#  Heat kernels of perturbed operators and index theory on $G$-proper
manifolds

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 and let $X$ be a
$G$-proper manifold without boundary. We give a detailed account of both the
large and small time behaviour of the heat-kernel of perturbed Dirac
operators, as a map from the positive real line to the Lafforgue algebra. As a
first application, we prove that certain delocalized eta invariants associated
to perturbed Dirac operators are well defined. We then prove index formulas
relating these delocalized eta invariants to Atiyah-Patodi-Singer delocalized
indices on $G$-proper manifolds with boundary. We apply these results to the
definition of rho-numbers associated to $G$-homotopy equivalences between
closed $G$-proper manifolds and to the study of their bordism properties.

###### 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 Index classes and cyclic cocycles
    1. 2.1 Definition of some algebras
    2. 2.2 Orbital integrals
    3. 2.3 $G$-proper manifolds with boundary
    4. 2.4 Manifolds with boundary: Lafforgue algebra of pseudodifferential operators and index classes
    5. 2.5 Cyclic cocycles associated to orbital integrals on manifolds with boundary
  3. 3 Large time behaviour of the heat kernel
    1. 3.1 The heat kernel of a perturbed Dirac operator
    2. 3.2 Large time behaviour of the Connes-Moscovici projector $V(t(D+C))$ on $G$-proper manifolds without boundary.
    3. 3.3 Large time behaviour on manifolds with boundary: the perturbed case
  4. 4 Small time behavior of the heat kernel
  5. 5 Modified delocalized eta invariants
    1. 5.1 Delocalized eta invariants associated to the gap case
    2. 5.2 Delocalized eta invariants associated to smoothing perturbations
  6. 6 The delocalized APS index theorem in the gap case
  7. 7 The delocalized index theorem corresponding to smoothing perturbations
  8. 8 Numeric rho invariants on $G$-proper manifolds
    1. 8.1 Rho numbers associated to $G$-homotopy equivalences
    2. 8.2 Bordism properties

##  1\. Introduction

Let $G$ be a connected, linear real reductive group. Let $K$ be a maximal
compact subgroup of $G$. This article complements our article Higher orbital
integrals, rho numbers and index theory [PPST]. There, among other
contributions, we gave a detailed study of the convergence problem for the
delocalized eta invariant associated to an equivariant $L^{2}$-invertible
Dirac operator on a G-proper manifold without boundary and showed that under
an $L^{2}$-invertibility assumption on the boundary operator, such an
invariant enters as a boundary correction term in a delocalized Atiyah-Patodi-
Singer index theorem.

More precisely, we established in [PPST] the following two results:

######  1.1 Theorem.

Let $(X,\mathbf{h})$ be a cocompact $G$-proper manifold without boundary and
let $D$ be a $G$-equivariant $L^{2}$-invertible Dirac-type operator
(associated to a unitary Clifford action and a Clifford connection). Let $g\in
G$ be a semi-simple element and let $\tau_{g}$ the associated orbital integral
on Lafforgue’s algebra111We actually worked with the Harish-Chandra Schwartz
algebra in [PPST] and the rapid decay algebra in [PP1, PP2]. Our proofs in
[PPST, PP1, PP2] generalize directly to Lafforgue’s algebra
$\mathcal{L}_{s}(G)$ which is a dense (Banach) subalgebra of $C^{*}_{r}(G)$
stable under holomorphic functional calculus, c.f. [Lafforgue].
$\mathcal{L}_{s}(G)$ (see Definition 2.3). Let $\tau^{X}_{g}$ be the
associated trace on the algebra of Lafforgue integral operators
$\mathcal{L}^{\infty}_{G,s}(X)$, a subalgebra of the Roe $C^{*}$-algebra
$C^{*}(X)^{G}$. Then the integral

(1.2)
$\eta_{g}(D):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}(D\exp(-tD^{2})\frac{dt}{\sqrt{t}}$

converges.

######  1.3 Theorem.

Let $(Y_{0},\mathbf{h}_{0})$ be a $G$-proper manifold with boundary with
compact quotient and let $(Y,\mathbf{h})$ be the associated manifold with
cylindrical ends. We assume that $D_{\partial}$ 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)

for the pairing of the index class $\operatorname{Ind}(D_{\infty})$ 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:

(1.4)
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\int_{(Y_{0})^{g}}c^{g}{\rm
AS}_{g}(D_{0})-\frac{1}{2}\eta_{g}(D_{\partial})\,,$

where $c^{g}$ is a cut-off function for the fixed-point manifold $(Y_{0})^{g}$
and where the Atiyah-Segal delocalized integrand ${\rm AS}_{g}(D_{0})$
appears.

This second statement was previously established by different methods in
[HWW2] but under the rather restrictive hypothesis that $G/Z_{G}(g)$ is
compact.

In this article we investigate under which conditions we can lift the
invertibility assumption on the boundary operator $D_{\partial}$. As in
previous work on the subject, our position is that such an assumption can
indeed be lifted, provided we are able to find a suitable perturbation of
$D_{\partial}$ that makes it $L^{2}$-invertible. The main issue is of course
the following: what kind of perturbation can one allow? Indeed, we certainly
want a perturbation for which we are able to prove a delocalized Atiyah-
Patodi-Singer index theorem. We consider the following two situations:

  1. (1)

the operator $D$ is such that $0$ is isolated in the spectrum. Under this
assumption we prove that the delocalized eta invariant $\eta_{g}(D)$ is well
defined; moreover, as a preparation for the Atiyah-Patodi-Singer we shall
establish later in the article, we consider $\eta_{g}(\Theta)$, $\Theta$ a
positive real number, defined in terms of a global 0-order perturbation
$D_{\Theta}$ of the operator; we study the limit of $\eta_{g}(\Theta)$ as
$\Theta\downarrow 0$ and prove that it is equal to the sum of $\eta_{g}(D)$
and the term $\tau^{X}_{g}(\Pi_{\operatorname{Ker}D})$, which we prove to be
finite;

  2. (2)

the delocalized eta invariant $\eta_{g}(D+C)$ with $C$ an element in
$\mathcal{L}^{c}_{G}(X,E)$, the $G$-equivariant Lafforgue integral operators
of $G$-compact support, making the operator $D+C$ $L^{2}$-invertible.

Crucial for all these results is a very detailed study of the large and short
time behaviour of the heat kernel of a perturbed Dirac operator. As we shall
see, results that are well known in the compact case often require alternative
and not so obvious arguments.

Once all these results are established on a $G$-proper manifold without
boundary, we investigate in Section 6 how these two invariants fit into
suitable Atiyah-Patodi-Singer index theorems for Dirac operator with non-
invertible boundary operator. We now state these results.

First we analyze the case in which 0 is isolated in the spectrum of the
boundary operator $D_{\partial}$. Following the notation in Theorem 1.3, we
prove the following result:

######  1.5 Theorem.

Assume that the boundary operator on $\partial Y$ satisfies

$\exists\;\delta>0\;\;\text{such that}\;\;{\rm
spec}_{L^{2}}(D_{\partial})\cap(-\delta,\delta)=\\{0\\}.$

Then for $\Theta\in(0,\delta)$ there exists a well defined smooth index class
$\operatorname{Ind}_{\infty}(D_{\Theta})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y,E))=K_{0}(C^{*}(Y_{0}\subset Y,E)^{G})$,
independent of $\Theta\in(0,\delta)$. Moreover, for its pairing with the
0-degree cyclic cocycle $\tau_{g}^{Y}$ the following formula holds:

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

Next, we have the following result, connected with the delocalized eta
invariant in (2); this can be seen as the main contribution of this article:

######  1.6 Theorem.

Let $C_{\partial}$ be a Lafforgue integral operator of $G$-compact support
such that $D_{\partial}+C_{\partial}$ is $L^{2}$-invertible. Then there exists
a well defined index class $\operatorname{Ind}(D,C_{\partial})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))=K_{0}(C^{*}(Y_{0}\subset Y,E)^{G}$.
Moreover, for the delocalized index
$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D,C_{\partial})\rangle$ the
following formula holds

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

We end the paper by defining rho-invariants associated to $G$-homotopy
equivalences and by studying their bordism properties, employing crucially the
Hilsum-Skandalis perturbation and Theorem 1.6.

The paper is organized as follows. In Section 2 we recall the results in
[PPST]. In Section 3 we give a very detailed study of the large time behaviour
of the heat kernel of a perturbed operator, both on closed manifolds and on
manifolds with boundary. These results, necessarily of a technical nature,
play a crucial role in our analysis. In Section 5 we introduce and study the
two delocalized eta invariants that have been briefly introduced above. In
Section 6 we prove the index theorems corresponding to the first eta
invariant. Section 7 is devoted to the rather intricate proof of Theorem 1.6,
the delocalized APS index theorem corresponding to a smoothing perturbation.
Finally in Section 8 we introduce rho-numbers associated to $G$-homotopy
equivalences and study their bordism-invariance 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.

##  2\. Index classes and cyclic cocycles

In this section we shall briefly recall the main characters of our paper
[PPST]. We shall be brief and refer the reader to that paper for details.

###  2.1. Definition of some algebras

We 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. Let $\mathcal{L}_{G}^{c}(X)$ be the algebra of
$G$-equivariant smoothing operators of $G$-compact support; this decomposition
of $X$ induces an isomorphism

(2.1)
$\mathcal{L}_{G}^{c}(X)\cong\left(C^{\infty}_{c}(G)\hat{\otimes}\Psi^{-\infty}(S)\right)^{K\times
K}$

and more generally

(2.2)
$\mathcal{L}_{G}^{c}(X,E)\cong\left(C^{\infty}_{c}(G)\hat{\otimes}\Psi^{-\infty}(S,E|_{S})\right)^{K\times
K},$

in the presence of a $G$-equivariant vector bundle $E$. (We shall often
expunge the vector bundle $E$ from the notation.)

######  2.3 Definition.

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

$v_{t}(f)\colon=\sum_{g\in
G}\left\\{(1+\|g\|)^{t}\cdot\Xi^{-1}(g)\cdot|f(g)|\right\\},$

where $\Xi(g)$ denotes the Harish-Chandras’s spherical function. Moreover, we
define

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

Throughout the article, we will fix a semisimple element $x\in G$, and a
sufficiently large $s\gg 0$, and work with $\mathcal{L}_{s}(G)$ and
$\mathcal{L}_{G,s}^{\infty}(X)$.

###  2.2. Orbital integrals

Let us fix a semisimple element $g\in G$. Under this assumption we know that
$G/Z_{g}$, with $Z_{g}$ denoting the centralizer of $g$, has a $G$-invariant
Haar measure $d(hZ_{g})$. Recall that the orbital integral of a function $f\in
C_{c}(G)$ is, by definition,

$\tau_{g}(f):=\int_{G/Z_{g}}f(hgh^{-1})d(hZ_{g}),$

which extends to a continuous trace homomorphism

(2.4) $\tau_{g}:\mathcal{L}_{s}(G)\to\mathbb{C}\,.$

Following [Hochs-Wang-KT], we define

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

with $c$ a cut-off function for the action of $G$ on $X$ and ${\rm tr}$
denoting the vector-bundle fiberwise trace. Then $\tau_{g}^{X}$ defines a
continuous trace

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

###  2.3. $G$-proper manifolds with boundary

Let now $Y_{0}$ be a manifold with boundary; we assume again that $G$ acts
properly and cocompactly on $Y_{0}$. We denote by $X$ the boundary of $Y_{0}$.
We endow $Y_{0}$ 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. We denote by $c_{0}$ a cut-off
function for the action of $G$ on $Y_{0}$; since the action is cocompact, this
can be chosen to be 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 $Y$. 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. Clearly for the boundary $X=\partial Y_{0}$ we have

$X\cong G\times_{K}S.$

Notice that $Y\cong G\times_{K}Z$ with $Z$ the $b$-manifold associated to
$Z_{0}$. As in [PPST] we assume that

  1. (1)

the symmetric space $G/K$,

  2. (2)

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

  3. (3)

the $K$-slice $Z_{0}$

are all even dimensional.

###  2.4. Manifolds with boundary: Lafforgue algebra of pseudodifferential
operators and index classes

Let $(Y_{0},\mathbf{h}_{0})$ be a cocompact $G$ proper manifold with boundary
$X$ as in the previous subsection and let $(Y,\mathbf{h})$ be the associated
$b$-manifold. We denote by $Z_{0}$ a slice for $Y_{0}$ and by $Z$ the
associated $b$-manifold.

Using the $b$-calculus with $\epsilon$-bounds on $Z$ and the algebra
${\mathcal{L}}^{\infty}_{G,s}(Y)$ one can employ Abels’ theorem [Abels] in
order to define algebras of operators on $Y$ fitting into a short exact
sequence of operators:

(2.7)
$0\to\mathcal{L}^{c}_{G}(Y)\to{}^{b}\mathcal{L}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}({\rm
cyl}(\partial Y))\to 0.$

with $\operatorname{cyl}(\partial Y)=\mathbb{R}\times\partial Y$. Here $I$
denotes the indicial operator. See [PP2] and references therein. Notice that
we have and we shall omit the $\epsilon$ from the notation of the calculus
with bounds.

Employing the Lafforgue’s algebra $\mathcal{L}^{\infty}_{s}(G)$ instead of
$C^{\infty}_{c}(G)$ we can define as in [PP2] the three algebras
$\mathcal{L}^{\infty}_{G,s}(Y)$, ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ and
${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm cyl}(\partial Y))$ fitting
into the short exact sequence

(2.8)
$0\to\mathcal{L}^{\infty}_{G,s}(Y)\to{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y))\to 0.$

This extends (2.7). Recall at this point, see [PP2], that
$\mathcal{L}^{\infty}_{G,s}(Y)$ is dense and holomorphically closed in the Roe
algebra $C^{*}(Y_{0}\subset Y)^{G}$.

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

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

Let $D$ be as above and let $Q^{\sigma}$ be a symbolic $b$-parametrix for $D$.
Following Melrose we can use a true $b$-parametrix
$Q^{b}=Q^{\sigma}-Q^{\prime}$ for $D$ in order to define a Connes-Skandalis
index class

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

This class can also be obtained à la Connes-Moscovici, by improving the
parametrix $Q:=\frac{I-\exp(-\frac{1}{2}D^{-}D^{+})}{D^{-}D^{+}}D^{+}$ to a
true $b$-parametrix. This defines a projector $V^{b}(D)$, a $2\times 2$ matrix
with entries in (the unitalization of) $\mathcal{L}^{\infty}_{G,s}(Y)$ and one
can prove that $[V^{b}(D)]-[e_{1}]=[P^{b}_{Q}]-[e_{1}]$ in
$K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))$.

We can also consider the plain Connes-Moscovici projector $V(D)$,

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

It is a $2\times 2$ matrix with entries in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y_{0})$; similarly, the Connes-Moscovici
projector $V(D^{\operatorname{cyl}})$ is a $2\times 2$ matrix with entries in
${}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm cyl}(\partial Y))$. It was
proved in [PP2], following [moriyoshi-piazza], that these two projectors
define a smooth relative index class
$\operatorname{Ind}_{\infty}(D,D_{\partial})\in
K_{0}({}^{b}\mathcal{L}^{\infty}_{G,s}(Y_{0}),{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y)))$; indeed
$\operatorname{Ind}_{\infty}(D,D_{\partial}):=[V(D),e_{1},p_{t}]$ with
$p_{t}$, $t\in[1,+\infty]$, defined as follows

(2.11)
$p_{t}:=\begin{cases}V(t(D_{\operatorname{cyl}}))\;\;\quad\text{if}\;\;\;t\in[1,+\infty)\\\
e_{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if }\;\;t=\infty\end{cases}$

Moreover, the class $\operatorname{Ind}_{\infty}(D)$ is sent to the class
$\operatorname{Ind}_{\infty}(D,D_{\partial})$ through the excision isomorphism
$\alpha_{{\rm exc}}$.Notice that in (2.11) we use implicitly the large time
behaviour of the heat kernel and, more generally, of the Connes-Moscovici
projector, discussed in detail in [PPST]. (This point was treated very briefly
in [PP2].)

###  2.5. Cyclic cocycles associated to orbital integrals on manifolds with
boundary

As explained in [PPST], the orbital integral $\tau^{g}$ defines a trace-
homomorphism

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

exactly as in the closed case

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

with $dx$ denoting now the $b$-volume form associated to the $b$-metric
$\mathbf{h}$. The 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 by $\langle\,,\,\rangle$, we have in our case

(2.14)
$\langle\cdot\,,\cdot\,\rangle:HC^{0}(\mathcal{L}^{\infty}_{G,s}(Y))\times
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\to\mathbb{C}$

and thus a homomorphism

(2.15)
$\langle\tau^{Y}_{g},\cdot\rangle:K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))\to\mathbb{C}.$

Following the relative cyclic cohomology approach in [moriyoshi-piazza, GMPi,
PP2] we defined in [PPST] a relative cyclic 0-cocycle
$(\tau^{Y,r}_{g},\sigma_{g})$. (The $r$ on the right hand side stands for
regularized, given that the $b$ is the differential in cyclic cohomology.)

The 1-cocycle $\sigma^{\partial Y}_{g}$ can in fact be defined on any proper
$G$-manifold $X$ without boundary; it is the following 1-cochain on
${}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}({\rm cyl}(X))$

(2.16)
$\sigma^{X}_{g}(A_{0},A_{1})=\frac{i}{2\pi}\int_{\mathbb{R}}\tau_{g}^{X}(\partial_{\lambda}I(A_{0},\lambda)\circ
I(A_{1},\lambda))d\lambda\,,$

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

Going back now to a $b$-manifold $Y$ with boundary $\partial Y$, we have also
defined in [PPST] the functional
$\tau^{Y,r}_{g}(\cdot):{}^{b}\mathcal{L}^{c}_{G}(Y)\to\mathbb{C}$:

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

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 $c$. We proved in [PPST] that

  * •

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

  * •

the pair $(\tau^{Y,r}_{g},\sigma^{\partial Y}_{g})$ extends continuously to a
relative 0-cocycle for
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y));$

  * •

the following formula holds:

(2.17)
$\langle\tau^{Y}_{g},\operatorname{Ind}_{\infty}(D)\rangle=\langle(\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}),\operatorname{Ind}_{\infty}(D,D_{\partial})\rangle\,.$

##  3\. Large time behaviour of the heat kernel

In this section we wish to extend our study of the large time behaviour of the
heat kernel in [PPST] to the case of perturbed Dirac operators, first on
cocompact $G$-proper manifolds without boundary and then on cocompact
$G$-proper manifolds with boundary. As we shall see, classic properties of the
heat kernel of a perturbed Dirac operator on compact manifolds require a
rather delicate analysis in order to be established in the $G$-proper case.

In this and the next section, we will freely use properties of the Bochner
integrals with values in Fréchet algebras, c.f. [Thomas, Wiciak].

###  3.1. The heat kernel of a perturbed Dirac operator

Instead of assuming that $D$ is $L^{2}$-invertible, as in [PPST], we assume
now that there is a self-adjoint element $C\in\mathcal{L}_{G}^{c}(X)$ (c.f.
Equation (2.1)) such that $D+C$ is $L^{2}$-invertible. Thus there exists $a>0$
such that

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

First of all, we need to make sense of $\exp(-t(D+C)^{2})$ as an element
$\mathcal{L}^{\infty}_{G,s}(X)$. Observe that $(D+C)^{2}=D^{2}+A$, with
$A=C^{2}+DC+CD$. By its definition, $C^{2}$ is an element in
$\mathcal{L}^{c}_{G}(X)$. For the operators $DC$ and $CD$, we proceed as
follows. First we consider the operator

$D_{{\rm split}}:=D_{G,K}\hat{\otimes}1+1\hat{\otimes}D_{S}.$

We know that elements of $\mathcal{L}^{\infty}_{G,s}(X)$ can be viewed as
$K\times K$-invariant Lafforgue’s functions on $G$ with value in smoothing
operators $\Psi^{-\infty}(S)$. Notice that $D_{G,K}$ is an elliptic first
order $G$-invariant differential operator on $G/K$, and $D_{S}$ is an elliptic
operator on the slice $S$. By the very definition of $\mathcal{L}^{c}_{G}(X)$,
$D_{G,K}C$ and $CD_{G,K}$ are in $\mathcal{L}^{c}_{G}(X)$. And it follows from
the definition of $\Psi^{-\infty}(S)$ that $D_{S}C$ and $CD_{S}$ are both in
$\mathcal{L}^{c}_{G}(X)$. In summary, both $D_{{\rm split}}C$ and $CD_{{\rm
split}}$ are in $\mathcal{L}^{c}_{G}(X)$. On the other hand we know that
$D-D_{{\rm split}}=R$, with $R$ a 0th-order $G$-equivariant differential
operator. Using the arguments in Lemma 4.17 in [PPST] we see that $RC$ and
$CR$ are both in $\mathcal{L}^{c}_{G}(X)$ and thus we conclude that $DC$ and
$CD$ are in $\mathcal{L}^{c}_{G}(X)$. Summarizing, $(D+C)^{2}=D^{2}+A$ with
$A\in\mathcal{L}^{c}_{G}(X)$.

Following [BGV, Ch. 9, Appendix] we consider the Volterra series

$\displaystyle Q_{t}$
$\displaystyle:=\sum_{k=0}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma$
$\displaystyle=e^{-tD^{2}}+\sum_{k=1}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma.$

Let $p$ be any seminorm on $\mathcal{L}^{\infty}_{G,s}(N)$. We know that
$e^{-tD^{2}}A\in\mathcal{L}^{\infty}_{G,s}(N)$; clearly we have
$p(e^{-tD^{2}}A)\leq C(p)p(A)$ so that for any $k\geq 1$ we have

$p\left(\sum_{k=1}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma\right)\leq\frac{(C(p))^{k+1}p(A)^{k}}{k!}\,.$

Thus the series converges in the Fréchet space
$\mathcal{L}^{\infty}_{G,s}(N)$.

###  3.2. Large time behaviour of the Connes-Moscovici projector $V(t(D+C))$
on $G$-proper manifolds without boundary.

We now tackle the large time behaviour of the heat operator
$\exp(-t(D+C)^{2})$ and of the Connes-Moscovici projector $V(tD)$. Let
$\gamma$ be a suitable path in the complex plane, missing the spectrum of
$(D+C)^{2}$. For example, we can take $\gamma$ to be the path given by the
union of the two straight half-lines ${\rm Im}z=\pm m{\rm Re}z+b$ in the half-
plane ${\rm Re}z>0$; here $m={\rm tg}\phi>0$, $m$ small, and $b>0$ is smaller
than the bottom of the spectrum of $(D+C)^{2}$.

We start with the following technical but elementary lemma.

######  3.2 Lemma.

Let $f$ be an analytic function on $[0,\infty)$ that is of rapid decay, i.e.
$\forall m,n$, there is an constant $M_{m,n}>0$ such that
$\operatorname{max}_{x\in[0,\infty)}|x^{m}\frac{\partial^{n}f(x)}{\partial
x^{n}}|<M_{m,n}$ for all $x\in[0,\infty)$. There is an analytic function $g$
on $[0,\infty)$ of rapid decay such that $f=g^{\prime}$.

###### Proof.

Consider $g_{0}(x)=\int_{0}^{x}f(t){\rm d}t$. As $f(t)$ is of rapid decay, the
integral $\int_{0}^{\infty}f(t){\rm d}t$ is finite. Set
$g(x)=g_{0}(x)-\int_{0}^{\infty}f(t){\rm d}t$. Notice that
$\frac{\partial^{n}g(x)}{\partial x^{n}}=\frac{\partial^{n}f(x)}{\partial
x^{n}}$ for $n\geq 1$. To prove the rapid property of $g$, it suffices to
check $\operatorname{max}_{x\in[0,\infty]}|x^{m}g(x)|<C_{m}$ for some $C_{m}$.
We prove this by observing that there exists $N>0$ such that when $x>N$,
$|f(x)|<\frac{M_{m}}{x^{m+1}}$, and

$x^{m}g(x)=x^{m}\int_{x}^{\infty}f(t){\rm d}t.$

When $t>N$, $f(t)$ is bounded by $\frac{M_{m}}{x^{m+1}}$ and

$|x^{m}g(x)|\leq x^{m}\int_{x}^{\infty}|f(t)|{\rm d}t\leq
x^{m}\int_{x}^{\infty}\frac{M_{m}}{t^{m+1}}{\rm dt}\leq\frac{M_{m}}{m}.$

∎

For a fixed function $f$ satisfying the assumption of Lemma 3.2, we choose
$\gamma$ a suitable path in the domain of $f$ missing the spectrum of
$-(D+C)^{2}$. Note that both $e^{-z^{2}}$ and
$e^{-\frac{z}{2}}\frac{1-e^{-z}}{z}$ satisfy the assumptions of the Lemma and
for these we can fix the $\gamma$ as above.

Write now

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

Applying Lemma 3.2 and integration by parts, we choose $k$ very large, larger
than, say, $2(\ell+j)$ and obtain the following expression for $f(-tD^{2})$,

$f(-tD^{2})=\frac{(k-1)!}{2\pi
i}t^{1-k}\int_{\gamma}g(-t\lambda^{2})(D^{2}-\lambda)^{-k}{\rm d}\lambda.$

To study the large $t$ behaviour of $f(-tD^{2})$, we start with the following
proposition.

######  3.3 Proposition.

If the operator $D+C$, $C\in\mathcal{L}^{c}_{G}(X)$, is $L^{2}$-invertible,
then

$(D+C)^{-1}=B+A,\quad\text{with}\quad
B\in\Psi^{-1}_{G,c}(X),\;\;A\in\mathcal{L}^{\infty}_{G,s}(X)$

###### Proof.

We know that there exists $B\in\Psi^{-1}_{G,c}$ such that

$(D+C)B=1-K,\quad K\in\mathcal{L}^{c}_{G}(X)$

where we denote the identity by 1. Indeed, we simply take
$B\in\Psi^{-1}_{G,c}$ such that $DB=1-R$, $R\in\Psi^{-\infty}_{G,c}$ and then
we have that $(D+C)B=1-R-CB$ and we know that $CB\in\mathcal{L}^{c}_{G}(X)$ by
[PPST, Lemma 4.17]; it now suffices to take $K:=R+CB$.

Claim: there exists $B^{\prime}\in\mathcal{L}^{\infty}_{G,s}(X)$ such that

$(D+C)(B-B^{\prime})=1-K^{\prime},\quad
K^{\prime}\in\mathcal{L}^{\infty}_{G,s}(X),\quad\|K^{\prime}\|<1/2$

where the norm is the $L^{2}$-operator-norm, that is, the norm in the algebra
of bounded operators $\mathcal{B}(L^{2})$.

######  3.4 Lemma.

In $\mathcal{B}(L^{2})$ we have that $\|f(-T(D+C)^{2})\|\to 0$ as
$T\to+\infty$ and $\|f(-\delta(D+C)^{2})\|\to||f||_{C([0,\infty))}$ as
$\delta\downarrow 0$. Furthermore, for any $K\in\mathcal{L}^{c}_{G}(X)$,
$\|\exp(-\delta(D+C)^{2})K-K\|\to 0$, as $\delta\downarrow 0$.

###### Proof.

For the first result, we use the fact that the function $f(T\lambda)$
converges to zero in the sup norm on ${\rm
spec}_{L^{2}}\big{(}-(D+C)^{2}\big{)}$. It follows from spectral calculus that
$f(-T(D+C)^{2})$ converges to zero in operator norm. For the second result, by
spectral calculus, the norm $||f(-\delta(D+C)^{2})||$ is equal to the sup norm
of the function $|f(-\delta\lambda^{2})|$ on ${\rm spec}_{L^{2}}(D+C)^{2}$,
which converges to $||f||_{C([0,\infty))}$ as $\delta\to 0$.

We next follow the arguments in [gilkey-book, Lemma 1.7.5 (c)] to prove
$\|\exp(-\delta(D+C)^{2})K-K\|\to 0$. For a properly chosen contour $\gamma$,
we compute

(3.5) $\begin{split}e^{-\delta(D+C)^{2}}K-e^{-\delta}K=&\frac{1}{2\pi
i}\int_{\gamma}e^{-t\lambda}\Big{(}\big{(}(D+C)^{2}-\lambda\big{)}^{-1}+\lambda^{-1}\Big{)}K{\rm
d}\lambda\\\ =&\frac{1}{2\pi
i}\int_{\gamma}e^{-t\lambda}\lambda^{-1}\big{(}(D+C)^{2}-\lambda\big{)}^{-1}(D+C)^{2}K{\rm
d}\lambda.\end{split}$

For $t\in(0,1)$, replace $\lambda$ by $t^{-1}\mu$ and shift the contour
$t\gamma$ to $\gamma$. We have the above integral equal to the following
expression

(3.6) $\frac{1}{2\pi
i}\int_{\gamma}e^{-\mu}\mu^{-1}\big{(}(D+C)^{2}-t^{-1}\mu\big{)}^{-1}(D+C)^{2}K{\rm
d}\mu.$

As $K\in\mathcal{L}^{c}_{G}(X)$, $(D+C)^{2}K\in\mathcal{L}^{c}_{G}(X)$. By the
spectral calculus, the operator norm of $\big{(}(D+C)^{2}-\lambda\big{)}^{-1}$
has the following bound,

$\|\big{(}(D+C)^{2}-\lambda\big{)}^{-1}\|\leq M\frac{1}{|\lambda|},$

for sufficiently large $\lambda$, and

(3.7) $\|\big{(}(D+C)^{2}-t^{-1}\mu\big{)}^{-1}\|\leq M\frac{t}{|\mu|}.$

Combining the estimate (3.7) and Equations (3.5) and (3.6), we conclude with
the following bound

$\|\exp(-\delta(D+C)^{2})K-\exp^{-\delta}K\|\leq\frac{1}{2\pi}t\int_{\gamma}|\mu^{-2}\exp^{-\mu}{\rm
d}\mu|\|(C+D)^{2}K\|\leq\delta M^{\prime}\|(C+D)^{2}K\|,$

and then

$\begin{split}\|\exp(-\delta(D+C)^{2})K-K\|&\leq\|\exp(-\delta(D+C)^{2})K-\exp^{-\delta}K\|+(1-e^{-\delta})\|K\|\\\
&\leq\delta M^{\prime}\|(D+C)^{2}K\|+(1-e^{-\delta})\|K\|\to 0,\ \delta\to
0.\end{split}$

as required. ∎

We now want to prove the claim. For $N>>0$ and $\delta>0$ small consider

$B^{\prime}:=-\int_{\delta}^{N}(D+C)\exp(-t(D+C)^{2})dt\circ K\,.$

This operator is in $\mathcal{L}^{\infty}_{G,s}(X)$. Then

$(D+C)(B-B^{\prime})=1-K+((D+C)^{2}\circ\int_{\delta}^{N}\exp(-t(D+C)^{2})dt\circ
K)$

But

$-(D+C)^{2}\circ\int_{\delta}^{N}\exp(-t(D+C)^{2})dt\circ
K=(\exp(-N(D+C)^{2})\circ K-\exp(-\delta(D+C)^{2}))\circ K$

Thus

$(D+C)(B-B^{\prime})=1-(K-\exp(-\delta(D+C)^{2})\circ
K)-\exp(-N(D+C)^{2})\circ K$

On the right hand side we have the identity minus an element in
$\mathcal{L}^{\infty}_{G,s}(X)$; moreover, using Lemma 3.4, we see that the
$L^{2}$-operator norm of this term, viz.

$\|-(K-\exp(-\delta(D+C)^{2})\circ K)-\exp(-N(D+C)^{2})\circ K\|$

is small if $N$ is big enough and $\delta$ is small enough. We have completed
the proof of the claim.

With the above established claim, we can take

$\left(1-((K+\exp(-\delta(D+C)^{2}))\circ K)+\exp(-N(D+C)^{2})\circ
K)\right)^{-1}$

which is an operator of the type $1+F$, with
$F\in\mathcal{L}^{\infty}_{G,s}(X)$, given that
$\mathcal{L}^{\infty}_{G,s}(X)$ is holomorphically closed.

It follows from the claim that

$(D+C)^{-1}=(B-B^{\prime})+(B-B^{\prime})\circ F=B+(-B^{\prime}+B\circ
F-B^{\prime}\circ F),\quad\text{with}\quad
F\in\mathcal{L}^{\infty}_{G,s}(X)\,.$

This shows that $(D+C)^{-1}$ is the sum of an element in $\Psi^{-1}_{G,c}$ and
of an element in $\mathcal{L}^{\infty}_{G,s}(X)$. This completes the proof of
Proposition 3.3. ∎

Given Proposition 3.3, we now go back to the large time behaviour of
$f(-t(D+C)^{2})$.

Recall that $(D+C)^{2}=D^{2}+A$, with $A=C^{2}+DC+CD$,
$A\in\mathcal{L}^{c}_{G,s}(X)$. Notice that $A$ is self-adjoint. We consider
the associated resolvent $((D+C)^{2}-\lambda)^{-1}$. In [PP2], for unperturbed
Dirac laplacians, this was analyzed first for $\lambda$ finite, see [PP2,
Proposition 2.2], and then for $\lambda$ large, using parameter-dependent
pseudodifferential operators. Assume $|{\rm Im}\lambda|>0$. We write

$(D^{2}+A-\lambda)=\left(1+A(D^{2}-\lambda)^{-1}\right)(D^{2}-\lambda)$

and get

$(D^{2}+A-\lambda)^{-1}=(D^{2}-\lambda)^{-1}\left(1+A(D^{2}-\lambda)^{-1}\right)^{-1}.$

By [PP2, Proposition 2.2, 2.10] we know222As already remarked, in [PP2] the
first two authors worked with a different smooth subalgebra, but the same
argument extends to $\mathcal{L}^{\infty}_{G,s}(X)$. that

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

with $B(\lambda)\in\Psi^{-2}_{G,c}(X)$ and of uniform $G$-compact support and
$C(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$. Moreover $C(\lambda)\to 0$ in
$\mathcal{L}^{\infty}_{G,s}(X)$ as $|\lambda|\to+\infty$. In fact, as already
observed in [PPST] we can easily see that $C(\lambda)$ is rapidly decreasing
in $\lambda$, with values in $\mathcal{L}^{\infty}_{G,s}(X)$. Indeed,
$(D^{2}-\lambda)^{-1}=B(\lambda)(1+F(\lambda))$ with
$1+F(\lambda)=(1+R(\lambda))^{-1}$ and $R(\lambda)$ the remainder of the
symbolic parametrix $B(\lambda)$. We argued in [PP2, Proposition 2.2, 2.10]
that $F(\lambda)$ is going to 0 in $\mathcal{L}^{\infty}_{G,s}(X)$. However,
by the elementary identity

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

we understand that $F(\lambda)$ is in fact rapidly decreasing with values in
$\mathcal{L}^{\infty}_{G,s}(X)$. Summarizing,
$(D^{2}-\lambda)^{-1}=B(\lambda)+C(\lambda)$ with
$B(\lambda)\in\Psi^{-2}_{G,c}(X)$ and of uniform $G$-compact support and
$C(\lambda)$ rapidly decreasing in $\lambda$ with values in
$\mathcal{L}^{\infty}_{G,s}(X)$. Thus

$(D^{2}+A-\lambda)^{-1}=\left(B(\lambda)+C(\lambda)\right)\left(1+A(B(\lambda)+C(\lambda))\right)^{-1}$

The term $A(B(\lambda)+C(\lambda))$ is clearly in
$\mathcal{L}^{\infty}_{G,s}(X)$, because we know, by [PPST, Lemma 4.17], that
for $\ell\leq 0$ composition defines a continuous map
$\Psi^{\ell}_{G,c}(X)\times\mathcal{L}^{\infty}_{G,s}(X)\to\mathcal{L}^{\infty}_{G,s}(X)$.
On the other hand, $\mathcal{L}^{\infty}_{G,s}(X)$ is holomorphically closed
and so $(1+A(B(\lambda)+C(\lambda)))^{-1}$ is equal to $1+M(\lambda)$ with
$M(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$. We have therefore proved that
for $\lambda$ finite, $|{\rm Im}\lambda|>0$,

(3.8) $(D^{2}+A-\lambda)^{-1}=B(\lambda)+N(\lambda)$

with $B(\lambda)\in\Psi^{-2}_{G,c}(X)$ and of uniform $G$-compact support and
$N(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$. The case $|{\rm Im}\lambda|=0$,
on the other hand, is treated using Proposition 3.3. Let us now look at the
large $\lambda$ behaviour. As already remarked we know that $C(\lambda)$ is
rapidly decreasing in $\lambda$ as an element in
$\mathcal{L}^{\infty}_{G,s}(X)$; by the same reasoning given there we have
that the same property is true for $M(\lambda)$ and thus, using the fact that
$\mathcal{L}^{\infty}_{G,s}(X)$ is a Fréchet algebra, we conclude that also
for $N(\lambda)$ we can state that $N(\lambda)$ is rapidly decreasing in
$\lambda$ as an element in $\mathcal{L}^{\infty}_{G,s}(X)$. Similarly

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

with $F(\lambda)\in\Psi^{-2k}_{G,c}(X,\Lambda)$, of $G$-compact support
uniformly in $\lambda$, and $G(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$ and
going to 0 rapidly in $\lambda$ as an element in
$\mathcal{L}^{\infty}_{G,s}(X)$. At this point we can proceed exactly as in
[PPST, Subsection 5.1] and show that if $p$ is a seminorm on
$\mathcal{L}^{\infty}_{G,s}(X)$ then

$p(\left(\int_{\gamma}g(-t\lambda)F(\lambda)d\lambda\right)\rightarrow
0\quad\text{as}\quad t\to+\infty$

and

$p(\left(\int_{\gamma}g(-t\lambda)G(\lambda)d\lambda\right)\rightarrow
0\quad\text{as}\quad t\to+\infty$

so that

$p\left(\int_{\gamma}g(-t\lambda)(D^{2}-\lambda)^{-k}d\lambda\right)\rightarrow
0\quad\text{as}\quad t\to+\infty\,.$

The details are very similar to those given in [PPST] for the unperturbed
case, and therefore omitted.
We summarise our discussion in the following proposition:

######  3.9 Proposition.

If $D+C$ is $L^{2}$-invertible and $f$ satisfies the assumption of Lemma 3.2,
then

(3.10) $f(-t(D+C)^{2})\rightarrow
0\quad\text{in}\quad\mathcal{L}^{\infty}_{G,s}(X)\quad\text{as}\quad
t\to+\infty.$

In particular, the Connes-Moscovici projector $V(t(D+C))-e_{1}$ converges to 0
in $\mathcal{L}^{\infty}_{G,s}(X)$ as $t\to+\infty$.

###  3.3. Large time behaviour on manifolds with boundary: the perturbed case

We now tackle the case of invertible perturbed Dirac operators on manifolds
with boundary (the unperturbed invertible case is treated in [PPST]). We
consider $(Y_{0},\mathbf{h}_{0})$, 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 $Y$. Let $C\in{}^{b}\mathcal{L}^{c}_{G}(Y)$ and
assume that $C$ is self adjoint as a bounded operator on $L^{2}(Y)$. We assume
that $D+C$ is $L^{2}$-invertible. We also make the natural assumption that the
indicial family $I(C,\lambda)$ is rapidly decreasing as a function with values
in $\mathcal{L}^{c}_{G}(\partial Y)$ as $|{\rm Re}(\lambda)|\to+\infty$
whenever $|{\rm Im}(\lambda)|$ ranges in a finite interval. In fact, we shall
apply what follows to a particular situation that we describe now and for
simplicity we only treat this special situation, given that is enough for our
geometric applications. Recall from333The map $\sigma$ was denoted by
$\varphi$ in [PPST]. [PPST, Sec. 5.2], we can construct a section
$\sigma:{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y))\to{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ of the indicial homomorphism
$I:{}^{b}\mathcal{L}^{\infty}_{G}(Y)\to{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}(\operatorname{cyl}(\partial
Y))$; $\sigma$ is defined by using a suitable cut-off function equal to 1 on
the boundary. The simple situation we will be dealing with, is when
$C=\sigma(C_{\operatorname{cyl}})$, with $C_{\operatorname{cyl}}$ obtained by
inverse Mellin transform of $\alpha(\lambda)C_{\partial}$, with
$C_{\partial}\in\mathcal{L}^{c}_{G}(\partial Y)$ and $\alpha(\lambda)$ an
entire function which is rapidly decreasing in ${\rm Re}\lambda$ on any
horizontal strip $|{\rm Im}\lambda|<N$. From now on, we put ourselves in this
particular situation.

We thus consider the perturbed operator $D+C$. We want first of all to make
sense of the heat kernel $\exp(-(D+C)^{2})$ as an element in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$. Next we want to prove that if $D+C$ is
$L^{2}$-invertible then

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

Arguing as we have done in the closed case we have that $(D+C)^{2}=D^{2}+A$
with $A\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$. We consider again the Volterra
series

$\displaystyle Q_{t}$
$\displaystyle:=\sum_{k=0}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma$
$\displaystyle=e^{-tD^{2}}+\sum_{k=1}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma.$

Let $p$ be any seminorm on ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$. We know that
$e^{-tD^{2}}\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$, see [PP2]; thus
$e^{-tD^{2}}A\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$; since
$p(e^{-tD^{2}}A)\leq C(p)p(A)$ we have

$p\left(\sum_{k=1}^{\infty}(-t)^{k}\int_{\Delta_{k}}e^{-\sigma_{0}tD^{2}}Ae^{-\sigma_{1}tD^{2}}\cdots
Ae^{-\sigma_{k}tD^{2}}d\sigma\right)\leq\frac{(C(p))^{k+1}p(A)^{k}}{k!}$

and thus $\exp(-(D+C)^{2})\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$.

Next we tackle the large time behaviour of the heat kernel under the
assumption that there exists $a>0$ such that

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

To this end we first need to investigate the structure of the inverse
$(D+C)^{-1}$. Let us fix $\delta\in(0,a)$. We shall work with the calculus
with bounds
${}^{b}\Psi^{*}_{G,c}(Y)+{}^{b}\mathcal{L}^{\infty,\delta}_{G}(Y)+\mathcal{L}^{\infty,\delta}_{G,s}(Y)$
but we shall not write the $\delta$ in our notation.

######  3.12 Proposition.

Let $C\in{}^{b}\mathcal{L}^{c}_{G,s}(Y)$ be self adjoint and assume that $D+C$
is $L^{2}$-invertible. If
$B\in{}^{b}\Psi^{-1}_{G,c}(Y)+{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ is a full
parametrix for $D+C$ then there exist
$B^{\prime}\in\mathcal{L}^{\infty}_{G,s}(Y)$ and
$F\in\mathcal{L}^{\infty}_{G,s}(Y)$, with $\mathcal{L}^{\infty}_{G,s}(Y)$
denoting as usual the residual algebra, such that

$(D+C)^{-1}=(B-B^{\prime})+(B-B^{\prime})\circ F\,.$

Consequently,

(3.13) $(D+C)^{-1}=B+G\quad\text{with}\quad
B\in{}^{b}\Psi^{-1}_{G,c}(Y)+{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\quad\text{and}\quad
G\in\mathcal{L}^{\infty}_{G,s}(Y)\,.$

###### Proof.

The proof is very similar to the proof of Proposition 3.3, once we use that

  * •

$\mathcal{L}^{\infty}_{G,s}(Y)$ is an ideal in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ and is holomorphically closed in
$C^{*}(Y_{0}\subset Y)^{G}\subset\mathcal{B}(L^{2})$;

  * •

by the analogue of [PPST, Lemma 4.17] in the $b$-context (already observed in
[PP2, Proposition 5.22]) there is a continuous module structure

${}^{b}\Psi^{0}_{G,c}(Y)\times{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\rightarrow{}^{b}\mathcal{L}^{\infty}_{G,s}(Y).$

  * •

by an analogue of Lemma 3.4 for ${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$, we can
see that $\exp(-t(D+C)^{2})\circ K$ is well defined in
${}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ and interpolates between $K$, when
$t\downarrow 0$, and the 0-operator, when $t\uparrow+\infty$, as a bounded
operator on $L^{2}$.

∎

We now study the resolvent $((D^{2}+A)-\mu)^{-1}$, $\mu$ in the image of
$\gamma$. The trick of writing
$(D^{2}+A-\mu)^{-1}=(D^{2}-\mu)^{-1}\left(1+A(D^{2}-\mu)^{-1}\right)^{-1}$ is
not very useful on manifolds with boundary, because $A(D^{2}-\mu)^{-1}$ does
not belong to a holomorphically closed subalgebra (indeed, this operator is
not residual). We proceed differently .
For $\mu$ is a finite region we can use Proposition 3.12 since

$((D^{2}+A)-\mu)^{-1}=((D+C)+i\sqrt{\mu})^{-1}((D+C)-i\sqrt{\mu})^{-1}$

and the operators on the right hand side have a structure like the one
appearing in (3.13). For $|\mu|$ large we proceed as follows. Consider the
true parametrix of $(D^{2}+A)-\mu)$ which is

$B(\mu):=B^{{\rm sym}}(\mu)-\sigma(I(D^{2}+A-\mu)^{-1}I(R^{{\rm sym}}(\mu)))$

with $B^{{\rm sym}}(\mu)$ a symbolic parametrix in ${}^{b}\Psi^{-2}_{G,c}(Y)$
and defining a $b$-pseudodifferential operator with parameter of order $-2$
and $R^{{\rm sym}}(\mu)$ the associated remainder, an element in
${}^{b}\Psi^{-\infty}_{G,c}(Y)$ and defining a $b$-pseudodifferential operator
with parameter of order $-\infty$ . The remainder of the true parametrix is

(3.14) $R(\mu):=R^{{\rm
sym}}(\mu)-(D^{2}+A-\mu)\sigma(I(D^{2}+A-\mu)^{-1}I(R^{{\rm sym}}(\mu)))$

an element in the residual algebra $\mathcal{L}^{\infty}_{G,s}(Y)$. Thus we
can write $(D^{2}+A-\mu)B(\mu)=1-R(\mu)$ with
$R(\mu)\in\mathcal{L}^{\infty}_{G,s}(Y)$. The $L^{2}$-norm of $R(\mu)$ is less
than or equal to the sum of the $L^{2}$-norms of the operators appearing on
the right hand side of (3.14). As $R^{{\rm sym}}(\mu)$ is a
$b$-pseudodifferential operator with parameter of order $-\infty$ we see that
its norm can be made smaller than $1/4$ for $|\mu|$ large enough. Let us
analyse the second term on the right hand side of (3.14). The operator
$I(D^{2}+A-\mu)^{-1}I(R^{{\rm sym}}(\mu))$ has a Schwartz kernel given in
projective coordinates by

(3.15)
$K(s,\cdot,\cdot)=\int_{\mathbb{R}}s^{i\lambda}(D^{2}_{\partial}+\lambda^{2}+I(A,\lambda)-\mu)^{-1}\circ
I(R^{{\rm sym}}(\mu),\lambda)d\lambda$

Observe that $I(R^{{\rm sym}}(\mu),\lambda)$ defines a function of
$\lambda\in\mathbb{C}$ and $\mu\in\gamma$ with values in
$\mathcal{L}^{\infty}_{G,s}(\partial Y)$; we consider it as a
$\mathcal{L}^{\infty}_{G,s}(\partial Y)$-valued function on
$(\mathbb{R}\times(-i\delta,i\delta))\times\gamma$ and observe crucially that
is rapidly decreasing as $|{\rm Re}(\lambda)|+|\mu|\to+\infty$. We can write

$(D^{2}_{\partial}+\lambda^{2}+I(A,\lambda)-\mu)^{-1}=(D^{2}_{\partial}+\lambda^{2}-\mu)^{-1}\circ(1+I(A,\lambda)(D^{2}_{\partial}+\lambda^{2}-\mu)^{-1})^{-1}$

We now make use of the particular structure of $C$; thus we can assume that
$I(A,\lambda)=\alpha(\lambda)A_{\partial}$, with $\alpha(\lambda)$ an entire
function which is rapidly decreasing in ${\rm Re}\lambda$ on any horizontal
strip $|{\rm Im}\lambda|<N$, in particular in the strip
$\mathbb{R}\times(-i\delta,i\delta)\subset\mathbb{C}$, and
$A_{\partial}\in\mathcal{L}^{\infty}_{G,s}(\partial Y)$. This means that the
kernel appearing under the sign of integral in the right hand side of (3.15)
can be written as

$\left((D^{2}_{\partial}+\lambda^{2}-\mu)^{-1}\circ(1+(\alpha(\lambda)A_{\partial})(D^{2}_{\partial}+\lambda^{2}-\mu)^{-1})^{-1}\right)\circ
I(R^{{\rm sym}}(\mu),\lambda).$

From the structure of $(D^{2}_{\partial}+\lambda^{2}-\mu)^{-1}$, discussed in
[PP2], and the joint rapid decay of $I(R^{{\rm sym}}(\mu),\lambda)$ we
understand that this operator is rapidly decreasing on
$(\mathbb{R}\times(-i\delta,i\delta))\times\gamma$ as $|{\rm
Re}\lambda|+|\mu|\to+\infty$. This means that, as in the unperturbed case,
$\mu\to\sigma(I(D^{2}+A-\mu)^{-1}I(R^{{\rm sym}}(\mu))$ is rapidly decreasing
with values in ${}^{b}\mathcal{L}_{G,s}^{\infty}(Y)$. This property is
unchanged by the composition with $D^{2}+A-\mu$. Notice in particular that
also this second summand in (3.14) has norm $<1/4$. We conclude that in the
right hand side of $(D^{2}+A-\mu)B(\mu)=1-R(\mu)$ we have an operator $R(\mu)$
that has $L^{2}$-norm smaller than $1/2$ and that is in fact rapidly
decreasing with values in $\mathcal{L}^{\infty}_{G,s}(Y)$; since the latter is
holomorphically closed, we have that $(1-R(\mu))^{-1}$ is equal to $1+L(\mu)$
and $L(\mu)\in\mathcal{L}^{\infty}_{G,s}(Y)$; moreover, $L(\mu)$ is rapidly
decreasing in Fréchet topology as $|\mu|\to+\infty$.
Summarizing, for $\mu\in\gamma$, $|\mu|$ large,

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

with $B(\mu)=B^{{\rm sym}}(\mu)-\sigma(I(D^{2}+A-\mu)^{-1}I(R^{{\rm
sym}}(\mu)))$ and $L(\mu)$ converging to $0$ rapidly as $|\mu|\to+\infty$. A
similar structure holds for $(D^{2}+A-\mu)^{-k}$. Now we can write
$\exp(-t(D+C)^{2})\equiv\exp(-t(D^{2}+A))$ as

(3.16) $\begin{split}\int_{\gamma}e^{-t\mu}B^{{\rm
sym}}(\mu)d\mu-\int_{\gamma}e^{-t\mu}\sigma(I(D^{2}+A-\mu)^{-k}I(R^{{\rm
sym}}(\mu)))d\mu+\int_{\gamma}e^{-t\mu}B^{{\rm sym}}(\mu)\circ L(\mu)d\mu-\\\
\int_{\gamma}e^{-t\mu}\sigma(I(D^{2}+A-\mu)^{-k}I(R^{{\rm sym}}(\mu)))\circ
L(\mu)d\mu.\end{split}$

with $B^{{\rm sym}}(\mu)$ of order $-2k$. Using the information we have just
gathered about the operators involved in these 4 integrals and reasoning
precisely as in the unperturbed case treated in [PPST] we can conclude that
the following proposition holds:

######  3.17 Proposition.

Let $C\in{}^{b}\mathcal{L}^{c}_{G}(Y)$ be self-adjoint,
$C=\sigma(C_{\operatorname{cyl}})$, with $C_{\operatorname{cyl}}$ obtained by
inverse Mellin transform of $\alpha(\lambda)C_{\partial}$, with
$C_{\partial}\in\mathcal{L}^{c}_{G}(\partial Y)$ and $\alpha(\lambda)$ an
entire function which is rapidly decreasing in ${\rm Re}\lambda$ on any
horizontal strip $|{\rm Im}\lambda|<N$. Let $D+C$ be $L^{2}$-invertible. Then

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

Proceeding as in the closed case, considering thus a function $f$ as in Lemma
3.2, we can establish more generally the following.

######  3.19 Proposition.

Let $C\in{}^{b}\mathcal{L}^{c}_{G}(Y)$ be self-adjoint and of the type
explained in the statement of the previous Proposition. If $D+C$ is
$L^{2}$-invertible then for the Connes-Moscovici projector it holds that

(3.20) $V(t(D+C))-e_{1}\to
0\quad\mbox{in}~{}{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)\quad\mbox{as}~{}t\to+\infty.$

##  4\. Small time behavior of the heat kernel

In this section, we will study the small time behavior of the heat kernel for
the perturbed Dirac operator. Following the previous section, we consider a
self-adjoint element $C\in\mathcal{L}_{G}^{c}(X)$ such that $D+C$ is
$L^{2}$-invertible.

In Lemma 3.4, we showed that for any $K\in\mathcal{L}_{G}^{c}(X)$, the
operator norm of $\exp(-\delta(D+C)^{2})K-K$ converges to zero as $\delta$
goes to $0$. In this section, we strengthen this result to the following one.

######  4.1 Proposition.

For any $K\in\mathcal{L}_{G}^{c}(X)$ we have that $K\exp(-\delta D^{2})$
converges to $K$ in $\mathcal{L}_{G,s}^{\infty}(X)$ as $\delta\to 0$.
Consequently, it directly follows from the Volterra series expansion of
$\exp(-\delta(D+C)^{2})$ that for any $C\in\mathcal{L}_{G}^{c}(X)$,

$K\exp(-\delta(D+C)^{2})\to K,\ \text{in}\ \mathcal{L}_{G,s}^{\infty}(X)\text{
as }\delta\downarrow 0.$

###### Proof.

We can write

(4.2) $\begin{split}Ke^{-\delta D^{2}}=&K\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}\big{(}D^{2}-\lambda\big{)}^{-1}{\rm
d}\lambda.\end{split}$

We need to show that for any seminorm $p$ on $\mathcal{L}^{\infty}_{G,s}(X)$,

$p(Ke^{-\delta D^{2}}-K)\to 0\text{ as }\delta\to 0.$

We start the proof by observing that left multiplication by a smoothing
operator $K\in\mathcal{L}_{G}^{c}(X)$ defines a continuous map on
$\mathcal{L}^{\infty}_{G,s}(X)$. A similar statement holds for
$\Psi^{-2k}_{G,c}(X)$: multiplication by $K\in\mathcal{L}_{G}^{c}(X)$ is
continuous with respect to the seminorms in $\Psi^{-2k}_{G,c}(X)$. By standard
properties of the Bochner integral under the application of a bounded operator
and using the integration by parts trick to replace $(D^{2}-\lambda)^{-1}$ by
$(D^{2}-\lambda)^{-k}$, we have

$\displaystyle K\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}(D^{2}-\lambda)^{-1}{\rm d}\lambda$
$\displaystyle=t^{1-k}\frac{(k-1)!}{2\pi
i}K\int_{\gamma}e^{-\delta\lambda}(D^{2}-\lambda)^{-k}{\rm d}\lambda$
$\displaystyle=t^{1-k}\frac{(k-1)!}{2\pi
i}K\left(\int_{\gamma}e^{-\delta\lambda}B(\lambda){\rm
d}\lambda+\int_{\gamma}e^{-\delta\lambda}C(\lambda){\rm d}\lambda\right)$
$\displaystyle=t^{1-k}\frac{(k-1)!}{2\pi
i}\left(\int_{\gamma}e^{-\delta\lambda}KB(\lambda){\rm
d}\lambda+\int_{\gamma}e^{-\delta\lambda}KC(\lambda){\rm d}\lambda\right)$
$\displaystyle=t^{1-k}\frac{(k-1)!}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}K(D^{2}-\lambda)^{-k}{\rm d}\lambda$
$\displaystyle=\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}K(D^{2}-\lambda)^{-1}{\rm d}\lambda$

where we have expressed $(D^{2}-\lambda)^{-k}$ as $B(\lambda)+C(\lambda)$ with
$B(\lambda)\in\Psi^{-2k}_{G,c}(X)$ and of uniform $G$-compact support and
$C(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$ and converging to zero rapidly as
$|\lambda|\to\infty$. Write now

$\big{(}D^{2}-\lambda\big{)}^{-1}=F(\lambda)+E(\lambda)$

with $F(\lambda)\in\Psi^{-2}_{G,c}(X)$ and of uniform $G$-compact support and
$E(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$ and converging to zero rapidly as
$|\lambda|\to\infty$. We then obtain the following expression for
$K\exp(-\delta D^{2})$,

$Ke^{-\delta D^{2}}=\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}KF(\lambda){\rm d}\lambda+\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}KE(\lambda){\rm d}\lambda.$

In the following, we study the above two integrals separately.

Part I. $E(\lambda)$ integral. We observe that
$p(e^{-\delta\lambda}KE(\lambda))\leq p(KE(\lambda))$. As
$E(\lambda)\in\mathcal{L}^{\infty}_{G,s}(X)$ converges to 0 rapidly, we have
that for every seminorm $p$ on $\mathcal{L}_{G,s}^{\infty}(X)$,

$\int_{\gamma}p\big{(}KE(\lambda)\big{)}{\rm d}\lambda<\infty.$

It follows from the dominated convergence theorem that

$\lim_{\delta\to 0}p\left(\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}KE(\lambda){\rm d}\lambda-\frac{1}{2\pi
i}\int_{\gamma}KE(\lambda){\rm d}\lambda\right)=0$

which is what we wanted to show.
Part II. $F(\lambda)$ integral. We need to use a $G$-equivariant version of
the pseudodifferential calculus [PPT, Section 1.3] for the special case that
the Lie groupoid is a connected real reductive Lie group $G$. More precisely,
there is a $G$-invariant symbol function $b(\lambda,x,\xi)$ on $T^{*}X$ such
that outside a $G$-compact neighborhood of the zero section in $T^{*}X$,

(4.3) $||b(\lambda,x,\xi)||\leq C(||\xi||^{2}+||\lambda||^{2})^{-1},\ $

for sufficiently large $|\lambda|$ and constant $C$ independent of $x$, $\xi$,
and $\lambda$.

Fix $\nu>0$ sufficiently small, i.e. smaller than the injective radius of the
Riemannian metric on $X$. Let $\chi(x,y)$ be a $G$-invariant smooth function
on $X\times X$ such that $\chi(x,y)$ vanishes when the distance between $x$
and $y$ is larger than $\nu$ and $\chi(x,y)$ takes value 1 when the distance
between $x$ and $y$ is less than $\nu/2$.

The operator $F(\lambda)$ has the following expression, for $f\in
C^{\infty}_{c}(X)$,

$F(f)(x)=\int_{X}\int_{T_{x}^{*}X}\chi(x,y)e^{i\langle\xi,\exp^{-1}_{x}(y)\rangle}b(\lambda,x,\xi)f(y){\rm
d}y{\rm d}\xi,$

where $\exp_{x}:T_{x}X\to X$ is the exponential map at $x$, which is bijective
for $v\in T_{x}X$ with $||v||$ less than the injective radius of $X$.

The operator $K$ has the following form

$K(f)(x)=\int_{X}k(x,y)f(y){\rm d}y,$

for a smooth $G$-equivariant function $k(x,y)$ on $X\times X$; moreover,
because of the $G$-compact support condition there is a positive $\mu>0$ such
that $k(x,y)$ vanishes when the distance between $x$ and $y$ is larger than
$\mu$.

The composition of $K$ and $F$ can be computed as follows,

$K\circ
F(f)(x)=\int_{X}\int_{T_{x}^{*}X}\int_{X}k(x,y)\chi(y,z)e^{i\langle\xi,\exp^{-1}_{y}(z)\rangle}b(\lambda,y,\xi)f(z){\rm
d}\xi{\rm d}y{\rm d}z.$

The proof of the following properties is by direct computation and left to the
reader.

  1. (1)

As the integration over $y$ is on a compact set, the integration by parts
trick on $y$ allows us to replace the symbol function $b(\lambda,y,\xi)$ by a
new one $\tilde{b}(\lambda,y,\xi)$ such that

$||\tilde{b}(\lambda,y,\xi)||\leq\frac{C}{(||\xi||^{2}+1)^{-2M}(||\xi||^{2}+||\lambda||^{2})}$

where $M$ can be made arbitrarily large, e.g. larger than the dimension of
$X$. It follows from the above decay property of $\tilde{b}(\lambda,y,\xi)$
that the following integral

$l(\lambda,x,z):=\int_{T_{x}^{*}X}\int_{X}k(x,y)\chi(y,z)e^{i\langle\xi,\exp^{-1}_{y}(z)\rangle}b(\lambda,y,\xi){\rm
d}\xi{\rm d}y$

converges absolutely.

  2. (2)

$\forall\lambda$, $l(\lambda,x,z)$ is a $G$-equivariant smooth function on
$X\times X$ that vanishes when the distance between $x$ and $z$ is larger than
$\mu+\nu$.

  3. (3)

It follows from estimate (4.3), that for $s,t\geq 0$, there is $C_{s,t}>0$
such that

$\sup_{x,z\in X\times
X}||(D_{x}^{s}D_{z}^{t}l)(\lambda,x,z)||\leq\frac{C_{s,t}}{||\lambda||^{2}},$

where $D_{x}$ and $D_{z}$ are partial differentiations along the $x$ and $z$
directions.

By (1) and (2), we conclude that the composition $K\circ F$ belongs to
$\mathcal{L}^{c}_{G}(X)\subset\mathcal{L}^{\infty}_{G,s}(X)$. Furthermore, by
(3), for any seminorm $p$ on $\mathcal{L}^{\infty}_{G,s}(X)$, we have the
bound $p(KF(\lambda))\leq C_{p}/||\lambda||^{2}$ and accordingly the
convergent Bochner integral as an element in $\mathcal{L}^{\infty}_{G,s}(X)$,

$\int_{\gamma}KF(\lambda){\rm d}\lambda.$

The dominated convergence theorem for the Bochner integral gives that

$\lim_{\delta\to 0}p\left(\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}KF(\lambda){\rm d}\lambda-\frac{1}{2\pi
i}\int_{\gamma}KF(\lambda){\rm d}\lambda\right)=0.$

Summarizing Part I and II, we conclude that in
$\mathcal{L}^{\infty}_{G,s}(X)$,

$\begin{split}\lim_{\delta\to 0}Ke^{-\delta D^{2}}&=\lim_{\delta\to
0}\left(\frac{1}{2\pi i}\int_{\gamma}e^{-\delta\lambda}KF(\lambda){\rm
d}\lambda\right)+\lim_{\delta\to 0}\left(\frac{1}{2\pi
i}\int_{\gamma}e^{-\delta\lambda}KE(\lambda){\rm d}\lambda\right)\\\
&=\frac{1}{2\pi i}\int_{\gamma}KF(\lambda){\rm d}\lambda+\frac{1}{2\pi
i}\int_{\gamma}KE(\lambda){\rm d}\lambda\\\ &=\frac{1}{2\pi
i}\int_{\gamma}K(F+E)(\lambda){\rm d}\lambda.\end{split}$

We conclude that $Ke^{-\delta D^{2}}$ is a convergent sequence in
$\mathcal{L}^{\infty}_{G,s}(X)$; since we have proved that for the
$L^{2}$-operator norm we do have

$\lim_{\delta\downarrow 0}\|Ke^{-\delta D^{2}}-K\|=0$

we conclude that $Ke^{-\delta D^{2}}$ converges to $K$ in
$\mathcal{L}^{\infty}_{G,s}(X)$∎

##  5\. Modified delocalized eta invariants

Let $(X,\mathbf{h})$ be a cocompact $G$-proper manifold of odd dimension,
without boundary. As we have assumed $G/K$ is of even dimension in Sec. 2, the
slice $S$ is of odd dimension. In this section we shall prove results on
delocalized eta invariants associated to perturbed Dirac operators. More
precisely we shall consider:

  * •

the delocalized eta invariant $\eta_{g}(\Theta)$, defined in terms of a
perturbation $D_{\Theta}$ of the operator $D$ on $X$, defined under a gap
assumption at $0$ for $D$; we shall also study the limit of $\eta_{g}(\Theta)$
as $\Theta\downarrow 0$;

  * •

the delocalized eta invariant $\eta_{g}(D+C)$ with $C$ an element in
$\mathcal{L}^{c}_{G}(X,E)$ making the operator $D+C$ invertible.

We use the same notations as in the previous subsection.

###  5.1. Delocalized eta invariants associated to the gap case

######  5.1 Definition.

For any $\Theta>0$, let us introduce the perturbed Dirac operator

$D_{\Theta}=D+\Theta.$

Let $\kappa^{\Theta}_{t}$ be the smoothing kernel of the following operator

$D_{\Theta}\cdot\exp(-tD^{2}_{\Theta}).$

For any small number $s>0$, we split the integral

(5.2)
$\begin{split}\int_{s}^{\infty}\tau^{X}_{g}(D_{\Theta}\exp(-tD_{\Theta}^{2})\frac{dt}{\sqrt{t}}=&\int_{s}^{\infty}\int_{X}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\
=&\int_{s}^{\infty}\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\
+&\int_{s}^{\infty}\int_{N^{c}(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}},\end{split}$

where $N(r)$ is a properly chosen $r$-neighborhood of the fixed point
submanifold $X^{g}$ in $X$, c.f. [PPST, Proposition 5.27 and Lemma 5.30].

Note that the integral

$\int_{s}^{\infty}\tau^{X}_{g}(D_{\Theta}\exp(-tD_{\Theta}^{2})\frac{dt}{\sqrt{t}}$

might be divergent as $s\to 0$. The reason is that $D_{\Theta}$ is no longer
associated to a Clifford connection and thus

$\int_{s}^{\infty}\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}$

will in general tend to infinity as $s\to 0$.

######  5.3 Proposition.

For the terms in the righthand side of (5.2), the following holds

  1. (1)

The integrand of the first summand has the asymptotic expansion [Zhangwp,
Section 2]:

$\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\sim t^{-\frac{\dim
X+1}{2}}\left(\sum_{i=0}^{\infty}a_{i}(r)\cdot t^{i}\right)$

We define

$A(r)\colon={\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}$

as the coefficient of $s^{0}$ in this asymptotic expansion.

  2. (2)

The integral

$\int_{N^{c}(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy$

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

$\begin{split}B(r)\colon=&\lim_{s\to
0}\int_{s}^{\infty}\int_{N^{c}(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\
=&\int_{0}^{\infty}\int_{N^{c}(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\end{split}$

is well defined.

###### Proof.

Part (i) follows from the similar arguments as in [PPST, Lemma 5.28] via
asymptotic expansion [Zhangwp, Section 2]. And Part (ii) follows from the same
arguments as in [PPST, Proposition 5.25]. We leave the details to the reader.
∎

######  5.4 Definition.

For the perturbed Dirac operator $D_{\Theta}$, we define the _regularized eta
invariant_ as

(5.5) $\begin{split}\eta_{g}(\Theta)=&{\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\tau^{X}_{g}(D_{\Theta}\exp(-tD_{\Theta}^{2}))\frac{dt}{\sqrt{t}}\\\
=&A(r)+B(r).\end{split}$

######  5.6 Lemma.

The regularized eta invariant $\eta_{g}(\Theta)$ is independent of the choice
of $r$.

###### Proof.

Suppose that $r^{\prime}>r>0$ are two small constants. We want to show that

$A(r)+B(r)=A(r^{\prime})+B(r^{\prime}).$

By definition, we have

$\begin{split}A(r^{\prime})=&{\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\ =&{\rm
LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\int_{N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\ +&{\rm
LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\int_{N(r^{\prime})\setminus
N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}.\end{split}$

Because

$\int_{N(r^{\prime})\setminus N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy$

is of exponential decay as $t\to 0$, the following limit is finite,

$\begin{split}&{\rm LIM}_{s\downarrow
0}\int_{s}^{\infty}\int_{N(r^{\prime})\setminus N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\
=&\int_{0}^{\infty}\int_{N(r^{\prime})\setminus N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}<+\infty.\end{split}$

Thus,

$A(r^{\prime})=A(r)+\int_{0}^{\infty}\int_{N(r^{\prime})\setminus
N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}.$

On the other hand, we compute

$\begin{split}B(r)=&\int_{0}^{\infty}\int_{N^{c}(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\\\
=&B(r^{\prime})+\int_{0}^{\infty}\int_{N(r^{\prime})\setminus
N(r)}\int_{G}c_{G}(h)c(hy){\rm
tr}(\kappa^{\Theta}_{t}(y,gy)g)\;dhdy\frac{dt}{\sqrt{t}}\end{split}$

This shows that

$A(r)+B(r)=A(r^{\prime})+B(r^{\prime}).$

∎

######  5.7 Lemma.

If $0$ is isolated in the $L^{2}$-spectrum of $D$, then

  1. (1)

the delocalized eta invariant

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

is well defined;

  2. (2)

the projection operator $\Pi_{\ker D}\in\mathcal{L}^{\infty}_{G,s}(X)$, in
particular, $\tau^{X}_{g}(\Pi_{\ker D})$ is finite.

###### Proof.

We are assuming that $D$, and thus $D^{2}$, has a gap at $0$. Then there
exists $\sigma>0$ such that ${\rm
spec}_{L^{2}}(D^{2})\cap[-2\sigma,2\sigma]=\\{0\\}$. Let $\gamma$ be the path
given by the union of the two straight lines ${\rm Im}z=\pm m{\rm Re}z$,
$m={\rm tg}\phi>0$, $m$ small, for $|z|>\sigma$ and of the portion of the
circle of radius $\sigma$ joining the points

$\\{\sigma\cos\phi+i\sigma\sin\phi,\sigma\cos\phi-i\sigma\sin\phi\\}$

and passing through the point $-\sigma\equiv-\sigma+i0$ in the negative real
axis. We know that

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

Consider now $\gamma_{\sigma}$, the path obtained by taking the union of the
two straight lines ${\rm Im}z=\pm m{\rm Re}z$, $m={\rm tg}\phi>0$ for
$|z|>\sigma$ and of the portion of the circle of radius $\sigma$ joining the
points $\\{\sigma\cos\phi+i\sigma\sin\phi,\sigma\cos\phi-i\sigma\sin\phi\\}$
and passing through the point $\sigma\equiv\sigma+i0$ in the positive real
axis. If we denote by $C_{\sigma}$ the circle of radius $\sigma$ centred at
the origin, then

$\exp(-tD^{2})=\frac{1}{2\pi
i}\int_{C_{\sigma}}e^{-t\lambda}(D^{2}-\lambda)^{2}d\lambda+\frac{1}{2\pi
i}\int_{\gamma_{\sigma}}e^{-t\lambda}(D^{2}-\lambda)^{2}d\lambda.$

It is well known that

$\frac{1}{2\pi
i}\int_{C_{\sigma}}e^{-t\lambda}(D^{2}-\lambda)^{2}d\lambda=\Pi_{\ker
D^{2}}\equiv\Pi_{\ker D}\,.$

See [Shubin-Book]. Thus

(5.8) $\exp(-tD^{2})=\Pi_{\ker D}+\frac{1}{2\pi
i}\int_{\gamma_{\sigma}}e^{-t\lambda}(D^{2}-\lambda)^{2}d\lambda.$

This establishes that

$\Pi_{\ker D}\,,\text{ which is equal to }\frac{1}{2\pi
i}\int_{C_{\sigma}}e^{-t\lambda}(D^{2}-\lambda)^{2}d\lambda\,,\text{ is in
}\mathcal{L}^{\infty}_{G,s}(M)$

(proof as in [PP2, Proposition 2.9]) and that $\exp(-tD^{2})\to\Pi_{\ker D}$
in $\mathcal{L}^{\infty}_{G,s}(M)$ and exponentially, as $t\to+\infty$, where
the techniques introduced in the proof of Proposition 3.9 must be used. In
particular $\tau^{X}_{g}(\Pi_{\ker D})$ is finite. From (5.8) we also see that

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

converges. Since we know that

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

does converge, see [PPST, Proposition 5.29], we conclude that $\eta_{g}(D)$ is
indeed well defined. ∎

######  5.9 Proposition.

Let $(X,\mathbf{h})$ be an odd dimensional $G$-proper manifold and let $D$ be
a $G$-equivariant Dirac operator. If $0$ is isolated in the $L^{2}$-spectrum
of $D$ and $\Theta>0$ is not in the $L^{2}$-spectrum of $D$ then

(5.10) $\eta_{g}(\Theta):={\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{\infty}\tau^{N}_{g}((D_{\Theta})\exp(-t(D+\Theta)^{2}))\frac{dt}{\sqrt{t}}$

is well-defined. Moreover if $\Theta>0$ belongs to the gap around $0$, then
$\lim_{\Theta\downarrow 0}\eta_{g}(\Theta)$ exists and we have

(5.11) $\lim_{\Theta\downarrow
0}\eta_{g}(\Theta)=\eta_{g}(D)+\tau^{X}_{g}(\Pi_{\ker D}).$

###### Proof.

We shall adapt to the present context an argument due to Melrose, see
[Melrose-Book, Prop. 8.38]. Let us break

$\eta_{g}(\Theta)={\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{1}\tau^{X}_{g}((D_{\Theta})\exp(-sD_{\Theta}^{2}))\frac{dt}{\sqrt{t}}+\frac{1}{\sqrt{\pi}}\int_{1}^{\infty}\tau^{X}_{g}((D_{\Theta})\exp(-sD_{\Theta}^{2}))\frac{dt}{\sqrt{t}}.$

Because $D_{\Theta}$ is $L^{2}$-invertible, we know that the second summand is
finite. For the first summand, we need to take the regularized limit, ${\rm
LIM}_{s\downarrow 0}$, because $D_{\Theta}$ is a Dirac-type operator but it is
not associated to a Clifford connection. In particular,
$\eta_{g}(\Theta):=\eta_{g}(D_{\Theta})$.

Recall that the heat kernel and its asymptotic expansion for small time, are
$C^{1}$ in $\Theta$, for finite times. Thus, we have that.

(5.12) $\lim_{\Theta\downarrow 0}{\rm LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{1}\tau^{X}_{g}(D_{\Theta}\exp(-tD_{\Theta}^{2}))\frac{dt}{\sqrt{t}}={\rm
LIM}_{s\downarrow
0}\frac{1}{\sqrt{\pi}}\int_{s}^{1}\tau^{X}_{g}(D\exp(-tD^{2}))\frac{dt}{\sqrt{t}}.$

Again by [PPST, Proposition 5.29], the righthand side is equal to

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

On the other hand, by the tracial property of $\tau_{g}^{X}$ and Duhamel’s
principle, we compute that

$\displaystyle\frac{d}{d\Theta}\left(\int_{1}^{T}\tau^{X}_{g}\left(D_{\Theta}\exp(-tD_{\Theta}^{2})\right)\frac{dt}{\sqrt{t}}\right)$
$\displaystyle=$ $\displaystyle\int_{1}^{T}\tau^{X}_{g}\left[\left(1-2t\cdot
D_{\Theta}^{2}\right)\exp(-tD_{\Theta}^{2})\right]\frac{dt}{\sqrt{t}}$
$\displaystyle=$ $\displaystyle
2T^{1/2}\tau^{X}_{g}\left(\exp(-TD_{\Theta}^{2})\right)-2\tau^{X}_{g}\left(\exp(-D_{\Theta}^{2})\right).$

Changing $\Theta$ in $\phi$ and integrating in $\phi$, from $\phi=0$ to
$\phi=\Theta$, we obtain:

(5.13)
$\displaystyle\int_{1}^{T}\tau^{X}_{g}\left(D_{\Theta}\exp(-tD_{\Theta}^{2})\right)\frac{dt}{\sqrt{t}}$
$\displaystyle=$
$\displaystyle\int_{1}^{T}\tau^{X}_{g}\left(D\exp(-tD^{2})\right)\frac{dt}{\sqrt{t}}+2\int_{0}^{\Theta}T^{1/2}\tau^{X}_{g}\left(\exp(-TD_{\phi}^{2})\right)d\phi-2\int_{0}^{\Theta}\tau^{X}_{g}\left(\exp(-D_{\phi}^{2})\right)d\phi.$

We want to take the limit as $T\to+\infty$ and then $\Theta\downarrow 0$. By
Lemma 5.7 the first summand on the righthand side converges to

$\int_{1}^{\infty}\tau^{X}_{g}\left(D\exp(-tD^{2})\right)\frac{dt}{\sqrt{t}}<+\infty.$

and it is obviously independent of $\Theta$. It is clear that

$\lim_{\Theta\downarrow
0}\int_{0}^{\Theta}\tau^{X}_{g}\left(\exp(-D_{\phi}^{2})\right)d\phi=0.$

The difficult term is

$\lim_{\Theta\downarrow
0}\lim_{T\to+\infty}\int_{0}^{\Theta}T^{1/2}\tau^{X}_{g}\left(\exp(-TD_{\phi}^{2})\right)d\phi.$

Note that $D_{\phi}$ is a lower order invertible perturbation of $D$. For
$\phi>0$ small, write

$\exp(-TD_{\phi}^{2})=\exp\left(-T(D^{2}+2\phi D)\right)\exp(-T\phi^{2}).$

Here the operator $D^{2}+2\phi D$ is a generalized laplacian. Because $0$ is
isolated in the spectrum of $D$, we can choose $\Theta$ small enough such that
for each $\phi\in[0,\Theta]$, $0$ is also isolated in the spectrum of
$D^{2}+2\phi D$. In particular,

$\ker(D^{2}+2\phi D)=\ker(D).$

By the decomposition of the heat kernel,

(5.14) $\exp\left(-T(D^{2}+2\phi D)\right)=\Pi_{\ker D}+R(T,\phi)$

with $R(T,\phi)$ going to 0 exponentially in $\Psi^{-\infty}_{G}(X)$. We
conclude that

(5.15)
$\displaystyle\lim_{T\to+\infty}\int_{0}^{\Theta}T^{1/2}\tau^{X}_{g}\left(\exp(-TD_{\phi}^{2})\right)d\phi$
$\displaystyle=$
$\displaystyle\lim_{T\to+\infty}\int_{0}^{\Theta}T^{1/2}e^{-T\phi^{2}}\cdot\tau^{X}_{g}\left(\exp\left(-T(D^{2}+2\phi
D)\right)\right)d\phi$ $\displaystyle=$
$\displaystyle\lim_{T\to+\infty}\int_{0}^{\Theta}T^{1/2}e^{-T\phi^{2}}\cdot\tau^{X}_{g}\left(\Pi_{\ker
D}\right)d\phi$ $\displaystyle=$
$\displaystyle\frac{\sqrt{\pi}}{2}\cdot\tau^{X}_{g}\left(\Pi_{\ker D}\right).$

Combining the above computation, we conclude that

$\lim_{\Theta\downarrow
0}\eta_{g}(\Theta)=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}((D)\exp(-sD^{2}))\frac{dt}{\sqrt{t}}+\tau^{X}_{g}\left(\Pi_{\ker
D}\right)=\eta_{g}(D)+\tau^{X}_{g}(\Pi_{\ker D}).$

∎

###  5.2. Delocalized eta invariants associated to smoothing perturbations

We know consider a perturbed Dirac operator $D+C$ with $C$ an element in
$\mathcal{L}^{\infty}_{G,s}(X,E)$ making the operator $D+C$ invertible.

######  5.16 Proposition.

Let $(X,\mathbf{h})$ be an odd dimensional cocompact $G$-proper manifold
without boundary and $\mathbf{h}$ a $G$-invariant Riemannian metric. Let $D$
be a Dirac-type operator associated to the Clifford connection. If
$C\in\mathcal{L}^{c}_{G}(X)$, has the structure explained in Proposition 3.17
and is such that $D+C$ is $L^{2}$-invertible then

$\eta_{g}(D+C):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}((D+C)\exp(-t(D+C)^{2})\frac{dt}{\sqrt{t}}$

is well defined.

###### Proof.

We made sense of $\exp(-t(D+C)^{2})$ as an element
$\mathcal{L}^{\infty}_{G,s}(X)$ and proved that $\exp(-t(D+C)^{2}))$ is
converging to $0$ in $\mathcal{L}^{\infty}_{G,s}(X)$ as $t\to+\infty$. A
similar estimates show that $(D+C)\exp(-t(D+C)^{2})$ is of $o(t)$ as
$t\to\infty$. (c.f. Proposition 3.9). This implies that

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

converges at $t=+\infty$.

On the other hand, by construction, $\exp(-t(D^{2}+A))-\exp(-tD^{2})$ is
$O(t)$ in $\mathcal{L}^{\infty}_{G,s}(X)$. Since $C\exp(-tD^{2})\to C$ as
$t\to 0$ in $\mathcal{L}^{\infty}_{G,s}(X)$ by Proposition 4.1, we conclude
that

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

is also convergent as the integral

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

converges ([PPST, Theorem 5.1]). The proof of the proposition is complete. ∎

##  6\. The delocalized APS index theorem in the gap case

In this section we shall discuss an extension of the 0-degree delocalized APS
index theorem to the case in which the boundary operator has a spectral gap.

We assume that

(6.1) $\exists\;a>0\;\;\text{such that}\;\;{\rm
spec}_{L^{2}}(D_{\partial})\cap(-a,a)=\\{0\\}$

Let $D$ be the $b$-Dirac operator on $Y$ associated to $D_{0}$ on $Y_{0}$, a
$\mathbb{Z}_{2}$-graded odd Dirac operator, product-type near the boundary.
Fix $\Theta\in(0,a)$ and consider the $\mathbb{Z}_{2}$-graded odd $b$-operator
given by

$D^{+}_{\Theta}:=x^{-\Theta}D^{+}x^{\Theta}\,,\quad
D^{-}_{\Theta}:=(x^{-\Theta}D^{+}x^{\Theta})^{*}=x^{\Theta}D^{-}x^{-\Theta}\,.$

This is a $0$-th order perturbation of $D$; indeed

(6.2) $D^{+}_{\Theta}\equiv
D^{+}+x^{-\Theta}[D,x^{\Theta}]=D^{+}+x^{-\Theta}{\rm cl}(d(x^{\Theta}))\,.$

and similarly for $D^{-}_{\Theta}$. It is known that

(6.3) $(D^{\pm}_{\Theta})_{\partial}=D_{\partial}+\Theta\,;$

as we shall now explain. By [Melrose-Book, Prop. 5.8],

$(x^{-\Theta}D^{+}x^{\Theta})_{\partial}=I(D^{+},-i\Theta)$

and it is well known that $I(D^{+},-i\Theta)=i(-i\Theta)+D_{\partial}$ which
is $\Theta+D_{\partial} $; a similar computation holds for $D^{-}_{\Theta}$:

$(x^{\Theta}D^{-}x^{-\Theta})_{\partial}=I(D^{-},i\Theta);$

moreover $I(D^{-},i\Theta)=-i(i\Theta)+D_{\partial}$ which is again equal to
$D_{\partial}+\Theta$ as claimed. This means that $D_{\Theta}$ has an
invertible boundary operator, provided $\Theta\in(0,a)$. We define the index
class in the gap case, as the index class associated to $D_{\Theta}$. This
does not depend on $\Theta$, for $\Theta\in(0,a)$.There is a well defined
delocalized APS numeric index given by
$\langle\operatorname{Ind}_{\infty}(D_{\Theta}),\tau^{Y}_{g}\rangle$.
Proceeding as before, using in particular the relative index class, excision
and the relative cyclic cocycle $(\tau^{Y,r}_{g},\sigma^{\partial Y}_{g})$ we
find that for each $s>0$

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

Proceeding as for the eta integrand for for $D_{X}+\Theta$, see Definition 5.1
and Proposition 5.9, we can use the short time analysis of the heat kernel on
the $b$-manifold $(Y,\mathbf{h})$ in order to prove that

${\rm LIM}_{s\downarrow
0}\left(\tau^{Y,r}_{g}(e^{-s^{2}(D^{-}_{\Theta}D^{+}_{\Theta})})-\tau^{Y,r}_{g}(e^{-s^{2}(D^{+}_{\Theta}D^{-}_{\Theta})})\right)$

is well defined. Thus, following Melrose, we have:

$\displaystyle\langle\operatorname{Ind}_{\infty}(D_{\Theta}),\tau^{Y}_{g}\rangle$
$\displaystyle={\rm LIM}_{s\downarrow
0}\left(\tau^{Y,r}_{g}(e^{-s^{2}(D^{-}_{\Theta}D^{+}_{\Theta})})-\tau^{Y,r}_{g}(e^{-s^{2}(D^{+}_{\Theta}D^{-}_{\Theta})})\right)$
$\displaystyle-{\rm LIM}_{s\downarrow
0}\frac{1}{2\sqrt{\pi}}\int_{s}^{\infty}\tau^{\partial
Y}_{g}((D_{\partial}+\Theta)\exp(-s(D_{\partial}+\Theta)^{2}))\frac{dt}{\sqrt{t}}$
$\displaystyle={\rm LIM}_{s\downarrow
0}\left(\tau^{Y,r}_{g}(e^{-s^{2}(D^{-}_{\Theta}D^{+}_{\Theta})})-\tau^{Y,r}_{g}(e^{-s^{2}(D^{+}_{\Theta}D^{-}_{\Theta})})\right)-\eta_{g}(\Theta)$

where the definition of $\eta_{g}(\Theta)$, the modified eta invariant
associated to $D_{\partial}+\Theta$, has been used. See Definition 5.4. Now,
the first summand in the right hand side of the above equation depends
continuously on $\Theta\in(0,a)$ and has a limit as $\Theta\downarrow 0$ equal
to

$\int_{Y_{0}^{g}}c_{0}^{g}{\rm AS}_{g}(D_{0})\,.$

See (1.4). For the second summand we recall Proposition 5.9:

$\lim_{\Theta\downarrow 0}\eta(\Theta)=\eta_{g}(D_{\partial})+\tau^{\partial
Y}_{g}(\Pi_{\ker D_{\partial}}).$

We can thus conclude that the following theorem holds:

######  6.4 Theorem.

Let $G$ be a connected, linear real reductive group and let $g\in G$ be a
semisimple element. Let $(Y_{0},\mathbf{h}_{0})$ be an even dimensional
cocompact $G$-proper manifold with boundary, with metric which is product type
near the boundary. Let $D_{0}$ be a $G$-equivariant Dirac operator, product
type near the boundary. We assume that the associated operator on $\partial Y$
satisfies

$\exists\;a>0\;\;\text{such that}\;\;{\rm
spec}_{L^{2}}(D_{\partial})\cap(-a,a)=\\{0\\}.$

Let $(Y,\mathbf{h})$ be the $b$-manifold associated to
$(Y_{0},\mathbf{h}_{0})$ and let $D$ be the $b$-operator associated to
$D_{0}$. Then for $\Theta\in(0,a)$ there exists a well defined smooth index
class $\operatorname{Ind}_{\infty}(D_{\Theta})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y,E))=K_{0}(C^{*}(Y_{0}\subset Y,E)^{G})$
which is independent of $\Theta\in(0,a)$ and denoted briefly
$\operatorname{Ind}_{\infty}(D)$. For the pairing of this index class with the
0-degree cyclic cocycle $\tau_{g}^{Y}$ the following formula holds:

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

##  7\. The delocalized index theorem corresponding to smoothing perturbations

In this section we shall prove an index formula for Dirac operators that admit
a smoothing perturbation of the boundary operator that makes it invertible.

As in the previous section, we do not assume that $D_{\partial}$ is
$L^{2}$-invertible; instead we assume the existence of an operator
$C_{\partial}\in\mathcal{L}^{c}_{G}(X)\equiv\mathcal{L}^{c}_{G}(\partial Y)$
such that $D_{\partial}+C_{\partial}$ is $L^{2}$-invertible. By cobordism
invariance of the index one can in fact prove that such a perturbation always
exists, but we shall not enter into a discussion of this result just now. We
know then that

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

is well defined. Our goal in this section is to show that this delocalized eta
invariant is the boundary correction term in a delocalized APS index theorem.
Geometric applications will be given in the next section.

We show first of all that we can lift the perturbation
$C_{\partial}\in\mathcal{L}^{c}_{G}(X)$ to a perturbation
$C_{Y}\in{}^{b}\mathcal{L}^{c}_{G}(Y)$. We follow closely [Mel-P1]. Recall the
short exact sequence (2.7) of algebras

$0\to\mathcal{L}^{c}_{G}(Y)\to{}^{b}\mathcal{L}^{c}_{G}(Y)\xrightarrow{I}{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}({\rm
cyl}(\partial Y))\to 0.$

Using a suitable cut-off function near the boundary, we can define a section
$\sigma:{}^{b}\mathcal{L}^{c}_{G,\mathbb{R}}({\rm cyl}(\partial
Y))\rightarrow{}^{b}\mathcal{L}^{c}_{G}(Y)$ of the indicial homomorphism. Thus
we can define an operator on $Y$ by defining a translation invariant operator
on ${\rm cyl}(\partial Y)$ and then applying $\sigma$ to it. Starting with
$C\in\mathcal{L}^{c}_{G}(X)$, with $X=\partial Y_{0}\equiv\partial Y$, we thus
want to define a translation invariant operator on ${\rm cyl}(\partial Y)$; by
inverse Mellin transform, we can equivalently define the indicial family of
such an operator.

Let $\rho\in C^{\infty}_{c}(\mathbb{R})$ be non negative, even and have
integral 1, so that $\rho_{\epsilon}(t):=\epsilon^{-1}\rho(t/\epsilon)$
approximates $\delta_{0}$ as $\epsilon\downarrow 0$. The Fourier-Laplace
transform

$\widehat{\rho}_{\epsilon}(z)=\int_{\mathbb{R}}e^{-itz}\rho_{\epsilon}(t)dt,\quad
z\in\mathbb{C},$

is therefore entire, even, real for real $z$ and has value at $0$ equal to 1.
Moreover, it is rapidly decreasing in ${\rm Re}z$ on any horizontal strip
$|{\rm Imz}|<N$ and as $\epsilon\downarrow 0$ it approximates the constant
function 1 uniformly on any compact subset of $\mathbb{C}$.

We consider the holomorphic family of operators on $X=\partial Y$ given by

$S(\epsilon,z):=\widehat{\rho}_{\epsilon}(z)C_{\partial}$

and consider the inverse Mellin transform for $z=\lambda\in\mathbb{R}$. By
applying suitable identifications near the boundary, this defines an element
$C^{+}_{\rm cyl}(\epsilon)$ in
${}^{b}\mathcal{L}^{\infty}_{G,s_{0},\mathbb{R}}({\rm cyl}(\partial Y))$ where
we do not write the bundles as usual; see [Mel-P1, Equation (8.5)]. By
applying $\sigma:{}^{b}\mathcal{L}^{\infty}_{G,s_{0},\mathbb{R}}({\rm
cyl}(\partial Y))\rightarrow{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$ to this
element we define finally

$C^{+}_{Y}(\epsilon)\,,\quad
C^{-}_{Y}(\epsilon):=(C^{+}_{Y}(\epsilon))^{*}\,.$

We have now defined an operator

$D+C_{Y}(\epsilon):=\left(\begin{array}[]{cc}0&D^{-}+C^{-}_{Y}(\epsilon)\\\
D^{+}+C^{+}_{Y}(\epsilon)&0\end{array}\right).$

The indicial family $I(D^{+}+C^{+}_{Y}(\epsilon),\lambda)$,
$\lambda\in\mathbb{R}$ is $L^{2}$-invertible for $\epsilon$ small; indeed, up
to identifications near the boundary,

$I(D^{+}+C^{+}_{Y}(\epsilon),\lambda)=i\lambda+S(\epsilon,\lambda)+D_{\partial}$

which is invertible by construction at $\lambda=0$ and it is invertible if
$\lambda\in\mathbb{R}\setminus\\{0\\}$ given that $D_{\partial}$ is self-
adjoint and $\epsilon$ is small, see [Mel-P1, Lemma 8]. From now on we fix
$\epsilon$ small as to ensure that
$I(D^{+}+C^{+}_{Y}(\epsilon),\lambda)=i\lambda+S(\epsilon,\lambda)+D_{\partial}$
is $L^{2}$-invertible for all $\lambda\in\mathbb{R}$. We then set

(7.1)
$R(\lambda):=S(\epsilon,\lambda):=\widehat{\rho}_{\epsilon}(\lambda)C_{\partial}$

and remark that

(7.2) $R(0)=C_{\partial}\quad\text{and}\quad R(\lambda)=R(-\lambda)$

We write $C_{Y}$ instead of $C_{Y}(\epsilon)$.
Consequently we have a well-defined index class

$\operatorname{Ind}(D+C_{Y})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))=K_{0}(C^{*}(Y)^{G})$

that can be defined à la Connes-Skandalis through a true parametrix or,
equivalently, through the heat kernel and the improved Connes-Moscovici
projector $V^{b}(D+C_{Y})$. This index class does not depend on $\epsilon$,
for $\epsilon$ small.

Notation. Following [PiazzaSchick_BCrho] we shall also denote this index class
by $\operatorname{Ind}(D,C_{\partial})$.

We also have the Connes-Moscovici projectors $V(D+C_{Y})$ and $V(D_{{\rm
cyl}}+C_{\rm cyl})$ that define together a relative class

$[V(D+C_{Y}),e_{1},r_{t}]\in
K_{0}({}^{b}\mathcal{L}^{\infty}_{G,s}(Y),{}^{b}\mathcal{L}^{\infty}_{G,s,\mathbb{R}}({\rm
cyl}(\partial Y)))$

with $e_{1}:=\begin{pmatrix}0&0\\\ 0&1\end{pmatrix}$ and $r_{t}$,
$t\in[1,+\infty]$, defined as follows

(7.3) $r_{t}:=\begin{cases}V(t(D_{\operatorname{cyl}}+C_{{\rm
cyl}}))\;\;\quad\text{if}\;\;\;t\in[1,+\infty)\\\
e_{1}\;\;\;\;\;\;\;\;\;\;\;\;\;\,\text{ if }\;\;t=\infty.\end{cases}$

Here we have used Proposition 3.17 to see that this is well-defined.
Proceeding as in [PPST] we then have that

(7.4)
$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D+C_{Y})\rangle=\langle[\tau^{Y,r}_{g},\sigma^{\partial
Y}_{g}],[V(D+C_{Y}),e_{1},r_{t}],\rangle$

We shall use this equality in order to prove the following

######  7.5 Theorem.

Let $C_{\partial}\in\mathcal{L}^{c}_{G}(\partial Y)$ be such that
$D_{\partial}+C_{\partial}$ is $L^{2}$-invertible. Then there exists a well
defined index class $\operatorname{Ind}(D,C_{\partial})\in
K_{0}(\mathcal{L}^{\infty}_{G,s}(Y))=K_{0}(C^{*}(Y_{0}\subset Y,E)^{G})$.
Moreover, for the delocalized index
$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D,C_{\partial})\rangle$ the
following formula holds

(7.6)
$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D,C_{\partial})\rangle=\int_{(Y_{0})^{g}}c^{g}{\rm
AS}_{g}(D_{0})-\frac{1}{2}\eta_{g}(D_{\partial}+C_{\partial})$

with

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

###### Proof.

We have already discussed the first part of the statement. We thus concentrate
on the proof of (7.6). Let us consider the right hand-side.

Notation. To simplify the notation we set

$A^{\pm}:=D^{\pm}+C^{\pm}_{Y}\,\quad A^{\pm}_{{\rm cyl}}:=D^{\pm}_{{\rm
cyl}}+C^{\pm}_{{\rm cyl}}\,,\quad B(\lambda)=D_{\partial}+R(\lambda)$

where we recall that
$R(\lambda)=\widehat{\rho}_{\epsilon}(\lambda)C_{\partial}$, so that
$R(0)=C_{\partial}$ and $R(\lambda)=R(-\lambda)$.

By definition of relative pairing we have:

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

We now rescale the operator on $Y$ and consider $s(D+C_{Y})$. This defines the
same index class $\operatorname{Ind}(D,C_{\partial})$ and so

$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D,C_{\partial})\rangle=\tau^{Y,r}_{g}(e^{-s^{2}A^{-}A^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}A^{+}A^{-}})+\int_{s}^{\infty}\sigma^{\partial
Y}_{g}([\dot{r}_{t},r_{t}],r_{t})dt\,.$

Observe that

$s^{2}A^{-}A^{+}=s^{2}D^{-}D^{+}+s^{2}F^{-,+}\quad\text{with}\quad
F^{-,+}\in{}^{b}\mathcal{L}^{\infty}_{G,s}(Y)$

and similarly for $s^{2}A^{+}A^{-}$. By a classic Volterra series argument we
obtain that

$\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}A^{-}A^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}A^{+}A^{-}})=\lim_{s\downarrow
0}\tau^{Y,r}_{g}(e^{-s^{2}D^{-}D^{+}})-\tau^{Y,r}_{g}(e^{-s^{2}D^{+}D^{-}})$

and by [PPST, Proposition 4.47] we thus have

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

Consequently, from the $s$-independence of the index class we obtain the
existence of the integral

$\int_{0}^{\infty}\sigma^{\partial Y}_{g}([\dot{r}_{t},r_{t}],r_{t})dt$

and the formula

$\langle[\tau^{Y}_{g}],\operatorname{Ind}(D,C_{\partial})\rangle=\int_{(Y_{0})^{g}}c^{g}{\rm
AS}_{g}(D_{0})+\int_{0}^{\infty}\sigma^{\partial
Y}_{g}([\dot{r}_{t},r_{t}],r_{t})dt\,.$

We consider $\int_{0}^{\infty}\sigma^{\partial
Y}_{g}([\dot{r}_{t},r_{t}],r_{t})dt$ where we recall that

$\int_{0}^{\infty}\sigma^{\partial
Y}_{g}([\dot{r}_{t},r_{t}],r_{t})dt=\frac{i}{2\pi}\int_{\mathbb{R}}\tau^{\partial
Y}_{g}(\partial_{\lambda}(I([\dot{r}_{t},r_{t}],\lambda))\circ
I(r_{t},\lambda))d\lambda.$

We can integrate by parts in $\lambda$ on the right hand side, given that the
two terms are rapidly decreasing in $\lambda$. Thus we consider

(7.8) $\frac{i}{2\pi}\int_{0}^{\infty}\left(\int_{\mathbb{R}}\tau^{\partial
Y}_{g}((I([\dot{r}_{t},r_{t}],\lambda)\circ\partial_{\lambda}(I(r_{t},\lambda))d\lambda\right)dt.$

######  7.9 Lemma.

The integral (7.8) is equal to

$-\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\tau^{\partial
Y}_{g}(D_{\partial}+C_{\partial})\exp(-t(D_{\partial}+C_{\partial})^{2})\frac{dt}{\sqrt{t}}.$

###### Proof.

We are considering the integral (7.8)

$\int_{0}^{\infty}\left(\int_{\mathbb{R}}\tau^{\partial
Y}_{g}((I([\dot{r}_{t},r_{t}],\lambda)\circ\partial_{\lambda}(I(r_{t},\lambda))d\lambda\right)dt.$

We write down the indicial family of $r_{t}$. By definition

$r_{t}=\left(\begin{array}[]{cc}e^{-t^{2}A_{\operatorname{cyl}}^{-}A_{\operatorname{cyl}}^{+}}&e^{-\frac{t^{2}}{2}A_{\operatorname{cyl}}^{-}A_{\operatorname{cyl}}^{+}}\left(\frac{I-e^{-t^{2}A_{\operatorname{cyl}}^{-}A_{\operatorname{cyl}}^{+}}}{t^{2}A_{\operatorname{cyl}}^{-}A_{\operatorname{cyl}}^{+}}\right)tA_{\operatorname{cyl}}^{-}\\\
e^{-\frac{t^{2}}{2}A_{\operatorname{cyl}}^{+}A_{\operatorname{cyl}}^{-}}tA^{+}&I-e^{-t^{2}A_{\operatorname{cyl}}^{+}A_{\operatorname{cyl}}^{-}}\end{array}\right)$

where we recall that $A^{\pm}_{{\rm cyl}}:=D^{\pm}_{{\rm cyl}}+C^{\pm}_{{\rm
cyl}}$. Recall, also, that we are using the notation
$B(\lambda)=D_{\partial}+R(\lambda)$ where, once again, we do not write the
$\epsilon$ in $R$ because we have fixed it. The indicial family of $r_{t}$ is
given by

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

We set

$r(t,\lambda):=I(r_{t},\lambda)\,.$

We now consider the integrand in (7.8):

Claim. We have the equality

(7.10) $\tau_{g}^{\partial
Y}\left([\partial_{t}r(t,\lambda),r(t,\lambda)]\circ\partial_{\lambda}r(t,\lambda)\right)=-2it\tau_{g}\left(e^{-t^{2}(\lambda^{2}+B(\lambda)^{2})}(B(\lambda)-\lambda\partial_{\lambda}B(\lambda))\right)$

Proof of the claim.
Computing the $t$-derivative of $r(t,\lambda)$ and we see that the $2\times
2$-matrix $Q(t,\lambda):=\left[\partial_{t}r(t,\lambda),r(t,\lambda)\right]$
has entries that consist of products of operators of the form $B(\lambda)\pm
i\lambda$, $(\lambda^{2}+B(\lambda)^{2})^{\pm 1}$ and heat-kernels
$e^{-\frac{t^{2}}{2}(\lambda^{2}+B(\lambda)^{2})}$. In particular, the
operators all commute with each other since they are obtained by applying
functional calculus to the operator $B(\lambda)\pm i\lambda$.

The $\lambda$-derivative of $r(\lambda,t)$ is more involved and introduces the
operator $\partial_{\lambda}B(\lambda)$ which does not necessarily commute
with other terms: to compute the derivative of the heat kernel we use
Duhamel’s formula

$\partial_{\lambda}r_{11}(t,\lambda)=\partial_{\lambda}e^{-t^{2}(\lambda^{2}+B(\lambda)^{2})}=\int_{0}^{t}e^{(t-s)^{2}(\lambda^{2}+B(\lambda)^{2})}\partial_{\lambda}(\lambda^{2}+B(\lambda)^{2})e^{s^{2}(\lambda^{2}+B(\lambda)^{2})}ds.$

Also to compute the $\lambda$-derivative of
$(\lambda^{2}+B(\lambda)^{2})^{-1}$ we use the identity

$(\lambda^{2}+B(\lambda)^{2})^{-1}=\int_{0}^{\infty}e^{-s(\lambda^{2}+B(\lambda)^{2})}ds,$

which holds true because $\lambda^{2}+B(\lambda)^{2}$ is invertible, and
subsequently use Duhamel’s formula once again. If we now remark that in the
computation the only source of noncommutativity is
$\partial_{\lambda}B(\lambda)$ and since there is at most one of them in each
term, we can use cyclicity of the trace $\tau_{g}$ to obtain the identities

$\displaystyle\tau_{g}\left(\ldots\partial_{\lambda}e^{-t^{2}(\lambda^{2}+B(\lambda)^{2})}\ldots\right)$
$\displaystyle=-2t^{2}\tau_{g}\left(\ldots
e^{-t^{2}(\lambda^{2}+B(\lambda)^{2})}\ldots(\lambda+B(\lambda)\partial_{\lambda}B(\lambda))\right),$
$\displaystyle\tau_{g}\left(\ldots\partial_{\lambda}(\lambda^{2}+B(\lambda^{2}))^{k}\ldots\right)$
$\displaystyle=2k\tau_{g}\left(\ldots(\lambda^{2}+B(\lambda^{2}))^{k-1}\ldots(\lambda+B(\lambda)\partial_{\lambda}B(\lambda))\right),\quad
k\in\mathbb{Z},$

where the $\ldots$ denotes terms commuting with
$\partial_{\lambda}B(\lambda)$. We therefore see that in the computation we
can always put the terms with $\partial_{\lambda}B(\lambda)$ on the right hand
side in the trace. With this convention, the computation reduces to a
straightforward, but tedious, computation in which we can assume that all
operators commute with each other. This can even be done using e.g.
Mathematica and the final outcome is:

$\tau_{g}\left(\left[\partial_{t}r(t,\lambda),r(t,\lambda)\right]\partial_{\lambda}(t,\lambda)\right)=-2it\tau_{g}\left(e^{-t^{2}(\lambda^{2}+B(\lambda)^{2})}(B(\lambda)-\lambda\partial_{\lambda}B(\lambda))\right).$

This finishes the proof of the claim.

To finish the proof of the Theorem, we adapt an argument from [loya, §3], in
turn inspired by [Mel-P1]. We have the following analogue of [loya, Lemma
3.10]:

######  7.11 Lemma.

Let $A(r,\lambda)$ be a continuous and rapidly decreasing function on
$[0,1]\times\mathbb{R}$ with values $\mathcal{L}^{\infty}_{G,s}(X)$. The
integrals

$\int_{\mathbb{R}}\tau_{g}^{X}\left(A(r,\lambda)e^{-t^{2}(\lambda^{2}+B(r\lambda)^{2})}\right)d\lambda,\qquad\int_{\mathbb{R}}\tau_{g}^{X}\left(B(r\lambda)e^{-t^{2}(\lambda^{2}+B(r\lambda)^{2})}\right)d\lambda,$

converge absolutely and are $\mathcal{O}(t^{-1})$ as $t\downarrow 0$ and
vanish exponentially as $t\to\infty$, both uniformly in $r\in[0,1]$.

###### Proof.

The arguments around (3.8), together with the Volterra expansion for the
perturbed heat kernel allow us to apply the same arguments as in [loya] once
we take into account that $\tau_{g}^{X}$ is a continuous trace on
$\mathcal{L}^{\infty}_{G,s}(X)$. ∎

Next, we define

$\displaystyle\zeta(r):=\frac{1}{\pi}\int_{0}^{\infty}\zeta(r,t)dt=\frac{1}{\pi}\int_{0}^{\infty}\int_{\mathbb{R}}\tau_{g}\left(e^{-t^{2}(\lambda^{2}+B(r\lambda)^{2})}t(B(r\lambda)-\lambda\partial_{\lambda}B(r\lambda))\right)d\lambda
dt.$

Remark that $\lambda\partial_{\lambda}B(r\lambda)$ is rapidly decreasing
function of $(r,\lambda)$ with values in $\mathcal{L}^{\infty}_{G,s}(X)$, so
the integrals above converge by Lemma 7.11. We can write the $t$-integrand as

$\zeta(r,t)=\int_{\mathbb{R}}\tau_{g}\left(L(\tilde{B})e^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\right)d\lambda,$

where $\tilde{B}(r,t,\lambda):=tB(r\lambda)$ and
$L:=t\partial_{t}-\lambda\partial_{\lambda}$. We then compute the
$r$-derivative of $\zeta(r,t)$ as

$\frac{d\zeta(r,t)}{dr}=\underbrace{\int_{\mathbb{R}}\tau_{g}\left(L\left(\frac{d\tilde{B}}{dr}\right)e^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\right)d\lambda}_{{\rm(I)}}+\underbrace{\int_{\mathbb{R}}\tau_{g}\Bigg{(}L\left(\tilde{B}\right)\frac{d}{dr}e^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\Bigg{)}d\lambda}_{\rm(II)}$

In the above formula, we can move the differential inside by dominated
convergence: Lemma 7.11 tells us that (I) converges absolutely. For (II) we
use Eq. (7.12) below together with the fact that
$\frac{d\tilde{B}}{dr}\tilde{B}$ and $\tilde{B}\frac{d\tilde{B}}{dr}$ are
rapidly decreasing in $\lambda$ in $\mathcal{L}^{\infty}_{G,s}(X)$ to obtain
absolute convergence by the same Lemma.

The first term (I) we work out using an integration by parts:

$\displaystyle{\rm(I)}$
$\displaystyle=\frac{d}{dt}\left[\int_{\mathbb{R}}\tau_{g}\left(t\frac{d\tilde{B}}{dr}e^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\right)d\lambda\right]-\int_{\mathbb{R}}\tau_{g}\left(\frac{d\tilde{B}}{dr}Le^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\right)d\lambda$
$\displaystyle=\frac{d}{dt}\left[\int_{\mathbb{R}}\tau_{g}\left(t\frac{d\tilde{B}}{dr}e^{-t^{2}\lambda^{2}-\tilde{B}^{2}}\right)d\lambda\right]+\int_{\mathbb{R}}e^{-t^{2}\lambda^{2}}\int_{0}^{1}\tau_{g}\left(\frac{d\tilde{B}}{dr}e^{-(t-s)^{2}B^{2}}\left(L(\tilde{B})\tilde{B}+\tilde{B}L(\tilde{B})\right)e^{s^{2}B^{2}}\right)dsd\lambda,$

by Duhamel’s formula combined with the fact that $L(e^{-t^{2}\lambda^{2}})=0$.
We can also write out the second term (II) using Duhamel’s formula:

(7.12)
${\rm(II)}=-\int_{\mathbb{R}}e^{-t^{2}\lambda^{2}}\int_{0}^{1}\tau_{g}\left(L(\tilde{B})e^{-(t-s)^{2}B^{2}}\left(\frac{d\tilde{B}}{dr}\tilde{B}+\tilde{B}\frac{d\tilde{B}}{dr}\right)e^{s^{2}B^{2}}\right)dsd\lambda.$

After this, putting the two terms (I) and (II) back together, we find that the
terms invoking Duhamel’s formula combine to a commutator:

$\displaystyle\int_{\mathbb{R}}e^{-t^{2}\lambda^{2}}\int_{0}^{1}\tau_{g}\left[\frac{d\tilde{B}}{dr},e^{-(t-s)^{2}B^{2}}\left(L(\tilde{B})\tilde{B}+\tilde{B}L(\tilde{B})\right)e^{s^{2}B^{2}}\right]dsd\lambda=0,$

by the trace property of $\tau_{g}$. This results in the following formula for
the derivative of $\zeta(r,t)$:

$\frac{d\zeta(r,t)}{dr}=\frac{d}{dt}\left[\int_{\mathbb{R}}\tau_{g}\left(t^{2}\frac{dB(r\lambda)}{dr}e^{-t^{2}(\lambda^{2}-B(r\lambda)^{2})}\right)d\lambda\right].$

With this formula we see that

$\displaystyle\frac{d\zeta(r)}{dr}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\frac{d\zeta(r,t)}{dr}dt$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\frac{d}{dt}\left[\int_{\mathbb{R}}\tau_{g}\left(t^{2}\frac{dB(r\lambda)}{dr}e^{-t^{2}(\lambda^{2}-B(r\lambda)^{2})}\right)d\lambda\right]dt=0,$

where we have used Lemma 7.11 in the final identity to evaluate the boundary
terms. It follows that $\zeta(r)$ is constant and therefore
$\zeta(0)=\zeta(1)$. But $\zeta(1)$ is exactly $(2\pi i)^{-1}$ times the right
hand side of Eq. (7.10). On the other hand we find, after a Gaussian
integration,

$\zeta(0)=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}e^{-t(D_{\partial}+C_{\partial})^{2}}(D_{\partial}+C_{\partial})\frac{dt}{\sqrt{t}}.$

This finishes the proof of Lemma 7.9. ∎

With the Lemma, the proof of Theorem 7.5 is now also complete. ∎

##  8\. Numeric rho invariants on $G$-proper manifolds

In our previous work [PPST] we have defined (higher) rho numbers associated to
a $G$-invariant positive scalar curvature metric on a $G$-proper cocompact
spin manifold. In this section we shall introduce rho numbers associated to
$G$-equivariant homotopy equivalences. We shall only give the main ideas and
do not insist on details, see the remark after (8.5) for an explanation.

###  8.1. Rho numbers associated to $G$-homotopy equivalences

We consider a closed $G$-proper manifold $X$ without boundary, $G$ a connected
linear real reductive, $g\in G$ a semisimple element, $D_{X}$ a
$G$-equivariant Dirac operator. We assume the existence of a perturbation
$B\in\mathcal{L}^{c}_{G}(X,E)$ with the property that there exists $a>0$ such
that

${\rm spec}_{L^{2}}(D_{X}+B)\cap[-2a,2a]=\emptyset\,.$

We consider

$\eta_{g}(D_{X}+B):=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\tau^{X}_{g}((D_{X}+B)\exp(-t(D_{X}+B)^{2})\frac{dt}{\sqrt{t}}\;;$

we have proved in Proposition 5.16 that this integral is well defined.

A fundamental geometric example where this number is relevant is obtained by
considering $\mathbf{f}:X_{1}\to X_{2}$, an oriented $G$-homotopy equivalence
between two cocompact oriented $G$-proper manifolds. Using [Fukumoto] we know
that there exists a bounded perturbation $B_{\mathbf{f}}$ of the signature
operator on $X:=X_{1}\sqcup(-X_{2})$, where $-X_{2}$ is the same manifold as
$X_{2}$ with the opposite orientation, that makes it invertible. Moreover, by
adapting for example the work of Spessato [spessato], we can prove that this
perturbation $B_{\mathbf{f}}$ is in $C^{*}(X,\Lambda^{*}X)^{G}$. Hence, by
density, we conclude that there exists a perturbation
$B_{\mathbf{f}}\in\mathcal{L}^{c}_{G}(X,\Lambda^{*}X)$ such that $D^{{\rm
sign}}_{X}+B_{\mathbf{f}}$ is $L^{2}$-invertible. We can thus define the rho-
number of the homotopy equivalence $\mathbf{f}$ as

(8.1) $\rho_{g}(\mathbf{f}):=\eta_{g}(D^{{\rm sign}}_{X}+B_{\mathbf{f}})\,.$

######  8.2 Remark.

We expect this number to be well defined, independent of the choices that
enter into the definition of the Hilsum-Skandalis perturbation. The analogous
result for Galois covering is discussed, for example [Wahl].

###  8.2. Bordism properties

The delocalized APS index theorem proved in this article can be used in order
to study the bordism properties of these rho invariants.

######  8.3 Definition.

Consider a cocompact proper $G$-manifold $(W,\mathbf{h})$, possibly with
boundary. Let $D_{W}$ be a Dirac operator on $W$, defined by a unitary and
Clifford connection. Let $g\in G$ be semisimple. We shall say that $g$ is
geometrically-simple on $W$ if

$\int_{W^{g}}c^{g}{\rm AS}_{g}(W,S)=0\,.$

with ${\rm AS}_{g}(W,S)$ the usual Atiyah-Segal integrand appearing as the
local term in a 0-degree delocalized APS-index theorem for $D_{W}$.

As explained in [PPST], the following proposition provides many examples of
geometrically-simple elements $g$ on an arbitrary $G$-proper manifold $W$.

######  8.4 Proposition.

If $g$ is non-elliptic, that is, does not conjugate to a compact element, then
every element of the conjugacy class $C(g):=\\{hgh^{-1}|h\in G\\}$ in $G$ does
not have any fixed point on $W$.

We now consider the case of the signature operator. We can again give the
definition of $g$ geometrically simple, employing this time the
$L$-differential form.

Let $Y$ be a cocompact orientable $G$-proper manifold. A $G$-structure over
$Y$ is a cocompact orientable $G$-proper manifold $M$ together with an
oriented $G$-homotopy equivalence $M\xrightarrow{\mathbf{f}}Y$. Two
$G$-structures over $Y$, $(M_{1}\xrightarrow{\mathbf{f}_{1}}Y)$ and
$(M_{2}\xrightarrow{\mathbf{f}_{2}}Y)$, are $G$-h-cobordant if there exists an
orientable $G$-proper manifold with boundary $W$ and an oriented $G$-homotopy
equivalence $\mathbf{F}:W\to Y\times[0,1]$ restricting to
$(M_{1}\xrightarrow{\mathbf{f}_{1}}Y)$ and
$(M_{2}\xrightarrow{\mathbf{f}_{2}}Y)$ over the boundary.

Consider such a $G$-h-cobordism $\mathbf{F}:W\to Y\times[0,1]$. By combining
the work of Fukumoto [Fukumoto] and Wahl [Wahl] we expect that it should be
possible to prove that there exists a well-defined signature index class
associated to $X:=W\sqcup-(Y\times[0,1])$ (for this it would suffice that
$\mathbf{F}|_{\partial}$ is a $G$-homotopy equivalence) and that, moreover,
this index class is equal to zero (this is because $\mathbf{F}$ is a global
$G$-homotopy equivalence). Moreover, we expect this vanishing to be a
consequence of the existence of a global perturbation that belongs to
${}^{b}\mathcal{L}^{\infty}_{G,s}(X,\Lambda^{*}X)$ and that gives a suitable
extension of the perturbations $B_{\mathbf{f}_{1}}$ and $B_{\mathbf{f}_{2}}$
on $\partial X$ associated to $\mathbf{f}_{1}$ and $\mathbf{f}_{2}$.

Once this technicalities have been resolved we shall be in the position of
applying our delocalized APS index theorem and thus see that that if $g\in G$
is geometrically-simple on $W\sqcup-(Y\times[0,1])$, for example $g$ is in the
conjugacy class of a non-elliptic element, then

(8.5) $\rho_{g}(\mathbf{f}_{1})=\rho_{g}(\mathbf{f}_{2}).$

######  8.6 Remark.

It is unclear whether such a result, the $G$-h-cobordism invariance of the rho
number in (8.1), can have immediate geometric applications, as in the case of
free proper $\Gamma$-manifolds, with $\Gamma$-discrete. This is the reason why
we have been rather sketchy in this last section. Indeed, to have such
geometric applications would seem to require the development of a
$G$-equivariant surgery sequence for a non-compact Lie group $G$, certainly a
non-trivial task. See for example [Do-S] for the case in which $G$ is compact.
We leave this task to future research.

## References