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

Abstract

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. Then an equivariant Dirac type operator on M under a suitable boundary condition has an equivariant index in the Ktheory of the reduced group C * -algebra C * r G of G, which is a common generalisation of the Baum-Connes analytic assembly map and the (equivariant) Atiyah-Patodi-Singer index. Using a trace on a suitable subalgebra of C * r G, defined by the orbital integral over the conjugacy class of an element g ∈ G, we prove an equivariant Atiyah-Patodi-Singer index theorem in this setting. We first state general analytic conditions under which 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 a connected, linear, real semisimple Lie group, and g is a semisimple element. In the classical case, where M is compact and G is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah-Patodi-Singer index theorem. Contents

Higher rho invariant and delocalized eta invariant at infinity

“…Let F be a fundamental domain of X with respect to the G-action. Compare the following definition to Definition 2.5 of [13].…”

“…The advantage of the Engel-Samukarş smooth subalgebra is that it allows us to do analysis in a noncocompact setting and to apply the method of Hochs, Wang and Wang (cf. [13]) to develop the index formula for the Dirac operator on noncocompact complete manifold with uniform positive scalar curvature metric at infinity, while the Connes-Moscovici smooth subalgebra is necessary for us to apply the theory of Xie and Yu on delocalized trace (cf. [29]) to study the higher rho invariant at infinity and define the delocalized eta invariant at infinity, where a cocompact setting is sufficient.…”

“…Here tr g is the following trace map tr g (A) = h∈ g F A(x, hx)dx, on G-equivariant Schwartz kernels A ∈ C ∞ ( M × M ), where F is a fundamental domain of M under the G-action. One can also define tr g for continuous group, see [13] for example. There is also a higher generalization of the pairing between tr g and K-theory of geometric C * -algebras, introduced by [17], [7], [19], and [23], which is to consider cyclic cocycles.…”

“…In the meanwhile, Equation (1.2) in Theorem 1.1 is established by applying the method in [13], which is invented when Hochs, Wang and Wang was to develop a new approach to obtain a refinement of the Atiyah-Patodi-Singer index theorem. See Section 3.6 for details.…”

A general Simonenko local principle and Fredholm condition for isotypical components

“…Nevertheless, the, now well known, Atiyah-Singer's result states that if P is G-transversally elliptic then π α (P ) is Fredholm for any α ∈ Ĝ, [5,54]. This allows directly to define an index for G-transversally elliptic operators as an element of the K-homology of C * G, the group C * -algebra of G. Furthermore, with little more work, Atiyah and Singer showed that this index is, in fact, a Ad-invariant distribution on G. See also [7,9,34,36,37] for related results and [8,13,45] for index theorems on G-transversally elliptic operators using equivariant cohomology. The Fredholm property of this restrictions to isotypical component was the starting point for the study carried out in [12].…”