# Nonlocal elliptic problems associated with actions of discrete groups on
manifolds with boundary

Boltachev A.V., Savin A.Yu

###### Abstract

Given a manifold with boundary endowed with an action of a discrete group on
it, we consider the algebra of operators generated by elements in the Boutet
de Monvel algebra of pseudodifferential boundary value problems and shift
operators acting on functions on the manifold and its boundary. Under certain
conditions on the group action, we construct a Chern character for elliptic
elements in this algebra with values in a de Rham type cohomology of the fixed
point manifolds for te group action and obtain an index formula in terms of
this Chern character.

##  1 Introduction

The aim of this work is to solve the index problem for nonlocal elliptic
boundary value problems associated with actions of discrete groups on
manifolds with boundary.

The index problem in elliptic theory (see [1]) consists in computing indices
of elliptic operators in terms of topological invariants of the principal
symbol of operators and topological invariants of manifolds on which the
operators are defined.

Index formulas are known in many geometric situations. In particular, many
authors developed index theory for elliptic boundary value problems in the
framework of classical boundary value problems (see [2, 3, 4]) and for Boutet
de Monvel algebra of pseudodifferential boundary value problems with boundary
and coboundary operators (see [5, 6, 7, 8, 9, 10]).

Firstly, we note that obtaining index formulas in the case of
pseudodifferential boundary value problems was of significant difficulty
because principal symbols of the boundary value problems in question are
operator functions defined on the cotangent bundle of the boundary of the
manifold and it was necessary to construct topological invariants which take
into account this operator function. These difficulties were overcame in [8],
where a topological index of elliptic problems in Boutet de Monvel algebra was
defined using special regularized traces on the algebra of principal symbols.

Secondly, we note that all known proofs of the index formulas for elliptic
boundary value problems use stable homotopies (these homotopies are
constructed using $K$-theory of the $C^{*}$-algebra of symbols), which allow
to reduce the boundary value problem to such a form that its index is equal to
the index of an operator on the double of the manifold with boundary. In this
case, the index is easy to calculate using Atiyah-Singer formula (see [3, 7,
8, 10]).

Noncommutative geometry of Connes [11] contributes greately to the development
of index theory. In noncommutative geometry, one usually considers algebras of
operators, the principal symbols of which generate essentially noncommutative
algebras. The algebras of crossed product type associated with group actions
on manifolds (see [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]) appear in many
applications. The corresponding equations on the manifold in question include
pseudodifferential operators as well as shift operators along the orbits of
the group action. For such operators, ellipticity conditions were obtained
which provide Fredholm solvability of the problem in Sobolev spaces (see
[15]). However, index formulas were obtained only in the case of smooth closed
manifolds (see [20, 22, 19, 23]). As a first step towards obtaining index
formulas for boundary value problems associated with group actions on
manifolds with boundary, a classification (up to stable homotopies) of
elliptic boundary value problems was obtained and $K$-groups of the
corresponding symbol algebra were calculated in [24].

In the present paper, we construct the topological index for elliptic boundary
value problems on manifolds with boundary endowed with an isometric action of
a discrete group of polynomial growth in the sense of Gromov [25]. More
precisely, for the principal symbol of such a problem we construct the Chern
character with values in a de Rham type cohomology of the cotangent bundle of
fixed point submanifolds. The definition of Chern character uses traces on the
algebra of noncommutative differential forms from [20] and on regularized
traces from [8] on the algebra of symbols of Boutet de Monvel operators.
Unlike the paper [8], where a topological index is constructed as a number, we
improve this construction and construct a Chern character for the symbol in
suitable cohomology groups, and the topological index is obtained from it by
taking product with the Todd class of the manifold and integrating over the
fundamental cycle.

Let us briefly describe the contents of the paper. In Sec. 2, we recall the
definitions related to the Boutet de Monvel algebra of pseudodifferential
boundary value problems on a manifold with boundary. In Sec. 3, we study the
algebra generated by Boutet de Monvel operators and shift operators associated
with isometric actions of discrete groups of polynomial growth on manifolds
with boundary. We state an ellipticity condition for the elements in this
algebra, their Fredholm property is established. After that we solve the index
problem. To this end, we firstly (in Sec. 4) construct cohomology of de Rham
type for the manifolds, whose boundary is the total space of a fibration (the
cotangent bundle of a manifold with boundary has this structure and this
structure implicitly appears in [8]). Furthermore, in Sec. 5, we introduce
noncommutative differential forms on the boundary and regularized traces on
them. These constructions enable us to define the Chern character for elliptic
problems as a cohomology class on the cotangent bundle of the manifold of
fixed points of the group action, considered as a manifold with fibered
boundary (in the sense of Sec. 4). Next, in Sec. 6, we give the definition of
the Todd class of the manifold and state the index theorem. The proof of the
index theorem is given in Secs. 7 and 8. To this end, we establish a homotopy
classification of elliptic boundary value problems, which enables us to make a
homotopy of the boundary value problem to a quite simple boundary value
problem in a neighborhood of the boundary of the manifold. Next, for such a
problem the index formula can be checked by a reduction of the operator to the
double and application of the index formula in [20] on the double.

The work of the first author was supported by RFBR grant 19-01-00574, the work
of the second author was supported by RFBR grant 19-01-00447.

##  2 Boutet de Monvel algebra

Let us recall the main properties of the Boutet de Monvel algebra (for more
detailed exposition see [5, 7, 26, 27]).

Let $M$ be a smooth compact manifold with boundary $X$. We suppose that $M$ is
endowed with a Riemannian metric. In a neighborhood of the boundary we will
use local coordinates $x=(x^{\prime},x_{n})$ on $M$, where ${\rm dim}M=n$,
$x^{\prime}=(x_{1},\ldots,x_{n-1})$ are the local coordinates on $X$, and
$x_{n}$ is a defining function of the boundary, i.e., the boundary is locally
defined by the equation $x_{n}=0$, while $M$ is defined by the inequality
$x_{n}\geq 0$. Let us consider Boutet de Monvel operators of zero order and
type. We write such operators as follows

$\mathcal{D}=\left(\begin{array}[]{cc}A+G&C\\\
B&A_{X}\end{array}\right):\begin{array}[]{c}L^{2}(M)\\\ \oplus\\\
L^{2}(X)\end{array}\longrightarrow\begin{array}[]{c}L^{2}(M)\\\ \oplus\\\
L^{2}(X)\end{array},$ (1)

where

  * •

$A$ is a pseudodifferential operator ($\psi$DO) of order zero on $M$, the
principal symbol of which satisfies the transmission property (see below);

  * •

$A_{X}$ is a $\psi$DO of order zero on $X$;

  * •

$B,C$ and $G$ are boundary, coboundary and Green operators, respectively.

Recall that a classical symbol $a=a(x^{\prime},x_{n},\xi^{\prime},\xi_{n})$
with an asymptotic expansion $a\leavevmode\nobreak\ a_{m}+a_{m-1}+...$
satisfies the transmission property, if its order $m\in\mathbb{Z}$ and for all
$k\in\mathbb{Z}_{+}$ and arbitrary multi-index
$\alpha=(\alpha_{1},...,\alpha_{n-1})\in\mathbb{Z}^{n-1}_{+}$ the following
equality is satisfied for its homogeneous components $a_{l}$:

$D^{k}_{x_{n}}D^{\alpha}_{\xi^{\prime}}a_{l}(x^{\prime},0,0,1)=e^{i\pi(l-|\alpha|)}D^{k}_{x_{n}}D^{\alpha}_{\xi^{\prime}}a_{l}(x^{\prime},0,0,-1),$

where

$D^{\alpha}_{\xi^{\prime}}=\left(-i\frac{\partial}{\partial\xi_{1}}\right)^{\alpha_{1}}\cdot...\cdot\left(-i\frac{\partial}{\partial\xi_{n-1}}\right)^{\alpha_{n-1}}.$

The principal symbol of operator (1) is the pair
$\sigma(\mathcal{D})=(\sigma_{M}(\mathcal{D}),\sigma_{X}(\mathcal{D}))$, where
the first component is called the interior symbol and is a function

$\sigma_{M}(\mathcal{D})=\sigma(A)\in C^{\infty}(S^{*}M),$ (2)

where $S^{*}M=\\{(x,\xi)\in T^{*}M,|\xi|=1\\}$ is the cosphere bundle of $M$,
where $\sigma(A)$ is the principal symbol of a $\psi$DO $A$. The second
component is called the boundary symbol and is an operator function

$\sigma_{X}(\mathcal{D})\in
C^{\infty}(S^{*}X,\mathcal{B}(L^{2}(\mathbb{R}_{+})\oplus\mathbb{C}))\simeq
C^{\infty}(S^{*}X,\mathcal{B}(\overline{H}_{+}\oplus\mathbb{C}))$

on the cosphere bundle $S^{*}X$ of the boundary, where $\mathcal{B}$ is the
algebra of bounded operators,

$H_{+}=\mathcal{F}(\mathcal{S}(\overline{\mathbb{R}}_{+}))$

stands for the space of images with the respect to the Fourier transform
$\mathcal{F}_{x_{n}\to\xi_{n}}$ of the Schwartz space
$\mathcal{S}(\overline{\mathbb{R}}_{+})$ of smooth rapidly decaying at
infinity functions on $\overline{\mathbb{R}}_{+}$. Denote the norm closure of
$H_{+}\subset L^{2}(\mathbb{R})$ by $\overline{H}_{+}$. Similarly, we define
the space $H_{-}$ as

$H_{-}=\mathcal{F}(\mathcal{S}(\overline{\mathbb{R}}_{-})).$

Next, we define the projection $\Pi_{+}:H_{+}\oplus H_{-}\to H_{+}$ to the
first summand and the continuous functional

$\begin{array}[]{ccc}\Pi^{\prime}:H_{+}\oplus
H_{-}&\longrightarrow&\mathbb{C},\vspace{2mm}\\\
u(\xi_{n})&\longmapsto&\lim_{x_{n}\to 0+}\mathcal{F}^{-1}_{\xi_{n}\to
x_{n}}(u(\xi_{n})).\end{array}$

Note that if $u\in L^{1}(\mathbb{R})\cap(H_{+}\oplus H_{-})$, then

$\Pi^{\prime}u=\frac{1}{2\pi}\int_{\mathbb{R}}u(\xi_{n})d\xi_{n}.$ (3)

Let us denote the algebra of boundary symbols $\sigma_{X}(\mathcal{D})$ by
$\Sigma_{X}\subset
C^{\infty}(S^{*}X,\mathcal{B}(\overline{H}_{+}\oplus\mathbb{C}))$. To describe
the elements of the algebra $\Sigma_{X}$, we consider smooth functions

  * •

$b(x^{\prime},\xi^{\prime},\xi_{n})\in C^{\infty}(S^{*}X,H_{-})$;

  * •

$c(x^{\prime},\xi^{\prime},\xi_{n})\in C^{\infty}(S^{*}X,H_{+})$;

  * •

$g(x^{\prime},\xi^{\prime},\xi_{n},\eta_{n})\in C^{\infty}(S^{*}X,H_{+}\otimes
H_{-})$;

  * •

$q(x^{\prime},\xi^{\prime})\in C^{\infty}(S^{*}X)$.

Here the spaces $H_{\pm}$ and their topological tensor products are considered
as Fréchet spaces. An arbitrary boundary symbol $a_{X}\in\Sigma_{X}$ is a
smooth operator family

$a_{X}(x^{\prime},\xi^{\prime})=\left(\begin{array}[]{cc}\Pi_{+}a(x^{\prime},0,\xi^{\prime},\xi_{n})+\Pi^{\prime}_{\eta_{n}}g(x^{\prime},\xi^{\prime},\xi_{n},\eta_{n})&c(x^{\prime},\xi^{\prime},\xi_{n})\\\\[5.69054pt]
\Pi^{\prime}_{\xi_{n}}b(x^{\prime},\xi^{\prime},\xi_{n})&q(x^{\prime},\xi^{\prime})\end{array}\right):\begin{array}[]{c}\overline{H}_{+}\\\
\oplus\\\
\mathbb{C}\end{array}\longrightarrow\begin{array}[]{c}\overline{H}_{+}\\\
\oplus\\\ \mathbb{C}\end{array}$ (4)

with the parameters $(x^{\prime},\xi^{\prime})\in S^{*}X$. Here
$a(x^{\prime},0,\xi^{\prime},\xi_{n})$ is the restriction to the boundary of a
symbol homogeneous of order zero with the transmission property on $M$. The
function $a(x^{\prime},0,\xi^{\prime},\xi_{n})$ is called the principal symbol
of the boundary symbol $a_{X}$. The operator of type (4) acts on the pair
$h\in\overline{H}_{+},v\in\mathbb{C}$ as follows:

$a_{X}(x^{\prime},\xi^{\prime})\left(\begin{matrix}h\\\
v\end{matrix}\right)=\left(\begin{matrix}\Pi_{+}(a(x^{\prime},0,\xi^{\prime},\xi_{n})h(\xi_{n}))+\Pi^{\prime}_{\eta_{n}}(g(x^{\prime},\xi^{\prime},\xi_{n},\eta_{n})h(\eta_{n}))+c(x^{\prime},\xi^{\prime},\xi_{n})v\\\\[5.69054pt]
\Pi^{\prime}_{\xi_{n}}(b(x^{\prime},\xi^{\prime},\xi_{n})h(\xi_{n}))+q(x^{\prime},\xi^{\prime})v\end{matrix}\right)$
(5)

It is known that smooth families in (4) and (5) form an algebra.

Let us denote the algebra of matrices (1) by
$\Psi_{B}(M)\subset\mathcal{B}(L^{2}(M)\oplus L^{2}(X))$. This algebra is
called the Boutet de Monvel algebra.

######  Theorem 1.

([7], Section 2.2.4.4, Corollary 2) The symbol mapping

$\begin{array}[]{ccc}\Psi_{B}(M)&\longrightarrow&C^{\infty}(S^{*}M)\oplus
C^{\infty}(S^{*}X,\mathcal{B}(L^{2}(\mathbb{R}_{+})\oplus\mathbb{C}))\\\
\mathcal{D}&\longmapsto&(\sigma_{M}(\mathcal{D}),\sigma_{X}(\mathcal{D}))\end{array}$

is well defined and continuously extends to a monomorphism of $C^{*}$-algebras

$\overline{\Psi_{B}(M)}\longrightarrow C(S^{*}M)\oplus
C(S^{*}X,\mathcal{B}(L^{2}(\mathbb{R}_{+})\oplus\mathbb{C})).$

The regularized trace $\operatorname{Tr}^{\prime}$ of a boundary
symbol$a_{X}\in\Sigma_{X}$ is defined by the formula

$\operatorname{Tr}^{\prime}a_{X}(x^{\prime},\xi^{\prime})=\Pi^{\prime}_{\eta_{n}}g(x^{\prime},\xi^{\prime},\eta_{n},\eta_{n})+q(x^{\prime},\xi^{\prime}).$
(6)

The mapping $\operatorname{Tr}^{\prime}$ does not possess the trace property.
More precisely, the following trace defect formula was obtained in [8, Section
2.4, Lemma 2.1]: given $a_{X,1},a_{X,2}\in\Sigma_{X}$, we have

$\operatorname{Tr}^{\prime}[a_{X,1},a_{X,2}]=-i\Pi^{\prime}\left(\frac{\partial
a_{1}(\xi_{n})}{\partial\xi_{n}}a_{2}(\xi_{n})\right)=i\Pi^{\prime}\left(a_{1}(\xi_{n})\frac{\partial
a_{2}(\xi_{n})}{\partial\xi_{n}}\right),$ (7)

where $a_{1},a_{2}$ are the principal symbols of $a_{X,1},a_{X,2}$
respectively.

##  3 $\Gamma$-Boutet de Monvel operators. Fredholm property

### Group actions and shift operators.

Let $\Gamma$ be a discrete finitely generated group of isometries
$\gamma\colon M\to M$, which preserve the boundary $\gamma(X)=X$. Given
$\gamma\in\Gamma$, we define the shift operator

$T_{\gamma}:L^{2}(M)\oplus L^{2}(X)\longrightarrow L^{2}(M)\oplus
L^{2}(X),\quad(u(x),v(x^{\prime}))\longmapsto(u(\gamma^{-1}(x)),v(\gamma^{-1}(x^{\prime}))).$

This operator is unitary if we equip the $L^{2}$-spaces with the norms,
defined by the volume forms associated with the Riemannian metric. The mapping
$\gamma\mapsto T_{\gamma}$ defines a unitary representation of $\Gamma$ on
$L^{2}(M)\oplus L^{2}(X)$.

It is known that compositions $T_{\gamma}\mathcal{D}T_{\gamma}^{-1}$, where
$\mathcal{D}$ is a Boutet de Monvel operator and $\gamma\in\Gamma$, is also a
Boutet de Monvel operator. Moreover, the interior and boundary symbols of
$T_{\gamma}\mathcal{D}T_{\gamma}^{-1}$ are equal to

$\sigma_{M}(T_{\gamma}\mathcal{D}T_{\gamma}^{-1})(x,\xi)=\sigma_{M}(\mathcal{D})(\partial\gamma^{-1}(x,\xi)),\quad\sigma_{X}(T_{\gamma}\mathcal{D}T_{\gamma}^{-1})(x^{\prime},\xi^{\prime})=\sigma_{X}(\mathcal{D})(\partial\gamma^{-1}(x^{\prime},\xi^{\prime})).$

Here the actions of $\Gamma$ on $M$ and $X$ are lifted to the bundles
${T}^{*}M$ and $T^{*}X$ using codifferentials
$\partial\gamma=(d\gamma^{t})^{-1}$ of the corresponding diffemorphisms (here
$d\gamma$ is the differential of $\gamma$, while $d\gamma^{t}$ is its dual
mapping of the cotangent bundle).

###  $\Gamma$-Boutet de Monvel operators.

Let us recall the definition of smooth crossed products (see [28] or [22]).
Let $\mathcal{A}$ be a Fréchet algebra with seminorms $\|\cdot\|_{m}$,
$m\in\mathbb{N}$, and $\Gamma$ be a group of polynomial growth in the sense of
Gromov (see [25]), acting on $\mathcal{A}$ by automorphisms
$a\mapsto\gamma(a)$, where $a\in\mathcal{A}$ and $\gamma\in\Gamma$. Then the
smooth crossed product denoted by $\mathcal{A}\rtimes\Gamma$ is the vector
space of functions $f:\Gamma\to\mathcal{A}$, which rapidly decay at infinity
in the sence of the following estimates:

$\|f(\gamma)\|_{m}\leq C_{m,N}(1+|\gamma|)^{-N}$

for all $N,m\in\mathbb{N}$ and $\gamma\in\Gamma$, where the constant $C_{m,N}$
does not depend on $\gamma$. Here $|\gamma|$ is the length $\gamma$ in the
word metric on $\Gamma$. Finally, we assume that the action of $\Gamma$ on
$\mathcal{A}$ satisfies the following property: given $m\in\mathbb{N}$, there
exists $k\in\mathbb{N}$ and a real polynomial $P(z)$ such that the following
inequality

$\|\gamma(a)\|_{m}\leq P(|\gamma|)\|a\|_{k}$

holds for all $a$ and $\gamma$. The product in $\mathcal{A}\rtimes\Gamma$ is
defined by the formula:

$\\{f_{1}(\gamma)\\}\cdot\\{f_{2}(\gamma)\\}=\left\\{\sum_{\gamma_{1}\gamma_{2}=\gamma}f_{1}(\gamma_{1})\gamma_{1}(f_{2}(\gamma_{2}))\right\\}.$
(8)

It can be shown that the right hand side in (8) is an element of
$\mathcal{A}\rtimes\Gamma$, i.e., this space is an algebra.

The action of $\Gamma$ on Fréchet algebras $C^{\infty}(S^{*}M),\Sigma_{X}$ and
$\Psi_{B}(M)$ satisfies the conditions above, and the following smooth crossed
products are defined:
$C^{\infty}(S^{*}M)\rtimes\Gamma,\Sigma_{X}\rtimes\Gamma$ and
$\Psi_{B}(M)\rtimes\Gamma$. Elements in the smooth crossed product
$\Psi_{B}(M)\rtimes\Gamma$ define operators

$\left\\{\mathcal{D}_{\gamma}\right\\}_{\gamma\in\Gamma}\in\Psi_{B}(M)\rtimes\Gamma\quad\longmapsto\quad\sum_{\gamma\in\Gamma}\mathcal{D}_{\gamma}T_{\gamma}:L^{2}(M)\oplus
L^{2}(X)\to L^{2}(M)\oplus L^{2}(X)$ (9)

######  Definition 1.

Operators in (9) are called $\Gamma$-Boutet de Monvel operators.

######  Definition 2.

The symbol of operator (9) is a pair
$\sigma(\mathcal{D})=(\sigma_{M}(\mathcal{D}),\sigma_{X}(\mathcal{D}))$, which
consists of the interior and the boundary symbols

$\sigma_{M}(\mathcal{D})=\\{\sigma(A_{\gamma})\\}_{\gamma\in\Gamma}\in
C^{\infty}(S^{*}M)\rtimes\Gamma,\quad\sigma_{X}(\mathcal{D})=\\{\sigma_{X}(\mathcal{D}_{\gamma})\\}_{\gamma\in\Gamma}\in\Sigma_{X}\rtimes\Gamma.$
(10)

The operators (9) form an algebra and symbols (10) enjoy the composition
formula: given $\mathcal{D}_{1},\mathcal{D}_{2}\in\Psi_{B}(M)\rtimes\Gamma$ we
have
$\sigma_{M}(\mathcal{D}_{1}\mathcal{D}_{2})=\sigma_{M}(\mathcal{D}_{1})\sigma_{M}(\mathcal{D}_{2})$
and
$\sigma_{X}(\mathcal{D}_{1}\mathcal{D}_{2})=\sigma_{X}(\mathcal{D}_{1})\sigma_{X}(\mathcal{D}_{2})$.

### Operators, acting between ranges of projections.

A matrix $\Gamma$-Boutet de Monvel operator is a triple
$(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ such that

  * •

$\mathcal{D}\in{\rm Mat}_{N}(\Psi_{B}(M)\rtimes\Gamma)$ is a matrix with the
components in $\Psi_{B}(M)\rtimes\Gamma$;

  * •

$\mathcal{P}_{j}\in{\rm Mat}_{N}(C^{\infty}(M)\rtimes\Gamma\oplus
C^{\infty}(X)\rtimes\Gamma),\;j=1,2;$

  * •

the following relation is satisfied
$\mathcal{P}_{2}\mathcal{D}\mathcal{P}_{1}=\mathcal{D}.$

Then we define the operator

$\mathcal{D}:\mathcal{P}_{1}(L^{2}(M,\mathbb{C}^{N})\oplus
L^{2}(X,\mathbb{C}^{N}))\longrightarrow\mathcal{P}_{2}(L^{2}(M,\mathbb{C}^{N})\oplus
L^{2}(X,\mathbb{C}^{N})).$ (11)

Denote operator (11) by $(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$.

We also introduce the notation

$P_{j}=\sigma_{M}(\mathcal{P}_{j}),\qquad
P^{\prime}_{j}=\sigma_{X}(\mathcal{P}_{j}).$

######  Definition 3.

Operator $(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ is elliptic, if there
exists a matrix operator $(\mathcal{R},\mathcal{P}_{2},\mathcal{P}_{1})$ such
that the following equalities hold

$\sigma(\mathcal{D})\sigma(\mathcal{R})=\sigma(\mathcal{P}_{2}),\qquad\sigma(\mathcal{R})\sigma(\mathcal{D})=\sigma(\mathcal{P}_{1}).$

######  Theorem 2.

An elliptic operator (11) has Fredholm property.

The proof is standard (see, for instance, [15]).

##  4 Two de Rham complexes for manifolds with fibered boundary

Let us consider a Riemannian manifold $M$ with boundary $\partial M$ and
suppose that the boundary is a total space of a locally trivial fiber bundle
$\pi:\partial M\to X$ with the fiber $F$. Then the pair $(M,\pi)$ is called a
manifold with fibered boundary.

The embedding $i:\partial M\to M$ induces the restriction mapping
$i^{*}:\Omega^{*}(M)\to\Omega^{*}(\partial M)$ of the differential forms to
the boundary. The projection $\pi$ defines the induced mapping
$\pi^{*}:\Omega^{*}(X)\to\Omega^{*}(\partial M)$ and the mapping

$\pi_{*}:\Omega_{c}^{*}(\partial
M)\longrightarrow\Omega_{c}^{*-\nu}(X),\quad\nu=\operatorname{dim}F,$

of integration of compactly supported differential forms over the fiber $F$
(see, for instance, [29]). Here we suppose that the fibers of $\pi$ have
orientation continuously depending on the point of the base. Let us recall the
definition of the integral of form over the fiber.

######  Definition 4.

The integral of a form $\omega\in\Omega_{c}^{k}(\partial M)$ over the fibers
of $\pi:\partial M\to X$ is a differential form
$\pi_{*}{\omega}\in\Omega_{c}^{k-\nu}(X)$ such that

$\int\limits_{X}{\Big{(}\pi_{*}{\omega}\Big{)}\wedge\omega_{1}}=\int_{\partial
M}{\omega}\wedge\pi^{*}\omega_{1}$

for all differential forms $\omega_{1}$ on $X$. The following properties are
valid:

  1. 1.

$\pi_{*}(\omega\wedge\pi^{*}\omega_{1})=(\pi_{*}\omega)\wedge\omega_{1},$ for
all forms $\omega\in\Omega_{c}^{k}(\partial
M),\omega_{1}\in\Omega_{c}^{k}(X)$;

  2. 2.

$d(\pi_{*}\omega)=(-1)^{\nu}\pi_{*}(d\omega)$, for all forms
$\omega\in\Omega_{c}^{k}(\partial M)$.

Let us consider the graded morphism

$(\Omega_{c}^{*}(M),d)\stackrel{{\scriptstyle\pi_{*}i^{*}}}{{\longrightarrow}}(\Omega_{c}^{*-\nu}(X),d),\quad
d\pi_{*}i^{*}=(-1)^{\nu}\pi_{*}i^{*}d$ (12)

of the de Rham complexes on $M$ and $X$. Denote the cone of $\pi_{*}i^{*}$ by
$(\Omega^{*}(M,\pi),\partial)$, where

$\Omega_{c}^{j}(M,\pi)=\Omega_{c}^{j}(M)\oplus\Omega_{c}^{j-\nu-1}(X),\quad\partial=\left(\begin{array}[]{cc}d&0\\\
\pi_{*}i^{*}&(-1)^{\nu+1}d\end{array}\right).$ (13)

Note that

$\partial^{2}=\left(\begin{array}[]{cc}d&0\\\
\pi_{*}i^{*}&(-1)^{\nu+1}d\end{array}\right)\left(\begin{array}[]{cc}d&0\\\
\pi_{*}i^{*}&(-1)^{\nu+1}d\end{array}\right)=\left(\begin{array}[]{cc}d^{2}&0\\\
\pi_{*}i^{*}d+(-1)^{\nu+1}d(\pi_{*}i^{*})&(-1)^{2\nu+2}d^{2}\end{array}\right)\\\
=\left(\begin{array}[]{cc}0&0\\\
\pi_{*}i^{*}d+(-1)^{\nu+1}d\pi_{*}i^{*}&0\end{array}\right)=0.$

The last equality follows from (12).

We denote the cohomology groups of the complex
$(\Omega_{c}^{*}(M,\pi),\partial)$ by $H_{c}^{*}(M,\pi)$.

Also, we consider the complex
$(\widetilde{\Omega}^{*}(M,\pi),\widetilde{\partial})$:

$\widetilde{\Omega}^{j}(M,\pi)=\\{(\omega,\omega_{X})\in\Omega^{j}(M)\oplus\Omega^{j}(X)\;|\;i^{*}\omega=\pi^{*}\omega_{X}\\},\quad\widetilde{\partial}=\left(\begin{array}[]{cc}d&0\\\
0&d\end{array}\right).$ (14)

Note that $di^{*}\omega=d\pi^{*}\omega_{X}$ since
$i^{*}\omega=\pi^{*}\omega_{X}$. Hence, $i^{*}d\omega=\pi^{*}d\omega_{X}$,
since $i^{*}$ and $\pi^{*}$ are the induced mappings on differential forms,
i.e., the mapping $\widetilde{\partial}$ is well defined. We denote the
cohomology groups of the complex
$(\widetilde{\Omega}^{*}(M,\pi),\widetilde{\partial})$ by
$\widetilde{H}^{*}(M,\pi)$.

Component-wise exterior multiplication of differential forms gives the product

$\wedge:\Omega_{c}^{j}(M,\pi)\times\widetilde{\Omega}^{k}(M,\pi)\longrightarrow\Omega_{c}^{j+k}(M,\pi).$

This operation enjoys the Leibniz rule

$\partial(a\wedge b)=\partial a\wedge
b+(-1)^{j}a\wedge\widetilde{\partial}b,\quad
a\in\Omega_{c}^{j}(M,\pi),b\in\widetilde{\Omega}^{k}(M,\pi).$ (15)

Indeed, given

$a=\left(\begin{array}[]{c}\omega\\\
\omega_{X}\end{array}\right)\quad\text{and}\quad
b=\left(\begin{array}[]{c}\omega^{\prime}\\\
\omega^{\prime}_{X}\end{array}\right),$

we have

$\partial(a\wedge
b)=\partial\left(\begin{array}[]{c}\omega\wedge\omega^{\prime}\\\
\omega_{X}\wedge\omega^{\prime}_{X}\end{array}\right)=\left(\begin{array}[]{c}d(\omega\wedge\omega^{\prime})\\\
\pi_{*}i^{*}(\omega\wedge\omega^{\prime})+(-1)^{\nu+1}d(\omega_{X}\wedge\omega^{\prime}_{X})\end{array}\right)\\\
=\left(\begin{array}[]{c}d\omega\wedge\omega^{\prime}+(-1)^{j}\omega\wedge
d\omega^{\prime}\\\ \pi_{*}(i^{*}\omega\wedge
i^{*}\omega^{\prime})+(-1)^{\nu+1}d\omega_{X}\wedge\omega^{\prime}_{X}+(-1)^{\nu+1}(-1)^{j-\nu-1}\omega_{X}\wedge
d\omega^{\prime}_{X}\end{array}\right)\\\
=\left(\begin{array}[]{c}d\omega\wedge\omega^{\prime}\\\
\pi_{*}i^{*}(\omega)\wedge\omega^{\prime}_{X}+(-1)^{\nu+1}d\omega_{X}\wedge\omega^{\prime}_{X}\end{array}\right)+\left(\begin{array}[]{c}(-1)^{j}\omega\wedge
d\omega^{\prime}\\\ (-1)^{j}\omega_{X}\wedge
d\omega^{\prime}_{X}\end{array}\right)\\\
=\partial\left(\begin{array}[]{c}\omega\\\
\omega_{X}\end{array}\right)\wedge\left(\begin{array}[]{c}\omega^{\prime}\\\
\omega^{\prime}_{X}\end{array}\right)+(-1)^{j}\left(\begin{array}[]{c}\omega\\\
\omega_{X}\end{array}\right)\wedge\widetilde{\partial}\left(\begin{array}[]{c}\omega^{\prime}\\\
\omega^{\prime}_{X}\end{array}\right).$

It follows from the Leibniz rule (15) that $\wedge$ defines a product in
cohomology

$\wedge:H_{c}^{j}(M,\pi)\times\widetilde{H}_{c}^{k}(M,\pi)\longrightarrow
H_{c}^{j+k}(M,\pi).$ (16)

Finally, we suppose that $M$ and $X$ are oriented manifolds and their
orientations are compatible with the orientation of the fibers in the
following way. Denote $n={\rm dim}M$ and choose as positively oriented the
form $(-1)^{n}dt_{1}\wedge\ldots\wedge dt_{\nu}\wedge dy_{1}\wedge\ldots\wedge
dy_{k}\wedge dx_{n}$, where $t_{1},\ldots,t_{\nu}$ are some positively
oriented coordinates on the fiber, $y_{1},\ldots,y_{n-\nu-1}$ are some
positively oriented coordinates on $X$, while $x_{n}\geq 0$ is a defining
function of the boundary.

We define the linear functional

$\begin{array}[]{rcc}\displaystyle\langle\cdot,[M,\pi]\rangle:H_{c}^{*}(M,\pi)&\longrightarrow&\mathbb{C}\vspace{2mm}\\\
(\omega,\omega_{X})&\longmapsto&\displaystyle\int_{M}\omega-(-1)^{n}\int_{X}\omega_{X}.\end{array}$

To prove that this functional is well defined, we choose forms
$\omega=f(t,y,x_{n})dt_{1}\wedge\ldots\wedge dy_{n-\nu-1}$ and
$\omega_{X}=g(y)dy_{1}\wedge\ldots\wedge dy_{n-\nu-1}$ and compute

$\langle\partial(\omega,\omega_{X}),[M,\pi]\rangle=\int\limits_{M}d\omega-(-1)^{n}\int\limits_{X}\left(\pi_{*}i^{*}\omega+(-1)^{\nu+1}d\omega_{X}\right)=\int\limits_{\partial
M}i^{*}\omega-(-1)^{n}\int\limits_{X}\left(\pi_{*}i^{*}\omega\right)\\\
=(-1)^{n}\\!\\!\\!\\!\int\limits_{\mathbb{R}^{\nu}\times\mathbb{R}^{n-\nu-1}}\\!\\!\\!\\!\\!\\!\\!f(t,y,0)dt_{1}...dy_{n-\nu-1}-(-1)^{n}\\!\\!\\!\\!\int\limits_{\mathbb{R}^{n-\nu-1}}{\left(\int\limits_{\mathbb{R}^{\nu}}{f(t,y,0)dt_{1}...dt_{\nu}}\right)dy_{1}...dy_{n-\nu-1}}=0.$

##  5 Chern character of elliptic symbols

### Noncommutative differential forms. Regularized trace.

In this section, we define the Chern character for symbols of elliptic
operators (11). The definition uses noncommutative differential forms which we
now define. Denote by

$\widetilde{\Sigma}_{X}\subset
C^{\infty}(T^{*}X,\mathcal{B}(\overline{H}_{+}\oplus\mathbb{C}))$

the algebra of boundary symbols on $T^{*}X$.

Let us consider the action of $\Gamma$ on the algebras $\Omega(T^{*}M)$ and
$\widetilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)$ of differential
forms with compact support and corresponding smooth crossed products

$\Omega(T^{*}M)\rtimes\Gamma\text{ and
}\left(\widetilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)\right)\rtimes\Gamma.$

These crossed products are differential graded algebras with respect to the
gradings defined by the gradings of the differential forms and the
differentials defined by the exterior differential on differential forms.

Given $\gamma\in\Gamma$, we define the mappings (cf. [20, 30])

$\tau^{\gamma}:\Omega(T^{*}M)\rtimes\Gamma\longrightarrow\Omega(T^{*}M^{\gamma}),$
(17)
$\tau^{\gamma}_{X}:\left(\widetilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)\right)\rtimes\Gamma\longrightarrow\Omega(T^{*}X^{\gamma}).$
(18)

To define these mappings, we introduce some notations. We denote the closure
of $\Gamma$ in the compact Lie group of isometries of $M$ by
$\overline{\Gamma}$. This closure is a compact Lie group. Let
$C^{\gamma}\subset\overline{\Gamma}$ be the centralizer111We recall that the
centralizer of $\gamma$ is a subgroup of elements which commute with $\gamma$.
of $\gamma$. The centralizer is a closed Lie subgroup in $\overline{\Gamma}$.
The elements of centralizer are denoted by $h$, while the induced Haar measure
on the centralizer is denoted as $dh$. Next, for an element
$\gamma^{\prime}\in\langle\gamma\rangle$ in the conjugacy class of $\gamma$,
we fix an arbitrary element $z=z(\gamma,\gamma^{\prime})$, which conjugates
$\gamma$ and $\gamma^{\prime}=z\gamma z^{-1}$. Any such element defines the
diffeomorphism $\partial z:T^{*}M^{\gamma}\to T^{*}M^{\gamma^{\prime}}$ for
the corresponding fixed point manifolds.

We define the functional in (17) by

$\tau^{\gamma}(\omega)=\sum_{\gamma^{\prime}\in\langle\gamma\rangle}\;\;\;\int_{C^{\gamma}}\Bigl{.}h^{*}\bigl{(}{z}^{*}\omega(\gamma^{\prime})\bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh,\quad\text{where
}\omega\in\Omega(T^{*}M)\rtimes\Gamma,$ (19)

and the functional (18) by

$\tau^{\gamma}_{X}(\omega_{X})=\sum_{\gamma^{\prime}\in\langle\gamma\rangle}\;\;\;\int_{C^{\gamma}}\operatorname{Tr}_{X}\left(\Bigl{.}h^{*}\bigl{(}{z}^{*}\omega_{X}(\gamma^{\prime})\bigr{)}\bigr{|}_{T^{*}X^{\gamma}}\right)dh,\quad\text{where
}\omega_{X}\in\left(\widetilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)\right)\rtimes\Gamma.$
(20)

Here

$\operatorname{Tr}_{X}\left(\sum_{I}\omega_{I}(t)dt^{I}\right)=\sum_{I}\operatorname{Tr}^{\prime}(\omega_{I}(t))dt^{I},$

where $\operatorname{Tr}^{\prime}:\widetilde{\Sigma}_{X}\to
C^{\infty}(T^{*}X)$ is the regularized trace defined earlier in (6).

######  Proposition 1.

The following assertions hold:

  1. 1.

The summands in (19) and (20) do not depend on the choice of the elements $z$.

  2. 2.

Functionals (19) and (20) have the following properties:

$\tau^{\gamma}(\omega_{1}\wedge\omega_{2})=(-1)^{\deg\omega_{1}\deg\omega_{2}}\tau^{\gamma}(\omega_{2}\wedge\omega_{1}),\text{
for all }\omega_{1},\omega_{2}\in\Omega(T^{*}M)\rtimes\Gamma,$ (21)
$d\tau^{\gamma}_{X}(\omega)=\tau^{\gamma}_{X}(d\omega),\quad\text{for all
}\omega\in\left(\widetilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)\right)\rtimes\Gamma.$
(22)

###### Proof.

Let us prove 1. Indeed, let $z_{1}$ be another element such that
$\gamma^{\prime}=z_{1}\gamma z_{1}^{-1}$. Then for the functional (19) we
have:

$\int_{C^{\gamma}}\Bigl{.}h^{*}\bigl{(}z_{1}^{*}\omega(\gamma^{\prime})\bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh=\int_{C^{\gamma}}\Bigl{.}(z_{1}h)^{*}\bigl{(}\omega(\gamma^{\prime})\bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh=\int_{C^{\gamma}}\Bigl{.}h^{*}\bigl{(}z^{*}\omega(\gamma^{\prime})\bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh.$

Here in the last equality we made the change of variable $z_{1}h=zh^{\prime}$
in the integral and used the invariance of the Haar measure. Equality (20) is
proved similarly.

Now let us move on to part 2. Firstly, we show that $\tau^{\gamma}$ is a
graded trace, i.e., equality (21) holds. It suffices to prove this property
for the following forms $\omega_{1},\omega_{2}$:

$\omega_{1}(\gamma)=\left\\{\begin{matrix}a,&\gamma=\gamma_{1},\\\
0,&\gamma\neq\gamma_{1},\end{matrix}\right.\qquad\omega_{2}(\gamma)=\left\\{\begin{matrix}b,&\gamma=\gamma_{2},\\\
0,&\gamma\neq\gamma_{2}.\end{matrix}\right.$

Then

$(\omega_{1}\wedge\omega_{2})(\gamma)=\left\\{\begin{matrix}0,&\gamma\neq\gamma_{1}\gamma_{2},\\\
a\wedge{\gamma_{1}^{*}}^{-1}b,&\gamma=\gamma_{1}\gamma_{2}.\end{matrix}\right.$

Since functional (19) does not depend on the choice of $z$, we can consider
$z=e\in\Gamma$.

On the one hand, for $\gamma=\gamma_{1}\gamma_{2}$ we have

$\tau^{\gamma}(\omega_{1}\wedge\omega_{2})=\int_{C^{\gamma}}\Bigl{.}h^{*}\Bigl{(}a\wedge{\gamma_{1}^{*}}^{-1}b\Bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh=\int_{C^{\gamma}}\Bigl{.}(h^{*}a\wedge
h^{*}{\gamma_{1}^{*}}^{-1}b)\Bigr{|}_{T^{*}M^{\gamma}}dh=\int_{C^{\gamma}}\Bigl{.}(h^{*}a\wedge\gamma^{*}h^{*}{\gamma_{1}^{*}}^{-1}b)\Bigr{|}_{T^{*}M^{\gamma}}dh,$
(23)

since the form is integrated over ${T^{*}M^{\gamma}}$ and over this manifold
we have $\gamma^{*}h^{*}=h^{*}$. Since $h\in C^{\gamma}$ in (23), we have
$h^{*}\gamma^{*}=\gamma^{*}h^{*}$. We then transform (23) as

$\tau^{\gamma}(\omega_{1}\wedge\omega_{2})=\int_{C^{\gamma}}\Bigl{.}h^{*}a\wedge
h^{*}\gamma^{*}{\gamma_{1}^{*}}^{-1}b\Bigr{|}_{T^{*}M^{\gamma}}dh=\int_{C^{\gamma}}\Bigl{.}h^{*}\Bigl{(}a\wedge\gamma_{2}^{*}b\Bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh,$
(24)

since $\gamma=\gamma_{1}\gamma_{2}$ and
$\gamma^{*}{\gamma_{1}^{*}}^{-1}=\gamma_{2}^{*}$.

On the other hand, for $z=\gamma_{2}$, we have:

$\tau^{\gamma}(\omega_{2}\wedge\omega_{1})=\int_{C^{\gamma}}\Bigl{.}h^{*}\Bigl{(}z^{*}\Bigl{(}b\wedge{\gamma_{2}^{*}}^{-1}a\Bigr{)}\Bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh=(-1)^{{\rm
deg}\omega_{1}{\rm
deg}\omega_{2}}\int_{C^{\gamma}}\Bigl{.}h^{*}\Bigl{(}z^{*}\Bigl{(}{\gamma_{2}^{*}}^{-1}a\wedge
b\Bigr{)}\Bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh$ $=(-1)^{{\rm deg}\omega_{1}{\rm
deg}\omega_{2}}\int_{C^{\gamma}}\Bigl{.}h^{*}\Bigl{(}a\wedge\gamma_{2}^{*}b\Bigr{)}\Bigr{|}_{T^{*}M^{\gamma}}dh=(-1)^{{\rm
deg}\omega_{1}{\rm deg}\omega_{2}}\tau^{\gamma}(\omega_{1}\wedge\omega_{2}).$

Here in the last equality we used (24). Now we prove equality (22):

$\tau^{\gamma}_{X}(d\omega_{X})=\sum_{\gamma^{\prime}\in\langle\gamma\rangle}\;\int_{C^{\gamma}}\operatorname{Tr}_{X}\left(\Bigl{.}h^{*}\bigl{(}{z}^{*}(d\omega_{X}(\gamma^{\prime}))\bigr{)}\bigr{|}_{T^{*}X^{\gamma}}\right)dh\\\
=\sum_{\gamma^{\prime}\in\langle\gamma\rangle}\;\int_{C^{\gamma}}\operatorname{Tr}_{X}\left(\Bigl{.}d\bigl{(}h^{*}\bigl{(}{z}^{*}(\omega_{X}(\gamma^{\prime}))\bigr{)}\bigr{)}\bigr{|}_{T^{*}X^{\gamma}}\right)dh=d\sum_{\gamma^{\prime}\in\langle\gamma\rangle}\;\int_{C^{\gamma}}\operatorname{Tr}_{X}\left(\Bigl{.}h^{*}\bigl{(}{z}^{*}(\omega_{X}(\gamma^{\prime}))\bigr{)}\bigr{|}_{T^{*}X^{\gamma}}\right)dh\\\
=d\tau^{\gamma}_{X}(\omega_{X}).$

∎

### Definition of the Chern character.

Consider an elliptic operator $(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$.
For brevity, we frequently denote it by $\mathcal{D}$. We extend the interior
symbols $\sigma_{M}(\mathcal{D}),\sigma_{M}(\mathcal{R})$ of the original
operator and its almost inverse to $T^{*}M$ as smooth symbols, which have the
transmission property. We extend the boundary symbols
$\sigma_{X}(\mathcal{D})$ and $\sigma_{X}(\mathcal{R})$ to $T^{*}X$ as smooth
symbols. We denote such extensions by

$a,r\in C^{\infty}(T^{*}M)\rtimes\Gamma,\qquad
a_{X},r_{X}\in\widetilde{\Sigma}_{X}\rtimes\Gamma.$

Suppose that these extensions are compatible, i.e., the principal symbol of
the boundary symbol is equal to the restriction of the interior symbol to the
boundary and the following equalities hold:

$a=P_{2}aP_{1},\quad r=P_{1}rP_{2},\quad
a_{X}=P^{\prime}_{2}a_{X}P^{\prime}_{1},\quad
r_{X}=P^{\prime}_{1}r_{X}P^{\prime}_{2}.$

We define the noncommutative connections

$\nabla_{P_{j}}=P_{j}\cdot d\cdot P_{j}\text{ on }T^{*}M\quad\text{ and
}\quad\nabla_{P^{\prime}_{j}}=P^{\prime}_{j}\cdot d^{\prime}\cdot
P^{\prime}_{j}\text{ on }T^{*}X,\text{ where }j=1,2,$

and also the connection on $T^{*}M$

$\widetilde{\nabla}_{P_{1}}=\nabla_{P_{1}}+r\nabla a,\text{ where }\nabla
a\equiv\nabla_{P_{2}}a-a\nabla_{P_{1}}.$ (25)

Similarly, we define the connection on $T^{*}X$

$\widetilde{\nabla}_{P^{\prime}_{1}}=\nabla_{P^{\prime}_{1}}+r_{X}\nabla^{\prime}a_{X},\text{
where
}\nabla^{\prime}a_{X}=\nabla_{P^{\prime}_{2}}a_{X}-a_{X}\nabla_{P^{\prime}_{1}}.$

######  Lemma 1.

The curvatures of connections $\widetilde{\nabla}_{P_{1}}$ and
$\widetilde{\nabla}_{P^{\prime}_{1}}$ are equal to

$\widetilde{\Omega}_{P_{1}}\overset{\mathrm{def}}{=}(\widetilde{\nabla}_{P_{1}})^{2}=\nabla_{P_{1}}^{2}+\nabla_{P_{1}}r\nabla
a+(r\nabla a)^{2},$
$\widetilde{\Omega}_{P^{\prime}_{1}}\overset{\mathrm{def}}{=}(\widetilde{\nabla}_{P^{\prime}_{1}})^{2}=\nabla_{P^{\prime}_{1}}^{2}+\nabla_{P^{\prime}_{1}}r_{X}\nabla^{\prime}a_{X}+(r_{X}\nabla^{\prime}a_{X})^{2}.$

###### Proof.

Indeed, we have

$\widetilde{\Omega}_{P_{1}}u=(\widetilde{\nabla}_{P_{1}})^{2}u=(\nabla_{P_{1}}+r\nabla
a)^{2}u=(\nabla_{P_{1}}^{2}+\nabla_{P_{1}}r\nabla a+r(\nabla
a)\nabla_{P_{1}}+(r\nabla a)^{2})u$
$=(\nabla_{P_{1}}^{2}+\nabla_{P_{1}}r\nabla a+(r\nabla a)^{2})u,$
$\widetilde{\Omega}_{P^{\prime}_{1}}v=(\widetilde{\nabla}_{P^{\prime}_{1}})^{2}v=(\nabla_{P^{\prime}_{1}}+r_{X}\nabla^{\prime}a_{X})^{2}v=(\nabla_{P^{\prime}_{1}}^{2}+\nabla_{P^{\prime}_{1}}r_{X}\nabla^{\prime}a_{X}+r_{X}(\nabla^{\prime}a_{X})\nabla_{P^{\prime}_{1}}+(r_{X}\nabla^{\prime}a_{X})^{2})v$
$=(\nabla_{P^{\prime}_{1}}^{2}+\nabla_{P^{\prime}_{1}}r_{X}\nabla^{\prime}a_{X}+(r_{X}\nabla^{\prime}a_{X})^{2})v.$

∎

Let us define the differential forms with compact supports

$\operatorname{ch}_{T^{*}M}^{\gamma}\sigma(\mathcal{D})\in\Omega^{ev}_{c}(T^{*}M^{\gamma}),\quad\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})\in\Omega^{ev}_{c}(T^{*}X^{\gamma})$
(26)

on the cotangent bundles of the submanifolds of fixed points by the formulas

$\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})=\tau^{\gamma}\left(e^{-\widetilde{\Omega}_{P_{1}}/2\pi
i}(P_{1}-ra)\right)-\tau^{\gamma}\left(P_{2}e^{-\nabla^{2}_{P_{2}}/2\pi
i}-ae^{-\widetilde{\Omega}_{P_{1}}/2\pi i}r\right)$ (27)
$\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})=\tau^{\gamma}_{X}\left(e^{-\widetilde{\Omega}_{P^{\prime}_{1}}/2\pi
i}(P^{\prime}_{1}-r_{X}a_{X})\right)-\tau^{\gamma}_{X}\left(P^{\prime}_{2}e^{-\nabla^{2}_{P^{\prime}_{2}}/2\pi
i}-a_{X}e^{-\widetilde{\Omega}_{P^{\prime}_{1}}/2\pi i}r_{X}\right).$ (28)

Here and below we denote the extensions of mappings (17) and (18) to the
matrix algebras over the corresponding crossed products again by
$\tau^{\gamma},\tau_{X}^{\gamma}$. The extensions are obtained as the
compositions of the matrix trace and mappings (17) and (18). Since
$\tau^{\gamma}$ is a graded trace, (27) can be written as

$\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})=\tau^{\gamma}\left(e^{-\widetilde{\Omega}_{P_{1}}/2\pi
i}P_{1}-P_{2}e^{-\nabla^{2}_{P_{2}}/2\pi i}\right).$

The boundary $\partial(T^{*}M^{\gamma})\simeq T^{*}X^{\gamma}\times\mathbb{R}$
is fibered over $T^{*}X^{\gamma}$ with the fiber $\mathbb{R}$. We denote the
corresponding projection by $\pi^{\gamma}:\partial(T^{*}M^{\gamma})\to
T^{*}X^{\gamma}$ and the embedding $\partial(T^{*}M^{\gamma})\subset
T^{*}M^{\gamma}$ by $i_{\gamma}$. Hence, the pair
$(T^{*}M^{\gamma},\pi^{\gamma})$ is a manifold with fibered boundary in the
sense of Sec. 4.

######  Proposition 2.

Given $\gamma\in\Gamma$, we have

$d\left(\operatorname{ch}_{T^{*}M}^{\gamma}\sigma(\mathcal{D})\right)=0,$ (29)
$d^{\prime}\left(\operatorname{ch}_{T^{*}X}^{\gamma}\sigma(\mathcal{D})\right)=\pi^{\gamma}_{*}i_{\gamma}^{*}\left(\operatorname{ch}_{T^{*}M}^{\gamma}\sigma(\mathcal{D})\right).$
(30)

In other words, the pair
$(\operatorname{ch}_{T^{*}M}^{\gamma}\sigma(\mathcal{D}),-\operatorname{ch}_{T^{*}X}^{\gamma}\sigma(\mathcal{D}))$
is closed in the complex
$(\Omega_{c}^{*}(T^{*}M^{\gamma},\pi^{\gamma}),\partial)$, see (13), and we
denote its cohomology class by

$\operatorname{ch}^{\gamma}\sigma(\mathcal{D})\in
H^{ev}(T^{*}M^{\gamma},\pi^{\gamma}).$

This class does not depend on the choice of the elements $a,r,a_{X},r_{X}$ and
does not change under homotopies of elliptic symbols.

###### Proof.

1\. Equality (29) can be proven in a standard way (see, for instance, [31])

$d\left(\operatorname{ch}_{T^{*}M}^{\gamma}\sigma(\mathcal{D})\right)=d\tau^{\gamma}\left(e^{-\widetilde{\nabla}^{2}_{P_{1}}/2\pi
i}P_{1}\right)-d\tau^{\gamma}\left(P_{2}e^{-{\nabla^{2}_{P_{2}}}/2\pi
i}\right)\\\
=\tau^{\gamma}\left(\widetilde{\nabla}_{P_{1}}\left(e^{-\widetilde{\nabla}^{2}_{P_{1}}/2\pi
i}P_{1}\right)\right)-\tau^{\gamma}\left(\nabla_{P_{2}}\left(P_{2}e^{-{\nabla^{2}_{P_{2}}}/2\pi
i}\right)\right)\\\
=\tau^{\gamma}\left[\widetilde{\nabla}_{P_{1}},e^{-\widetilde{\nabla}^{2}_{P_{1}}/2\pi
i}P_{1}\right]-\tau^{\gamma}\left[\nabla_{P_{2}},P_{2}e^{-{\nabla^{2}_{P_{2}}}/2\pi
i}\right]=0.$

The last equality holds, since the commutators used in it are equal to zero.

2\. Next, let us prove equality (30). First, we calculate the right hand side
in (30). We denote the restriction of the curvature form
$\widetilde{\Omega}_{P_{1}}$ to $\partial T^{*}M$ by
$\Omega_{1}=\widetilde{\Omega}_{P_{1}}|_{\partial T^{*}M}$. It is clear that
$\Omega_{1}$ is equal to the curvature form for the pair of restrictions
$(a|_{\partial T^{*}M},r|_{\partial T^{*}M})$. Also we denote by
$\Omega_{2}=\Omega_{2}(\xi_{n})=\widetilde{\Omega}_{P_{1}}|_{\partial T^{*}X}$
the family of curvature forms for the restrictions $(a|_{\partial
T^{*}M\cap\\{\xi_{n}={\rm const}\\}},r|_{\partial T^{*}M\cap\\{\xi_{n}={\rm
const}\\}})$, where $\xi_{n}$ is considered as a parameter. We have

$i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})=\tau^{\gamma}\left(e^{-\Omega_{1}/2\pi
i}P_{1}-P_{2}e^{-{\nabla^{2}_{P_{2}}}/2\pi i}\right),$
$\pi^{\gamma}_{*}i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})=-\frac{1}{2\pi
i}\int_{\mathbb{R}}{\tau^{\gamma}\left(\left(\frac{\partial}{\partial\xi_{n}}\lrcorner\Omega_{1}\right)e^{-\Omega_{2}/2\pi
i}\right)d\xi_{n}}.$ (31)

Here $\frac{\partial}{\partial\xi_{n}}\lrcorner\Omega$ is the substitution of
the vector field into the differential form. By Lemma 1, we have

$\frac{\partial}{\partial\xi_{n}}\lrcorner\Omega_{1}=\frac{\partial}{\partial\xi_{n}}\lrcorner\left(\nabla_{P_{1}}^{2}+\nabla_{P_{1}}(r\nabla
a)+(r\nabla a)^{2}\right).$ (32)

We now substitute $\partial/\partial\xi_{n}$ into each of the summands in
(32). To this end, we represent the connections in the following form:

$d=d\xi_{n}\frac{\partial}{\partial\xi_{n}}+d^{\prime},\quad\nabla_{P_{j}}=P_{j}d^{\prime}P_{j}+P_{j}d\xi_{n}\frac{\partial}{\partial\xi_{n}}\equiv\nabla^{\prime}_{P_{j}}+P_{j}d\xi_{n}\frac{\partial}{\partial\xi_{n}},\nabla=\nabla^{\prime}+d\xi_{n}\frac{\partial}{\partial\xi_{n}}.$

For the first summand in (32), we have

$\frac{\partial}{\partial\xi_{n}}\lrcorner\nabla_{P_{1}}^{2}=\frac{\partial}{\partial\xi_{n}}\lrcorner(P_{1}dP_{1})^{2}=0,$
(33)

since $P_{1}$ does not depend on $\xi_{n}$. For the second summand in (32), we
have

$\nabla_{P_{1}}(r\nabla
a)=\left(\nabla^{\prime}_{P_{1}}+P_{1}d\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)\left(r\nabla^{\prime}a+r\frac{\partial
a}{\partial\xi_{n}}d\xi_{n}\right)\\\
=\nabla^{\prime}_{P_{1}}(r\nabla^{\prime}a)+\nabla^{\prime}_{P_{1}}\left(r\frac{\partial
a}{\partial\xi_{n}}d\xi_{n}\right)+\left(P_{1}d\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)(r\nabla^{\prime}a)+\left(P_{1}d\xi_{n}\frac{\partial}{\partial\xi_{n}}\right)\left(r\frac{\partial
a}{\partial\xi_{n}}d\xi_{n}\right)\\\
=\nabla^{\prime}_{P_{1}}(r\nabla^{\prime}a)+\nabla^{\prime}_{P_{1}}\left(r\frac{\partial
a}{\partial\xi_{n}}\right)d\xi_{n}+d\xi_{n}\frac{\partial}{\partial\xi_{n}}(r\nabla^{\prime}a).$
(34)

We now substitute $\partial/\partial\xi_{n}$ into (34):

$\frac{\partial}{\partial\xi_{n}}\lrcorner\nabla_{P_{1}}(r\nabla
a)=-\nabla^{\prime}_{P_{1}}\left(r\frac{\partial
a}{\partial\xi_{n}}\right)+\frac{\partial}{\partial\xi_{n}}(r\nabla^{\prime}a)\\\
=-\left(\nabla^{\prime}r\right)\frac{\partial
a}{\partial\xi_{n}}-r\nabla^{\prime}\frac{\partial
a}{\partial\xi_{n}}+\frac{\partial
r}{\partial\xi_{n}}\nabla^{\prime}a+r\nabla^{\prime}\frac{\partial
a}{\partial\xi_{n}}=-\left(\nabla^{\prime}r\right)\frac{\partial
a}{\partial\xi_{n}}+\frac{\partial r}{\partial\xi_{n}}\nabla^{\prime}a.$ (35)

For the third summand in (32), we get

$\frac{\partial}{\partial\xi_{n}}\lrcorner\left(\left(r\nabla^{\prime}a+r\frac{\partial
a}{\partial\xi_{n}}d\xi_{n}\right)\left(r\nabla^{\prime}a+r\frac{\partial
a}{\partial\xi_{n}}d\xi_{n}\right)\right)=r\frac{\partial
a}{\partial\xi_{n}}r\nabla^{\prime}a-(r\nabla^{\prime}a)r\frac{\partial
a}{\partial\xi_{n}}.$ (36)

Substituting (33), (35), (36) into (31), we obtain

$\pi^{\gamma}_{*}i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})=\frac{i}{2\pi}\int_{\mathbb{R}}{\tau^{\gamma}\left(\left(\frac{\partial
r}{\partial\xi_{n}}\nabla^{\prime}a-\left(\nabla^{\prime}r\right)\frac{\partial
a}{\partial\xi_{n}}+\left[r\frac{\partial
a}{\partial\xi_{n}},r\nabla^{\prime}a\right]\right)e^{-\Omega_{2}/2\pi
i}\right)d\xi_{n}}.$ (37)

3\. Now we calculate the left hand side in (30).

######  Lemma 2.

For each form
$\omega^{\prime}\in\operatorname{Mat}_{N}((\tilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X))\rtimes\Gamma)$
such that $\omega^{\prime}=P^{\prime}_{1}\omega^{\prime}P^{\prime}_{1}$, we
have

$d^{\prime}\tau^{\gamma}_{X}(\omega^{\prime})=\tau^{\gamma}_{X}(\nabla_{P^{\prime}_{1}}\omega^{\prime}).$
(38)

###### Proof.

The difference between the left and right hand sides in (38) is equal to

$d^{\prime}\tau^{\gamma}_{X}(\omega^{\prime})-\tau^{\gamma}_{X}(\nabla_{P^{\prime}_{1}}\omega^{\prime})=\tau^{\gamma}_{X}(d^{\prime}(P^{\prime}_{1}\omega^{\prime})-P^{\prime}_{1}d^{\prime}\omega^{\prime})=\tau^{\gamma}_{X}((d^{\prime}P^{\prime}_{1})\omega^{\prime})\\\
=\tau^{\gamma}_{X}((d^{\prime}P^{\prime}_{1})P^{\prime}_{1}\omega^{\prime}P^{\prime}_{1})=\tau^{\gamma}_{X}(P^{\prime}_{1}d^{\prime}P^{\prime}_{1}P^{\prime}_{1}\omega^{\prime})=0,$
(39)

where in the last line we used the cyclic property:
$\tau^{\gamma}_{X}:\tau^{\gamma}_{X}(\omega^{\prime}P^{\prime}_{1})=\tau^{\gamma}_{X}(P^{\prime}_{1}\omega^{\prime})$.
The last equality holds, since $P^{\prime}_{1}$ acts as a scalar operator in
the variable $\xi_{n}$. In the last equality in (39) we used the identity
$P^{\prime}_{1}(d^{\prime}P^{\prime}_{1})P^{\prime}_{1}=0$ for the projection
$P^{\prime}_{1}$.

This completes the proof of the Lemma. ∎

Using Lemma 2, we obtain the following expression for the left hand side in
(30)

$d^{\prime}\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})=\tau^{\gamma}_{X}\left(\nabla_{P^{\prime}_{1}}\big{(}e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{)}\right)-\tau^{\gamma}_{X}\left(\nabla_{P^{\prime}_{2}}\big{(}P^{\prime}_{2}e^{-\nabla^{2}_{P^{\prime}_{2}}/2\pi
i}\big{)}\right)+d^{\prime}\tau^{\gamma}_{X}\left[a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}r_{X}\right].$ (40)

For the first summand in (40), we have

$\nabla_{P^{\prime}_{1}}\left(e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\right)=\big{[}\nabla_{P^{\prime}_{1}},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{]}=\big{[}(\nabla_{P^{\prime}_{1}}+r_{X}\nabla^{\prime}a_{X})-r_{X}\nabla^{\prime}a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{]}\\\
=\big{[}\widetilde{\nabla}_{P^{\prime}_{1}},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{]}-\big{[}r_{X}\nabla^{\prime}a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{]}=-\big{[}r_{X}\nabla^{\prime}a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\big{]}.$

For the second summand in (40), we obtain

$\nabla_{P^{\prime}_{2}}\left(P^{\prime}_{2}e^{-\nabla^{2}_{P^{\prime}_{2}}/2\pi
i}\right)=\big{[}\nabla_{P^{\prime}_{2}},P^{\prime}_{2}e^{-\nabla^{2}_{P^{\prime}_{2}}/2\pi
i}\big{]}=0.$

Substituting the last two formulas into (40), we get

$d^{\prime}\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})=-\tau^{\gamma}_{X}\left[r_{X}\nabla^{\prime}a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}P^{\prime}_{1}\right]+d^{\prime}\tau^{\gamma}_{X}\left[a_{X},e^{-\widetilde{\nabla}^{2}_{P^{\prime}_{1}}/2\pi
i}r_{X}\right].$ (41)

Next, we use an analogue of (7).

######  Lemma 3.

For each form
$\omega_{X,1},\omega_{X,2}\in\operatorname{Mat}_{N}(\tilde{\Sigma}_{X}\otimes_{C^{\infty}(X)}\Omega(T^{*}X)\rtimes\Gamma)$,
we have

$\tau^{\gamma}_{X}[\omega_{X,1},\omega_{X,2}]=-i\Pi^{\prime}\left(\tau^{\gamma}\left(\frac{\partial\omega_{X,1}}{\partial\xi_{n}}\omega_{X,2}\right)\right)=i\Pi^{\prime}\left(\tau^{\gamma}\left(\omega_{X,1}\frac{\partial\omega_{X,2}}{\partial\xi_{n}}\right)\right).$

Now let us calculate the traces of the commutators in (41) using Lemma 3:

$$$d^{\prime}\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})=i\Pi^{\prime}\tau^{\gamma}\left(\left(\frac{\partial}{\partial\xi_{n}}(r\nabla^{\prime}a)\right)e^{-\Omega_{2}/2\pi
i}P_{1}\right)-id^{\prime}\Pi^{\prime}\tau^{\gamma}\left(\frac{\partial
a}{\partial\xi_{n}}e^{-\Omega_{2}/2\pi i}r\right)\\\
=i\Pi^{\prime}\tau^{\gamma}\left(\left(\frac{\partial}{\partial\xi_{n}}(r\nabla^{\prime}a)\right)e^{-\Omega_{2}/2\pi
i}\right)-i\Pi^{\prime}\tau^{\gamma}\left[\widetilde{\nabla}^{\prime}_{P_{1}},r\frac{\partial
a}{\partial\xi_{n}}e^{-\Omega_{2}/2\pi i}\right]\\\
=i\Pi^{\prime}\tau^{\gamma}\left(\left(\frac{\partial}{\partial\xi_{n}}(r\nabla^{\prime}a)-\left[\widetilde{\nabla}^{\prime}_{P_{1}},r\frac{\partial
a}{\partial\xi_{n}}\right]\right)e^{-\Omega_{2}/2\pi i}\right)\\\
=i\Pi^{\prime}\tau^{\gamma}\left(\left(\frac{\partial
r}{\partial\xi_{n}}\nabla^{\prime}a+r\nabla^{\prime}\frac{\partial
a}{\partial\xi_{n}}-\nabla^{\prime}_{P_{1}}\left(r\frac{\partial
a}{\partial\xi_{n}}\right)-\left[r\nabla^{\prime}a,r\frac{\partial
a}{\partial\xi_{n}}\right]\right)e^{-\Omega_{2}/2\pi i}\right)\\\
=i\Pi^{\prime}\tau^{\gamma}\left(\left(\frac{\partial
r}{\partial\xi_{n}}\nabla^{\prime}a-\left(\nabla^{\prime}r\right)\frac{\partial
a}{\partial\xi_{n}}+\left[r\frac{\partial
a}{\partial\xi_{n}},r\nabla^{\prime}a\right]\right)e^{-\Omega_{2}/2\pi
i}\right).$ (42)

Using (3), equation (42) gives us

$d^{\prime}\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})=\frac{1}{2\pi}\int{\tau^{\gamma}\left(\left(\frac{\partial
r}{\partial\xi_{n}}\nabla^{\prime}a-\left(\nabla^{\prime}r\right)\frac{\partial
a}{\partial\xi_{n}}+\left[r\frac{\partial
a}{\partial\xi_{n}},r\nabla^{\prime}a\right]\right)e^{-\Omega_{2}/2\pi
i}\right)d\xi_{n}}.$ (43)

Since the expressions in (37) and (43) are equal, we have desired equality
(30).

4\. Consider compatible families of interior symbols
$a_{t},r_{t}overT^{*}M\times[0,1]$ and boundary symbols $a_{X,t},r_{X,t}$ over
$T^{*}X\times[0,1]$, which smoothly depend on $t$. For such pairs of symbols,
we consider the Chern forms
$\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})$ and
$\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})$. We
represent the form
$\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})$ as

$\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})=dt\wedge\alpha+\beta,$
(44)

where $\alpha(t),\beta(t)\in\Omega(T^{*}M)$ are smooth families of forms. Here

$\beta(t_{0})=\operatorname{ch}^{\gamma}_{T^{*}M\times\\{t=t_{0}\\}}\sigma(\mathcal{D}),\qquad\alpha=\frac{\partial}{\partial
t}\lrcorner\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}.$

By already proven statement 1 of the theorem, we have
$d\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})=0$. Using
expansion (44), we obtain

$d\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})=-dt\wedge
d\alpha+d\beta+dt\wedge\frac{\partial\beta}{\partial t}=0.$

Therefore, we obtain

$\frac{\partial\beta}{\partial t}=d\alpha,$

which gives us

$\beta(1)-\beta(0)=d\int\limits^{1}_{0}{\alpha(t)dt}.$

Now we use the expansion

$\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})=dt\wedge\alpha_{X}+\beta_{X},$
(45)

where $\alpha_{X}(t),\beta_{X}(t)\in\Omega(T^{*}X)$. Let us find
$\pi^{\gamma}_{*}i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})$.
We obtain

$i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})=dt\wedge
i^{*}_{\gamma}\alpha+i^{*}_{\gamma}\beta.$
$\pi^{\gamma}_{*}i^{*}_{\gamma}\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})=-dt\wedge\pi^{\gamma}_{*}i^{*}_{\gamma}\alpha+\pi^{\gamma}_{*}i^{*}_{\gamma}\beta.$
(46)

Let us now find
$d\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})$. Using
expansion (45), we obtain

$d\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})=-dt\wedge
d^{\prime}\alpha_{X}+dt\wedge\frac{\partial\beta_{X}}{\partial
t}+d^{\prime}\beta_{X}.$ (47)

By virtue of proven statement 1 of the theorem, the left hand sides in (46)
and (47) differ by a sign. That is why their right hand sides differ by a
sign:

$-dt\wedge\pi^{\gamma}_{*}i^{*}_{\gamma}\alpha+\pi^{\gamma}_{*}i^{*}_{\gamma}\beta=dt\wedge
d^{\prime}\alpha_{X}-dt\wedge\frac{\partial\beta_{X}}{\partial
t}-d^{\prime}\beta_{X},$

which gives us

$\frac{\partial\beta_{X}}{\partial
t}=d^{\prime}\alpha_{X}+\pi^{\gamma}_{*}i^{*}_{\gamma}\alpha.$

Integrating both sides in this equation, we obtain

$\beta_{X}(1)-\beta_{X}(0)=d^{\prime}\int\limits_{0}^{1}{\alpha_{X}(t)dt}+\pi^{\gamma}_{*}i^{*}_{\gamma}\int\limits_{0}^{1}{\alpha(t)dt}.$

Thus, we obtain

$\operatorname{ch}^{\gamma}_{T^{*}M}\sigma(\mathcal{D})(1)-\operatorname{ch}_{T^{*}M}\sigma(\mathcal{D})(0)=d\omega,$
(48)
$\operatorname{ch}^{\gamma}_{T^{*}X}\sigma(\mathcal{D})(1)-\operatorname{ch}_{T^{*}X}\sigma(\mathcal{D})(0)=d^{\prime}\omega_{X}+\pi^{\gamma}_{*}i^{*}_{\gamma}\omega,$
(49)

where

$\omega=\int\limits_{0}^{1}{\alpha(t)dt},\qquad\omega_{X}=\int\limits_{0}^{1}{\alpha_{X}(t)dt}.$

Equalities (48) and (49) imply that the difference

$(\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})(1),-\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})(1))-(\operatorname{ch}^{\gamma}_{T^{*}M\times[0,1]}\sigma(\mathcal{D})(0),-\operatorname{ch}^{\gamma}_{T^{*}X\times[0,1]}\sigma(\mathcal{D})(0))$

is a coboundary in the complex
$(\Omega(T^{*}M^{\gamma},\pi^{\gamma}),\partial)$. This proves the homotopy
invariance of the Chern character.

Let now $a_{1},r_{1},a_{X,1},r_{X,1}$ be different extensions of the elliptic
symbols to $T^{*}M$ and $T^{*}X$. Then we consider the homotopies

$a_{t}=a(1-t)+a_{1}\cdot t,\qquad r_{t}=r(1-t)+r_{1}\cdot t,$
$a_{X,t}=a_{X}(1-t)+a_{X,1}\cdot t,\qquad r_{X,t}=r_{X}(1-t)+r_{X,1}\cdot t,$

where $t\in[0,1]$. At $t=0$ we have the set $a,r,a_{X},r_{X}$, while at $t=1$
we have the set $a_{1},r_{1},a_{X,1},r_{X,1}$. Thus the homotopy invariance
gives independence of the choice of the extensions. ∎

##  6 Index theorem

To give the index formula, we need to define the necessary equivariant
characteristic classes. Firstly, we define the Todd forms on $M^{\gamma}$:

$\operatorname{Td}(T^{*}M^{\gamma}\otimes\mathbb{C})\overset{\mathrm{def}}{=}\operatorname{det}\nolimits\left(\frac{-\Omega^{\gamma}/2\pi
i}{1-\exp(\Omega^{\gamma}/2\pi i)}\right)\in\Omega^{ev}(M^{\gamma}),$

where $\Omega^{\gamma}$ is the curvature form of the Levi-Civita connection on
$M^{\gamma}$. The Todd form
$\operatorname{Td}(T^{*}X^{\gamma}\otimes\mathbb{C})$ on $X^{\gamma}$ is
defined in a similar way. The pair of these forms is closed in the complex
$(\widetilde{\Omega}^{*}(M^{\gamma},\pi^{\gamma}),\widetilde{\partial})$ (see
(14)) and its cohomology class is denoted by

$\operatorname{Td}^{\gamma}(T^{*}M\otimes\mathbb{C})\in\widetilde{H}^{ev}(M^{\gamma},\pi^{\gamma}).$

Next, let $N^{\gamma}$ be the normal bundle of $M^{\gamma}\subset M$. Then we
have the natural action of $\gamma$ on $N^{\gamma}$ and the following
differential forms on $M^{\gamma}$:

$\operatorname{ch}^{\gamma}\Lambda({N}^{\gamma}\otimes\mathbb{C})=\operatorname{Tr}_{\Lambda^{ev}(N^{\gamma})}\left(\gamma\exp(-\Omega/2\pi
i)\right)-\operatorname{Tr}_{\Lambda^{odd}(N^{\gamma})}\left(\gamma\exp(-\Omega/2\pi
i)\right)\Omega^{ev}(M^{\gamma}),$

where $\Omega$ is the curvature form of the exterior bundle
$\Lambda(N^{\gamma})$, $\gamma$ is considered as an endomorphism of the
subbundles $\Lambda^{ev/odd}(N^{\gamma})$ of even/odd forms and
$\operatorname{Tr}_{\Lambda^{ev/odd}(N^{\gamma})}$ is the fiber-wise trace
functional on endomorphisms of the bundles $\Lambda^{ev/odd}(N^{\gamma})$.
Similarly, let $N^{\gamma}_{X}$ be the normal bundle of $X^{\gamma}\subset X$.
Then one can define the form
$\operatorname{ch}^{\gamma}\Lambda({N}_{X}^{\gamma}\otimes\mathbb{C})$ on
$X^{\gamma}$ along the same lines. The pair
$(\operatorname{ch}^{\gamma}\Lambda({N}^{\gamma}\otimes\mathbb{C}),\operatorname{ch}^{\gamma}\Lambda({N}_{X}^{\gamma}\otimes\mathbb{C}))$
is closed in the complex
$(\widetilde{\Omega}^{*}(M^{\gamma},\pi^{\gamma}),\widetilde{\partial})$. We
denote its cohomology class by

$\operatorname{ch}^{\gamma}\Lambda(\mathcal{N}^{\gamma}\otimes\mathbb{C})\in\widetilde{H}^{ev}(M^{\gamma},\pi^{\gamma}).$

The last class is invertible since its zero degree component is a nonzero
complex number (see the proof in [1] or [20]).

######  Theorem 3.

Let $\mathcal{D}$ be an elliptic operator in the sense of Definition 3. Then
the following index formula holds:

$\operatorname{ind}\mathcal{D}=\sum_{\langle\gamma\rangle\subset\Gamma}\langle\operatorname{ch}^{\gamma}\sigma(\mathcal{D})\wedge\operatorname{Td}^{\gamma}(T^{*}M\otimes\mathbb{C})\wedge\operatorname{ch}^{\gamma}\Lambda(\mathcal{N}^{\gamma}\otimes\mathbb{C})^{-1},[T^{*}M^{\gamma},\pi^{\gamma}]\rangle,$
(50)

where the summation is over the conjugacy classes in $\Gamma$ and the series
converges absolutely.

To prove this theorem, we need to establish some auxiliary statements.

##  7 Homotopy classification

###  $\operatorname{Ell}$-groups.

Let us denote the Abelian group of stable homotopy classes of elliptic
$\Gamma$-Boutet de Monvel operators (11) by $\operatorname{Ell}(M,\Gamma)$. We
recall (for details see [32]) that two operators
$(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ and
$(\mathcal{D}^{\prime},\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2})$ are
called stably homotopic, if there exists a smooth homotopy of elliptic
operators $(\mathcal{D}_{t},\mathcal{P}_{1,t},\mathcal{P}_{2,t}),t\in[0,1]$
such that

$\left.(\mathcal{D}_{t},\mathcal{P}_{1,t},\mathcal{P}_{2,t})\right|_{t=0}=(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})\oplus\rm{Triv},\qquad\left.(\mathcal{D}_{t},\mathcal{P}_{1,t},\mathcal{P}_{2,t})\right|_{t=1}=(\mathcal{D}^{\prime},\mathcal{P}^{\prime}_{1},\mathcal{P}^{\prime}_{2})\oplus\rm{Triv}^{\prime},$

where $\rm{Triv},\rm{Triv}^{\prime}$ are some trivial operators. Here a
trivial operator is an elliptic operator of type (11), in which the matrix
operator $\mathcal{D}$ has components in the subalgebra

$(C^{\infty}(M)\oplus
C^{\infty}(X))\rtimes\Gamma\subset\Psi_{B}(M)\rtimes\Gamma.$ (51)

It can be proven in a standard way that stable homotopy is an equivalence
relation on the set of elliptic operators (11). Then the set of elliptic
operators $(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ considered modulo
stable homotopies is denoted by $\operatorname{Ell}(M,\Gamma)$. This set is an
Abelian group with respect to the direct sum of operators. Here the
equivalence class of operators
$(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$, where $\mathcal{D}$ is a
matrix operator over the algebra $(C^{\infty}(M)\oplus
C^{\infty}(X))\rtimes\Gamma$, defines the zero element of the group.

The aim of the current section is to obtain analogs of the Boutet de Monvel
theorem, see [5, 9, 10], which allows to construct homotopies of elliptic
boundary value problems to a simple form in a neighborhood of the boundary. To
state the corresponding result, we introduce some notations.

Firstly, we denote by $\operatorname{Ell}(M^{\circ},\Gamma)$ the group of
stable homotopy classes of triples
$(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ (see above), for which
$\mathcal{D}$ has the components in subalgebra (51) in some neigborhood of the
boundary $X\subset M$.

Secondly, for each projection
$P\in\operatorname{Mat}_{N}(C^{\infty}(X)\rtimes\Gamma)$, we define a special
boundary value problem. Consider the matrix $N\times N$ operator (cf. [3,
Corollary 20.3.1]) on $M$:

$\Lambda_{P}=\chi\left(\frac{\partial}{\partial
x_{n}}(2P-1)+\Lambda_{X}\right)+(1-\chi)\Lambda_{M},$ (52)

where $\Lambda_{M},\Lambda_{X}$ are non-negative elliptic $\psi$DOs of order 1
on $M$ and $X$ respectively, $\chi\in C^{\infty}(M)$ is a non-negative
function, identically equal to one in a neighborhood of the boundary and equal
to zero outside a slightly larger neighborhood of the boundary. Here we
suppose that $\Lambda_{X}$ commutes with the action of $\Gamma$ on $X$. Let us
consider the following boundary value problem for operator (52)

$\left(\begin{array}[]{c}\Lambda_{P}\\\
Pi^{*}\end{array}\right):H^{1}(M,\mathbb{C}^{N})\longrightarrow\begin{array}[]{c}L^{2}(M,\mathbb{C}^{N})\\\
\oplus\\\ PH^{1/2}(X,\mathbb{C}^{N}).\end{array}$ (53)

Similarly to [7], one can show that this boundary value problem is elliptic
and its Fredholm index is equal to zero. Now reduce problem (53) to a problem
in $L^{2}$ spaces. To this end, we denote by $\Lambda_{-}$ operator (53) for
$P=0$. Note that the latter operator is Fredholm without any boundary
conditions. Now let us define the zero order boundary value problem:

$\left(\begin{array}[]{c}\Lambda_{P}(\Lambda_{-})^{-1}\\\
\Lambda_{X}^{1/2}Pi^{*}(\Lambda_{-})^{-1}\end{array}\right):L^{2}(M,\mathbb{C}^{N})\longrightarrow\begin{array}[]{c}L^{2}(M,\mathbb{C}^{N})\\\
\oplus\\\ PL^{2}(X,\mathbb{C}^{N}).\end{array}$ (54)

This problem is Fredholm with zero index as the composition of
$\Lambda_{-}^{-1}$, problem (53) and operator $\Lambda_{X}^{1/2}$.

We denote problem (54) by
$[\mathcal{D}_{P},\mathcal{P}_{1,P},\mathcal{P}_{2,P}]$.

######  Theorem 4 (homotopy classification).

The mapping

$\begin{array}[]{ccc}\operatorname{Ell}(M^{\circ},\Gamma)\oplus
K_{0}(C^{\infty}(X)\rtimes\Gamma)&\longrightarrow&\operatorname{Ell}(M,\Gamma)\\\
{}[\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2}]\oplus[P]&\longmapsto&[\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2}]\oplus[\mathcal{D}_{P},\mathcal{P}_{1,P},\mathcal{P}_{2,P}]\end{array}$
(55)

defines the group isomorphism.

###### Proof.

1\. Let us move from Fréchet algebras
$C^{\infty}(M),C^{\infty}(X),\Sigma,\Sigma_{X},\Psi_{B}(M),\ldots$ to their
closures up to $C^{*}$-algebras denoted by
$C(M),C(X),\overline{\Sigma},\overline{\Sigma}_{X},\overline{\Psi}(M),\ldots$.

Denote by $\overline{\operatorname{Ell}}(M,\Gamma)$ the group of stable
homotopy classes of triples $(\mathcal{D},\mathcal{P}_{1},\mathcal{P}_{2})$ as
in (11), where the elements are chosen from $C^{*}$-algebras

$\mathcal{D}\in\operatorname{Mat}_{N}(\overline{\Psi}(M))\rtimes\Gamma,\quad\mathcal{P}_{1,2}\in\operatorname{Mat}_{N}(C(M)\oplus
C(X))\rtimes\Gamma.$

Similarly, we define the group
$\overline{\operatorname{Ell}}(M^{\circ},\Gamma)$.

Since the considered Fréchet algebras are spectral invariant in the
corresponding $C^{*}$-closures, the natural mappings

$\overline{\operatorname{Ell}}(M,\Gamma)\longrightarrow{\operatorname{Ell}}(M,\Gamma)\quad\text{and}\quad\overline{\operatorname{Ell}}(M^{\circ},\Gamma)\longrightarrow{\operatorname{Ell}}(M^{\circ},\Gamma)$
(56)

are isomorphisms of Abelian groups. Hence, to prove this theorem, it suffices
to establish the group isomorphism

$\overline{\operatorname{Ell}}(M^{\circ},\Gamma)\oplus
K_{0}(C(X)\rtimes\Gamma)\longrightarrow\overline{\operatorname{Ell}}(M,\Gamma).$
(57)

2\. Let us express the group $\overline{\operatorname{Ell}}(M,\Gamma)$ in
terms of the $K$-group of some $C^{*}$-algebra associated with the symbol
algebra. Namely, by the results in [32] we have the isomorphism of Abelian
groups

$\overline{\operatorname{Ell}}(M,\Gamma)\simeq
K_{0}\Bigl{(}\operatorname{Con}(C(M)\oplus
C(X)\to\overline{\Sigma})\rtimes\Gamma\Bigr{)},$ (58)

where

$\operatorname{Con}(A\to B)=\bigl{\\{}(a,b(t))\in A\oplus
C([0,1),B)\;\bigl{|}\;f(a)=b(0)\bigr{\\}}$

is the cone of a homomorphism $f:A\to B$ of some $C^{*}$-algebras $A$ and $B$.
The mapping $C(M)\oplus C(X)\to\overline{\Sigma}$ in (58) is a monomorphism,
which takes a pair of functions $f,g$ to the symbol $(f,{\rm
diag}(f|_{X},g))$.

For brevity, the $C^{*}$-algebra $\operatorname{Con}(C(M)\oplus
C(X)\to\overline{\Sigma})$ is denoted by $\mathcal{A}$.

3\. Denote by $\overline{\Sigma}_{0}\subset\overline{\Sigma}$ the ideal of all
symbols with zero interior symbol. We consider the commutative diagram

$\begin{array}[]{rcccccl}0\longrightarrow&\overline{\Sigma}_{0}&\longrightarrow&\overline{\Sigma}&\stackrel{{\scriptstyle\sigma_{int}}}{{\longrightarrow}}&C(S^{*}_{tr}M)&\longrightarrow
0\\\ &\uparrow&&\uparrow&&\uparrow\\\
0\longrightarrow&C(X)&\longrightarrow&C(M)\oplus
C(X)&\longrightarrow&C(M)&\longrightarrow 0,\end{array}$ (59)

where the space $S^{*}_{tr}M$ is obtained from the cosphere bundle $S^{*}M$ by
identifying pairs of points $(x^{\prime},0,0,\pm 1)$ on its boundary. The
diagram (59) gives the short exact sequence

$0\to{\rm
Con}(C(X)\to\overline{\Sigma}_{0})\rtimes\Gamma\longrightarrow\mathcal{A}\rtimes\Gamma\longrightarrow\operatorname{Con}(C(M)\to
C(S^{*}_{tr}M))\rtimes\Gamma\to 0$ (60)

of crossed products of cones of vertical mappings in (59). Then (60) gives a
periodic exact sequence of $K$-groups

$\ldots\to
K_{0}\bigl{(}\operatorname{Con}(C(X)\to\overline{\Sigma}_{0})\rtimes\Gamma\bigr{)}\longrightarrow
K_{0}(\mathcal{A}\rtimes\Gamma)\longrightarrow
K_{0}(\operatorname{Con}(C(M)\to C(S^{*}_{tr}M))\rtimes\Gamma)\\\
\longrightarrow
K_{1}\bigl{(}\operatorname{Con}(C(X)\to\overline{\Sigma}_{0})\rtimes\Gamma\bigr{)}\to\ldots$
(61)

4\. Now let us calculate the $K$-groups in (61).

######  Lemma 4.

We have group isomorphisms

$K_{*}\bigl{(}\operatorname{Con}(C(X)\to\overline{\Sigma}_{0})\rtimes\Gamma\bigr{)}\simeq
K_{*}(C_{0}(T^{*}X)\rtimes\Gamma),$ (62) $K_{*}(\operatorname{Con}(C(M)\to
C(S^{*}_{tr}M))\rtimes\Gamma)\simeq K_{*}(C_{0}(T^{*}M)\rtimes G)\oplus
K_{*}(C(X)\rtimes\Gamma).$ (63)

###### Proof.

To begin with, we construct isomorphism (62). The isomorphism of
$C^{*}$-algebras $\overline{\Sigma}_{0}\simeq C(S^{*}X,\mathcal{K})$ implies
the desired isomorphism:

$K_{*}\bigl{(}\operatorname{Con}(C(X)\to\overline{\Sigma}_{0})\rtimes\Gamma\bigr{)}\simeq
K_{*}\bigl{(}\operatorname{Con}(C(X)\to
C(S^{*}X,\mathcal{K}))\rtimes\Gamma\bigr{)}\\\ \simeq
K_{*}\bigl{(}\operatorname{Con}(C(X)\to C(S^{*}X))\rtimes\Gamma\bigr{)}\simeq
K_{*}\bigl{(}C_{0}(T^{*}X)\rtimes\Gamma\bigr{)},$

where we used the isomorphism of $C^{*}$-algebras $\operatorname{Con}(C(X)\to
C(S^{*}X))\simeq C_{0}(T^{*}X)$ endowed with $\Gamma$ actions.

Isomorphism (63) can be constructed similarly. ∎

5\. Using Lemma 4, we can write sequence (61) as

$\ldots\to K_{1}(C_{0}(T^{*}M)\rtimes\Gamma)\oplus
K_{1}(C(X)\rtimes\Gamma)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{0}(C_{0}(T^{*}X)\rtimes\Gamma)\longrightarrow
K_{0}(\mathcal{A}\rtimes\Gamma)\\\ \longrightarrow
K_{0}(C_{0}(T^{*}M)\rtimes\Gamma)\oplus
K_{0}(C(X)\rtimes\Gamma)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{1}(C_{0}(T^{*}X)\rtimes\Gamma)\longrightarrow\ldots$
(64)

Here the boundary mappings $\partial$ are the compositions

$K_{*}(C_{0}(T^{*}M)\rtimes\Gamma)\oplus
K_{*}(C(X)\rtimes\Gamma)\longrightarrow
K_{*}(C_{0}(T^{*}M)\rtimes\Gamma)\longrightarrow
K_{*+1}(C_{0}(T^{*}X)\rtimes\Gamma)$

of projections to the first summand and restriction to the boundary
$T^{*}M|_{X}\simeq T^{*}X\times\mathbb{R}$.

6\. Consider the exact sequence

$\to K_{1}(C_{0}(T^{*}M)\rtimes\Gamma)\oplus
K_{1}(C(X)\rtimes\Gamma)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{0}(C_{0}(T^{*}X)\rtimes\Gamma)\rightarrow
K_{0}(C_{0}(T^{*}M^{\circ})\rtimes\Gamma)\oplus K_{0}(C(X)\rtimes\Gamma)\\\
\longrightarrow K_{0}(C_{0}(T^{*}M)\rtimes\Gamma)\oplus
K_{0}(C(X)\rtimes\Gamma)\stackrel{{\scriptstyle\partial}}{{\longrightarrow}}K_{1}(C_{0}(T^{*}X)\rtimes\Gamma)\longrightarrow\ldots,$
(65)

which represents the direct sum of the exact sequence of the pair

$C_{0}(T^{*}M^{\circ})\rtimes\Gamma\subset C_{0}(T^{*}M^{\circ})\rtimes\Gamma$

and the sequence $0\to K_{*}(C(X)\rtimes\Gamma)\stackrel{{\scriptstyle
id}}{{\to}}K_{*}(C(X)\rtimes\Gamma)\to 0$. We consider (65) as upper row in
the commutative diagram

$\begin{array}[]{ccccc}K_{1}(C_{0}(T^{*}(M\cup
X))\\!\rtimes\\!\Gamma)&\stackrel{{\scriptstyle\partial}}{{\to}}K_{0}(C_{0}(T^{*}X)\\!\rtimes\\!\Gamma)\to\\!\\!\\!\\!&K_{0}(C_{0}(T^{*}M^{\circ}\cup
X)\\!\rtimes\\!\Gamma)&\to\\\ \downarrow&\downarrow&\downarrow&\\\
K_{1}(C_{0}(T^{*}(M\cup
X))\\!\rtimes\\!\Gamma)&\stackrel{{\scriptstyle\partial}}{{\to}}K_{0}(C_{0}(T^{*}X)\\!\rtimes\\!\Gamma)\to\\!\\!\\!\\!&K_{0}(\mathcal{A}\\!\rtimes\\!\Gamma)&\to\end{array}\\\
\begin{array}[]{cc}\to K_{0}(C_{0}(T^{*}(M\cup
X))\\!\rtimes\\!\Gamma)\stackrel{{\scriptstyle\partial}}{{\to}}\\!\\!\\!\\!&K_{1}(C_{0}(T^{*}X)\\!\rtimes\\!\Gamma)\\\
\downarrow&\downarrow\\\ \to K_{0}(C_{0}(T^{*}(M\cup
X))\\!\rtimes\\!\Gamma)\stackrel{{\scriptstyle\partial}}{{\to}}\\!\\!\\!\\!&K_{1}(C_{0}(T^{*}X)\\!\rtimes\\!\Gamma).\end{array}$
(66)

The vertical mappings in this diagram (except the middle one) are identity
mappings. Hence, using Lemma 4, we obtain from the diagram that the middle
mapping is an isomorphism:

$K_{0}(\mathcal{A}\rtimes\Gamma)\simeq
K_{0}(C_{0}(T^{*}M^{\circ})\rtimes\Gamma)\oplus K_{0}(C(X)\rtimes\Gamma).$

Isomorphisms (56) and (58) give the statement in Theorem 4. ∎

##  8 Proof of the index theorem

Let us now prove Theorem 3.

1\. We claim that the left and right hand sides of the index formula (50)
define homomorphisms of Abelian groups

$\operatorname{ind},\operatorname{ind}_{t}:\operatorname{Ell}(M,\Gamma)\longrightarrow\mathbb{C}.$
(67)

Indeed, the analytical index is invariant with respect to homotopies of
elliptic operators. It is equal to zero in the case of trivial elliptic
operators, since trivial operators are invertible. That is why the analytical
index $\operatorname{ind}$ does not change under stable homotopies and it
defines a group homomorphism

$\operatorname{ind}:\operatorname{Ell}(M,\Gamma)\to\mathbb{Z}.$

On the other hand, the topological index is also invariant with respect to
homotopies of elliptic symbols. It is equal to zero for the trivial operators,
since the Chern character of the symbol of such operators is equal to zero.

2\. By Theorem 4, it is sufficient to prove the equality of the indices a) in
the case of operators in $\operatorname{Ell}(M^{\circ},G)$ (i.e., operators
trivial in a neighborhood of the boundary); b) in the case of special boundary
value problems (54).

3\. Case a): since the operator is trivial in a neighborhood of the boundary,
it can be extended to the double of the manifold preserving the analytical
index. In this case (50) follows from the index theorem in [20].

4\. Case b): the analytical index is equal to zero for special boundary value
problems (54). The topological index is also equal to zero.

5\. By 3. and 4. functionals (67) are equal on all elements, which generate
the group $\operatorname{Ell}(M,\Gamma)$. Hence, these functionals are equal.
This completes the proof of the index theorem.

## References

  * [1] M. F. Atiyah and I. M. Singer.  The index of elliptic operators I.  Ann. of Math., 87:484–530, 1968.
  * [2] M. F. Atiyah and R. Bott.  The index problem for manifolds with boundary.  In Bombay Colloquium on Differential Analysis, pages 175–186, Oxford, 1964. Oxford University Press.
  * [3] L. Hörmander.  The Analysis of Linear Partial Differential Operators. III.  Springer–Verlag, Berlin Heidelberg New York Tokyo, 1985.
  * [4] J. T. Wloka, B. Rowley, and B. Lawruk.  Boundary Value Problems for Elliptic Systems.  Cambridge Univ. Press, Cambridge, 1995.
  * [5] L. Boutet de Monvel.  Boundary problems for pseudodifferential operators.  Acta Math., 126:11–51, 1971.
  * [6] G. I. Eskin.  Boundary Value Problems for Elliptic Pseudodifferential Equations.  Nauka, Moscow, 1973.  English transl.: Transl. Math. Monogr. 52 (1981), Am. Math. Soc. Providence.
  * [7] S. Rempel and B.-W. Schulze.  Index Theory of Elliptic Boundary Problems.  Akademie–Verlag, Berlin, 1982.
  * [8] B. V. Fedosov.  Index theorems.  In Partial differential equations, VIII, volume 65 of Encyclopaedia Math. Sci., pages 155–251. Springer, Berlin, 1996.
  * [9] S. T. Melo, R. Nest, and E. Schrohe.  $C^{\ast}$-structure and $K$-theory of Boutet de Monvel’s algebra.  J. Reine Angew. Math., 561:145–175, 2003.
  * [10] S. T. Melo, Th. Schick, and E. Schrohe.  A $K$-theoretic proof of Boutet de Monvel’s index theorem for boundary value problems.  J. Reine Angew. Math., 599:217–233, 2006.
  * [11] A. Connes.  Noncommutative geometry.  Academic Press Inc., San Diego, CA, 1994.
  * [12] A. Connes.  $C^{\ast}$ algèbres et géométrie différentielle.  C. R. Acad. Sci. Paris Sér. A-B, 290(13):A599–A604, 1980.
  * [13] A. Connes and M. Dubois-Violette.  Noncommutative finite-dimensional manifolds. I. Spherical manifolds and related examples.  Comm. Math. Phys., 230(3):539–579, 2002.
  * [14] A. Antonevich and A. Lebedev.  Functional-Differential Equations. I. $C^{*}$-Theory.  Number 70 in Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman, Harlow, 1994.
  * [15] A. Antonevich, M. Belousov, and A. Lebedev.  Functional differential equations. II. $C^{*}$-applications. Parts 1, 2.  Number 94, 95 in Pitman Monographs and Surveys in Pure and Applied Mathematics. Longman, Harlow, 1998.
  * [16] G. Landi and W. van Suijlekom.  Principal fibrations from noncommutative spheres.  Comm. Math. Phys., 260(1):203–225, 2005.
  * [17] A. Connes and G. Landi.  Noncommutative manifolds, the instanton algebra and isospectral deformations.  Comm. Math. Phys., 221(1):141–159, 2001.
  * [18] R. Ponge and H. Wang.  Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants.  J. Noncommut. Geom., 12(4):1573–1639, 2018.
  * [19] D. Perrot.  A Riemann-Roch theorem for one-dimensional complex groupoids.  Comm. Math. Phys., 218(2):373–391, 2001.
  * [20] V. E. Nazaikinskii, A. Yu. Savin, and B. Yu. Sternin.  Elliptic theory and noncommutative geometry, volume 183 of Operator Theory: Advances and Applications.  Birkhäuser Verlag, Basel, 2008.
  * [21] J. Rosenberg.  Noncommutative variations on Laplace’s equation.  Anal. PDE, 1(1):95–114, 2008.
  * [22] A. Yu. Savin and B.Yu. Sternin.  Index of elliptic operators for diffeomorphisms of manifolds.  J. of Noncommut. Geometry, 8(3):695–734, 2014.
  * [23] D. Perrot.  Local index theory for certain Fourier integral operators on Lie groupoids.  arXiv:1401.0225, 2014.
  * [24] A. Yu. Savin and B. Yu. Sternin.  Homotopy classification of elliptic problems associated with discrete group actions on manifolds with boundary.  Ufa Math. J., 8(3):122–129, 2016.
  * [25] M. Gromov.  Groups of polynomial growth and expanding maps.  Inst. Hautes Études Sci. Publ. Math., (53):53–73, 1981.
  * [26] G. Grubb.  Functional Calculus of Pseudo-Differential Boundary Problems.  Progress in Mathematics. Birkhäuser, Boston, 1986.
  * [27] E. Schrohe.  A short introduction to Boutet de Monvel’s calculus.  In Approaches to singular analysis (Berlin, 1999), volume 125 of Oper. Theory Adv. Appl., pages 85–116. Birkhäuser, Basel, 2001.
  * [28] L. B. Schweitzer.  Spectral invariance of dense subalgebras of operator algebras.  Internat. J. Math., 4(2):289–317, 1993.
  * [29] N. Berline, E. Getzler, and M. Vergne.  Heat Kernals and Dirac Operators.  Grundlehren der mathematischen Wissenschaften 298. Springer–Verlag, 1992\.
  * [30] A.Yu. Savin and B. Yu. Sternin.  On the index of noncommutative elliptic operators over ${C}^{*}$-algebras.  Sbornik. Mathematics, 201(3):377–417, 2010.
  * [31] W. Zhang.  Lectures on Chern-Weil theory and Witten deformations, volume 4 of Nankai Tracts in Mathematics.  World Scientific Publishing Co. Inc., River Edge, NJ, 2001.
  * [32] A. Savin.  Elliptic Operators on Manifolds with Singularities and ${K}$-homology.  K-theory, 34(1):71–98, 2005.