-Segal-Singer fixed point theorem to noncompact manifolds. Using $KK$-theory, we extend the equivariant index to the noncompact setting, and obtain a fixed point formula for it. The fixed point formula is the explicit cohomological expression from Atiyah-Segal-Singer's result. In the noncompact case, however, we show in examples that this expression yields characters of infinite-dimensional representations. In one example, we realise characters of discrete series representations on the regular elements of a maximal torus, in terms of the index we define. Further results are a fixed point formula for the index pairing between equivariant $K$-theory and $K$-homology, and a non-localised expression for the index we use, in terms of deformations of principal symbols. The latter result is one of several links we find to indices of deformed symbols and operators studied by various authors.

math.KT↗

arXiv2016

An equivariant index for proper actions III: the invariant and discrete series indices

We study two special cases of the equivariant index defined in part I of this series. We apply this index to deformations of Spin$^c$-Dirac operators, invariant under actions by possibly noncompact groups, with possibly noncompact orbit spaces. One special case is an index defined in terms of multiplicities of discrete series representations of semisimple groups, where we assume the Riemannian metric to have a certain product form. The other is an index defined in terms of sections invariant under a group action. We obtain a relation with the analytic assembly map, quantisation commutes with reduction results, and Atiyah-Hirzebruch type vanishing theorems. The arguments are based on an explicit decomposition of Spin$^c$-Dirac operators with respect to a global slice for the action.

math.DG↗

arXiv2016

An equivariant index for proper actions II: properties and applications

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 groups by Braverman. In this paper, we investigate properties and applications of this index. We prove that it has an induction property that can be used to deduce various other properties of the index. In the case of compact orbit spaces, we show how it is related to the analytic assembly map from the Baum-Connes conjecture, and an index used by Mathai and Zhang. We use the index to define a notion of K-homological Dirac induction, and show that, under conditions, it satisfies the quantisation commutes with reduction principle.

math.KT↗

arXiv2016

Spin-structures and proper group actions

We generalise Atiyah and Hirzebruch's vanishing theorem for actions by compact groups on compact Spin-manifolds to possibly noncompact groups acting properly and cocompactly on possibly noncompact Spin-manifolds. As corollaries, we obtain some vanishing results for $\hat A$-type genera.

math.DG↗

arXiv2015

On the Vergne conjecture

Consider a Hamiltonian action by a compact Lie group on a possibly noncompact symplectic manifold. We give a short proof of a geometric formula for decomposition into irreducible representations of the equivariant index of a Spin$^c$-Dirac operator in this context. This formula was conjectured by Michèle Vergne in 2006 and proved by Ma and Zhang in 2014.

math.SG↗

arXiv2015

Quantising proper actions on Spin$^c$-manifolds

Paradan and Vergne generalised the quantisation commutes with reduction principle of Guillemin and Sternberg from symplectic to Spin$^c$-manifolds. We extend their result to noncompact groups and manifolds. This leads to a result for cocompact actions, and a result for non-cocompact actions for reduction at zero. The result for cocompact actions is stated in terms of $K$-theory of group $C^*$-algebras, and the result for non-cocompact actions is an equality of numerical indices. In the non-cocompact case, the result generalises to Spin$^c$-Dirac operators twisted by vector bundles. This yields an index formula for Braverman's analytic index of such operators, in terms of characteristic classes on reduced spaces.

math.DG↗

arXiv2015

Geometric quantization and families of inner products

We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra $\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on $\mathfrak{g}$ instead of a single one makes it possible to handle both noncompact groups and manifolds, by extending Tian and Zhang's Witten deformation approach to the noncompact case.

math.DG↗