English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Consider a proper, isometric action by a unimodular locally compact group $G$on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For anequivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we definea numerical index $\operatorname{index}_g(D)$, in terms of a parametrix for $D$and a trace associated to $g$. We prove an equivariant Atiyah-Patodi-Singerindex theorem for this index. We first state general analytic conditions underwhich this theorem holds, and then show that these conditions are satisfied if$g=e$ is the identity element; if $G$ is a finitely generated, discrete group,and the conjugacy class of $g$ has polynomial growth; and if $G$ is aconnected, linear, real semisimple Lie group, and $g$ is a semisimple element.In the classical case, where $M$ is compact and $G$ is trivial, our argumentsreduce to a relatively short and simple proof of the originalAtiyah-Patodi-Singer index theorem. In part II of this series, we prove that,under certain conditions, $\operatorname{index}_g(D)$ can be recovered from a$K$-theoretic index of $D$ via a trace defined by the orbital integral over theconjugacy class of $g$.

An equivariant Atiyah-Patodi-Singer index theorem for proper actions I: the index formula