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Abstract
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Let
be a Lie group with finitely many connected components and let
be a maximal compact subgroup. We assume that
satisfies the rapid decay
(RD) property and that
has a nonpositive sectional curvature. As an example, we can take
to be a connected
semisimple Lie group. Let
be a
-proper manifold
with compact quotient
.
Building on work by Connes and Moscovici (1990) and
Pflaum et al. (2015), we establish index formulae for the
-higher indices of a
-equivariant Dirac-type
operator on
.
We use these formulae to investigate geometric properties of suitably defined higher genera on
. In particular, we
establish the
-homotopy
invariance of the higher signatures of a
-proper manifold and the
vanishing of the
-genera
of a
-spin
-proper manifold
admitting a
-invariant
metric of positive scalar curvature.
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Keywords
Lie groups, proper actions, group cocycles, van Est
isomorphism, cyclic cohomology, $K\mkern-2mu$-theory, index
classes, higher indices, higher index formulae, higher
signatures, $G$-homotopy invariance, higher genera,
positive scalar curvature
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Mathematical Subject Classification 2010
Primary: 58J20
Secondary: 19K56, 58J42
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Milestones
Received: 19 June 2018
Revised: 21 February 2019
Accepted: 12 March 2019
Published: 17 December 2019
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