# Higher orbital integrals, rho numbers and index theory

Paolo Piazza  Dipartimento di Matematica, Sapienza Università di Roma, I-00185
Roma, Italy<EMAIL_ADDRESS>,  Hessel Posthuma  Korteweg-de Vries
Institute for Mathematics, University of Amsterdam, 1098 XG Amsterdam, the
Netherlands<EMAIL_ADDRESS>,  Yanli Song  Department of Mathematics and
Statistics, Washington University, St. Louis, MO, 63130, U.S.A.
<EMAIL_ADDRESS>and  Xiang Tang  Department of Mathematics and Statistics,
Washington University, St. Louis, MO, 63130, U.S.A<EMAIL_ADDRESS>
###### Abstract.

Let $G$ be a connected, linear real reductive group. We give sufficient
conditions ensuring the well-definedness of the delocalized eta invariant
$\eta_{g}(D_{X})$ associated to a Dirac operator $D_{X}$ on a cocompact
$G$-proper manifold $X$ and to the orbital integral $\tau_{g}$ defined by a
semisimple element $g\in G$. Along the way, we give a detailed account of the
large time behaviour of the heat kernel and of its short time bahaviour near
the fixed point set of $g$. We prove that such a delocalized eta invariant
enters as the boundary correction term in an index theorem computing the
pairing between the index class and the 0-degree cyclic cocycle defined by
$\tau_{g}$ on a $G$-proper manifold with boundary. More importantly, we also
prove a higher version of such a theorem, for the pairing of the index class
and the higher cyclic cocycles defined by the higher orbital integral
$\Phi^{P}_{g}$ associated to a cuspidal parabolic subgroup $P<G$ with
Langlands decomposition $P=MAN$ and a semisimple element $g\in M$. We employ
these results in order to define (higher) rho numbers associated to
$G$-invariant positive scalar curvature metrics.

###### Key words and phrases:

Lie groups, proper actions, higher orbital integrals, delocalized cyclic
cocycles, index classes, relative pairing, excision, Atiyah-Patodi-Singer
higher index theory, delocalized eta invariants, higher delocalized eta
invariants.

###### 2010 Mathematics Subject Classification:

Primary: 58J20. Secondary: 58B34, 58J22, 58J42, 19K56.

###### Contents

  1. 1 Introduction
  2. 2 Notations
  3. 3 Geometric preliminaries.
    1. 3.1 Slice compatible metric on $Y_{0}$
  4. 4 Delocalized traces and the APS index formula
    1. 4.1 Orbital integrals and associated cyclic $0$-cocycles in the closed case
    2. 4.2 $0$-degree (relative) cyclic cocycles associated to orbital integrals
    3. 4.3 The $0$-degree delocalized APS index theorem on $G$-proper manifolds
  5. 5 Delocalized eta invariants for $G$-proper manifolds
    1. 5.1 Large time behaviour on manifolds without boundary
    2. 5.2 Large time behaviour on manifolds with boundary
    3. 5.3 Small time behavior
    4. 5.4 Short time limit and the Atiyah-Segal integrand
  6. 6 Higher delocalized cyclic cocycles
  7. 7 Toward a general higher APS index formula
  8. 8 Reduction
    1. 8.1 First reduction for manifolds without boundary
    2. 8.2 First reduction for manifold with boundary
    3. 8.3 Second reduction
  9. 9 An index theorem for higher orbital integral through reduction
  10. 10 Numeric rho invariants on $G$-proper manifolds
    1. 10.1 Rho numbers associated to delocalized 0-cocycles
    2. 10.2 Rho numbers associated to higher delocalized cocycles
    3. 10.3 Bordism properties
  11. A Dirac operator on $G$

##  1\. Introduction

This article is a contribution to higher index theory on $G$-proper manifolds,
with $G$ a connected, linear real reductive group. Before considering the case
of $G$-proper manifolds with $G$ a Lie group, it is worth spending a few words
on the pivotal example of Galois $\Gamma$-coverings. Let $\Gamma$ be a
finitely generated discrete group and let $\Gamma\to X\to X/\Gamma$ be a
Galois covering; let $D$ be a $\Gamma$-equivariant Dirac-type operator on a
spinc manifold $X$, acting on the sections of a $\Gamma$-equivariant vector
bundle $E$. We initially assume that $X/\Gamma$ is a smooth compact manifold
without boundary. We also assume that $\Gamma$ is Gromov hyperbolic or of
polynomial growth with respect to a word metric. The seminal work of Connes
and Moscovici [ConnesMoscovici] allows to define a pairing between
$HC^{*}(\mathbb{C}\Gamma,\langle e\rangle)$, the cyclic cohomology of
$\mathbb{C}\Gamma$ localized at the unit element, and the K-theory of the Roe
$C^{*}$-algebra $K_{*}(C^{*}(X,E)^{\Gamma})$. In particular, by applying this
pairing to the index class associated to $D$, $\operatorname{Ind}(D)\in
K_{*}(C^{*}(X,E)^{\Gamma})$, it is possible to define higher indices
associated to $D$, parametrized by the elements in
$HC^{*}(\mathbb{C}\Gamma,\langle e\rangle)$, a group which is in fact
isomorphic to $H^{*}(\Gamma,\mathbb{C})$. The higher index formula of Connes-
Moscovici provides a geometric formula for these higher indices, with very
interesting geometric applications to higher signatures and higher
$\widehat{A}$-genera. One can also pair the index class with
$HC^{*}(\mathbb{C}\Gamma,\langle x\rangle)$, the cyclic cohomology localized
at the conjugacy class $\langle x\rangle$, but this pairing turns out to be
identically zero.

Despite these interesting geometric applications the higher index invariants
we have just introduced will be inadequate whenever we have a geometric
situation in which the index class vanishes. This is the case, for example, if
$D$ is the spin Dirac operator associated to a metric of positive scalar
curvature. Thus, in geometric questions involving, for example, the moduli
space of metrics of positive scalar curvature $\mathcal{R}^{+}(M)/{\rm
Diffeo}(M)$, one is led to consider secondary invariants. One such invariant
is the delocalized eta invariant of Lott, $\eta_{\langle x\rangle}(D)$,
associated to an invertible Dirac operator $D$ and, initially, to a finite
conjugacy class $\langle x\rangle$ in $\Gamma$, see [lott-delocalized]. This
invariant was extended to conjugacy classes of polynomial growth in
[PiazzaSchick_PJM] and then, much more generally, to arbitrary conjugacy
classes of Gromov hyperbolic groups in the deep work of Puschnigg, see
[Puschnigg].

The geometric interest for such an invariant stems from its connection with
the Atiyah-Patodi-Singer index class on a Galois covering with boundary
$\Gamma\to Y\to Y/\Gamma$, once we assume the boundary operator associated to
$D$ to be invertible. Thanks to the higher Atiyah-Patodi-Singer index formula
of Leichtnam and Piazza and Wahl, see [LeichtnamPiazzaMemoires,
LeichtnamPiazzaBSMF, Wahl1], one can prove that the pairing of the Atiyah-
Patodi-Singer index class with the cyclic cocycle $[\tau_{\langle
x\rangle}]\in HC^{0}(\mathbb{C}\Gamma,\langle x\rangle)$,

(1.1) $\tau_{\langle
x\rangle}(\sum_{\gamma}\alpha_{\gamma}\gamma):=\sum_{\gamma\in\langle
x\rangle}\alpha_{\gamma}\gamma$

is well defined if $\Gamma$ is Gromov hyperbolic or of polynomial growth and
precisely equal to the delocalized eta invariant of Lott; in formulae

(1.2) $\langle\operatorname{Ind}(D),[\tau_{\langle
x\rangle}]\rangle=-\frac{1}{2}\eta_{\langle x\rangle}(D_{\partial})$

We call the formula appearing in (1.2) a 0-degree delocalized APS index
formula. Recent work of Chen-Wang-Xie-Yu [ChenWangXieYu], Sheagan [Sheagan]
and Piazza-Schick-Zenobi [PSZ] extend this pairing to all elements in
$HC^{*}(\mathbb{C}\Gamma,\langle x\rangle)$, $x$ any element in a Gromov
hyperbolic or polynomial growth group, obtaining correspondingly higher
delocalized Atiyah-Patodi-Singer index theorems for Gromov hyperbolic groups.
These express the pairing between $\tau\in HC^{*}(\mathbb{C}\Gamma,\langle
x\rangle)$ and the Atiyah-Patodi-Singer index class $\operatorname{Ind}(D)$ in
terms of certain higher rho numbers, secondary invariants of Dirac operators
that are particularly useful in studying, for example, metrics of positive
scalar curvature. See [PSZ] for these geometric applications. Notice that
these higher rho numbers appear here for the boundary operator $D_{\partial}$
on $\partial Y$ but they can be defined for any Galois covering $\Gamma\to
X\to X/\Gamma$, with $X$ without boundary.

We now turn to a $G$-proper manifold, initially without boundary, with compact
quotient and a $G$-equivariant Dirac operator $D$ acting on the sections of a
$G$-equivariant twisted spinor bundle $E$. Here $G$ is a unimodular Lie group.
Let $\mathcal{L}_{G}^{c}(X,E)$ be the algebra of $G$-equivariant smoothing
operators of $G$-compact support. There is a compactly supported index class
$\operatorname{Ind}_{c}(D)\in K_{*}(\mathcal{L}_{G}^{c}(X,E))$ and a
homomorphism $H^{*}_{{\rm diff}}(G)\to HC^{*}(\mathcal{L}_{G}^{c}(X,E))$, from
the differentiable cohomology of $G$, $H^{*}_{{\rm diff}}(G)$, to the cyclic
cohomology of $\mathcal{L}_{G}^{c}(X,E)$, $HC^{*}(\mathcal{L}_{G}^{c}(X,E))$.
We think to these cyclic cocycles in $HC^{*}(\mathcal{L}_{G}^{c}(X,E))$ as
localized at the identity element of $G$. The higher index theorem of Pflaum-
Posthuma-Tang [PPT] gives a formula for the pairing of
$\operatorname{Ind}_{c}(D)$ with the cyclic cocycle $\tau^{M}_{\varphi}$
associated to $[\varphi]\in H^{*}_{{\rm diff}}(G)$. Under additional
assumptions on the group $G$, satisfied for example by a connected, linear
real reductive group, this result was improved to a $C^{*}$-index theorem in
[PP1]. In the latter work a key role is played by a suitable dense
holomorphically closed subalgebra $\mathcal{L}_{G,s}^{\infty}(X,E)$
(Definition 4.3) of the Roe algebra $C^{*}(X,E)^{G}$ and by a smooth index
class $\operatorname{Ind}_{\infty}(D)\in\mathcal{L}_{G,c}^{\infty}(X,E)$.
Applications were given once again to higher signatures and higher
$\widehat{A}$-genera. This index theorem was extended to $G$-proper manifolds
with boundary in the recent paper [PP2]; this is a higher APS index theorem on
$G$-proper manifolds for cyclic cocyles localized at the identity element.

Let us now go back to a $G$-proper manifold without boundary $X$. In contrast
with the free case, in the proper case we also have delocalized index theorems
associated to delocalized cyclic cocycles. Let $g$ be a semisimple element and
set $Z:=Z_{G}(g)$. First of all we have the orbital integral
$\tau_{g}:C^{\infty}_{c}(G)\to\mathbb{C}$ associated to such a $g$:

(1.3) $\tau_{g}(f):=\int_{G/Z}f(xgx^{-1})d(xZ)\,.$

Assume $G$ is a connected, linear real reductive group. Then $\tau_{g}$
extends to the Lafforgue Schwartz algebra $\mathcal{L}_{s}(G)$ (Definition
4.3), a dense holomorphically closed subalgebra of $C^{*}_{r}G$, where it
defines a $0$-degree cyclic cocycle $[\tau_{g}]\in
HC^{0}(\mathcal{L}_{s}(G))$. The orbital integral (1.3) also defines a
0-degree cyclic cocycle $\tau^{X}_{g}$ on $\mathcal{L}^{c}_{G}(X,E)$:

(1.4) $\tau_{g}^{X}(T_{\kappa}):=\int_{G/Z}\int_{X}c(hgh^{-1}x){\rm
tr}(hgh^{-1}\kappa(hg^{-1}h^{-1}x,x))dx\,d(hZ),$

where $\kappa$ denotes the kernel of an element
$T_{\kappa}\in\mathcal{L}^{c}_{G}(X,E)$ and $c$ is a cut-off function for the
action of $G$ on $X$ and ${\rm tr}$ denotes the vector-bundle fiberwise trace.

The Lafforgue algebra $\mathcal{L}_{s}(G)$ defines an algebra of smoothing
operators $\mathcal{L}^{\infty}_{G,s}(X,E)$ which is a dense and
holomorphically closed subalgebra of the Roe algebra $C^{*}(X,E)^{G}$;
moreover $\tau_{g}^{X}$ extends from $\mathcal{L}^{c}_{G}(X,E)$ to
$\mathcal{L}^{\infty}_{G,s}(X,E)$ and defines on this algebra a 0-degree
cyclic cocycle $[\tau_{g}^{X}]\in HC^{0}(\mathcal{L}^{\infty}_{G,s}(X,E))$. We
prove the following index formula for any semi-simple element $g$,

(1.5)
$\langle\operatorname{Ind}_{\infty}(D),\tau^{X}_{g}\rangle=\int_{X^{g}}c^{g}{\rm
AS}_{g}(D).$

In [Hochs-Wang-HC], Hochs and Wang partially developed the above formula by
considering a splitting Dirac operator, c.f. (3.6). We give a more detailed
approach to (1.5) in this paper. For more on the subtleties involved in the
proof of this index formula we refer the reader to Section 5.

This formula establishes a 0-degree delocalized index theorem. In Equation
(1.5), $X^{g}$ is the fixed point submanifold associated to the $g$-action on
$X$; $c^{g}$ is a compactly supported cutoff function on $X^{g}$ for the
action of $Z$ on $X^{g}$. Below, we recall the explicit differential form
expression for $\operatorname{AS}_{g}(D)$. Consider the following curvature
forms.

  * •

Since we are assuming that $E$ is a $G$-equivariant twisted spinor bundle on
$X$, we can write

$E=\mathcal{E}\otimes W,$

where $\mathcal{E}$ is the spinor bundle associated to the Spinc-structure on
$X$ and $W$ is an auxiliary $G$-equivariant vector bundle on $X$.

We define $R^{W}$ to be the curvature form of the Hermitian connection on $W$;

  * •

$R^{\mathcal{N}}$, the curvature form associated to the Hermitian connection
on $\mathcal{N}_{X^{g}}\otimes\mathbb{C}$, where $\mathcal{N}_{X^{g}}$ is the
normal bundle of the $g$-fixed point submanifold $X^{g}$ in $X$;

  * •

$R^{L}$, the curvature form associated to the Hermitian connection on
$L_{\operatorname{det}}|_{X^{g}}$ ($L_{\operatorname{det}}$ is the determinant
line bundle of the Spinc-structure on $X$ and
$L_{\operatorname{det}}|_{X^{g}}$ is its restriction to $X^{g}$);

  * •

$R_{X^{g}}$, the Riemannian curvature form associated to the Levi-Civita
connection on the tangent bundle of $X^{g}$.

The ${\rm AS}_{g}(D)$ in Equation (1.5) is, by definition, the following
expression:

(1.6) ${\rm AS}_{g}(D):=\frac{\widehat{A}\big{(}\frac{R_{X^{g}}}{2\pi
i}\big{)}\operatorname{tr}\big{(}g\exp(\frac{R^{W}}{2\pi
i})\big{)}\exp(\operatorname{tr}(\frac{R^{L}}{2\pi
i}))}{\operatorname{det}\big{(}1-g\exp(-\frac{R^{\mathcal{N}}}{2\pi
i})\big{)}^{\frac{1}{2}}}.$

This will also be denoted by ${\rm AS}_{g}(X,E)$ or, if there is no confusion
on the vector bundle $E$, simply by ${\rm AS}_{g}(X)$.

One might wonder if there is a higher delocalized index theorem; the answer is
affirmative and it is work of Song-Tang and Hochs-Song-Tang as we shall now
explain. Let $P<G$ a cuspidal parabolic subgroup and $P=MAN$ its Langlands
decomposition. Let $g\in M$ be a semisimple element. Song and Tang in [st]
have defined a higher delocalized cyclic cocycle $[\Phi^{P}_{g}]$ on the
Lafforgue Schwartz algebra $\mathcal{L}_{s}(G)$. For
$m=\operatorname{dim}(A)$, $\Phi^{P}_{g}$ is an $m$-cyclic cocycle on
$\mathcal{L}_{s}(G)$ generalizing the orbital integral, Equation (1.3). When
the metric on a $G$ proper cocompact manifold $X$ without boundary is slice
compatible (i.e. there is a slice $Z_{0}$ equipped with a $K$ action such that
$X=G\times_{K}Z_{0}$ and the metric on $X$ is obtained from a $K$-invariant
metric on $Z_{0}$ and $G$-invariant metric on $G/K$), the pairing between
$\Phi^{P}_{g}$ and the $K$-theory index $\operatorname{Ind}_{\infty}(D)$ is
computed by Hochs-Song-Tang in [hst]. More precisely, the index pairing has
the following topological formula,

(1.7)
$\begin{split}\langle\operatorname{Ind}_{\infty}(D),\Phi^{P}_{g}\rangle=&\int_{(X/AN)^{g}}c^{g}_{X/AN}{\rm
AS}(X/AN)_{g}\end{split}$

In the above formula, $X/AN$ is the quotient of $X$ with respect to the $AN<G$
action; the property of the Langlands decomposition implies that the group $M$
acts on the quotient $X/AN$; $(X/AN)^{g}$ is the fixed point submanifold of
the $g$ action on $X/AN$; $c^{g}_{X/AN}$ is a smooth compactly supported
function on $(X/AN)^{g}$ defined in the same way as $c^{g}$ in Equation (1.5)
; the action of the Dirac operator $D$ on the $AN$-invariant sections of $E$
defines a Dirac type operator $D_{X/AN}$ on $X/AN$; the characteristic classes
are defined in a similar but slightly different way from those in Equation
(1.6) for the $M$ action on $X/AN$ (see Equation (9.12) for its explicit
expression).

We can finally state the main goals of this article:

(i) state and prove a (higher) delocalized Atiyah-Patodi-Singer index theorem
on G-proper manifolds with boundary;

(ii) define consequently (higher) rho numbers on $G$-proper manifolds without
boundary and study their properties.

We state now in detail our main results; these deal both with cocompact
$G$-proper manifolds with boundary and without boundary.

We begin by stating the results on manifolds with boundary; these are obtained
by combining the results of Hochs, Song and Tang, the use of (an extension of)
Melrose’s $b$-calculus to the present context developed in [PP2] and relative
K-theory and relative cyclic cohomology techniques, as developed in
[moriyoshi-piazza, GMPi, PP2].

First we give an improvement, through the above techniques, of a result
already proved in [HWW2]. Let $G$ be a connected, linear real reductive group.
Let $g$ be a semisimple element. Let $Y_{0}$ be a $G$-proper manifold with
boundary with compact quotient and let $Y$ be the associated manifold with
cylindrical ends. We fix a $G$-equivariant slice-compatible metric. Let
$D_{0}$ be an equivariant Dirac operator on $Y_{0}$, acting on the sections of
an equivariant vector bundle $E_{0}$. We denote by $D$ and $E$ the associated
objects on $Y$.

######  1.8 Theorem.

(0-degree delocalized APS on G-proper manifolds)
We assume that $D_{\partial Y}$ is $L^{2}$-invertible. Then:

  * 1)

there exists a dense holomorphically closed subalgebra
$\mathcal{L}^{\infty}_{G,s}(Y,E)$ of the Roe algebra $C^{*}(Y_{0}\subset
Y,E)^{G}$ and a smooth index class $\operatorname{Ind}_{\infty}(D)\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y,E))\cong K_{0}(C^{*}(Y_{0}\subset
Y,E)^{G})$;

  * 2)

the delocalized eta invariant

$\eta_{g}(D_{\partial Y}):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y})\frac{dt}{\sqrt{t}}$

exists and for the pairing of the index class $\operatorname{Ind}_{\infty}(D)$
with the $0$-cocycle $\tau^{Y}_{g}\in HC^{0}(\mathcal{L}^{\infty}_{G,s}(Y,E))$
defined by the orbital integral $\tau_{g}$ the following delocalized 0-degree
index formula holds:

(1.9)
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\int_{(Y_{0})^{g}}c^{g}{\rm
AS}_{g}(D_{0})-\frac{1}{2}\eta_{g}(D_{\partial Y})\,,$

where the integrand $c^{g}{\rm AS}_{g}(D_{0})$ in the above integral is
defined as that in Equation (1.5).

This theorem is an improvement with respect to [HWW2] because we do not assume
that $G/Z_{G}(g)$ is compact. Moreover, the two proofs are different.

Consider now $Y/AN$, an $M$-proper manifold, which has a slice decomposition
given by $M\times_{K\cap M}Z=:Y_{M}$. The following theorem is one of the main
results of this paper:

######  1.10 Theorem.

Suppose that the metric is slice compatible (Definition 3.5). Assume that
$D_{\partial Y}$ is $L^{2}$-invertible and consider the higher index
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle$. The following
formula holds:

(1.11)
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle=\int_{(Y_{0}/AN)_{g}}c^{g}_{Y_{0}/AN}{\rm
AS}(Y_{0}/AN)_{g}-\frac{1}{2}\eta_{g}(D_{\partial Y_{M}})$

with

$\eta_{g}(D_{\partial
Y_{M}})=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{\partial
Y_{M}}_{g}(D_{\partial Y_{M}}\exp(-tD^{2}_{\partial
Y_{M}})\frac{dt}{\sqrt{t}},$

where the integrand $c^{g}_{Y_{0}/AN}{\rm AS}(Y_{0}/AN)_{g}$ is defined in the
same way as that in Equation (1.7). We refer to Equation (9.12) for its
explicit expression. We regard $\eta_{g}(D_{\partial Y_{M}})$ as a higher
delocalized eta invariant associated to $P$ and $g$.

######  1.12 Remark.

We observe that in Theorem 1.10, when $P$ is not maximal cuspidal parabolic
subgroup (see Definition 6.3), the pairing
$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle$ vanishes,
similarly to [hst, Theorem 2.1]; moreover, each term on the right side of
Equation (1.11) vanishes because of the existence of extra symmetry (see
Remark 9.14).

In these two theorems the well-definedness of the (higher) delocalized eta
invariant for the boundary operator is a consequence of the delocalized
Atiyah-Patodi-Singer index theorem. In Section 5 we pass to the general case
of a cocompact $G$ proper manifold without boundary (thus not necessarily
equal to the boundary of a cocompact $G$ proper manifold with boundary). Some
discussion about this question was given in [HWW1, Proposition 4.1 and Section
4.4]. We make a very detailed study on the most general hypothesis under which
the delocalized eta invariant is well defined and establish the following
result:

######  1.13 Theorem.

Let $(X,\mathbf{h})$ be a cocompact $G$-proper manifold without boundary and
let $D$ be a $G$-equivariant Dirac-type operator (associated to a unitary
Clifford action and a Clifford connection). Let $g\in G$ be a semi-simple
element. If $D$ is $L^{2}$-invertible, then the integral

(1.14)
$\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}(D\exp(-tD^{2})\frac{dt}{\sqrt{t}}$

converges.

The proof, which is rather involved, is divided into two parts: the small $t$
and the large $t$ integrability. The former is based on a very detailed study
of the Schwartz kernel of $D\exp(-tD^{2})$ and the way it behaves near the
fixed-point set of $g$; the latter is based on a detailed study of the large
time behavior of the heat kernel. Our analysis of the short time behaviour
also gives, as a byproduct, a rigorous proof of the Hoch-Wang index formula in
[Hochs-Wang-HC]. We end the paper by defining (higher) rho-invariants
associated to metrics of positive scalar curvature and by studying their
properties.

This article uses quite some background knowledge from diverse fields of
mathematics such as noncommutative geometry, analysis on manifolds with
boundaries and representation theory. For the benefit of the reader we give a
quick, but certainly incomplete, guide to the literature for the necessary
background in these subjects

  * •

The basic tools from noncommutative geometry needed for this paper are cyclic
cohomology and its pairing with $K$-theory, as can be found in [bookconnes,
Ch. III]. For manifolds with boundary the formalism of relative cyclic
cohomology is particularly useful, for which we refer to [LMP, moriyoshi-
piazza]

  * •

The basic ingredient from the theory of representations of reductive Lie
groups for this paper is the so-called Harish—Chandra algebra of functions on
such groups [Knapp]. The connection between representation theory and
noncommutative geometry is rooted in the Connes—Kasparov conjecture proved by
Wassermann in [Wa:Connes-Kasparov], see also [Lafforgue].

  * •

A standard reference for index theory on manifolds with boundary using the so-
called $b$-calculus is the book by Melrose [Melrose-Book]. Important for the
construction of the relative cyclic cocycles of this paper is the construction
of the $b$-trace and the so-called “defect-formula”; these two topics are
discussed in detail in [Melrose-Book]. Quick introductions to the $b$-calculus
are given in the Appendix of [Mel-P1] and also in the surveys [MP, Grieser,
Loya]. This material is adapted to the present context of G-proper manifolds
in Section 4B and Section 6 of [PP2].

The paper is organized as follows. In Section 3 we give a few geometric
preliminaries; these will play a major role throughout the paper. Section 4 is
devoted to a proof of the 0-degree delocalized Atiyah-Patodi-Singer, using
relative cyclic 0-cocycles associated to orbital integrals. In Section 5 we
prove Theorem 1.13 above (existence of the delocalized eta invariant in the
non-bounding case); crucial in the proof is the study of the large time
behaviour of the heat kernel, as an element in the algebra of integral
operators associated to the Lafforgue algebra, and of the short time behaviour
near a fixed point-set; the latter can also be used to give a detailed proof
of the index formula (1.5). In Section 6 we recall the higher cyclic cocycles
$\Phi^{P}_{g}$ introduced in [st] and we introduce the cyclic cocycles they
define on $G$-proper manifolds; we also introduce the relative version of
these cocycles on manifolds with boundary. In Section 7 and Section 8 we
explain an approach to a general higher delocalized APS index theorem, using
again the interplay between cyclic cocycles and relative cyclic cocycles
associated to $\Phi^{P}_{g}$. In Section 8 we do prove such a theorem in the
slice compatible case by adapting to manifolds with boundary a reduction
procedure due to Hochs, Song and Tang [hst]. Finally in Section 10 we
introduce (higher) rho numbers and study their properties.

Acknowledgements. We would like to thank Pierre Albin, Jean-Michel Bismut,
Peter Hochs, Yuri Kordyukov, Xiaonan Ma, Shu Shen, and Weiping Zhang for
inspiring discussions. Piazza was partially supported by the PRIN Moduli
spaces and Lie Theory of MIUR (Ministero Istruzione Università Ricerca); Song
was partially supported by NSF Grant DMS-1800667 and DMS-1952557; Tang was
partially supported by NSF Grant DMS-1800666 and DMS-1952551. Part of the
research was carried out within the online Research Community on
Representation Theory and Noncommutative Geometry sponsored by the American
Institute of Mathematics.

##  2\. Notations

We use the following notations throughout this paper:

  * •

$G$ = connected, linear real reductive group with maximal compact subgroup
$K$;

  * •

$Y_{0}$ = cocompact proper $G$-manifold with boundary;

  * •

$\mathbf{h}_{0}$ a $G$-invariant metric on $Y_{0}$, product-type near the
boundary;

  * •

$Z_{0}$ = a $K$-slice of $Y_{0}$ so that $Y_{0}$ is diffeomorphic to
$G\times_{K}Z_{0}$; and abusing notation, we will write
$Y_{0}=G\times_{K}Z_{0}$;

  * •

$Y$ = the $G$-proper $b$-manifold associated to $Y_{0}$;

  * •

$\mathbf{h}$ the $G$-invariant $b$-metric on $Y$ associated to
$\mathbf{h}_{0}$;

  * •

$Z$ = the $b$-manifold associated to the slice $Z_{0}$; $Z$ is a $K$-slice of
$Y$ so that $Y=G\times_{K}Z$;

  * •

$X$ = cocompact proper $G$-manifold without boundary;

  * •

$S$ = a $K$-slice of $X$ so that $X=G\times_{K}S$;

  * •

$E$ = $G$-equivariant twisted spinor bundle so that $E=\mathcal{E}\otimes W$,
where $\mathcal{E}$ is the spinor bundle associated Spinc-structure and $W$ is
an auxiliary $G$-equivariant vector bundle;

  * •

$C^{\infty}_{c}(G)$ = space of compactly supported smooth functions on $G$;

  * •

$\mathcal{L}_{s}(G)$ = space of Lafforgue’s Schwartz functions on $G$
associated to the norm $\nu_{s}$;

  * •

$\mathcal{L}_{G}^{c}(X)$ = the space of $G$-equivariant $G$-compactly
supported smoothing operators;

  * •

${}^{b}\mathcal{L}_{G}^{c}(Y)$ = the space of $G$-equivariant $G$-compactly
supported $b$-smoothing operators in the $b$-calculus with $\epsilon$-bounds;

  * •

$\mathcal{L}_{G,s}^{\infty}(X)$ = the space of $G$-Lafforgue Schwartz
smoothing operators associated to the norm $\nu_{s}$;

  * •

$\mathcal{L}_{G,s}^{\infty}(Y)$ = the space of $G$-Lafforgue Schwartz residual
smoothing operators in the $b$-calculus associated to $\nu_{s}$ with
$\epsilon$-bounds ;

  * •

${}^{b}\mathcal{L}_{G,s}^{\infty}(Y)$= the space of $G$-Lafforgue Schwartz
$b$-smoothing operators in the $b$-calculus associated to $\nu_{s}$ with
$\epsilon$-bounds;

  * •

$C^{*}_{r}(G)$=reduced group $C^{*}$-algebra of $G$;

  * •

$\tau_{g}$ = orbital integral on $\mathcal{L}_{s}(G)$ associated to the
conjugacy class of $g$;

  * •

$\tau_{g}^{X}$ = trace on $\mathcal{L}_{G,s}^{\infty}(X)$ associated to the
orbital integral $\tau_{g}$;

  * •

$\tau_{g}^{Y}$ = trace on $\mathcal{L}_{G,s}^{\infty}(Y)$ associated to the
orbital integral $\tau_{g}$.

##  3\. Geometric preliminaries.

Let $Y_{0}$ be a manifold with boundary, $G$ a connected linear real reductive
Lie group acting properly and cocompactly on $Y_{0}$. We denote by $X$ the
boundary of $Y_{0}$. There exists a collar neighbourhood $U$ of the boundary
$\partial Y_{0}$, $U\cong[0,2]\times\partial Y_{0}$, which is $G$-invariant
and such that the action of $G$ on $U$ is of product type. We assume that
$Y_{0}$ is endowed with a $G$-invariant metric $\mathbf{h}_{0}$ which is of
product type near the boundary. We let $(Y_{0},\mathbf{h}_{0})$ be the
resulting Riemannian manifold with boundary; in the collar neighborhood
$U\cong[0,2]\times\partial Y_{0}$ the metric $\mathbf{h}_{0}$ can be written,
through the above isomorphism, as $dt^{2}+\mathbf{h}_{X}$, with
$\mathbf{h}_{X}$ a $G$-invariant Riemannian metric on $X=\partial Y_{0}$. We
denote by $c_{0}$ a cut-off function for the action of $G$ on $Y_{0}$; since
the action is cocompact, this is a compactly supported smooth function. We
consider the associated manifold with cylindrical ends
$\widehat{Y}:=Y_{0}\cup_{\partial Y_{0}}\left((-\infty,0]\times\partial
Y_{0}\right)$, endowed with the extended metric $\widehat{\mathbf{h}}$ and the
extended $G$-action. We denote by $(Y,\mathbf{h})$ the $b$-manifold associated
to $(\widehat{Y},\widehat{\mathbf{h}})$. We shall often treat
$(\widehat{Y},\widehat{\mathbf{h}})$ and $(Y,\mathbf{h})$ as the same object.
We denote by $c$ the obvious extension of the cut-off function $c_{0}$ for the
action of $G$ on $Y_{0}$ (constant along the cylindrical end); this is a cut-
off function of the extended action of $G$ on $Y$. If $x$ is a boundary
defining function for the cocompact $G$-manifold $Y_{0}$, then the $b$-metric
$\mathbf{h}$ has the following product-structure near the boundary $X$:

$\frac{dx^{2}}{x^{2}}+\mathbf{h}_{X}$

We remark at this point that our arguments will actually apply to the more
general case of exact $b$-metrics, or, equivalently, manifolds with asymptotic
cylindrical ends. We shall not insist on this point.

In this article we shall be interested in the case in which $Y_{0}$ admits a
$G$-invariant Spinc-structure. Let $E_{0}$ be the $G$-equivariant spinor
bundle associated to a Spinc-structure on $Y_{0}$ twisted by an auxiliary
$G$-equivariant vector bundle. In particular, $E_{0}$ has a $G$-equivariant
$\text{Cliff}(TY_{0})$-module structure. Let $\nabla^{E_{0}}$ be a _Clifford
connection_ on $E_{0}$, that is

(3.1) $\left[\nabla^{E_{0}}_{V},c(W)\right]=c(\nabla^{TY_{0}}_{V}W),\quad
V,W\in C^{\infty}(Y_{0},TY_{0})$

where $c$ denotes the Clifford action and $\nabla^{TY_{0}}$ is the Levi-Civita
connection. The Dirac operator associated to the Clifford connection is given
by the following composition

(3.2) $D_{Y_{0}}\colon
C^{\infty}(Y_{0},E_{0})\xlongrightarrow{\nabla^{E_{0}}}C^{\infty}(Y_{0},T^{*}Y_{0}\otimes
E_{0})\cong C^{\infty}(Y_{0},TY_{0}\otimes
E_{0})\xlongrightarrow{c}C^{\infty}(Y_{0},E_{0}).$

######  3.3 Assumption.

Suppose that $G$ is a connected reductive Lie group with maximal compact
subgroup $K$. We denote by

$\mathfrak{r}\colon=\text{dim}\ G/K\ (\text{mod}\
2),\quad\mathbf{t}\colon=\text{dim}\ Y_{0}\ (\text{mod}\ 2).$

The index class

$\operatorname{Ind}_{\infty}(D_{Y_{0}})\in
K_{\mathbf{t}}\left(\mathcal{L}_{G,s}^{\infty}\left(Y,E\right)\right)\cong
K_{\mathbf{t}}(C^{*}_{r}G).$

Recall that $K_{\mathfrak{r}+1}(C^{*}_{r}G)=0$, see [Lafforgue]. We assume
that

$\text{dim}\ G/K=\text{dim}\ Y_{0}\ (\text{mod}\ 2).$

Otherwise, the index class is written as

$\operatorname{Ind}_{\infty}(D_{Y_{0}})\in
K_{\mathbf{t}}(C^{*}_{r}G)=K_{\mathfrak{r}+1}(C^{*}_{r}G)=0.$

Without loss of generality, we can further assume

$\text{dim}\ G/K=\text{dim}\ Y_{0}=0\ (\text{mod}\ 2);$

otherwise, we can simply replace $G$ (and $Y_{0}$) by $G\times\mathbb{R}$
(respectively $Y_{0}\times\mathbb{R}$). To sum up, we assume that

  1. (1)

the symmetric space $G/K$;

  2. (2)

the $G$-manifold $Y_{0}$.

are all even dimensional.

For any cuspidal parabolic subgroup $P=MAN$ with $m=\dim(A)$, the $m$-cyclic
cocycle introduced below in (6.1), $\Phi^{P}_{g}$, is an $m$-cyclic cocycle on
$\mathcal{L}_{s}(G)$. The pairing

$\langle\Phi^{P}_{Y,g},\operatorname{Ind}_{\infty}(D)\rangle$

is automatically zero if $m$ is odd. Thus it suffices to consider those
cuspidal parabolic subgroups such that $m=\dim(A)$ is even. In particular, for
the maximal parabolic subgroup $P=MAN$, it is always this case under our
assumption.

###  3.1. Slice compatible metric on $Y_{0}$

Let us fix a slice $Z_{0}$ for the $G$ action on $Y_{0}$; thus

$Y_{0}\cong G\times_{K}Z_{0}$

with $K$ a maximal compact subgroup of $G$ and $Z_{0}$ a smooth compact
manifold with boundary, denoted by $S=\partial Z_{0}$, endowed with a
$K$-action. Consequently, $Y\cong G\times_{K}Z$ with $Z$ the $b$-manifold
associated to $Z_{0}$ and the boundary

$X=\partial Y_{0}\cong G\times_{K}S.$

Choose a $K$-invariant inner product on the Lie algebra $\mathfrak{g}$ of $G$,
so that we have the so called Cartan decomposition
$\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ where $\mathfrak{k}$ is the Lie
algebra of $K$ and $\mathfrak{p}$ its orthogonal complement. We have an
isomorphism

(3.4) $TY_{0}\cong G\times_{K}(\mathfrak{p}\oplus TZ_{0}).$

Here we abuse the notation $\mathfrak{p}$ to denote the trivial vector bundle
$Z_{0}\times\mathfrak{p}\to Z_{0}$.

######  3.5 Definition.

(slice compatible metrics) Given a slice $Z_{0}$, we shall say that a
$G$-invariant metric on $Y_{0}$ is slice compatible with $Z_{0}$ if it is
constructed from a $K$-invariant metric on $Z_{0}$ and a $K$-invariant metric
on $\mathfrak{p}$ via the above isomorphism $TY_{0}\cong
G\times_{K}(\mathfrak{p}\oplus TZ_{0})$. We shall say that the $G$-invariant
metric $\mathbf{h}_{0}$ on $Y_{0}$ is slice-compatible if there is a slice
$Z_{0}$ such that it is slice compatible with $Z_{0}$.

When $Y_{0}$ has a $G$-equivariant Spinc-structure, we can construct its
spinor bundle as follow. We can assume, up to the passage to double covers,
that the adjoint representation ${\rm Ad}:K\to SO(\mathfrak{p})$ admits a lift
$\widetilde{{\rm Ad}}:K\to{\rm Spin}(\mathfrak{p})$. We then obtain a
$G$-invariant Spinc-structure $P^{G/K}:=G\times_{K}{\rm Spin}(\mathfrak{p})\to
G/K$. Assume now that $Z_{0}$ admits a $K$-invariant Spinc-structure. Then,
proceeding as in [Hochs2009, Hochs-Mathai], we obtain a $G$-invariant Spinc-
structure on $M$. Because

$0\to G\times_{K}\mathfrak{p}\to TY_{0}\to G\times_{K}TZ_{0}\to 0,$

the two out of three lemma of Spinc-structure shows that every $G$-invariant
Spinc-structure on $Y_{0}$ is induced from a $K$-invariant Spinc-structure on
$Z_{0}$. The vector bundle $E_{0}$ induces a $K$-equivariant vector bundle
$E_{Z_{0}}$ such that

$E_{0}\cong G\times_{K}(S_{\mathfrak{p}}\otimes E_{Z_{0}})$

and $E_{Z_{0}}$ admits a $K$-equivariant $\text{Cliff}(TZ_{0})$-module
structure and $S_{\mathfrak{p}}$ is the spinor bundle along the $\mathfrak{p}$
direction. We call the above a _slice compatible Spin c-structure_. Throughout
this paper, we only need to assume that the Spinc-structure on $Y_{0}$ is
slice compatible in section 8.

We decompose

$L^{2}(Y_{0},E_{0})\cong\left[L^{2}(G)\otimes S_{\mathfrak{p}}\otimes
L^{2}(Z_{0},E_{Z_{0}})\right]^{K}.$

We can define a _split Dirac operator_ $D_{\text{split}}$ by the following
formula

(3.6) $D_{\text{split}}=D_{G,K}\hat{\otimes}1+1\hat{\otimes}D_{Z_{0}},$

where $D_{G,K}$ is the Spinc-Dirac operator on $L^{2}(G)\otimes
S_{\mathfrak{p}}$, and $D_{Z_{0}}$ is a $K$-equivariant Dirac operator on
$E_{Z_{0}}$, and $\hat{\otimes}$ means the graded tensor product. We point out
here that the two Dirac operators $D_{Y_{0}}$ and $D_{\text{split}}$ are
different in general. We would like to thank Jean-Michel Bismut and Xiaonan Ma
for pointing out this difference to us. It was incorrectly stated in [HS] and
used in the articles111The list of references using this wrong property might
be incomplete. [MGW, hst, Hochs-Wang-HC, HWW1, HWW2] that these two Dirac
operators are identical. In the appendix, we consider the example where
$Y_{0}=G$ and give an explicit formula for the two Dirac operators, showing in
particular that their difference is different from zero. Nevertheless, as the
operators have the same principal symbol they give the same index class
provided the $G$ proper manifold has no boundary; thus in this case

$\operatorname{Ind}_{\infty}(D_{Y_{0}})=\operatorname{Ind}_{\infty}(D_{\text{split}})\in
K_{0}(C^{*}_{r}G).$

The split Dirac operator $D_{\text{split}}$ has been extensively studied in
[hst, HWW1, HWW2]. It is important to point out that the connection used in
the definition of $D_{\text{split}}$ might not be the Clifford connection (see
(3.1)); consequently, the short time behaviour of the heat kernel is ill
behaved for $D^{2}_{\text{split}}$. The main goal of this paper is to study
delocalized APS index theory, where the choice of connection is even more
crucial, given that it affects the invertibility properties of the boundary
operator. Hence, in this paper it is crucial that we work with the Dirac
operator introduced in Equation (3.2).

##  4\. Delocalized traces and the APS index formula

In this section we want to tackle the case of a delocalized APS index theorem
on $G$-proper manifolds with boundary, that is, 0-degree delocalized cyclic
cocycles. Similar results have been discussed in [HWW1, HWW2]. Our treatment
is different, centred around the interplay between absolute and relative
cyclic cohomology and the $b$-calculus; moreover our treatment allows us to
get sharper results compared to [HWW2] in the case of a connected linear real
reductive group222In [HWW2], the authors consider general locally compact
topological groups. $G$. More precisely, in Theorem 1.8 we only assume that
$g$ is a semisimple element of $G$ to obtain the index formula (1.9), while in
[HWW2, Theorem 2.1], the authors require that $G/Z_{g}$ is compact333Notice
that for the numeric $g$-index associated to $D$ a formula is proved in [HWW1]
under the same hypothesis given here; on the other hand, the $g$-index is
proved to be equal to the pairing of an index class with a 0-degree cyclic
cocycle (this is the number we consider in the present article) only under the
additional assumption that $G/Z_{g}$ is compact..

###  4.1. Orbital integrals and associated cyclic $0$-cocycles in the closed
case

Consider first a co-compact $G$-proper manifold without boundary $X$. We know
that there exists a compact submanifold $S\subset X$ on which the $G$-action
restricts to an action of a maximal compact subgroup $K\subset G$, so that the
natural map

$G\times_{K}S\to X,\quad[g,x]\mapsto g\cdot x,$

is a diffeomorphism. This decomposition of $X$ induces an isomorphism

(4.1)
$\mathcal{L}_{G}^{c}(X)\cong\left(C^{\infty}_{c}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K}$

and more generally

(4.2)
$\mathcal{L}_{G}^{c}(X,E)\cong\left(C^{\infty}_{c}(G)\hat{\otimes}\Psi^{-\infty}(S,E|_{S})\right)^{K\times
K},$

in the presence of a $G$-equivariant vector bundle $E$. (We shall often
expunge the vector bundle $E$ from the notation.) On the left hand side we
have the smoothing operators defined by $G$-invariant smooth kernels of
$G$-compact support; on the right hand side we use the (unique) completion of
the algebraic tensor product between the two algebras given that they are both
nuclear.

######  4.3 Definition.

For $s\in[0,\infty)$, define the Lafforgue algebra $\mathcal{L}_{s}(G)$ to be
the completion of $C_{c}(G)$ with respect to the norm $\nu_{s}$ defined as
follows

$\nu_{s}(f)\colon=\sup_{g\in
G}\left\\{(1+\|g\|)^{s}\cdot\Xi^{-1}(g)\cdot|f(g)|\right\\},$

where $\Xi(g)$ denotes Harish-Chandra’s spherical function.

The family of Banach algebras $\\{\mathcal{L}_{s}(G)\\}_{s\geq 0}$ satisfies
the following properties [Lafforgue]:

  1. (1)

For every $s\in[0,\infty)$, $\mathcal{L}_{s}(G)$ is a dense subalgebra of
$C^{*}_{r}(G)$ stable under holomorphic calculus.

  2. (2)

For $0\leq s_{1}<s_{2}$, $\|f\|_{s_{1}}\leq\|f\|_{s_{2}}$, $\forall
f\in\mathcal{L}_{s_{2}}(G)$. Hence,
$\mathcal{L}_{s_{2}}(G)\subset\mathcal{L}_{s_{1}}(G)$. Define

$\mathcal{L}(G)\colon=\cap_{s\geq 0}\mathcal{L}_{s}(G).$

  3. (3)

For any semisimple element $x\in G$, there exists $d_{0}>0$, such that
$\forall s>d_{0}$, the orbital integral

$\int_{G/Z_{x}}f(gxg^{-1})\;dgZ_{x}$

is convergent for all $f\in\mathcal{L}_{s}(G)$.

We observe that Harish-Chandra’s Schwartz algebra is contained in the algebra
$\mathcal{L}_{s}(G)$ for any $s>0$.

Throughout the paper, for a fixed $x\in G$, we work with a Lafforgue Schwartz
algebra $\mathcal{L}_{s}(G)$ for a sufficiently large $s$ so that the orbital
integral is a well defined continuous linear functional on
$\mathcal{L}_{s}(G)$.

Let us consider now a cocompact $G$-proper manifold without boundary $X\cong
G\times_{K}S$. The algebras $\mathcal{L}_{G,s}^{\infty}(X)$ and
$\mathcal{L}^{\infty}_{G}(X)$ are defined as

$\mathcal{L}^{\infty}_{G,s}(X):=\left(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K},\
\mathcal{L}^{\infty}_{G}(X):=\left(\mathcal{L}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K}.$

There are similar algebras when we consider a $G$-equivariant vector bundle
$E$ on $X$; however, for notational simplicity, we shall expunge the vector
bundle $E$ from the notation. For

$\widetilde{k}\in\mathcal{L}^{\infty}_{G}(X):=\left(\mathcal{L}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K}$

or

$\widetilde{k}\in\mathcal{L}^{\infty}_{G,s}(X):=\left(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K}$

we consider the bounded operator $T_{\widetilde{k}}$ on $L^{2}(X)$ given by

(4.4)
$(T_{\widetilde{k}}e)(gs)=\int_{G}\int_{S}g\widetilde{k}(g^{-1}g^{\prime},s,s^{\prime})g^{\prime\,-1}e(g^{\prime}s^{\prime})ds^{\prime}dg^{\prime}.$

The operator $T_{\widetilde{k}}$ is an integral operator with $G$-equivariant
Schwartz kernel $\kappa$ given by

$\kappa(gs,g^{\prime}s^{\prime})=g\widetilde{k}(g^{-1}g^{\prime},s,s^{\prime})g^{\prime\,-1}\,.$

The map $\widetilde{k}\to T_{\widetilde{k}}$ is injective; moreover

$T_{\widetilde{k}}\circ
T_{\widetilde{k}^{\prime}}=T_{\widetilde{k}*\widetilde{k}^{\prime}}$

so that its image is a subalgebra of the $G$-equivariant bounded operators on
$L^{2}(X)$. Following an established abuse of notation we shall not
distinguish between these two algebras, thus identifying a smooth kernel with
the bounded operators it defines.

Following [PP2], there are isomorphisms that associate to a smooth
$G$-equivariant kernel $A$ on $X\times X$ a map
$\Phi_{A}:G\to\Psi^{-\infty}(S)$, with $\Phi_{A}$ equivariant with respect to
the natural $K\times K$ action on $\Psi^{-\infty}(S)$ and the action
$\alpha(k_{1},k_{2})(g):=k_{1}gk_{2}^{-1}$ on $G$. More precisely, by [PP2,
Prop. 1.7], there are isomorphisms

(4.5) $\displaystyle\mathcal{L}^{c}_{G}(X)$
$\displaystyle\cong\left\\{\Phi:G\to\Psi^{-\infty}(S),~{}\mbox{smooth,
compactly supported and }~{}K\times K~{}\mbox{invariant}\right\\},$ (4.6)
$\displaystyle\mathcal{L}^{\infty}_{G,s}(X)$
$\displaystyle\cong\left\\{\Phi:G\to\Psi^{-\infty}(S),~{}K\times
K~{}\mbox{invariant and}~{}g\mapsto
v_{s}(\|\Phi(g)\|_{\alpha})\;\;\text{bounded}\right\\}$

with $\alpha$ a multi-index indexing derivatives with respect to the spacial
variables of $S$, $\|\;\|_{\alpha}$ denoting the associated well-known
seminorm on $\Psi^{-\infty}(S)$ and $v_{s}(\,\,)$, denoting the Lafforgue
norm. The image of the product by convolution of $A$ and $B$ on the left hand
side is equal to $\Phi_{A}*\Phi_{B}$ on the right hand side, with

(4.7) $(\Phi_{A}*\Phi_{B})(g)=\int_{G}\Phi_{A}(gh^{-1})\circ\Phi_{B}(h)dh\,.$

Following again [Hochs-Wang-KT] we define for $T=T_{\widetilde{k}}$

(4.8) $\tau_{g}^{X}(T):=\int_{G/Z_{g}}\int_{X}c(hgh^{-1}x){\rm
tr}(hgh^{-1}\kappa(hg^{-1}h^{-1}x,x))dx\,d(hZ)$

with $c$ a cut-off function for the action of $G$ on $X$ and ${\rm tr}$
denoting the vector-bundle fiberwise trace. There are equivalent ways to write
the right-hand-side, see [Hochs-Wang-KT, Lemma 3.2]; for example if $c_{G}$ is
a cutoff function for the action of $Z_{g}$ on $G$ by right multiplication and
$c^{g}(x)=\int_{G}c_{G}(h)c(hgx)dh$ then the right hand side of (4.8) can be
written as

(4.9) $\int_{X}c^{g}(x){\rm tr}(\kappa(x,gx)g)dx.$

It is proved in [Hochs-Wang-KT, Lemma 3.4] that $\tau_{g}^{X}$ defines a
continuous trace

(4.10) $\tau_{g}^{X}:\mathcal{L}^{\infty}_{G,s}(X)\to\mathbb{C}\,.$

In fact, the two traces (4.8) and (4.10) are related by a homomorphism of
integration along the slice $S$, ${\rm
Tr}_{S}:\mathcal{L}^{\infty}_{G,s}(X)\to\mathcal{L}_{s}(G)$, such that

$\tau^{X}_{g}=\tau_{g}\circ{\rm Tr}_{S}$

(see [Hochs-Wang-KT, Section 3.3]). We see that $\operatorname{Tr}_{S}$
associates to $\Phi$ the function

$G\ni g\to\operatorname{Tr}(\Phi(g))\,.$

Using [PP2, Lemma 1.24] and the well known inequality
$|\operatorname{Tr}(T)|\leq\|T\|_{1}$ for a smoothing operator on a smooth
compact manifold444with $\|\;\;\|_{1}$ denoting the trace norm, we see that
$\operatorname{Tr}_{S}:\mathcal{L}^{\infty}_{G,s}(X)\to\mathcal{L}_{s}(G)$ so
defined is a continuous map. (Even though [PP2, Lemma 1.24] uses a slightly
different algebra on $G$ instead of $\mathcal{L}_{s}(G)$, the proof showing
that the trace norm $A\mapsto||A||_{1},~{}A\in\Psi^{-\infty}(S)$ is continuous
for the Fréchèt topology on $\Psi^{-\infty}(S)$ applies verbatim to show
continuity in our case.) Exactly the same results hold for the algebra
$\mathcal{L}^{\infty}_{G,s}(X)$.

Let now $D$ be an equivariant Dirac operator, of product type near the
boundary. We shall make the following assumption:

(4.11) $\text{the boundary operator}\;\;D_{\partial
Y}\;\;\text{is}\;\;L^{2}\text{-invertible}.$

The following Theorems sharpen the corresponding results in [PP2], (see in
particular Subsection 5.3, Proposition 5.27, Proposition 5.33 and Theorem 5.42
there).

######  4.12 Theorem.

Let $D$ be as above and let $Q^{\sigma}$ be a symbolic $b$-parametrix for $D$.
The Connes-Skandalis projector

(4.13)
$P^{b}_{Q}:=\left(\begin{array}[]{cc}{}^{b}S_{+}^{2}&{}^{b}S_{+}(I+{}^{b}S_{+})Q^{b}\\\
{}^{b}S_{-}D^{+}&I-{}^{b}S_{-}^{2}\end{array}\right)$

associated to a true $b$-parametrix $Q^{b}=Q^{\sigma}-Q^{\prime}$ with
remainders ${}^{b}S^{\pm}$ in $\mathcal{L}^{\infty}_{G,s}(Y_{0})$ is a
$2\times 2$ matrix with entries in $\mathcal{L}^{\infty}_{G,s}(Y)$ 555really
in a slightly extended algebra because of the identity appearing in the right
lower corner of the matrix. We thus have a well-defined smooth index class

(4.14) $\operatorname{Ind}_{\infty}(D):=[P^{b}_{Q}]-[e_{1}]\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\equiv K_{0}(C^{*}(Y_{0}\subset
Y)^{G})\;\;\;\text{with}\;\;\;e_{1}:=\left(\begin{array}[]{cc}0&0\\\
0&1\end{array}\right).$

######  4.15 Theorem.

  * 1)

The Connes-Moscovici projector $V(D)$,

$V(D):=\left(\begin{array}[]{cc}e^{-D^{-}D^{+}}&e^{-\frac{1}{2}D^{-}D^{+}}\left(\frac{I-e^{-D^{-}D^{+}}}{D^{-}D^{+}}\right)D^{-}\\\
e^{-\frac{1}{2}D^{+}D^{-}}D^{+}&I-e^{-D^{+}D^{-}}\end{array}\right),$

is a $2\times 2$ matrix with entries in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y_{0})$; the Connes-Moscovici projector
$V(D^{\operatorname{cyl}})$ is a $2\times 2$ matrix with entries in
${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm cyl}(\partial Y))$. These
two projectors define a smooth relative index class
$\operatorname{Ind}_{\infty}(D,D_{\partial})\in
K_{0}({}^{b}\mathcal{L}^{\infty}_{G,s}(Y_{0}),{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y)))$.

  * 2)

The projector $V^{b}(D)$ obtained by improving the parametrix
$Q:=\frac{I-\exp(-\frac{1}{2}D^{-}D^{+})}{D^{-}D^{+}}D^{-}$ defining $V(D)$ to
a true $b$-parametrix $Q^{b}$ is a $2\times 2$ matrix with entries in
$\mathcal{L}^{\infty}_{G,s}(Y)$ and defines the same smooth index class in
$K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))$ as the Connes-Skandalis projector of
Theorem 4.12

  * 3)

The class $\operatorname{Ind}_{\infty}(D)$ is sent to the class
$\operatorname{Ind}_{\infty}(D,D_{\partial})$ through the excision isomorphism
$\alpha_{{\rm exc}}$.

Summarizing: using the Connes-Moscovici projector(s) we have smooth index
classes $\operatorname{Ind}_{\infty}(D)\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))$ and
$\operatorname{Ind}_{\infty}(D,D_{\partial})\in
K_{0}({}^{b}\mathcal{L}^{\infty}_{G,s}(Y),{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y)))$, with the first one sent to the second one by the excision
isomorphism $\alpha_{{\rm exc}}$.

###### Proof.

The proof proceeds as in [PP2]. Recall, in particular, that the relative index
class is defined by the triple

(4.16) $(V(D),e_{1},q_{t})\,,\;\;t\in[1,+\infty]\,,\;\;\text{ with
}q_{t}:=\begin{cases}V(tD_{\operatorname{cyl}})\;\;\quad\text{if}\;\;\;t\in[1,+\infty)\\\
e_{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if }\;\;t=\infty\end{cases}$

and with $e_{1}:=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$. The well-definedness
of this class is a consequence of the large time behaviour of the heat kernel
on $b$-manifolds for invertible operators, treated in detail in the next
section. Notice that most of the arguments given in [PP2] use the closure
under holomorphic calculus of certain algebras. This applies unchanged whether
we use pseudodifferential operators based on the Lafforgue algebra
$\mathcal{L}_{s}(G)$ or the rapid decay algebra $H^{\infty}_{L}(G)$. The only
exception is the analogue of Lemma 2.7 in [PP2] that we state now explicitly,
in the present case, for the benefit of the reader:

######  4.17 Lemma.

Let $X$ be a cocompact $G$-proper manifold without boundary with a slice $S$.
Consider $\Psi^{-\infty}_{G,c}(X)$ and its extension
$\mathcal{L}^{\infty}_{G,s}(X):=(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}_{c}(S))^{K\times
K}$, an algebra of smoothing operators. Then the composition
$\Psi^{0}_{G,c}(X)\times\Psi^{-\infty}_{G,c}(X)\to\Psi^{-\infty}_{G,c}(X)$
extends to a continuous map

$\Psi^{0}_{G,c}(X)\times\mathcal{L}^{\infty}_{G,s}(X)\to\mathcal{L}^{\infty}_{G,s}(X).$

Proof of the Lemma
The proof is a variation of the corresponding one for
$(H^{\infty}_{L}(G)\hat{\otimes}\Psi_{c}^{-\infty}(S))^{K\times K}$ given in
[PP2, Lemma 2.7]. Recall that
$\Psi^{0}_{G,c}(X)=\big{(}\Psi^{0}_{G,c}(G)\hat{\otimes}\Psi^{0}_{c}(S)\big{)}^{K\times
K}$. As $\Psi^{-\infty}_{c}(S)$ is an ideal of $\Psi^{0}_{c}(S)$, we are
reduced to prove that the product

$\Psi^{0}_{c}(G)\times\mathcal{L}_{c}(G)\to\mathcal{L}_{c}(G),\ (A,f)\mapsto
A\ast f$

is well-defined and continuous with respect to $f$.

A general element $A$ of $\Psi^{0}_{c}(G)$ can be written as

$A=\operatorname{Op}(a)+K,$

where $K\in\Psi^{-\infty}_{G,c}(G)\cong C^{\infty}_{c}(G)$ and
$\operatorname{Op}(a)$ is the operator corresponding to a symbol $a\in
S^{0}(\mathfrak{g}^{*})$ of order zero with property

$|D_{\xi}^{\alpha}(a)|\leq C_{\alpha}(1+|\xi|)^{-|\alpha|}.$

It follows from the inclusion $C^{\infty}_{c}(G)\subset\mathcal{L}_{s}(G)$
that the product $(K,f)\mapsto K\ast f$ is well-defined and continuous for
$K\in\Psi^{-\infty}_{G,c}(G)$. We are left to show that the composition
$(\operatorname{Op}(a),f)\mapsto\operatorname{Op}(a)(f)$ belongs to
$\mathcal{L}(G)$ and is continuous. Recall that $\operatorname{Op}(a)(f)$ has
the following expression

$\operatorname{Op}(a)(f)(g):=\int_{G}\int_{\mathfrak{g}^{*}}\chi(h^{-1})e^{i\langle\xi,\exp^{-1}(h^{-1})\rangle}a(\xi)f(hg)d\xi
dh.$

Let $\Xi(g)$ be Harish-Chandra’s spherical function as before. We have

$\begin{split}&\left(1+\|g\|\right)^{s}\Xi(g)^{-1}\big{(}\operatorname{Op}(a)(f)(g)\big{)}\\\
=&\int_{G}\int_{\mathfrak{g}^{*}}\chi(h^{-1})e^{i\langle\xi,\exp^{-1}(h^{-1})\rangle}a(\xi)f(hg)\left(1+\|g\|\right)^{s}\Xi(g)^{-1}d\xi
dh\\\
=&\int_{G}\int_{\mathfrak{g}^{*}}\chi(h^{-1})e^{i\langle\xi,\exp^{-1}(h^{-1})\rangle}a(\xi)\frac{\Xi(hg)}{\Xi(g)}\Xi(hg)^{-1}f(hg)\left(1+\|g\|\right)^{s}d\xi
dh\end{split}$

By [Knapp, Lemma 12.5], for $h^{-1}$ in the support of the function $\chi$,
the support of which is compact, there is a constant $C>0$ such that

$\left|\frac{\Xi(hg)}{\Xi(g)}\right|<C,\ \forall g\in
G,h\in\operatorname{supp}(\chi).$

Peetre’s inequality gives,

$\left(1+\|g\|\right)^{s}\leq\left(1+\|h^{-1}\|\right)^{s}\cdot\left(1+\|hg\|\right)^{s}.$

It follows from the above two inequalities that we have that

$\lVert\left(1+\|g\|\right)^{s}\Xi(g)^{-1}\big{(}\operatorname{Op}(a)(f)(g)\big{)}\rVert$

is bounded by

$C\Big{\lVert}\int_{G}\int_{\mathfrak{g}^{*}}\chi(h^{-1}))e^{i\langle\xi,\exp^{-1}(h^{-1})\rangle}a(\xi)(1+\|h\|))^{s}d\xi
dh\Big{\rVert}\times\operatorname{sup}(1+\|hg\|)^{s}\big{\lVert}\Xi(hg)^{-1}f(hg)\big{\rVert}.$

The integral

$\Big{\lVert}\int_{G}\int_{\mathfrak{g}^{*}}\chi(h^{-1}))e^{i\langle\xi,\exp^{-1}(h^{-1})\rangle}a(\xi)(1+\|h\|)^{s}d\xi
dh\Big{\rVert}$

is finite as the integration of $h$ is over the support of $\chi$, which is
compact.

Summarizing the above estimates, we have proved that there is a constant
$\widetilde{C}>0$ such that

$\lVert|1+\|g\||^{s}\Xi(g)^{-1}\big{(}\operatorname{Op}(a)(f)(g)\big{)}\rVert\leq\widetilde{C}\operatorname{sup}(1+\|hg\|)^{s}\big{\lVert}\Xi(hg)^{-1}f(hg)\big{\rVert}.$

This proves that $\operatorname{Op}(a)(f)$ belongs to the Lafforgue algebra
$\mathcal{L}_{s}(G)$, and the map
$(\operatorname{Op}(a),f)\mapsto\operatorname{Op}(a)(f)$ is continuous with
respect to $f$. This proves the Lemma. ∎

###  4.2. $0$-degree (relative) cyclic cocycles associated to orbital
integrals

Let us now pass to cyclic cocycles associated to orbital integrals. Consider
for the time being a compact $b$-manifold $Y$ endowed with a $b$-metric
$\mathbf{h}_{Y}$ which has product structure
$\mathbf{h}_{Y}=dx^{2}/x^{2}+\mathbf{h}_{\partial Y}$ near the boundary. We
use the associated volume form in order to trivialize the relevant half-
density bundles. In the $b$-calculus with $\epsilon$-bounds we consider

$\operatorname{Ker}\left({}^{b}\Psi^{-\infty,\epsilon}(Y)\xrightarrow{I_{Y}}{}^{b}\Psi^{-\infty,\epsilon}_{\mathbb{R}}(\partial
Y\times\mathbb{R})\right)$

the kernel of the (surjective) indicial homomorphism.

It is well known, see Melrose’s book [Melrose-Book], that if $\epsilon<1$,
then

$\operatorname{Ker}(I_{Y})\subset\Psi^{-\infty,\epsilon}(Y)$

with $\Psi^{-\infty,\epsilon}(Y)$ denoting the residual operators; these are
smoothing kernels on the $b$-stretched product $Y^{2}_{b}$ that vanish at
order $\epsilon$ on all boundary faces, that is, the left boundary
$lb(Y^{2}_{b})$, the right boundary $rb(Y^{2}_{b})$ and the front face
$bf(Y^{2}_{b})$. See the figure below. In fact
$\operatorname{Ker}(I_{Y})=\rho_{bf(Y^{2}_{b})}{}^{b}\Psi^{-\infty,\epsilon}(Y)$
with $\rho_{bf(Y^{2}_{b})}$ a boundary defining function for the front face,
and by definition
$\rho_{bf(Y^{2}_{b})}{}^{b}\Psi^{-\infty,\epsilon}(Y)\subset\Psi^{-\infty,\epsilon}(Y)$
if $\epsilon<1$.

$lb(Y^{2}_{b})$$rb(Y^{2}_{b})$$\Delta_{b}$$bf(Y^{2}_{b})$

Because of this vanishing, the residual operators are trace class on
$L^{2}_{b}$ and the trace is obtained by integration over the lifted diagonal
$\Delta_{b}$ (because of the extra vanishing the integral with respect to the
$b$-volume form, which near the boundary can be written as
$\frac{dx}{x}dvol_{\partial X}$, is absolutely convergent). Put it
differently, this algebra of operators behaves very much as the smoothing
operators on a smooth compact manifold without boundary.

Consider now the $G$-proper case and

(4.18)
$\mathcal{L}^{\infty}_{G,s}(Y):=\operatorname{Ker}\left(I:{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\rightarrow{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}\partial
Y_{0})\right)\,.$

For exactly the same reason as above this algebra behaves very much like
$\left(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times K}$,
with $S$ now a slice for the action of $G$ on a cocompact $G$-proper manifold
without boundary $X$. This is in fact a great advantage of our method and
allows us to avoid completely ad hoc arguments.

We thus have a trace-homomorphism

(4.19) $\tau_{g}^{Y}:\mathcal{L}^{\infty}_{G,s}(Y)\to\mathbb{C}\,.$

exactly as in the closed case:

(4.20) $\tau_{g}^{Y}(T):=\int_{G/Z_{g}}\int_{Y}c(hgh^{-1}x){\rm
tr}(hgh^{-1}\kappa(hg^{-1}h^{-1}x,x))dx\,d(hZ)$

with $dx$ denoting now the $b$-volume form associated to the $b$-metric
$\mathbf{h}$. The proof of the well-definedness of (4.20) now proceeds as in
the case of manifold without boundary, using the fact that $\kappa$ is
residual and therefore vanishing of order $\epsilon$ at all boundary faces. In
fact: there is a homomorphism of integration along the slice $S$, ${\rm
Tr}_{S}:\mathcal{L}^{\infty}_{G,s}(Y,E)\to\mathcal{L}_{s}(G)$, which is
continuous and such that $\tau^{Y}_{g}=\tau_{g}\circ{\rm Tr}_{S}\,.$

The trace $\tau^{Y}_{g}$ defines a cyclic 0-cocycle on the algebra
$\mathcal{L}^{\infty}_{G,s}(Y)$. Using the pairing between $K$-theory and
cyclic cohomology, denoted $\langle\,,\,\rangle$, we have in our case

(4.21)
$\langle\cdot\,,\cdot\,\rangle:HC^{0}(\mathcal{L}^{\infty}_{G,s}(Y))\times
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\to\mathbb{C}$

and thus a homomorphism

(4.22)
$\langle\tau^{Y}_{g},\cdot\rangle:K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\to\mathbb{C}.$

######  4.23 Definition.

Let $D$ be a $G$-equivariant operator on $Y$ as in section 3. Assume that the
induced boundary operator is $L^{2}$-invertible, so that there is a well-
defined index class $\operatorname{Ind}_{\infty}(D)\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))$. The $g$-index of $D$ is, by definition,
the number $\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle$.

Our goal in this section is to give a formula for
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle$.

Following the relative cyclic cohomology approach in [moriyoshi-piazza, GMPi,
PP2] we want to find a relative cyclic 0-cocycle $(\tau^{Y,r}_{g},\sigma_{g})$
verifying

(4.24)
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\langle(\tau^{Y,r}_{g},\sigma_{g}),\operatorname{Ind}_{\infty}(D,D_{\partial})\rangle.$

(The $r$ on the right hand side stands for regularized.)

######  4.25 Proposition.

Let $X$ be a cocompact $G$-proper manifold without boundary. Define the
following 1-cochain on ${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(X))$

(4.26)
$\sigma^{X}_{g}(A_{0},A_{1})=\frac{i}{2\pi}\int_{\mathbb{R}}\tau_{g}^{X}(\partial_{\lambda}I(A_{0},\lambda)\circ
I(A_{1},\lambda))d\lambda\,,$

where the indicial family of
$A\in{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm cyl}(X))$, denoted
$I(A,\lambda)$, appears. Then $\sigma^{X}_{g}(\,,\,)$ is well-defined and a
cyclic 1-cocycle.

###### Proof.

Fourier transform identifies ${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(X))$ with holomorphic families

$\\{\mathbb{R}\times
i(-\epsilon,\epsilon)\ni\lambda\to\mathcal{L}^{\infty}_{G,s}(\partial X)\\}$

with values in the Fréchet algebra $\mathcal{L}^{\infty}_{G,s}(X)$, rapidly
decreasing in ${\rm Re}\lambda$. (Recall that we do not write $\epsilon$ in
the notation, but elements in our algebras are built from $b$-operators of
order $-\infty$ in the calculus with $\epsilon$-bounds.) It is then immediate
that the integral is absolutely convergent and depends continuously on
$A_{0}$, $A_{1}$. The fact that it is a cyclic 1-cocycle follows from the
tracial property of $\tau^{X}_{g}$ and integration by parts in $\lambda$. ∎

Let $Y$ be now a $b$-manifold and let $\partial Y$ be its boundary. Let
$\tau^{Y,r}_{g}$ be the functional on ${}^{b}\mathcal{L}^{c}_{G}(Y)$:

$\tau^{Y,r}_{g}(T):=\int_{G/Z_{g}}\int^{b}_{Y}c(hgh^{-1}y){\rm
tr}(hgh^{-1}\kappa(hg^{-1}h^{-1}y,y))dy\,d(hZ)$

where Melrose’s $b$-integral has been used, $dy$ denotes the $b$-density
associated to the $b$-metric $\mathbf{h}$ and where we recall that the cut-off
function $c_{0}$ on $Y_{0}$ is extended constantly along the cylinder to
define $c$. This is the regularization of $\tau^{Y}_{g}$ on a $b$-manifold,
for the time being on kernels of $G$-compact support (we shall deal with the
extension of $\tau^{Y,r}_{g}$ on all of ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$
momentarily). Observe that

$\tau^{Y,r}_{g}=\tau_{g}\circ{}^{b}{\rm Tr}_{S}$

with ${}^{b}{\rm Tr}_{S}:{}^{b}\mathcal{L}^{c}_{G}(Y)\to C^{\infty}_{c}(G)$
denoting $b$-integration along the slice $S$. More precisely, as in the closed
case, we have an isomorphism

${}^{b}\mathcal{L}^{c}_{G}(Y)\cong\left\\{\Phi:G\to{}^{b}\Psi^{-\infty,\epsilon}(S),~{}\mbox{smooth,
compactly supported and }~{}K\times K~{}\mbox{invariant}\right\\}$

and ${}^{b}\operatorname{Tr}_{S}$ associates to $\Phi$ the function
$G\ni\gamma\to{}^{b}\operatorname{Tr}(\Phi(\gamma))$. The continuity of this
map will be treated more generally in the proof of Proposition 4.28 below.

######  4.27 Proposition.

The pair $(\tau^{Y,r}_{g},\sigma^{\partial Y}_{g})$ defines a relative
0-cocycle for
${}^{b}\mathcal{L}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}({\rm
cyl}(\partial Y))$.

###### Proof.

As we are dealing with a 0-cochain, it suffices to show that for the
Hochschild $b$-differential of $\tau^{Y,r}_{g}$ the following formula holds:

$(b\tau^{Y,r}_{g})(A_{0},A_{1})=\sigma^{\partial Y}_{g}(I(A_{0}),I(A_{1}))$

where we recall that

$\sigma^{\partial
Y}_{g}(I(A_{0}),I(A_{1}))=\frac{i}{2\pi}\int_{\mathbb{R}}\tau_{g}^{\partial
Y}(\partial_{\lambda}I(A_{0},\lambda)\circ I(A_{1},\lambda))d\lambda\,.$

The left hand side $(b\tau^{Y,r}_{g})(A_{0},A_{1})$ is equal to
$\tau^{Y,r}_{g}[A_{0},A_{1}]$; if $A_{i}$, $i=0,1$, corresponds to
$\Phi_{i}:G\to{}^{b}\Psi^{-\infty}(S)$ then $\tau^{Y,r}_{g}[A_{0},A_{1}]$ is
equal to

$\int_{G/Z_{g}}\int_{G}{}^{b}\operatorname{Tr}(\Phi_{0}(h_{1}gh_{1}^{-1}h^{-1})\circ\Phi_{1}(h)dhd(h_{1}Z_{g})-\int_{G/Z_{g}}\int_{G}{}^{b}\operatorname{Tr}(\Phi_{1}h_{1}gh_{1}^{-1}h^{-1})\circ\Phi_{0}(h)dhd(h_{1}Z_{g})\,.$

Changing the order of integration and with a suitable change of coordinates in
the second summand, using the unimodularity of $G$, we can rewrite the above
expression as

$\int_{G/Z_{g}}\int_{G}{}^{b}\operatorname{Tr}[\Phi_{0}(h_{1}gx^{-1}h_{1}^{-1}),\Phi_{1}(h)]dhd(h_{1}Z_{g})\,.$

Now we can apply Melrose’s formula for the $b$-trace of a commutator and get

$\frac{i}{2\pi}\int_{G/Z_{g}}\int_{G}\int_{\mathbb{R}}\operatorname{Tr}\left(\partial_{\lambda}I(\Phi_{0}(h_{1}gh_{1}^{-1}h^{-1}),\lambda)\circ
I(\Phi_{1}(h),\lambda)\right)d\lambda dhd(h_{1}Z_{g})\,.$

We can interchange the order of integration without problems here (rapid decay
in $\lambda$ and compact support in $G$) and so we conclude reverting to
$I(A_{0},\lambda)$ and $I(A_{1},\lambda)$ that

$\displaystyle\tau^{Y,r}_{g}[A_{0},A_{1}]$
$\displaystyle=\frac{i}{2\pi}\tau_{g}\int_{\mathbb{R}}\operatorname{Tr}_{\partial
S}(\partial_{\lambda}I(A_{0},\lambda)\circ I(A_{1},\lambda))d\lambda$
$\displaystyle=\frac{i}{2\pi}\int_{\mathbb{R}}\tau_{g}^{\partial
Y}(\partial_{\lambda}I(A_{0},\lambda)\circ I(A_{1},\lambda))d\lambda$

This completes the proof. ∎

######  4.28 Proposition.

The pair $(\tau^{Y,r}_{g},\sigma^{\partial Y}_{g})$ extends continuously to a
relative 0-cocycle for

${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y)).$

Moreover, the following formula holds:

(4.29)
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\langle(\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}),\operatorname{Ind}_{\infty}(D,D_{\partial})\rangle\,.$

###### Proof.

We already know that $\sigma^{\partial Y}_{g}$ extends to a 1-cocycle on
$\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm cyl}(\partial Y))$. If we could
show that $\tau^{Y,r}_{g}$ extends to ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$
then by density and continuity we would get the first statement of the
proposition. The second would then follow as usual from the fact that

$\tau^{Y,r}_{g}|_{\mathcal{L}^{\infty}_{G,s}(Y)}=\tau^{Y}_{g}$

given that on residual operators the $b$-integral equals the ordinary
integral.

As in [PP2] we have an isomorphism

${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\cong\left\\{\Phi:G\to{}^{b}\Psi^{-\infty,\epsilon}(S),~{}K\times
K~{}\mbox{invariant and}~{}g\mapsto
v_{s}(\|\Phi(g)\|_{\alpha})\;\;\text{bounded}\right\\}$

where $\|\;\;\|_{\alpha}$ are now the $C^{\infty}$ seminorms of a smooth
kernel on the $b$-stretched product.
We want to show that the map

(4.30)
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\ni\Phi\longrightarrow{}^{b}\operatorname{Tr}(\Phi(\cdot))\in\mathcal{L}_{s}(G)$

is continuous. Here we can use a Proposition of Lesch-Moscovici-Pflaum [LMP,
Proposition 2.6], expressing the b-trace of a b-smoothing operator on a
compact $b$-manifold in terms of the trace of two residual operators, see for
example Proposition 7.1 in [PP2]. Using this Proposition and the arguments in
[PP2, Lemma 7.8], directly inspired in turn on those of [GMPi], we prove the
continuity of the map

${}^{b}\Psi^{-\infty,\epsilon}(S)\ni
T\longrightarrow{}^{b}\operatorname{Tr}(T)\in\mathbb{C}$

and thus of the map (4.30). ∎

###  4.3. The $0$-degree delocalized APS index theorem on $G$-proper manifolds

We now apply formula (4.29) to the index class associated to the Connes-
Moscovici projector

(4.31)
$V(D)=\left(\begin{array}[]{cc}e^{-D^{-}D^{+}}&e^{-\frac{1}{2}D^{-}D^{+}}\left(\frac{I-e^{-D^{-}D^{+}}}{D^{-}D^{+}}\right)D^{-}\\\
e^{-\frac{1}{2}D^{+}D^{-}}D^{+}&I-e^{-D^{+}D^{-}}\end{array}\right)$

On the left hand-side of (4.29) we have

$\langle\tau^{M}_{g},V^{b}(D)\rangle$

with $V^{b}(D)$ the modified Connes-Moscovici projector. Indeed, recall that
the Connes-Moscovici projector is simply the Connes-Skandalis projector for
the choice of parametrix

$Q:=\frac{I-\exp(-\frac{1}{2}D^{-}D^{+})}{D^{-}D^{+}}D^{-}$

This produces the remainders $I-QD^{+}=\exp(-\frac{1}{2}D^{-}D^{+})$,
$I-D^{+}Q=\exp(-\frac{1}{2}D^{+}D^{-})$ and $V^{b}(D)$ is obtained by
improving this parametrix to a true parametrix in the $b$-calculus, that is,
an inverse modulo residual operators.

On the right hand-side we have the pairing of the relative cocycle
$(\tau^{Y,r}_{g},\sigma^{\partial Y}_{g})$ with the relative class

(4.32) $(V(D),e_{1},q_{t})\,,\;\;t\in[1,+\infty]\,,\;\;\text{ with
}q_{t}:=\begin{cases}V(tD_{\operatorname{cyl}})\;\;\quad\text{if}\;\;\;t\in[1,+\infty)\\\
e_{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if }\;\;t=\infty\end{cases}$

and with $e_{1}:=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$. Let us concentrate
first on the right hand-side. By definition of relative pairing we have:

(4.33) $\langle(\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}),(V(D),e_{1},q_{t})\rangle=\tau^{Y,r}_{g}(e^{-D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-D^{+}D^{-}})+\int_{1}^{\infty}\sigma^{\partial
Y}_{g}([\dot{q}_{t},q_{t}],q_{t})dt\,.$

######  4.34 Proposition.

The term $\int_{1}^{\infty}\sigma^{\partial
Y}_{g}([\dot{q}_{t},q_{t}],q_{t})dt$, with $q_{t}:=V(tD_{\operatorname{cyl}})$
is equal to

$-\frac{1}{2}\left(\frac{1}{\sqrt{\pi}}\int_{1}^{\infty}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y}))\frac{dt}{\sqrt{t}}\right)\,.$

###### Proof.

By definition

$\sigma^{\partial
Y}_{g}([\dot{q}_{t},q_{t}],q_{t})dt=\frac{i}{2\pi}\int_{\mathbb{R}}\tau^{\partial
Y}_{g}(\partial_{\lambda}(I([\dot{q}_{t},q_{t}],\lambda))\circ
I(q_{t},\lambda))d\lambda.$

We can integrate by parts in $\lambda$ on the right hand side, given that the
two terms are rapidly decreasing in $\lambda$. Thus, taking into account the
multiplicative constants, we want to prove that

(4.35) $\int_{\mathbb{R}}\tau^{\partial
Y}_{g}((I([\dot{q}_{t},q_{t}],\lambda)\circ\partial_{\lambda}(I(q_{t},\lambda))d\lambda=\frac{i\sqrt{\pi}}{\sqrt{t}}\,\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y}))\,.$

We need to write down the indicial family of $q_{t}$. By definition

$q_{t}=\left(\begin{array}[]{cc}e^{-t^{2}D_{\operatorname{cyl}}^{-}D_{\operatorname{cyl}}^{+}}&e^{-\frac{t^{2}}{2}D_{\operatorname{cyl}}^{-}D_{\operatorname{cyl}}^{+}}\left(\frac{I-e^{-t^{2}D_{\operatorname{cyl}}^{-}D_{\operatorname{cyl}}^{+}}}{t^{2}D_{\operatorname{cyl}}^{-}D_{\operatorname{cyl}}^{+}}\right)tD_{\operatorname{cyl}}^{-}\\\
e^{-\frac{t^{2}}{2}D_{\operatorname{cyl}}^{+}D_{\operatorname{cyl}}^{-}}tD^{+}&I-e^{-t^{2}D_{\operatorname{cyl}}^{+}D_{\operatorname{cyl}}^{-}}\end{array}\right)$

and thus, denoting $D_{\partial Y}$ by $B$ we have

$I(q_{t},\lambda)=\left(\begin{array}[]{cc}e^{-t^{2}(\lambda^{2}+B^{2})}&e^{-\frac{t^{2}}{2}(\lambda^{2}+B^{2})}\left(\frac{I-e^{-t^{2}(\lambda^{2}+B^{2})}}{t^{2}(\lambda^{2}+B^{2})}\right)t(-i\lambda+B)\\\
e^{-\frac{t^{2}}{2}(\lambda^{2}+B^{2})}t(i\lambda+B)&I-e^{-t^{2}(\lambda^{2}+B^{2})}\end{array}\right).$

We set

$p(t,\lambda):=I(q_{t},\lambda)\,.$

We must compute

$\int_{\mathbb{R}}\tau_{g}^{\partial
Y}\left(\left(\partial_{t}p(t,\lambda)\circ
p(t,\lambda)-p(t,\lambda)\circ\partial_{t}p(t,\lambda)\right)\circ\partial_{\lambda}p(t,\lambda)\right)d\lambda$

and show that it equals

$\frac{i\sqrt{\pi}}{\sqrt{t}}\,\tau^{\partial Y}_{g}(D_{\partial
Y}\exp(-tD^{2}_{\partial Y})).$

This is a complicated computation; however, we remark that all operators
appearing in the $2\times 2$ matrices

$\partial_{t}p(t,\lambda)\,,\quad
p(t,\lambda)\quad\text{and}\quad\partial_{\lambda}p(t,\lambda)$

are given by operators obtained by functional calculus for $B$ and thus, in
particular, they commute. This means that we can deal with this computation in
a formal way and so, with some help by Mathematica, formula (4.35) does
follow. We omit the elementary but lengthy details. ∎

Thanks to this Proposition we have that

(4.36) $\langle(\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}),(V(D),e_{1},q_{t})\rangle=\tau^{Y,r}_{g}(e^{-D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-D^{+}D^{-}})-\frac{1}{2}\int_{1}^{\infty}\frac{1}{\sqrt{\pi}}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y}))\frac{dt}{\sqrt{t}}\,.$

As the last step we replace $D$ by $sD$; in the equality

$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\langle(\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}),\operatorname{Ind}_{\infty}(D,D_{\partial Y})\rangle$

the left hand side $\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle$
remains unchanged whereas the right hand side becomes

$\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})-\frac{1}{2}\int_{s}^{\infty}\frac{1}{\sqrt{\pi}}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y})\frac{dt}{\sqrt{t}}\,.$

Summarizing, for each $s>0$ we have

(4.37)
$\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})=\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle+\frac{1}{2}\int_{s}^{\infty}\frac{1}{\sqrt{\pi}}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y})\frac{dt}{\sqrt{t}}\,.$

Now we take the limit as $s\downarrow 0$. It is a general principle, explained
in detail in [Melrose-Book], that the Getzler rescaling applies to the heat
kernel in the $b$-context; needless to say, the geometry here is more
complicated that in the case of a compact $b$-manifold endowed with a product
$b$-metric. Still, we shall prove in the next section the following
proposition, where all structures are assumed to be product-like near the
boundary.

######  4.38 Proposition.

Let $(Y_{0},\mathbf{h}_{0})$ be a cocompact $G$-proper manifold. Let
$(Y,\mathbf{h})$ be the associated $b$-manifold. Let $D_{0}$ be a
G-equivariant Dirac operator defined in (3.2) and let $D$ be the associated
$b$-differential operator. Let $g$ be a semisimple element and let $Y_{0}^{g}$
the fixed point set of $g$. Then the limit $\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})$
exists and we have

$\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})=\int_{Y_{0}^{g}}c^{g}_{0}{\rm
AS}_{g}(D_{0})$

with $c^{g}_{0}{\rm AS}_{g}(D_{0})$ defined in Equation (1.6).

Assuming the last Proposition we can infer that the limit

$\lim_{s\downarrow
0}\frac{1}{2}\int_{s}^{\infty}\frac{1}{\sqrt{\pi}}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y})\frac{dt}{\sqrt{t}}$

exists and equals

$\int_{Y_{0}^{g}}c^{g}{\rm
AS}_{g}(D_{0})-\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle\,.$

We conclude that we have proved the following

######  4.39 Theorem.

(0-degree delocalized APS)
Let $G$ be connected, linear real reductive group. Let $g$ be a semisimple
element. Let $Y_{0}$, $Y$, $D$, $D_{\partial Y}$ as above. Assume that
$D_{\partial Y}$ is $L^{2}$-invertible. Then

$\eta_{g}(D_{\partial Y}):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{\partial
Y}_{g}(D_{\partial Y}\exp(-tD^{2}_{\partial Y})\frac{dt}{\sqrt{t}}$

exists and for the pairing of the index class
$\operatorname{Ind}_{\infty}(D)\in K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\equiv
K_{0}(C^{*}(Y_{0}\subset Y)^{G})$ with the $0$-cocycle $\tau^{Y}_{g}\in
HC^{0}((\mathcal{L}^{\infty}_{G,s}(Y)))$ the following delocalized 0-degree
APS index formula holds:

$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\int_{Y_{0}^{g}}c^{g}{\rm
AS}_{g}(D_{0})-\frac{1}{2}\eta_{g}(D_{\partial Y})\,,$

where the integrand $c^{g}{\rm AS}_{g}(D_{0})$ is defined in the same way as
the one in Equation (1.6).

##  5\. Delocalized eta invariants for $G$-proper manifolds

In the previous section we have obtained the well-definedness of
$\eta_{g}(D_{\partial Y})$, with $D_{\partial Y}$ being $L^{2}$-invertible, as
a byproduct of the proof of the delocalized APS index theorem for 0-degree
cocycles. In fact, one can show that $\eta_{g}(D)$ is well defined on a
cocompact $G$-proper manifold even if $D$ does not arise as a boundary
operator. This is the content of the next theorems, partially discussed also
in [HWW1, HWW2]. The main result of this section is the following theorem.

######  5.1 Theorem.

Let $(X,\mathbf{h})$ be a cocompact $G$-proper manifold without boundary
endowed with a $G$-equivariant Spinc-structure and let $D$ be the Dirac-type
operator defined in (3.2). Let $g$ be a semi-simple element. If $D$ is
$L^{2}$-invertible, then the integral

(5.2)
$\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}(D\exp(-tD^{2})\frac{dt}{\sqrt{t}}$

converges.

###### Proof.

We split the proof into two parts.

In Proposition 5.8, we study the large time behavior of the heat kernel and
prove the following integral

$\frac{1}{\sqrt{\pi}}\int_{1}^{\infty}\tau^{X}_{g}(D\exp(-tD^{2}))\frac{dt}{\sqrt{t}}$

converges.

In Proposition 5.31, we study the small time behavior of the heat kernel and
prove the following integral

$\frac{1}{\sqrt{\pi}}\int^{1}_{0}\tau^{X}_{g}(D\exp(-tD^{2}))\frac{dt}{\sqrt{t}}$

converges.

We complete the proof of the theorem by combining the above two results. ∎

Recall, following [PP2], that for all $t>0$

$\exp(-tD^{2})=\frac{1}{2\pi i}\int_{\gamma}e^{-t\mu}(D^{2}-\mu)^{-1}d\mu$

is an element in the Fréchet algebra $\mathcal{L}^{\infty}_{G,s}(X,E)$ with
any $s>0$. It was stated in [PP2], but without a proper proof, that if $D$ is
$L^{2}$-invertible then $\exp(-tD^{2})$ converges exponentially to $0$ in
$\mathcal{L}^{\infty}_{G,s}(X,E)$. The corresponding statement for
$b$-manifolds was also stated there. The next two subsections provide detailed
proofs of these two results.

###  5.1. Large time behaviour on manifolds without boundary

Assume now that there exists $a>0$ such that

(5.3) ${\rm Spec}_{L^{2}}(D)\cap[-2a,2a]=\emptyset\,.$

We can and we shall choose $\gamma$ so that ${\rm
Spec}_{L^{2}}(D)\cap\gamma(z)=\emptyset$ and ${\rm Re}\,\gamma(z)>a$ for every
$z$. We want to show that $\exp(-tD^{2})\in\mathcal{L}^{\infty}_{G,s}(X)$ is
exponentially converging to $0$ in $\mathcal{L}^{\infty}_{G,s}(X)$ as
$t\to+\infty$.

By [PP2, Proposition 5.21],

$(D^{2}-\mu)^{-1}=B(\mu)+C(\mu)$

where

  * •

$B(\mu)\in\Psi^{-2}_{G,c}(X)$ is a symbolic parametrix for $D^{2}-\mu$ and
defines a pseudodifferential operator with parameter of order $-2$;

  * •

$C(\mu)\in\mathcal{L}^{\infty}_{G,s}(X)$, and $C(\mu)$ goes to 0 in the
Fréchet topology of $\mathcal{L}^{\infty}_{G,s}(X)$, as $|\mu|\to+\infty$.

In fact, we can improve this result and see easily that $C(\mu)$ is rapidly
decreasing in $|\mu|$, with values in $\mathcal{L}^{\infty}_{G,s}(X)$. Indeed,
$(D^{2}-\mu)^{-1}=B(\mu)(1+F(\mu))$ with $1+F(\mu)=(1+R(\mu))^{-1}$ and
$R(\mu)$ the remainder of the symbolic parametrix $B(\mu)$. By the
pseudodifferential calculus with parameter we know that $R(\mu)$ is rapidly
decreasing in $|\mu|$ (it is a smoothing operator with parameter). Using the
elementary identity

$1-(1+R(\mu))^{-1}=R(\mu)[(1+R(\mu))^{-1}]$

we understand that $F(\mu)$ is in fact rapidly decreasing in $|\mu|$ with
values in $\mathcal{L}^{\infty}_{G,s}(X)$. Thus $C(\mu):=B(\mu)\circ F(\mu)$
is also rapidly decreasing in $|\mu|$, with values in
$\mathcal{L}^{\infty}_{G,s}(X)$; here Lemma 4.17 has been used. By the same
analysis carried out in [PP2] and above we also know that for all $\mu$ and
$k\geq 1$

$(D^{2}-\mu)^{-k}=F(\mu)+G(\mu)$

with $F(\mu)\in\Psi^{-2k}_{G,c}(X,\Lambda)$, of $G$-compact support uniformly
in $\mu$, and $G(\mu)\in\mathcal{L}^{\infty}_{G,s}(X)$ and rapidly decreasing
in $\mathcal{L}^{\infty}_{G,s}(X)$ as $|\mu|\to+\infty$.
Let $p_{s,\alpha}$ be a seminorm on the Fréchet algebra
$\mathcal{L}^{\infty}_{G,s}(X):=(\mathcal{L}_{s}(G)\hat{\otimes}\Psi^{-\infty}(S))^{K\times
K}$. This depends on $s$ through the Banach norm on $\mathcal{L}_{s}(G)$ and
on the multiindex $\alpha$ through a seminorm $\|\cdot\|_{\alpha}$ on the
smoothing operators on the slice. Let $|\alpha|=\ell$. Consider now
$(D^{2}-\mu)^{-k}$ with $k>{\rm max}(\dim X,\ell/2)$. Observe that

$\exp(-tD^{2})=\frac{(k-1)!}{2\pi
i}t^{1-k}\int_{\gamma}e^{-t\mu}(D^{2}-\mu)^{-k}d\mu$

We want to show that

$p_{s,\alpha}\left(\int_{\gamma}e^{-t\mu}(D^{2}-\mu)^{-k}d\mu\right)\rightarrow
0\quad\text{as}\quad t\to+\infty\,.$

We thus consider the decomposition

$(D^{2}-\mu)^{-k}=F(\mu)+G(\mu)$

and

$\int_{\gamma}e^{-t\mu}F(\mu)d\mu,\quad\int_{\gamma}e^{-t\mu}G(\mu)d\mu.$

Consider the first integral. Using the arguments employed by Shubin in
[Shubin-Book, Ch. XII, Sect. 11] in order to discuss the properties of complex
powers, see also Vassout [Vassout, Ch. 4] and Gilkey[gilkey-book, Lemma
1.8.1], we see that this first integral is in $\mathcal{L}^{c}_{G}(X)$, which
is a subalgebra of $\mathcal{L}^{\infty}_{G,s}(X)$; the integrand, on the
other hand, is in $\Psi^{-2k}_{G,c}(X,\Lambda)$ and to such an integrand we
can apply the seminorm $p_{s,\alpha}$, given that it is contained in
$(\mathcal{L}_{s}(G)\hat{\otimes}C^{2k}(S\times S))^{K\times K}$. From the
properties of the Bochner integral we thus have that

$p_{s,\alpha}\left(\int_{\gamma}e^{-t\mu}F(\mu))d\mu\right)\leq\int_{\gamma}e^{-t\mu}p_{s,\alpha}(F(\mu))d\mu$

Now, $F(\mu)$ is a pseudodifferential operator with parameter of order $(-2k)$
of $G$-compact support uniformly in $\mu$. We claim that
$p_{m,\alpha}(F(\mu))$ can be bounded by a negative power of $|\mu|$. To see
this we take the associated element
$\Phi_{F(\cdot)}:G\to\Psi^{-2k}(S,\Lambda)$ under the isomorphism (4.5). We
can equivalently show that the seminorm $p^{\Phi}_{s,\alpha}$ defined by
$v_{s}$ and $\|\cdot\|_{\alpha}$ can be bounded by a negative power of $\mu$
when applied to the element $\Phi_{F(\cdot)}$. Now, $\Phi_{F(\cdot)}$ is a
compactly supported function on $G$ with values in $\Psi^{-2k}(S,\Lambda)$.
Because of the compactness of the support in $G$ we only need to understand
the statement for the elements in $\Psi^{-2k}(S,\Lambda)$ for $k$ large;
however this is well known, see for example [gilkey-book, Lemma 1.7.4]. It is
at this point clear that

(5.4) $A(t):=\int_{\gamma}e^{-t\mu}p_{m,\alpha}(F(\mu))d\mu$

goes to $0$ as $t\to+\infty$. Indeed, as we can choose the path $\gamma$ so
that ${\rm Re}\gamma(z)>a$ for all $z$, the result is immediate from the
stated properties of $p_{m,\alpha}(F(\mu))$.
Consider now the second term,

$p_{s,\alpha}\left(\int_{\gamma}e^{-t\mu}G(\mu)d\mu\right)$

Here the integrand is already an element in $\mathcal{L}^{\infty}_{G,s}(X)$,
going to $0$ as $|\mu|$ goes to $+\infty$. In this case we can bring the
seminorm under the sign of integral. Thus we are considering

(5.5) $B(t):=\int_{\gamma}e^{-t\mu}p_{s,\alpha}(G(\mu))d\mu$

Since we know that $p_{s,\alpha}(G(\mu))$ goes to $0$ as $|\mu|$ goes to
$+\infty$ (in fact, rapidly), using once again in a crucial way the fact that
${\rm Re}\gamma(z)>a$, we can conclude that $B(t)$ goes to $0$ as
$t\to+\infty$.
Notice that in fact, by elementary manipulations, writing $e^{-t\mu}$ as
$e^{-t\mu/2}\circ e^{-t\mu/2}$ we can prove that the convergence is weighted
exponential, with weight $a$ (and $a$ as in (5.3)). See [PP2, Remark 2.11].

We summarize our discussion in the following proposition:

######  5.6 Proposition.

If $D$ is $L^{2}$-invertible then

(5.7) $\exp(-tD^{2})\rightarrow 0\quad\text{weighted exponentially
in}\quad\mathcal{L}^{\infty}_{G,s}(Y)\quad\text{as}\quad t\to+\infty.$

In general for a Schwartz function $f$ on $\mathbb{R}$, the operators
$f(tD^{2})$and $Df(tD^{2})$ both converge to 0 weighted exponentially in
$\mathcal{L}^{\infty}_{G,s}(Y)$ as $t\to+\infty$. In particular, the Connes-
Moscovici projector $V(tD)-e_{1}$ converges to 0 weighted exponentially in
$\mathcal{L}^{\infty}_{G,s}(Y)$ as $t\to+\infty$.

###### Proof.

For the heat kernel we have already given a very detailed proof. For a general
Schwartz function $f$ on $\mathbb{R}$, we can directly generalize the above
analysis on $\exp(-tD^{2})$ via a similar estimate for
$D(D^{2}-\lambda)^{-2k}$ to show that $Df(-tD^{2})$ converges to 0 weighted
exponentially in $\mathcal{L}^{\infty}_{G,s}(Y)$ using the following integral
formulas

$f(tD^{2})=\frac{1}{2\pi i}\int_{\gamma}f(t\mu)(D^{2}-\mu)^{-1}d\mu,\ \ \ \
Df(tD^{2})=\frac{1}{2\pi i}\int_{\gamma}f(t\mu)D(D^{2}-\mu)^{-1}d\mu.$

The property about the Connes-Moscovici projector follows from the observation
that every component of the projector is of the form $f(tD^{2})$ or
$Df(tD^{2})$ for some Schwartz function $f$ on $\mathbb{R}$. ∎

######  5.8 Proposition.

Let $(X,\mathbf{h})$ be a cocompact $G$-proper manifold without boundary
endowed with a $G$-equivariant Spinc-structure and let $D$ be the Dirac-type
operator defined in (3.2). Let $g$ be a semi-simple element. If $D$ is
$L^{2}$-invertible, then the integral

(5.9)
$\frac{1}{\sqrt{\pi}}\int_{1}^{\infty}\tau^{X}_{g}(D\exp(-tD^{2})\frac{dt}{\sqrt{t}}$

converges.

###### Proof.

Consider the Schwartz function $f(x)=\exp^{-|x|}$ on $\mathbb{R}$. It follows
from Proposition 5.6 that the operator $Df(-tD^{2})=D\exp(-tD^{2})$ converges
weighted exponentially in $\mathcal{L}^{\infty}_{G,s}(Y)\quad\text{as}\quad
t\to+\infty$. Accordingly, the orbital integral $\tau^{X}_{g}(D\exp(-tD^{2}))$
converges to 0 weighted exponentially as $t\to+\infty$. This decay property
assures that integral (5.9) is well defined. ∎

###  5.2. Large time behaviour on manifolds with boundary

We now discuss the case of manifolds with boundary. We consider as before
$(Y_{0},\mathbf{h}_{0})$, a cocompact $G$ proper manifold with boundary $X$
and denote by $(Y,\mathbf{h})$ the associated $b$-manifold. We denote by
$Z_{0}$ a slice for $Y_{0}$ and by $Z$ the associated $b$-manifold. Let $D$ be
a Dirac operator on $(Y,\mathbf{h})$ and assume that $D_{\partial}$ is
$L^{2}$-invertible. The same method as in [PP2] can be directly generalized to
establish that

$\exp(-tD^{2})=\frac{1}{2\pi i}\int_{\gamma}e^{-t\mu}(D^{2}-\mu)^{-1}d\mu$

is an element in ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ for every $t>0$ and for
every $s>0$.

Assume now that there exists $a>0$ such that

${\rm Spec}_{L^{2}}(D)\cap[-2a,2a]=\emptyset\,.$

We can and we shall choose $\gamma$ so that ${\rm
Spec}_{L^{2}}(D)\cap\gamma(z)=\emptyset$ and ${\rm Re}\,\gamma(z)>a$ for every
$z$.

We want to show that $\exp(-tD^{2})\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ is
exponentially converging to $0$ in ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ as
$t\to+\infty$. By [PP2, Proposition 5.21],

$(D^{2}-\mu)^{-1}=B(\mu)+B(\mu)\circ L(\mu)$

with

$B(\mu)=B^{\sigma}(\mu)-\varphi((I(D^{2}-\mu)^{-1}I(R^{\sigma}(\mu)))$

In these formulae

  * •

$\varphi:{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y)))\to{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ is a section of the indicial
homomorphism
$I:{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\to{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$; $\varphi$ is defined by using a suitable cut-off function equal to 1 on
the boundary;

  * •

$B^{\sigma}(\mu)\in{}^{b}\Psi^{-2}_{G,c}(X)$ is a symbolic parametrix for
$D^{2}-\mu$ and defines a $b$-pseudodifferential operator with parameter of
order $-2$ (in the small $b$-calculus);

  * •

$R^{\sigma}(\mu)\in{}^{b}\Psi^{-\infty}_{G,c}(X)$ is the remainder of a
symbolic parametrix and defines a $b$-pseudodifferential operator with
parameter of order $-\infty$ (always in the small calculus);

  * •

$B(\mu)$ is a true parametrix for $D^{2}$ in the calculus with bounds.

Moreover, as proved in [PP2],
$\mu\to\varphi((I(D^{2}-\mu)^{-1}I(R^{\sigma}(\mu)))$ is rapidly decreasing
when ${\rm Re}(\mu)\to+\infty$ as a map with values in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ and
$L(\mu)\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ goes to 0, always in Fréchet
topology, as ${\rm Re}(\mu)\to+\infty$.
A similar formula can be written for $(D^{2}-\mu)^{-k}$ for any $k\geq 1$. We
consider, as for the case of closed manifolds, $k$ very large; we fix such a
$k$ and adopt the same notation as above. Write now

$\exp(-tD^{2})=\frac{(k-1)!}{2\pi
i}t^{1-k}\int_{\gamma}e^{-t\mu}(D^{2}-\mu)^{-k}d\mu.$

We can express the integral on the right hand side initially as the sum of 2
terms

$\int_{\gamma}e^{-t\mu}B_{\mu}d\mu\;+\;\int_{\gamma}e^{-t\mu}B_{\mu}\circ
L_{\mu}d\mu$

and then, using the analogue of the above expression for $(D^{2}-\mu)^{-k}$,
as the sum of 4 terms:

(5.10)
$\begin{split}\int_{\gamma}e^{-t\mu}B^{\sigma}(\mu)d\mu-\int_{\gamma}e^{-t\mu}\varphi(I(D^{2}-\mu)^{-k}I(R^{\sigma}(\mu)))d\mu+\int_{\gamma}e^{-t\mu}B^{\sigma}(\mu)\circ
L(\mu)d\mu-\\\
\int_{\gamma}e^{-t\mu}\varphi(I(D^{2}-\mu)^{-k}I(R^{\sigma}(\mu)))\circ
L(\mu)d\mu\end{split}$

where now $B^{\sigma}(\mu)\in{}^{b}\Psi^{-2k}_{G,c}(X)$ and defines a
$b$-pseudodifferential operator with parameter of order $-2k$. The large-time
behaviour of these four integrals can be treated as in the closed case.
Indeed, for the first term we know that the Schwartz kernel of
$B^{\sigma}(\mu)$ will be $C^{2k}$ across the $b$-diagonal and so, keeping in
mind that this term if of order $(-2k)$ as a pseudodifferential operator with
parameter, we can directly estimate the integral of the relevant seminorm
applied to $B^{\sigma}(\mu)$, as in the closed case, c.f. (5.4). For the
second summand it suffices to recall that
$\mu\to\varphi(I(D^{2}-\mu)^{-1}I(R^{\sigma}_{\mu}))$ is a rapidly decreasing
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$-valued map. Similarly, the operators in
the third integral define a ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$-valued map
going to $0$ in the Fréchet topology as ${\rm Re}(\mu)\to+\infty$, whereas the
operators in the fourth integral define a
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$-valued map rapidly decreasing as ${\rm
Re}(\mu)\to+\infty$. When we apply the seminorms to these families of
operators inside the integral and we perform the integration, we obtain
expressions in $t$ that clearly go to 0 when $t\to+\infty$; here we use again,
crucially, that ${\rm Re}\gamma(z)>a>0$. As this is precisely as in (5.5) we
omit the details. In fact, by a simple trick with the countour, we see also in
this case that the convergence is weighted exponential.

We summarize our discussion in the following proposition:

######  5.11 Proposition.

If $D$ is $L^{2}$-invertible then

(5.12) $\exp(-tD^{2})\rightarrow 0\quad\text{weighted exponentially
in}\quad{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\quad\text{as}\quad t\to+\infty.$

In general for a Schwartz function $f$ on $\mathbb{R}$, the operators
$f(tD^{2})$ and $Df(tD^{2})$ converge to 0 weighted exponentially in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ as $t\to+\infty$. In particular the
Connes-Moscovici projector $V(tD)-e_{1}$ converges to 0 weighted exponentially
in ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ as $t\to+\infty$.

###  5.3. Small time behavior

To study the small $t$ convergence of the eta integral, we will need the
following properties.

######  5.13 Lemma.

Let $X$ be a cocompact $G$-proper manifold without boundary with a
$G$-invariant complete Riemannian metric $\mathbf{h}_{X}$. Let $d_{X}(-,-)$ be
the associated distance function, and $d_{G}(-,-)$ be the distance function on
$G/K$. Suppose that $V$ is a compact set on $X$ and $g$ is any element in $G$.
There are constants $C_{0}>0,C_{1}>0$ such that

(5.14) $d_{X}(hgh^{-1}x,x)\geq C_{1}d_{G}(hgh^{-1}e,e)-C_{0},\ \forall x\in
V,h\in G.\quad$

###### Proof.

The slice theorem gives a fibration structure on $X=(G\times S)/K$ over $G/K$,
i.e. $\pi:X\to G/K$. For every $x\in X$, let $T_{x}X$ be the tangent space of
$X$ at $x$, and $V_{x}$ be the kernel of the map $\pi_{*}:T_{x}X\to
T_{\pi(x)}G/K$, and $H_{x}$ be the orthogonal complement of $V_{x}$ in
$T_{x}X$ with respect to the metric $\mathbf{h}_{X}$. The induced map
$\pi_{*}:H_{x}\to T_{\pi(x)}G/K$ is a linear isomorphism. Though the map
$\pi_{*}:H_{x}\to T_{\pi(x)}G/K$ might not be an isometry with respect to the
restricted metric $\mathbf{h}_{X}|_{H_{x}}$ and the metric $\mathbf{h}_{G/K}$
on $T_{\pi(x)}G/K$, there is a positive constant $c_{x}$ such that

$\mathbf{h}_{X}|_{H_{x}}\geq
c_{x}(\pi_{*})^{-1}\big{(}\mathbf{h}_{G/K}\big{)}.$

Running $x$ over every point in the fiber $\pi^{-1}(\pi(x))$, as the fiber is
compact, there is a constant $C_{1}>0$ such that

$\mathbf{h}_{X}|_{H_{y}}\geq
C_{1}(\pi_{*})^{-1}\big{(}\mathbf{h}_{G/K}\big{)},\ \forall
y\in\pi^{-1}(\pi(x)).$

As $G$ acts on $G/K$ transitively and the metric $\mathbf{h}_{X}$ and
$\mathbf{h}_{G/K}$ are $G$-invariant, the above estimate holds for all $x\in
X$, i.e.

$\mathbf{h}_{X}|_{H_{x}}\geq
C_{1}(\pi_{*})^{-1}\big{(}\mathbf{h}_{G/K}\big{)},\ \forall x\in X.$

The above comparison of the Riemannian metrics gives the following property on
the distance functions by the standard argument on path lengths

$d_{X}(hgh^{-1}x,x)\geq C_{1}d_{G}(hgh^{-1}\pi(x),\pi(x)).$

Choose $x_{0}$ to be a point in $\pi^{-1}([e])$, where $[e]$ the point in
$G/K$ associated to the coset of the identity element. The above inequality
shows

(5.15) $d_{X}(hgh^{-1}x_{0},x_{0})\geq C_{1}d_{G}(hgh^{-1}e,e).$

We apply the triangle inequality to obtain

(5.16) $d_{X}(hgh^{-1}x_{0},hgh^{-1}x)+d_{X}(hgh^{-1}x,x)+d_{X}(x,x_{0})\geq
d_{X}(hgh^{-1}x_{0},x_{0}).$

As $V$ is compact, there is a finite upper bound $C/2$ such that
$d(x,x_{0})\leq C/2$ for all $x\in V$. Combining this with Inequalities (5.16)
and (5.15), we conclude with the desired inequality, i.e.

$d_{X}(hgh^{-1}x,x)\geq C_{1}d_{G}(hgh^{-1}e,e)-C_{0}.$

∎

######  5.17 Remark.

It is obvious that Lemma 5.13 also holds if instead of $(X,\mathbf{h})$,
$V\subset X$ and, for (ii), $C(g)$ we consider:
$-$ $(Y_{0},\mathbf{h}_{0})$, a cocompact $G$-proper manifold with boundary,
with slice-compatible metric, product-type
near the boundary;
$-$ $V_{0}\subset Y_{0}$ a compact set in $Y_{0}$.

We also consider $(Y,\mathbf{h})$, the $G$-manifold with cylindrical ends
associated to $(Y_{0},\mathbf{h}_{0})$. Here we recall that there is a collar
neighbourhood of $\partial Y_{0}$ in $Y_{0}$ where the action of $G$ is of
product type (and thus extendable to a $G$-proper action on $(Y,\mathbf{h})$).
Notice that, consequently, the action of $G$ preserves the decomposition
$Y:=(-\infty,0]\times\partial Y_{0}\cup_{\partial Y_{0}}Y_{0}$. Consider the
compact subset $V_{0}\subset Y_{0}$ and $V_{0}\cap\partial Y_{0}$, denoted
$\partial V_{0}$ if non-empty. Consider finally $V:=(-\infty,0]\times\partial
V_{0}\cup_{\partial V_{0}}V_{0}$. Lemma 5.13 also holds if we consider now
$(Y,\mathbf{h})$, $V$. Indeed (5.14) does hold if $x\in V_{0}\subset Y_{0}$,
as we have just observed, and also holds if $x\in(-\infty,0]\times\partial
V_{0}\subset V\subset Y$ because the action of $G$ is of product type along
the cylinder.

The following property follows from [CareyGRS, Proposition 4.2, (i)] because a
cocompact $G$-proper manifold without boundary has bounded geometry.

######  5.18 Lemma.

Let $X$ be a cocompact $G$-proper manifold without boundary and let $D$ be a
Dirac-type operator (associated to a unitary Clifford action and a Clifford
connection) on a spinor bundle $\mathcal{E}$. Let $\kappa_{t}(x,y)$ be the
kernel function of the operator $D\exp(-tD^{2})$. There are constant
$\alpha,\beta>0$ such that

$||\kappa_{t}(x,y)||\leq\beta
t^{-\frac{n+1}{2}}\exp\left(-\alpha\frac{d_{X}(x,y)^{2}}{t}\right).$

where $||\cdot||$ is the operator norm from $\mathcal{E}_{x}$ to
$\mathcal{E}_{y}$.

######  5.19 Remark.

Consider the heat kernel $k_{t}$ associated to a Dirac-Laplacian on
$(Y,\mathbf{h})$, the $G$-proper manifold with cylindrical ends associated to
a cocompact $G$-proper manifold with boundary $(Y_{0},\mathbf{h}_{0})$. As
$(Y,\mathbf{h})$ is a complete manifold with bounded geometry we can apply
again [CareyGRS, Proposition 4.2, (i)] and conclude that there are constant
$\alpha_{0},\beta_{0}>0$ such that

(5.20)
$||k_{t}(x,y)||\leq\beta_{0}t^{-\frac{n}{2}}\exp\left(-\alpha\frac{d_{Y}(x,y)^{2}}{t}\right).$

The following estimate is proved by Harish-Chandra [Harish-Chandra-dis,
Theorem 6] (see also [Hochs-Wang-HC, Lemma 4.4]).

######  5.21 Lemma.

For a semisimple element $g$, when $t$ is close to 0, the integral

$\int_{G/Z_{g}}\exp\left(-\alpha\frac{d_{G}(hg^{-1}h^{-1}e,e)^{2}}{t}\right)$

is bounded. We will sometimes use $d_{G}(g)$ for $d_{G}(ge,e)$ in the
following of this article for abbreviation.

######  5.22 Lemma.

For a semisimple $g\in G$, the quotient $X^{g}/Z_{g}$ is compact.

###### Proof.

If $g$ is not conjugate to an element in $K$ (that is, $g$ is not elliptic)
then $X^{g}$ is empty (See Proposition 10.6). It suffices to work with
$g=k_{0}\in K$. By the slice theorem, we assume that $X=G\times_{K}S$. A point
$[(h,x)]$ belongs to $X^{k_{0}}$ if there is $k\in K$ such that
$k_{0}h=hk^{-1}$ and $kx=x$. Therefore,

$X^{k_{0}}=\\{[(h,x)]\in G\times_{K}S|h^{-1}k_{0}^{-1}h=k,kx=x\\}.$

Let $\pi$ be the quotient map from $G\times S$ to $G\times_{K}S$. The space
$\tilde{X}^{k_{0}}:=\\{(h,x)\in G\times S|h^{-1}k_{0}^{-1}h=k,kx=x\\}$ as a
subset of $G\times S$ is the preimage $\pi^{-1}(X^{k_{0}})$. As $\pi$ is a
principal $K$-bundle, $\tilde{X}^{k_{0}}$ is a manifold. As the $Z_{k_{0}}$
action on $X\times G$ commutes with the $K$ action, it is sufficient to prove
that $Z_{k_{0}}\backslash\tilde{X}^{k_{0}}$ is compact to conclude that
$Z_{k_{0}}\backslash X^{k_{0}}$ is compact.

Consider the right conjugacy action of $G$ on the conjugacy class
$C(k_{0}^{-1})=\\{hk_{0}^{-1}h^{-1}|h\in G\\}$. $C(k_{0}^{-1})$ is a closed
submanifold of $G$, and the $G$ action on $C(k_{0}^{-1})$ is transitive with
the stabilizer group at $k_{0}^{-1}$ being $Z_{k_{0}}$. We have that
$C(k_{0}^{-1})$ is diffeomorphic to $Z_{k_{0}}\backslash G$ and the map
$f:G\to C(k_{0}^{-1})$ mapping $g$ to $h^{-1}k_{0}^{-1}h$ is a fibration.

Consider the intersection $C(k_{0},K):=C(k_{0}^{-1})\cap K$, which is a
compact subset. Let $F(k_{0})$ be $f^{-1}\big{(}C(k_{0}^{-1})\cap K\big{)}$.
By the fibration property of $f$, $Z_{k_{0}}\backslash F(k_{0})$ is
homeomorphic to $C(k_{0},K):=C(k_{0}^{-1})\cap K$. For $C(k_{0},K)$, we
consider the closed space $I_{k_{0},K}:=\\{(k,x)|k\in C(k_{0},K),k_{0}x=x\\}$
of $K\times S$. The map $\tilde{f}:I_{k_{0},K}\to C(k_{0},K)$ mapping
$(k,x)\in I_{k_{0},K}$ to $k\in C(k_{0},K)$ is a continuous surjective map.

Let $F(k_{0})_{f}\times_{\tilde{f}}I_{k_{0},K}$ be the fiber product of
$F(k_{0})$ and $I_{k_{0},K}$ over the maps $f|_{F(k_{0})}$ and $\tilde{f}$. As
$f$ is a fibration, $F(k_{0})_{f}\times_{\tilde{f}}I_{k_{0},K}$ can be
identified with $\tilde{X}^{k_{0}}=\\{(g,x)|g\in C(k_{0},K),x\in S^{f(g)}\\}$.

With the above identification, the quotient of $\tilde{X}^{k_{0}}$ by
$Z_{k_{0}}$ is homeomorphic to

$Z_{k_{0}}\backslash\left(F(k_{0})_{f}\times_{\tilde{f}}I_{k_{0},K}\right)=\left(Z_{k_{0}}\backslash
F(k_{0})\right)_{f}\times_{\tilde{f}}I_{k_{0},K},$

which is homeomorphic to $I_{k_{0},K}$ as $Z_{k_{0}}\backslash F(k_{0})$ is
homeomorphic to $C(k_{0},K)$. As a closed subspace of $K\times S$,
$I_{k_{0},K}$ is compact. We conclude that
$Z_{k_{0}}\backslash\tilde{X}^{k_{0}}$ is compact. ∎

For an elliptic element $g$, the fixed point set could be non-empty. To tackle
this situation, let $c_{G}$ be a cut-off function associated to the right
$Z_{g}$ action on $G$. Following [Hochs-Wang-HC, Lemma 3.2], we have the
following equivalent expression for $\tau^{X}_{g}(D\exp(-tD^{2}))$,

(5.23)
$\begin{split}\tau^{X}_{g}(D\exp(-tD^{2}))&=\int_{G/Z_{g}}\int_{X}c(y){\rm
tr}(\kappa_{t}(y,hgh^{-1}y)hgh^{-1})dydh(Z_{g})\\\
&=\int_{G}c_{G}(h)\int_{X}c(y){\rm tr}(\kappa_{t}(y,hgh^{-1}y)hgh^{-1})dydh\\\
&=\int_{X}\int_{G}c_{G}(h)c(y){\rm
tr}(h\kappa_{t}(h^{-1}y,gh^{-1}y)gh^{-1})dhdy\\\
&=\int_{X}\int_{G}c_{G}(h)c(hy){\rm tr}(\kappa_{t}(y,gy)g)dhdy.\end{split}$

Recall that $V$ is the support of the cut-off function $c$. We apply Lemma
5.13 to $V$ and conclude that there is a constant $C_{0}>0$ such that

$d_{X}(hgh^{-1}x,x)\geq C_{1}d_{G}(hgh^{-1}e,e)-C_{0},\forall x\in V,h\in G.$

We introduce two subsets of the conjugacy class $C(g)$, i.e.

$K(g):=\\{g\in C(g)|d_{G}(ge,e)\leq 2C_{0}/C_{1}\\},\ K^{c}(g):=\\{g\in
C(g)|d_{G}(ge,e)>2C_{0}/C_{1}\\}.$

Define a map $\chi:G\to C(g)$ by $\chi(h):=hgh^{-1}$. Let $U(g)$ be the subset
of $G$ defined to be the preimage of $K(g)$. And let $U^{c}(g)$ be the
complement of $U(g)$ in $G$. As $g$ is semisimple, $C(g)$ is a closed subset
of $G$, and therefore the image of $K(g)$ in $G/K$ is a bounded closed subset.
Therefore, the image of $K(g)$ in $G/K$ is compact, and $K(g)$ is compact.
Observe that $U(g)$ is invariant under the right $Z_{g}$ action and the
quotient $U(g)/Z_{g}$ is diffeomorphic to $K(g)$. So $U(g)/Z_{g}$ is also
compact. We split the integral (5.23) into two parts,

$\begin{split}\tau^{X}_{g}(D\exp(-tD^{2}))&=\int_{X}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy\\\ &=\int_{X}\int_{U(g)}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy+\int_{X}\int_{U^{c}(g)}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy))dhdy.\end{split}$

######  5.24 Lemma.

Let $\widetilde{W}$ be a closed subset of $X$. The integral

(5.25)
$I^{t}_{\widetilde{W},U^{c}}(g):=\int_{\widetilde{W}}\int_{U^{c}(g)}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy$

is of exponential decay as $t\to 0$.

###### Proof.

It is sufficient to prove the property for $\widetilde{W}=X$. We rewrite the
integral (5.25) by applying the change of variable $x=hy$.

$\begin{split}I^{t}_{U^{c}}(g)&:=\int_{X}\int_{U^{c}(g)}c_{G}(h)c(x){\rm
tr}(\kappa_{t}(x,hgh^{-1}x)hgh^{-1})dhdx\\\
&=\int_{U^{c}(g)}c_{G}(h)\int_{V}c(x){\rm
tr}(\kappa_{t}(x,hgh^{-1}x)hgh^{-1})dhdx.\end{split}$

By Lemma 5.18, we have the following estimate for $|{\rm
tr}(\kappa_{t}(x,hgh^{-1}x)hgh^{-1})|$,

$\begin{split}||\kappa_{t}(x,hgh^{-1}x)||&\leq\beta
t^{-\frac{n+1}{2}}\exp\left(-\alpha\frac{d_{X}^{2}(x,hgh^{-1}x)}{t}\right),\\\
|{\rm
tr}(\kappa_{t}(x,hgh^{-1}x)hgh^{-1})|&\leq\tilde{\beta}t^{-\frac{n+1}{2}}\exp\left(-\alpha\frac{d_{X}^{2}(x,hgh^{-1}x)}{t}\right),\qquad\forall
x\in X.\end{split}$

Applying Lemma 5.13, we get

$d_{X}(x,hgh^{-1}x)\geq C_{1}d_{G}(hgh^{-1}e,e)-C_{0}.$

By the definition of $U^{c}(g)$, for $h\in U^{c}(g)$, we have

$d_{X}(x,hgh^{-1}x)\geq C_{0},\
d_{X}(x,hgh^{-1}x)\geq\frac{C_{1}}{2}d_{G}(hgh^{-1}e,e),$

and accordingly

$d_{X}(x,hgh^{-1}x)\geq\frac{C_{0}}{2}+\frac{C_{1}d_{G}(hgh^{-1}e,e)}{4},\
d_{X}(x,hgh^{-1}x)^{2}\geq\frac{C^{2}_{0}}{4}+\frac{C^{2}_{1}d_{G}(hgh^{-1}e,e)^{2}}{16}.$

Hence, we have the estimate

$|{\rm
tr}(\kappa_{t}(x,hgh^{-1}x)hgh^{-1})|\leq\tilde{\beta}t^{-\frac{n+1}{2}}\exp\left(-\frac{\alpha
C_{0}^{2}}{4t}\right)\exp\left(-\alpha
C_{1}^{2}\frac{d_{G}^{2}(hgh^{-1}e,e)}{16t}\right).$

As $V$ is compact, the integral

$\int_{V}c(x)dx$

is finite. And $I^{t}_{U^{c}}(g)$ can be bounded by

$\begin{split}|I^{t}_{U^{c}}(g)|\leq&\gamma
t^{-\frac{n+1}{2}}\exp\left(-\frac{\alpha
C_{0}^{2}}{4t}\right)\int_{U^{c}(g)}c(g)\exp\left(-\alpha
C_{1}^{2}\frac{d^{2}_{G}(hgh^{-1}e,e)}{16t}\right)dh\\\ \leq&\gamma
t^{-\frac{n+1}{2}}\exp\left(-\frac{\alpha
C_{0}^{2}}{4t}\right)\int_{G/Z_{g}}\left(-\alpha
C_{1}^{2}\frac{d^{2}_{G}(hgh^{-1}e,e)}{16t}\right)d(hZ_{g}).\end{split}$

This implies the exponential decay property of $I^{t}_{U^{c}}(g)$. ∎

Let $W$ be an open subset of $X$ containing the $g$ fixed point submanifold
$X^{g}$.

######  5.26 Lemma.

The integral

$I^{t}_{W^{c},U}(g):=\int_{W^{c}}\int_{U(g)}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy$

converges exponentially to 0 as $t\to 0$.

###### Proof.

Because $U(g)/Z_{g}$ is compact and $c_{G}$ is the cut-off function for the
$Z_{g}$-action on $G$, we know that $c_{G}$ is compactly supported on $U(g)$.
Moreover, since $c$ is compactly supported, the properness of the $G$ action
on $X$ implies that the function

$\tilde{c}(y):=\int_{U(g)}c_{G}(h)c(hy)dh$

is also compactly supported. Let $\tilde{V}$ be the support of $\tilde{c}$.
And $I^{t}_{W^{c},U}(g)$ can be expressed as

$\int_{W^{c}\cap\tilde{V}}\tilde{c}(y){\rm tr}(\kappa_{t}(y,gy)gdy.$

Using Lemma 5.18, we have the estimate

$|{\rm
tr}(\kappa_{t}(y,gy)g)|\leq\tilde{\beta}t^{-\frac{n+1}{2}}\exp\left(-\alpha\frac{d^{2}_{X}(y,gy)}{t}\right)\forall
y\in X.$

As $W^{c}\cap\tilde{V}$ is compact and does not contain any $g$ fixed point,
there is a positive $\epsilon_{0}$ such that

$d_{X}(y,gy)\geq\epsilon_{0},\ \forall y\in W^{c}\cap\tilde{V}.$

We can bound $I^{t}_{W^{c},U}(g)$ using the above estimate about $d(y,gy)$,

$|I^{t}_{W^{c},U}(g)|\leq\tilde{\beta}t^{-\frac{n+1}{2}}\exp\left(-\alpha\frac{\epsilon_{0}^{2}}{t}\right)\int_{W^{c}\cap\tilde{V}}|\tilde{c}(y)|dy.$

This gives the exponential decay property of $I^{t}_{W^{c},U}(g)$. ∎

We can summarize Lemma 5.24 and 5.26 in the following result.

######  5.27 Proposition.

Given any open set $W$ containing the $g$ fixed point submanifold $X^{g}$,
then the integral

$I^{t}_{W^{c}}(g):=\int_{W^{c}}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy=I^{t}_{W^{c},U}(g)+I^{t}_{W^{c},U^{c}}(g).$

decay exponentially to zero at $t\to 0$. When $W$ is assumed to be $g$
invariant, $I^{t}_{W^{c}}(g)$ has the following expression under change of
coordinates

$I^{t}_{W^{c}}(g)=\int_{W^{c}}\int_{G}c_{G}(h)c(hgx){\rm
tr}(\kappa_{t}(x,gx)g)dhdx.$

Let $W$ be a sufficiently small neighborhood of $X^{g}$. Using the
$Z_{g}$-invariant tubular neighborhood theorem, we can identify $W$ to be an
$r$-ball bundle $N(r)$ in the normal bundle $N$ of $X^{g}$ in $X$ with a
sufficiently small radius $r$. Define $c_{g}$, a smooth function on $X$, by

$c_{g}(x):=\int_{G}c_{G}(h)c(hgx)dh.$

######  5.28 Lemma.

The function $c_{g}$ is a cut-off function for the $Z_{g}$-action on $X$. In
particular, for any $x\in X$,

  1. (1)

the integral

$\int_{Z_{g}}c_{g}(hx)=1;$

  2. (2)

Let $q:X\to X/Z_{g}$ be the quotient map. The intersection of the support of
$c_{g}$ with $q^{-1}(V^{\prime})$ is compact for any compact subset
$V^{\prime}$ in $X/Z_{g}$.

######  5.29 Remark.

Part (1) of Lemma 5.28 is proved in [Hochs-Wang-HC, Lemma 4.11].

###### Proof.

For part (1), compute that

$\int_{Z_{g}}c_{g}(zx)dz=\int_{Z_{g}}\int_{G}c_{G}(h)c(hgzx)\;dhdz=\int_{Z_{g}}\int_{G}c_{G}(h^{\prime}z^{\prime-1})c(h^{\prime}x)\;dhdz=1$

where we substitute $h^{\prime}=hgz$ and $z^{\prime}=gz$.

For part (2), suppose that $V^{\prime}$ is a compact subset in $X/Z_{g}$. The
intersection of the support of $c_{g}$ with $q^{-1}(V^{\prime})$ consists is
the closure of the following set,

$Q_{g}(V^{\prime}):=\\{x\in X:c_{g}(x)\neq 0,q(x)\in V^{\prime}\\}.$

Recall that

$c_{g}(x)=\int_{G}c_{G}(h)c(hgx)dh$

Hence, $c_{g}(x)\neq 0$ if and only if there is some $h$ such

$c_{G}(h)\neq 0,\ {\text{and}}\ c(hgx)\neq 0.$

Using this characterization of $c_{g}(x)\neq 0$, we have

$Q_{g}(V^{\prime})=\\{x\in X:\exists h\in G,c_{G}(h)\neq 0,c(hgx)\neq
0,q(x)\in V^{\prime}\\}.$

Observe that $g$ commutes with $Z_{g}$. So $g$ acts on the quotient $X/Z_{g}$,
and the map $q:X\to X/Z_{g}$ is $g$-equivariant. Hence, substituting
$x^{\prime}=gx$, we obtain the following expression,

$Q_{g}(V^{\prime})=g^{-1}\left(\\{x^{\prime}\in X:\exists h\in G,c_{G}(h)\neq
0,c(hx^{\prime})\neq 0,q(x^{\prime})\in g^{-1}(V^{\prime})\\}\right).$

As $V^{\prime}$ is compact in $X/Z_{g}$, $g^{-1}(V^{\prime})$ is also compact.
We can assume without loss of generality that
$q^{-1}\big{(}g^{-1}(V^{\prime})\big{)}$ is of the form

$q^{-1}\big{(}g^{-1}(V^{\prime})\big{)}=Z_{g}V^{\prime\prime},$

where $V^{\prime\prime}$ is a compact subset of $X$.

Consider the set $G(V^{\prime\prime}):=\\{h|\exists x^{\prime}\in
V^{\prime\prime},c(hx^{\prime})\neq 0\\}$. Using the following properties

  * •

the set $V^{\prime\prime}$ is compact,

  * •

the support of $c$ is compact,

  * •

the $G$ action on $X$ is proper,

we conclude that $G(V^{\prime\prime})$ is relatively compact.

Consider the set $P_{g}(V^{\prime\prime}):=\\{h|c(hx^{\prime})\neq 0,\exists
x^{\prime}\in Z_{g}V^{\prime\prime}\\}$. We have the property that
$P_{g}(V^{\prime\prime})=G(V^{\prime\prime})Z_{g}$. Hence, we conclude from
the relative compactness of $G(V^{\prime\prime})$ that
$P_{g}(V^{\prime\prime})/Z_{g}$ is relatively compact.

Consider the set $R_{g}(V^{\prime\prime}):=\\{h|\exists x^{\prime}\in
Z_{g}V^{\prime\prime},c_{G}(h)\neq 0,c(hx^{\prime})\neq 0\\}$.
$R_{g}(V^{\prime\prime})$ is the intersection of $P_{g}(V^{\prime\prime})$
with the support of $c_{G}$. As $c_{G}$ is a cutoff function of the right
$Z_{g}$ action on $G$, we conclude from the relative compactness of
$P_{g}(V^{\prime\prime})/Z_{g}$ that $R_{g}(V^{\prime\prime})$ is compact.

Observe that the set

$g(Q_{g}(V))=\\{x^{\prime}\in X:\exists h\in G,c_{G}(h)\neq
0,c(hx^{\prime})\neq 0,x^{\prime}\in q{x^{\prime}}\in g^{-1}(V^{\prime})\\}$

is a subset of

$\\{x^{\prime}\in X:\exists h\in R_{g}(V^{\prime\prime}),c(gx^{\prime})\neq
0\\}.$

As $R_{g}(V^{\prime\prime})$ is compact, we conclude from the compactness of
the support of $c$ and properness of the $G$ action that
$g(Q_{g}(V^{\prime}))$ is relatively compact, and therefore
$Q_{g}(V^{\prime})$ is also relatively compact.

∎

The integral

$I^{t}_{N(r)}(g):=\int_{N(r)}\int_{G}c_{G}(h)c(hgy){\rm
tr}(\kappa_{t}(y,gy)g)dhdy$

can be expressed by

$\int_{N(r)}c_{g}(y){\rm tr}(\kappa_{t}(y,gy)g)dy.$

By Lemma 5.28, we know that the set $X^{g}/Z_{g}$ is compact. Thus
$N(r)/Z_{g}$ is relatively compact. It follows from Lemma 5.28 that $c_{g}$
has relative compact support in $N(r)$ and is a smooth function on $N(r)$ with
bounded derivatives. This allows us to use the method and estimates in the
proof of [Zhangwp, Equation (2.2)] to prove the following lemma.

######  5.30 Lemma.

The integral $I^{t}_{N(r)}(g)$ satisfies

$\lim_{t\to 0}\frac{|I^{t}_{N(r)}(g)|}{\sqrt{t}}<+\infty.$

###### Proof.

We use the local asymptotic expansion [Zhangwp, Equation (2.16)]

$g\kappa_{t}(y,g)=\frac{e^{-\frac{d_{X}(y,gy)^{2}}{4t}}}{(4\pi
t)^{\frac{n}{2}}}dg\left[\sum_{i}\left(((dg-I)y)_{i}e_{i}\right)\left(\sum_{i=0}^{[n/2]+2}U_{i}t^{i}+o(t^{[n/2]+2})\right)+\left(\sum_{i=0}^{[n/2]+2}V_{i}t^{i}+o(t^{[n/2]+2})\right)\right].$

In [Zhangwp], the author assumed to work with an isometry group action on a
closed manifold $M$. For our case, as the support $c_{g}$ in $N(r)$ is
relatively compact and $g$ is contained in the group $Z_{g}$ which acts
properly on $N(r)$, the author’s analysis in [Zhangwp, Section 2] in the
tubular neighborhood $N(r)$ is also valid in our case near ${\rm
supp}(c_{g})\cap N(r)$. We refer to [Zhangwp, Section 2] for the explanation
of the above formula.

We use the above formula to compute $c_{g}(y){\rm tr}(\kappa_{t}(y,gy)g)$,

$\begin{split}&c_{g}(y){\rm tr}(\kappa_{t}(y,gy)g)=\\\
&\frac{e^{-\frac{d(y,gy)^{2}}{4t}}}{(4\pi
t)^{\frac{n}{2}}}dg\left[\sum_{i}\left(((dg-I)y)_{i}e_{i}\right)\left(\sum_{i=0}^{[n/2]+2}c_{g}(y)U_{i}t^{i}+o(t^{[n/2]+2})\right)+\left(\sum_{i=0}^{[n/2]+2}c_{g}(y)V_{i}t^{i}+o(t^{[n/2]+2})\right)\right].\end{split}$

Apply [Zhangwp, Lemma 2.17] to the integral of each term in the above
expansion of $c_{g}(y){\rm tr}(\kappa_{t}(y,gy)g)$. We can conclude the
estimate in the lemma following the same argument as the one for [Zhangwp,
Equation (2.2)], that is,

$\lim_{t\to 0^{+}}\frac{1}{\sqrt{t}}\left|c_{g}(y){\rm
tr}(\kappa_{t}(y,gy)g)\right|<C$

for some constant $C>0$. ∎

Combing Proposition 5.27 and Lemma 5.30, we have reached the following
proposition.

######  5.31 Proposition.

Let $X$ be a cocompact $G$-proper manifold without boundary and let $D$ be a
Dirac-type operator defined in (3.2). Let $g$ be a semi-simple element in $G$.
The integral

$\frac{1}{\sqrt{\pi}}\int_{0}^{1}\tau^{X}_{g}(D\exp(-tD^{2})\frac{dt}{\sqrt{t}}$

converges.

###  5.4. Short time limit and the Atiyah-Segal integrand

If we replace $D\exp(-sD^{2})$ by $\exp(-sD^{2})$, a similar argument as in
Theorem 5.1 proves the topological formula stated in [Hochs-Wang-HC, Theorem
2.8] for the pairing between $\tau_{g}$ and the index element
$\operatorname{ind}_{G}(D)$ for a Dirac type operator $D$ on a $G$-proper
cocompact manifold $X$ without boundary, as we shall now explain. We know that

$\langle\operatorname{Ind}_{G}(D),\tau_{g}\rangle=\langle\operatorname{Ind}(D),\tau_{g}^{X}\rangle=\tau^{X}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{X}_{g}(e^{-s^{2}D^{+}D^{-}})\,.$

We compute the left hand side by taking the limit as $s\downarrow 0$ and using
the following

######  5.32 Theorem.

Let $(X,h)$ be a cocompact $G$-proper manifold without boundary. Assume that
$X$, $G/K$, and the slice $S$ are all even dimensional. Let $D$ be a
G-equivariant Dirac operator defined in (3.2). Let $g$ be a semi-simple
element and let $X^{g}$ be the fixed point set of $g$. Then the limit
$\lim_{s\downarrow
0}\tau_{g}(e^{-s^{2}D^{-}D^{+}})-\tau_{g}(e^{-s^{2}D^{+}D^{-}})$ exists and we
have

$\lim_{s\downarrow
0}\tau^{X}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{X}_{g}(e^{-s^{2}D^{+}D^{-}})=\int_{X^{g}}c^{g}{\rm
AS}_{g}(D)$

with $c^{g}{\rm AS}_{g}(D)$ defined in Equation (1.6). Consequently we obtain
the following formula

$\langle\operatorname{Ind}(D),\tau_{g}^{X}\rangle=\int_{X^{g}}c^{g}{\rm
AS}_{g}(D)\,.$

###### Proof.

Let $\kappa_{s}$ be the kernel of the heat operator $\exp(-sD^{2})$. Observe
that in term of supertrace,
$\tau^{X}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{X}_{g}(e^{-s^{2}D^{+}D^{-}})$ can be
written as the integral

(5.33)
$J(s):=\int_{G/Z_{g}}\int_{X}c(hgh^{-1}x)\operatorname{str}(hgh^{-1}\kappa_{s^{2}}(hg^{-1}h^{-1}x,x))dxd(hZ).$

By a similar computation as in Equation (5.23), we can express $J(s)$ defined
by Equation (5.33) as follows,

$J(s)=\int_{X}\int_{G}c_{G}(h)c(hy){\rm str}(\kappa_{s^{2}}(y,gy)g)dydh,$

where $c_{G}$ is the cut-off function associated to the right $Z_{g}$ action
on $G$. We can follow the exactly same strategy as in Section 5.3 to study the
limit of $J(s)$ as $s\to 0$. Below is a brief outline of steps.

  1. (1)

We observe that the analysis on the small time behavior the operator
$D\exp(-sD^{2})$ in Section 5.3 also holds for the operator $\exp(-sD^{2})$.
In particular, similar to Lemma 5.18, there are constant
$\alpha_{0},\beta_{0}>0$ such that

$||\kappa_{s}(x,y)||\leq\beta_{0}s^{-\frac{n}{2}}\exp\left(-\alpha\frac{d_{X}(x,y)^{2}}{s}\right).$

  2. (2)

Similar to Proposition 5.27, using the above Gaussian estimate of the heat
kernel $\kappa_{t}(x,y)$, we prove that given any $Z_{g}$-invariant open set
$W$ containing the $g$ fixed point submanifold $X^{g}$, the integral

$J_{W^{c}}(s):=\int_{W^{c}}\int_{G}c_{G}(h)c(hy)\operatorname{str}(\kappa_{s^{2}}(y,gy)g)dhdy=\int_{W^{c}}\int_{G}c_{G}(h)c(hgx)\operatorname{str}(\kappa_{s^{2}}(x,gx)g)dhdx$

decays exponentially to zero as $s\to 0$. The last equality holds because of
the invariance property of $W$.

  3. (3)

Following Lemma 5.30, we choose $W$ to be an $r$-ball bundle $N(r)$ in the
normal bundle $N$ of $X^{g}$ in $X$ with a sufficiently small radius $r$. For
the integral

$J_{N(r)}(s):=\int_{N(r)}\int_{G}c_{G}(h)c(hy)\operatorname{str}(\kappa_{s^{2}}(y,gy)g)dhdy,$

we follow the local analysis as in the proofs [BGV, Theorem 6.16] and [Hochs-
Wang-HC, Proposition 4.12] to get

$\lim_{s\to 0}J_{N(r)}(s)=\int_{X^{g}}c^{g}\operatorname{AS}_{g}(D).$

Combining the above (1)-(3), we reach the desired equality

$\lim_{s\downarrow
0}\tau^{X}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{X}_{g}(e^{-s^{2}D^{+}D^{-}})=\int_{X^{g}}c^{g}{\rm
AS}_{g}(D).$

∎

######  5.34 Remark.

Theorem 5.32 was originally stated by Hochs and Wang [Hochs-Wang-HC, Theorem
2.8] with a stronger assumption on the group element $g$. Our strategy of
proof is similar to the one in [Hochs-Wang-HC]. However, our proof does not
use the inequality (4.6) in [Hochs-Wang-HC] for the heat kernel
$\widetilde{\kappa}_{t}^{G,K}$, the proof of which is not clear to us.
Instead, we use the global Gaussian estimates for the heat kernel
$e^{-tD^{2}}$, c.f. Lemma 5.18, in order to establish, as a crucial step, the
exponential decay property of $\tau_{g}\big{(}\exp(-tD^{2})\big{)}$ outside a
neighborhood $W$ of the fixed point submanifold $X^{g}$ as in Proposition
5.27. Notice also that in [Hochs-Wang-HC] the split Dirac operator (3.6) is
used throughout and we know that this operator is not associated to a Clifford
connection.

We are now in the position to prove Proposition 4.38 that we restate here for
the benefit of the reader:

Let $(Y_{0},\mathbf{h}_{0})$ be a cocompact $G$-proper Riemannian manifold.
Let $(Y,\mathbf{h})$ be the associated $b$-manifold. Let $D_{0}$ be a
G-equivariant Dirac operator defined in (3.2) and let $D$ be the associated
$b$-differential operator. Let $g$ be a semisimple element and let $Y_{0}^{g}$
the fixed point set of $g$. Then the limit $\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})$
exists and we have

$\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})=\int_{Y_{0}^{g}}c^{g}_{0}{\rm
AS}_{g}(D_{0})$

with $c^{g}_{0}{\rm AS}_{g}(D_{0})$ defined in Equation (1.6).

###### Proof.

Let $\kappa_{t}$ be the kernel of the heat operator $\exp(-tD^{2})$. By
definition

$\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})$

is equal to

(5.35)
${}^{b}J(s):=\int_{G/Z_{g}}\int^{b}_{Y}c(hgh^{-1}y)\operatorname{str}(hgh^{-1}\kappa_{s^{2}}(hg^{-1}h^{-1}y,y))dyd(hZ)$

with $dy$ denoting the b-density associated to the $b$-metric $\mathbf{h}$. We
claim that the following expression holds:

(5.36) ${}^{b}J(s)=\int^{b}_{Y}\int_{G}c_{G}(h)c(hy){\rm
str}(\kappa_{s^{2}}(y,gy)g)dhdy$

where $c_{G}$ is the cut-off function associated to the right $Z_{g}$ action
on $G$. In order to prove (5.36) we make a preliminary remark. The spaces of
$b$-pseudodifferential operators ${}^{b}\mathcal{L}_{G}^{c}(Y,E)$ and
${}^{b}\mathcal{L}_{G,s}^{\infty}(Y,E)$ have been defined in terms of the
slice theorem, as projective tensor products. We could have defined these
operators directly, in terms of a $b$-stretched product $Y\times_{b}Y$.
Proceeding exactly as in [LeichtnamPiazzaMemoires, Chapter 4] we see that
$Y\times_{b}Y$ inherits an action of $G\times G$; in particular, it makes
sense to consider $L^{*}_{(h,h^{\prime})}\exp(-tD^{2})$ which is a smoothing
$b$-kernel (not $G$-invariant unless $h=h^{\prime}$). Coming back to the
claim, we can rewrite the right hand side of (5.35) as

$\int_{G/Z_{g}}\int^{b}_{Y}c(y)\,{\rm
str}(\kappa_{s^{2}}(y,hgh^{-1}y)hgh^{-1})dyd(hZ)$

which is in turn equal to

(5.37) $\int_{G}c_{G}(h)\int^{b}_{Y}c(y)\,{\rm
str}(\kappa_{s^{2}}(y,hgh^{-1}y)hgh^{-1})dydh.$

Consider the smoothing $b$-kernel

$\Theta_{g}(h):=L_{e,hgh^{-1}}^{*}(\exp(-(sD)^{2}));$

we have that

(5.38) $\int_{G}c_{G}(h)\int^{b}_{Y}c(y)\,{\rm
str}(\kappa_{s^{2}}(y,hgh^{-1}y)hgh^{-1})dydh=\int_{G}c_{G}(h)\int^{b}_{Y}c(y)\,{\rm
str}(\kappa(\Theta_{g}(h))(y,y)dydh.$

Write $\int^{b}_{Y}c(y)\,{\rm str}(\kappa(\Theta_{g}(h))(y,y)dy$ as

$\int_{Y_{0}}c_{0}(y_{0})\,{\rm
str}(\kappa(\Theta_{g}(h))(y_{0},y_{0})dy_{0}+\int^{b}_{(-\infty,0]_{t}\times\partial
Y_{0}}c_{0}(x)\,{\rm str}(\kappa(\Theta_{g}(h))(t,x,t,x)dtdx.$

Consider the function $F_{g}(h)$ on $Y$, so defined:

\- $F_{g}(h)(y):=c_{0}(y_{0})\,{\rm str}(\kappa(\Theta_{g}(h))(y_{0},y_{0})$
if $y=y_{0}\in Y_{0}$;

\- $F_{g}(h)(y):=-t\frac{d}{dt}\left(c_{0}(x)\,{\rm
str}(\kappa(\Theta_{g}(h))(t,x,t,x)\right)$ if
$y=(t,x)\in(-\infty,0]_{t}\times\partial Y_{0}$.
The function $F_{g}(h)$ is discontinuous at $\partial Y_{0}$, a set of measure
0, but it is otherwise smooth. By [LMP, Prop. 2.6] we have that

$\int_{G}c_{G}(h)\int^{b}_{Y}c(y)\,{\rm
str}(\kappa(\Theta_{g}(h))(y,y)dydh=\int_{G}\int_{Y}c_{G}(h)F_{g}(h)dydh.$

We know that the double integral on the right hand side is absolutely
convergent; we can now proceed as in [Hochs-Wang-HC], use Fubini’s theorem and
interchange the two integrals in the right hand side of the above formula,
obtaining

$\int_{Y}\int_{G}c_{G}(h)F_{g}(h)dydh\,.$

Going back to the $b$-integral we obtaining finally that (5.37) is equal to

$\int^{b}_{Y}\int_{G}c_{G}(h)c(y)\,{\rm
str}(\kappa_{s^{2}}(y,hgh^{-1}y)hgh^{-1})dydh$

which is easily seen to be equal to

$\int^{b}_{Y}\int_{G}c_{G}(h)c(hy){\rm str}(k_{s^{2}}(y,gy)g)dhdy$

as claimed. Summarizing:

(5.39)
$\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})=\int^{b}_{Y}\int_{G}c_{G}(h)c(hy){\rm
str}(k_{s^{2}}(y,gy)g)dhdy\,.$

We now consider the fixed point set $Y_{0}^{g}\subset Y_{0}$ and the
associated $Y^{g}\subset Y$. We fix a $Z_{g}$-invariant open set $W_{0}$
containing $Y_{0}^{g}$ and the associated $W\subset Y$ containing $Y^{g}$. We
choose $W_{0}$ to be an $r$-ball bundle $N_{0}(r)$ in the normal bundle
$N_{0}$ of $Y^{g}_{0}$ in $Y_{0}$ with a sufficiently small radius $r$. We can
then decompose the right hand side of (5.39) as:

(5.40) $\int^{b}_{W^{c}}\int_{G}c_{G}(h)c(hy){\rm
str}(k_{s^{2}}(y,gy)g)dhdy+\int^{b}_{W}\int_{G}c_{G}(h)c(hy){\rm
str}(k_{s^{2}}(y,gy)g)dhdy$

Recall that if $Z$ is a b-manifold obtained from a manifold with boundary
$Z_{0}$ and $\phi$ is a $b$-density obtained as an extension from $Z_{0}$ of a
density $\phi_{0}$ on $Z_{0}$, then

$\int^{b}_{Z}\phi=\int_{Z_{0}}\phi_{0}\,.$

In particular if $c_{0}$ is a cut-off function for the $G$ action on $Y_{0}$
and $c$ is obtained by extending constantly along the cylindrical end, then

$\int^{b}_{Y}c(y)d{\rm vol}_{b}=\int_{Y_{0}}c_{0}(y_{0})d{\rm vol}\,.$

Thanks to Remark 5.19 we know that there are constant $\alpha_{0},\beta_{0}>0$
such that

$||k_{t}(x,y)||\leq\beta_{0}t^{-\frac{n}{2}}\exp\left(-\alpha\frac{d_{Y}(x,y)^{2}}{t}\right).$

Using this global Gaussian estimate and proceeding exactly as in the closed
case we find that the first integral in (5.40) converges exponentially to $0$
(here we use the above remark about the cut-off function $c$; it is not
compactly supported but its $b$-integral is equal to the integral of a
compactly supported function). We are left with the second integral in (5.40).
In this case $Y^{g}_{0}$ is compact and $W_{0}:=N_{0}(r)$ is relatively
compact; put it differently, $Y^{g}$ and $W$ are obtained attaching cylinders
to compact (respectively relatively compact) spaces. We can thus analyze

$\int^{b}_{N(r)}\int_{G}c_{G}(h)c(hy){\rm str}(k_{s^{2}}(y,gy)g)dhdy$

using Getzler rescaling and find that the limit as $s\downarrow 0$ of this
double integral is equal to

$\int^{b}_{Y^{g}}c^{g}{\rm AS}_{g}(D).$

As all the structures are product-like near the boundary we see that

$\int^{b}_{Y^{g}}c^{g}{\rm AS}_{g}(D)=\int_{Y_{0}^{g}}c_{0}^{g}{\rm
AS}_{g}(D_{0})$

which is what we wanted to prove. ∎

##  6\. Higher delocalized cyclic cocycles

In this section we shall introduce higher delocalized cyclic cochains.

Let $K<G$ be a maximal compact subgroup and let $P<G$, $P=MAN$, be a cuspidal
parabolic subgroup of $G$. By the Iwasawa decomposition $G=KMAN$ we can write
an element $g\in G$ as

$g=\kappa(g)\mu(g)e^{H(g)}n\in KMAN=G.$

Let $\operatorname{dim}(A)=m$. By choosing coordinates of the Lie algebra
$\mathfrak{a}$ of $A$, consider the function

$H=(H_{1},\dots,H_{m}):G\to\mathfrak{a}.$

We define the following cyclic cochain on the algebra $\mathcal{L}_{s}(G)$,
the Lafforgue Schwartz algebra. For $f_{0},...,f_{m}\in\mathcal{L}_{s}(G)$ and
a semi-simple element $g\in M$, define $\Phi^{P}_{g}$ by the following
integral,

(6.1) $\displaystyle\Phi^{P}_{g}(f_{0},f_{1},$ $\displaystyle\dots,f_{m})$
$\displaystyle:=$ $\displaystyle\int_{h\in M/Z_{M}(g)}\int_{KN}\int_{G^{\times
m}}\sum_{\tau\in S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)$ $\displaystyle f_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}\big{)}f_{1}(g_{1})\dots f_{m}(g_{m})dg_{1}\cdots dg_{m}dkdndh,$

where $Z_{M}(g)$ is the centralizer of $g$ in $M$. The following result is
proved in the work of Song-Tang, [st, Theorem 3.5].

######  6.2 Proposition.

$\Phi^{P}_{g}$ is a cyclic cocycle on the Harish-Chandra Schwartz algebra
$\mathcal{L}_{s}(G)$.

######  6.3 Definition.

We say that a cuspidal parabolic subgroup $P=MAN$ is _maximal_ if one of the
following equivalent conditions holds

  * •

the dimension of $A$ is minimal;

  * •

the rank of $M$ is maximal;

  * •

the rank of $M$ equals the rank of $K$.

In [hst, Corollary 6.3], the authors showed that $\Phi^{P}_{g}$ is trivial in
the cyclic cohomology of $\mathcal{L}_{s}(G)$ unless $P$ is maximal.

Let $Y_{0}$ be an even dimensional cocompact $G$-proper manifold with boundary
and let $Y$ be the associated $b$-manifold. We shall now use $\Phi^{P}_{g}$ in
order to define a cyclic cocycle $\Phi^{P}_{Y,g}$ on the algebra
$\mathcal{L}^{\infty}_{G,s}(Y)$; subsequently we shall use $\Phi^{P}_{Y,g}$ in
order to define a relative cyclic cocycle
$(\Phi^{r,P}_{Y,g},\sigma^{P}_{\partial Y,g})$ for the indicial homomorphism
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$. These algebras involve the choice of an $\epsilon$ strictly smaller than
half of the width of the spectral gap for $D_{\partial Y}$. We fix such an
$\epsilon$ and we choose it in any case smaller than $1$.

Recall, see [PP2], that if $S$ is a slice for the action of $G$ on $Y$, then
we have an identification

$\mathcal{L}^{c}_{G}(Y)\cong\left\\{F:G\to\rho_{bf}{}^{b}\Psi^{-\infty,\epsilon}(S),~{}K\times
K~{}\mbox{equivariant, continuous and of compact support in}~{}G\right\\}.$

Recall that
$\rho_{bf}{}^{b}\Psi^{-\infty}(S)\subset\Psi^{-\infty,\epsilon}(S)$ if
$\epsilon<1$, which is in turn contained in the trace class operators on
$L^{2}_{b}(S)$.
We have

${}^{b}\mathcal{L}^{c}_{G}(Y)\cong\left\\{F:G\to{}^{b}\Psi^{-\infty,\epsilon}(S),~{}K\times
K~{}\mbox{equivariant, continuous and of compact support in}~{}G\right\\}.$

Finally, by Fourier transform we have an injection

${}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}(\operatorname{cyl}(\partial
Y))\hookrightarrow\mathcal{S}(\mathbb{R},\mathcal{L}^{c}_{G}(\partial Y))\,.$

######  6.4 Definition.

For a semisimple element $g\in M$ and
$A_{0},...,A_{m}\in{\mathcal{L}}^{c}_{G}(Y)$, define a cochain
$\Phi^{P}_{Y,g}$ on $\mathcal{L}^{c}_{G}(Y)$ by

$\displaystyle\Phi^{P}_{Y,g}(A_{0},A_{1},\dots,A_{m})$ $\displaystyle:=$
$\displaystyle\int_{h\in M/Z_{M}(g)}\int_{KN}\int_{G^{\times m}}\sum_{\tau\in
S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)$
$\displaystyle\operatorname{Tr}\Big{(}A_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}\big{)}\circ A_{1}(g_{1})\dots\circ A_{m}(g_{m})\Big{)}dg_{1}\cdots
dg_{m}dkdndh.$

Proceeding as in Song-Tang one can prove that this is in fact a cyclic
cocycle.

Using always $\Phi^{P}_{g}$ on $\mathcal{L}_{s}(G)$ we shall now define a
relative cyclic cocycle $(\Phi^{r,P}_{Y,g},\sigma^{P}_{\partial Y,g})$ for the
homomorphism
${{}^{b}\mathcal{L}}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$.

######  6.5 Definition.

For a semisimple element $g\in M$ and
$A_{0},...,A_{m}\in{{}^{b}\mathcal{L}}^{c}_{G}(Y)$, define a cochain
$\Phi^{r,P}_{Y,g}$ on ${{}^{b}\mathcal{L}}^{c}_{G}(M)$ by

$\displaystyle\Phi^{r,P}_{Y,g}$ $\displaystyle(A_{0},A_{1},\dots,A_{m})$
$\displaystyle:=$ $\displaystyle\int_{h\in M/Z_{M}(g)}\int_{KN}\int_{G^{\times
m}}\sum_{\tau\in S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)$
$\displaystyle{{}^{b}\operatorname{Tr}}\Big{(}A_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}\big{)}\circ A_{1}(g_{1})\dots\circ A_{m}(g_{m})\Big{)}dg_{1}\cdots
dg_{m}dkdndh.$

For
$B_{0},...,B_{m+1}\in{{}^{b}\mathcal{L}}^{c}_{G,\mathbb{R}}(\text{cyl}(Y))$,
define a cochain $\sigma_{\partial Y,g}$ on
${{}^{b}\mathcal{L}}^{c}_{G,\mathbb{R}}(\text{cyl}(Y))$ by

$\displaystyle\sigma^{P}_{\partial Y,g}(B_{0},...,B_{m+1})$ $\displaystyle:=$
$\displaystyle\int_{h\in
M/Z_{M}(g)}\int_{KN}\int_{G^{\times{m+1}}}\int_{\mathbb{R}}\sum_{\tau\in
S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)$
$\displaystyle{\operatorname{Tr}}\Big{(}\hat{B}_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m}g_{m+1})^{-1},\lambda\big{)}\circ\hat{B}_{1}(g_{1},\lambda)\circ\dots\circ\hat{B}_{m}(g_{m},\lambda)\circ\frac{\partial\hat{B}_{m+1}(g_{m+1},\lambda)}{\partial\lambda}\Big{)}$
$\displaystyle\qquad\qquad dg_{1}\cdots dg_{m+1}dkdndhd\lambda,$

where we have used the Fourier transform

${}^{b}\mathcal{L}_{G,s,\mathbb{R}}^{c}(\operatorname{cyl}(Y))\ni
A\longrightarrow\widehat{A}\in\mathscr{S}(\mathbb{R},\mathcal{L}^{c}_{G}(Y)).$

######  6.6 Proposition.

Let $Y_{0}$ be a proper $G$ manifold with boundary and $Y$ be the associated
$b$-manifold. For a semisimple element $g\in M$, we have the following
identities:

$\left(\begin{array}[]{cc}(b+B)&-I^{*}\\\
0&-(b+B)\end{array}\right)\left(\begin{array}[]{c}\Phi^{r,P}_{Y,g}\\\
\sigma^{P}_{\partial Y,g}\end{array}\right)=\left(\begin{array}[]{c}0\\\
0\end{array}\right).$

Consequently, the pair $(\Phi^{r,P}_{Y,g},\sigma^{P}_{\partial Y,g})$ defines
a relative cyclic cocycle for the homomorphism
${{}^{b}\mathcal{L}}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$.

###### Proof.

The proof proceeds analogous to the proof of [PP2, Prop. 6.7.]. For the
Hochschild differential, we compute

$\displaystyle b$ $\displaystyle\Phi^{r,P}_{Y,g}(A_{0},\ldots,A_{m+1})$
$\displaystyle=\int_{h\in
M/Z_{M}(x)}\int_{KN}\int_{G^{\times(m+1)}}\Bigg{(}{{}^{b}\operatorname{Tr}}\Big{(}A_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}(g^{\prime})^{-1}\big{)}\circ A_{1}(g^{\prime})\circ
A_{2}(g_{1})\circ\dots\circ A_{m+1}(g_{m})\Big{)}$
$\displaystyle+\sum_{i=1}^{m}(-1)^{i}\left[{{}^{b}\operatorname{Tr}}\Big{(}A_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}\big{)}\circ A_{1}(g_{1})\circ\ldots\circ
A_{i}(g_{i}(g^{\prime})^{-1})\circ A_{i+1}(g^{\prime})\circ\dots\circ
A_{m+1}(g_{m})\Big{)}\right]$
$\displaystyle+(-1)^{m+1}\left[{{}^{b}\operatorname{Tr}}\Big{(}A_{m+1}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m})^{-1}(g^{\prime})^{-1}\big{)}\circ A_{0}(g^{\prime})\circ
A_{2}(g_{1})\circ\dots\circ A_{m}(g_{m})\Big{)}\right]\Bigg{)}$
$\displaystyle\hskip 85.35826pt\times\sum_{\tau\in
S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)dg_{1}\cdots dg_{m}dkdndhdg^{\prime}$
$\displaystyle=(-1)^{m+1}\int_{h\in
M/Z_{M}(x)}\int_{KN}\int_{G^{\times(m+1)}}{{}^{b}\operatorname{Tr}}\Big{(}\left[A_{1}(g_{1})\circ\cdots\circ
A_{m+1}(g_{m+1}),A_{0}\big{(}khgh^{-1}nk^{-1}(g_{1}\dots
g_{m+1})^{-1}\big{)}\right]$ $\displaystyle\hskip
85.35826pt\times\sum_{\tau\in S_{m}}\mathrm{sgn}(\tau)\cdot
H_{\tau(1)}(g_{1}...g_{m}k)H_{\tau(2)}(g_{2}...g_{m}k)\dots
H_{\tau(m)}(g_{m},k)dg_{1}\cdots dg_{m+1}dkdndh$
$\displaystyle=I^{*}\sigma^{P}_{\partial Y,g}(A_{0},\ldots,A_{m+1}).$

In this computation we have used the fact that $\Phi^{P}_{g}$ is a cyclic
cocycle and Melrose’s formula for the $b$-trace. To show that
$B\Phi^{r,P}_{Y,g}=0$ we write out the differential