Pseudo-differential operators on groupoids

Abstract

We construct an algebra of pseudodifferential operators on each groupoid in a class that generalizes differentiable groupoids to allow manifolds with corners. We show that this construction encompasses many examples. The subalgebra of regularizing operators is identified with the smooth algebra of the groupoid, in the sense of non-commutative geometry. Symbol calculus for our algebra lies in the Poisson algebra of functions on the dual of the Lie algebroid of the groupoid. As applications, we give a new proof of the Poincar\'e-Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the Lie-Poisson structure on the dual A∗A^* of a Lie algebroid.

Full text

arXiv:funct-an/9702004v1 11 Feb 1997
PSEUDODIFFERENTIAL OPERATORS ON DIFFERENTIAL
GROUPOIDS
VICTOR NISTOR, ALAN WEINSTEIN, AND PING XU
Abstract. We construct an algebra of pseudodifferential operators on each
groupoid in a class that generalizes differentiable groupoids to allow manifolds
with corners. We show that this construction encompasses many examples.
The subalgebra of regularizing operators is identified with the smooth algebra
of the groupoid, in the sense of non-commutative geometry. Symbol calculus
for our algebra lies in the Poisson algebra of functions on the dual of the Lie
algebroid of the groupoid. As applications, we give a new proof of the Poincar´e-
Birkhoff-Witt theorem for Lie algebroids and a concrete quantization of the
Lie-Poisson structure on the dual A
∗
of a Lie algebroid.
Contents
Introduction 1
1. Preliminaries 3
2. Main definition 6
3. Differential operators and qua ntization 13
4. Examples 18
5. Distribution kernels 21
6. The action on sections of E 26
References 28
Introduction
Certain impo rtant applications of pseudodifferential operators require variants
of the original definition. Among the many examples one can find in the literature
are regular or adiabatic families of pseudo differe ntial operators [2, 40] and pseudo-
differential operators along the leaves of foliations [6, 8, 27, 2 8], on coverings [9, 29]
or on certain singular spaces [21, 22, 26, 25].
Since these classes of operators share many common features, it is natural to ask
whether they can be treated in a unified way. In this paper we shall suggest an
answer to this question. For any “almost differential” groupoid (a class which allows
manifolds with corners), we c onstruct an algebra of pseudodifferential operators.
We then show that our construction recovers (almost) all the classes desc ribed above
(for opera tors on manifolds with boundary our a lgebra is slightly smaller than the
Date: February 7, 2008.
Nistor is partially supported by NSF Young Investigator Award DMS-9457859 and a Sloan
research fellowship, pr eprints available from http://www.math.psu.edu/nistor/ .
Weinstein is partially supported by NSF Grants DMS-9309653 and DMS-9625122.
Xu is partially supported by NSF Grant DMS 95-04913.
1

2 V. NISTOR, A. WE INSTEIN, AND PING XU
one defined in [21]). We ex pect our results to have applications to analysis on
singular spaces, not only manifolds with co rners.
Our construction and results owe a great deal to the previous work of several
authors, especially Connes [6] and Melrose [21, 23, 20]. A hint of the direction we
take was given at the end of [37]. The basic idea o f our construction is to consider
families of pseudodifferential operators along the fibers of the domain (or source)
map of the groupoid. More precisely, for any almost differentiable groupoid (see
Definition 3) we consider the fibers G
x
= d
−1
(x) of the domain map d consisting of
all ar rows with domain x. It follows from the definition that these fibers are smoo th
manifolds (without corners). The calculus of pseudodifferential operators on smooth
manifolds is well understood and by now a classical subject, see for example [14].
We shall consider differentiable families of pseudodifferential operators P
x
on the
smooth manifolds G
x
. Right translation by g ∈ G defines an isomo rphism G
x
≡ G
y
where x is the domain of g and y is the range of g. We say that the family P
x
is invariant if P
x
transforms to P
y
under the diffeomorphisms above (for a ll g).
The algebra Ψ
∞
(G) of pseudodifferen tial operators on G that we shall consider
will consist of invariant differentiable families of operators P
x
as explained above
(the actual definition also involves a technical condition on the support of these
operators). See Definition 7 for details. The relation with the work of Melrose
relies on an alternative description of our algebra as an algebra of distributions
on G with suitable properties (compactly supp orted, and cono rmal with singular
support contained in the s et of units). This is contained in Theorem 7. The
difference between our theory and Melrose’s lies in the fact that he considers a
compactification of G as a manifold with corners, and his distributions are allowed to
extend to the compactification, with precis e behavior at the boundary. This is useful
for the analysis of these operators. In contrast, our work is purely algebraic (or
geometric, depending on whether one considers Lie algebroids as part of g e ometry
or algebra).
We now review the contents of the sections of this paper. In the first se c tion
we recall the definitions of a groupoid, Lie algebroid and, the less known definition
of a local Lie groupoid. We extend the definition of a Lie groupoid to include
manifolds with c orners. These groupoids are called almost differentiable groupoids.
The second section contains the definition of a pseudodifferential operator on a
groupoid (really a family of pseudodifferential operators, as explained above) and
the proof that they form an algebra, if a support condition is included. We also
extend this definition to include local Lie groupoids. This is useful in the third
section where we use this to give a new proof of the Poincar´e-Birkhoff-Witt theorem
for Lie algebroids. In the proces s o f proving this theorem we also exemplify our
definition of pseudodifferential operators on a n almost differentiable groupoid by
describing the differential operators in this class. As an application we give an
explicit construction of a deformation quantization of the Lie-Poisson structure on
A
∗
, the dual of Lie algebroid A. The section entitled “Examples” contains just
what the title suggests: for many particular examples of groupoids G we explicitly
describe the algebra Ψ
∞
(G) of pseudodifferential operators on G. This recovers
classes of operators that were previously defined using ad hoc constructions. Our
definition is often not only more general, but also simpler. This is the case for
operators along the leaves of foliations [8 , 27] or adiabatic families of operators.

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 3
Since one of our main themes is that the Lie algebras of vector fields which are
central in [24] are in fact the spa c e s of sections of Lie algebroids, we describe these
Lie algebroids explicitly in each of o ur examples.
In the sixth section of the paper, we describe the convolution kernels (called
reduced kernels) of opera tors in Ψ
∞
(G). Then we extend to our setting some fun-
damental results on principal symbols, by reducing to the classical results. This
makes our proofs short (and easy). Finally, the la st section treats the action of
Ψ
∞
(G) on functions on the units of G, and a few related topics.
The first author would like to thank Richard Melrose for several useful conver-
sations.
1. Preliminaries
In the following we allow manifolds to have corners. Thus by “manifold” we
shall mean a C
∞
manifold, p ossibly with corners, and by a “smo oth manifold”
we shall mean a manifold without corners. By definition, if M is a manifold with
corners then ever y po int p ∈ M has co ordinate neighborhoods diffeomorphic to
[0, ∞)
k
× R
n−k
. The transition functions betwe e n such coordinate neighborhoods
must be smooth everywhere (including on the boundary). We shall use the following
definition of submersions between manifolds (with corners).
Definition 1. A submersion between two manifolds with corners M and N is a
differentiable map f : M → N such that df
x
: T
x
M → T
f(x)
N is onto for any
x ∈ M, and if df
x
(v) is an inward pointing tangent vector to N , then v is an
inward pointing tangent vector to M .
The reason for introducing the definition above is that for any submersion f :
M → N, the set M
y
= f
−1
(y), y ∈ N is a smooth manifold, just as for submersions
of smooth manifolds.
We shall study groupoids endowed with various structures. ([32] is a general
reference for some of w hat follows.) We recall fir st tha t a small category is a
category whose class of morphisms is a set. The class of objects of a small category
is then a set as well.
Definition 2. A groupoid is a small category G in which every morphism is in-
vertible.
This is the shortest but least explicit definition. We are going to make this
definition more explicit in c ases of interest. The set of objects, or units, of G will
be denoted by
M = G
(0)
= Ob(G).
The set of mor phisms, or arrows, of G will be denoted by
G
(1)
= Mor(G).
We shall sometimes write G instead of G
(1)
by abuse of notation. For example,
when we consider a space of functions on G, we actually mean a spac e of functions
on G
(1)
. We will denote by d(g) [respectively r(g)] the domain [respectively, the
range] of the morphism g : d(g) → r(g). We thus obtain functions
d, r : G
(1)
−→ G
(0)
(1)

4 V. NISTOR, A. WE INSTEIN, AND PING XU
that will play an important role be llow. The multiplication operator µ : (g, h) 7→
µ(g, h) = gh is defined on the set of composable pairs of arrows G
(2)
:
µ : G
(2)
= G
(1)
×
M
G
(1)
:= {(g, h) : d(g) = r(h)} −→ G
(1)
.(2)
The inversion operation is a bijection ι : g 7→ g
−1
of G
(1)
. Denoting by u(x)
the identity morphism of the object x ∈ M = G
(0)
, we obtain an inclusion o f
G
(0)
into G
(1)
. We s e e that a groupoid G is completely determined by the spaces
G
(0)
and G
(1)
and the structural morphisms d, r, µ, u, ι. We sometimes write G =
(G
(0)
, G
(1)
, d, r, µ, u, ι). The structural maps satisfy the following pro perties:
(i) r(gh) = r(g), d(gh) = d(h) for any pair (g, h) ∈ G
(2)
, and the partially defined
multiplication µ is associative.
(ii) d(u(x)) = r(u(x)) = x, ∀x ∈ G
(0)
, u(r(g))g = g and gu(d(g)) = g, ∀g ∈ G
(1)
and u : G
(0)
→ G
(1)
is one-to-one.
(iii) r(g
−1
) = d(g), d(g
−1
) = r(g), gg
−1
= u(r(g)) and g
−1
g = u(d(g)).
Definition 3. An almost differentiable groupoid G = (G
(0)
, G
(1)
, d, r, µ, u, ι) is a
groupoid such that G
(0)
and G
(1)
are manifolds with corners, the structural maps
d, r, µ, u, ι are differentiable, and the domain map d is a submersion.
We observe that ι is a diffeomorphism and hence d is a submersion if and only
if r = d ◦ ι is a submersion. Also, it follows from the definition that each fiber
G
x
= d
−1
(x) ⊂ G
(1)
is a smooth manifold whose dimension n is constant on each
connected component of G
(0)
. T he ´etale groupoids considered in [5] are extreme
examples of differentiable groupoids (corresponding to dim G
x
= 0). If G
(0)
is
smooth (i.e. if it has no corners) then G
(1)
is also smooth and G becomes what is
known as a differentiable, or Lie groupoid.
1
We now introduce a few important geometric objects associated to an almost
differentiable groupoid.
The vertical tangent bundle (along the fibers of d) of an almost differentiable
groupoid G is
T
d
G = ker d
∗
=
[
x∈G
(0)
T G
x
⊂ T G
(1)
.(3)
Its restriction A(G) = T
d
G


G
(0)
to the set of units is the Lie algebro id of G [19, 30].
We denote by T
∗
d
G the dual of T
d
G a nd by A
∗
(G) the dual of A(G). In addition to
these bundles we shall also c onsider the bundle Ω
λ
d
of λ-densities along the fibers
of d. If the fibers of d have dimension n then Ω
λ
d
= |Λ
n
T
∗
d
G|
λ
= ∪
x
Ω
λ
(G
x
). By
invariance these bundles can be obtained as pull-backs of bundles on G
(0)
. For
example T
d
G = r
∗
(A(G)) and Ω
λ
d
= r
∗
(D
λ
) where D
λ
denotes Ω
λ
d
|
G
(0)
. If E is a
(smooth complex) vector bundle on the set of units G
(0)
then the pull-back bundle
r
∗
(E) on G will have right invariant connections obtained as follows. A connection
∇ o n E lifts to a connection on r
∗
(E). Its restriction to any fiber G
x
defines a linear
connection in the usual sense, which is denoted by ∇
x
. It is easy to see that these
connections are right invariant in the sense that
R
∗
g
∇
x
= ∇
y
, ∀g ∈ G such that r(g) = x and d(g) = y.(4)
The bundles considered above will thus have invariant connections.
1
Earlier terminology, such as in [19], used the term Li e groupoid only for differentiable
groupoids in which every pair of objects is connected by a morphism .

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 5
The bundle A(G), called the Lie algebroid of G, plays in the theory of almost
differentiable groupoids the rˆole Lie alge bras play in the theory of L ie groups. We
recall for the benefit of the reader the definition of a Lie algebroid [30].
Definition 4. A Lie algebroid A over a m anifold M is a vector bundle A over M
together with a Lie algebra structure on the space Γ(A) of smooth sections of A, and
a bundle map ρ : A → T P, extended to a map between sections of these bundles,
such that
(i) ρ([X, Y ]) = [ρ(X), ρ(Y )]; and
(ii) [X, fY ] = f[X, Y ] + (ρ(X)f)Y
for any smooth sections X and Y of A and any smooth function f on M .
Note that we allow the base M in the definition above to b e a manifold with
corners.
If G is an almost differentiable groupoid then A(G) will naturally have the struc-
ture of a Lie algebroid [19]. Let us recall how this structure is defined (the original
definition easily extends to include manifolds with corners). Clearly A(G) is a vec-
tor bundle. The right translation by an arrow g ∈ G defines a diffeomorphism
R
g
: G
r(g)
∋ g
′
→ g
′
g ∈ G
d(g)
. This allows us to talk about right invariant dif-
ferential geometric quantities as long as they are completely determined by their
restriction to all submanifolds G
x
. This is tr ue of functions and d–vertical vector
fields, and this is all that is needed in order to define the Lie alge broid structure on
A(G). The sections of A(G) are in one-to-one c orrespondence w ith vector fields X
on G that are d–vertical, in the sense that d
∗
(X(g)) = 0, and right invariant. The
condition d
∗
(X(g)) = 0 means that X is tangent to the submanifolds G
x
, the fibers
of d. The Lie bracket [X, Y ] of two d–vertical right–invariant vector fields X and Y
will also be d–vertical and right–invariant, and hence the Lie bracket induces a Lie
algebra structure on the sections o f A(G). To define the action of the sections of
A(G) o n functions on G
(0)
, observe that the right invariance property makes sense
also for functions on G and that C
∞
(G
(0)
) may be identified with the subspa c e of
right–invariant functions on G. If X is a right–invariant vector field on G a nd f is
a right–invariant function on G then X(f ) will still be a right invariant function.
This identifies the action of Γ(A(G)) on functions o n G
(0)
.
Not every Lie alg ebroid is the Lie algebroid of a Lie groupoid (see [1] for an
example). However, every Lie algebroid is associated to a local Lie groupoid [31].
The definition of a local Lie (or more generally, almost differentiable) groupoid
[10] is obtained by relaxing the condition that the multiplication µ be everywhere
defined on G
(2)
(see Equation (2)), and replacing it by the condition that µ be
defined in a neighborhood U of the set of units.
Definition 5 (van Est). An almost differentiable local groupoid L = (L
(0)
, L
(1)
) is
a pair of manifolds with corners together with structural morphisms d, r : L
(1)
→
L
(0)
, ι : L
(1)
→ L
(1)
, u : L
(0)
→ L
(1)
and µ : U → L
(1)
, where U is a neighborhood
of (u × u)(L
(0)
) = {(u(x), u(x))} in L
(2)
= {(g, h), d(g) = r(h)} ⊂ L
(1)
× L
(1)
.
The structural morphisms are required to be differentiable maps such that d is a
submersion, u is an embedding, and to satisfy the following properties:
(i) The products u(d(g))g, gu(r(g)), gg
−1
and g
−1
g are defin ed and coincide
with, respectively, g, g, u(r(g)) and u(d(g)); where we denoted g
−1
= ι(g) as usual.
(ii) If gh is defined, then h
−1
g
−1
is defined and equal to (gh)
−1
.
(iii) (Local associativity) If gg
′
, g
′
g
′′
and (gg
′
)g
′′
are defined then g(g
′
g
′′
) is also
defined and equal to (gg
′
)g
′′
.

6 V. NISTOR, A. WE INSTEIN, AND PING XU
The s e t U is the set of arrows for which the product gh = µ(g, h) is defined.
We see that the only difference between a groupoid and a lo c al groupoid L is
the fact that the condition d(g) = r(h) is necessary for the product gh = µ(g, h)
to be defined, but not sufficient in general. The product is defined as soon a s the
arrows g and h are “small enough”. A consequence of this definition is that the
right multiplication by an a rrow g ∈ L
(1)
defines only a diffeomorphism
U
g
−1
∋ g
′
→ g
′
g ∈ U
g
(5)
of an open (and poss ibly empty) subset U
g
−1
of L
y
, y = r(g) to an open subset
U
g
⊂ L
x
, x = d(g). This will not affect the considerations above, however, so we
can associate a Lie algebroid A(L) to any almost differentiable local group oid L.
In the following, when considering groupoids, we shall sometimes refer to them
as global groupoids, in order to stress the difference between gro upoids and local
groupoids.
2. Main definition
Consider a complex vecto r bundle E on the spac e of units G
(0)
of an almost
differentiable groupoid G. Denote by r
∗
(E) its pull-back to G
(1)
. Right translations
on G define linea r isomorphisms
U
g
: C
∞
(G
d(g)
, r
∗
(E)) → C
∞
(G
r(g)
, r
∗
(E))(6)
(U
g
f)(g
′
) = f(g
′
g) ∈ (r
∗
E)
g
′
which makes sense bec ause (r
∗
E)
g
′
= (r
∗
E)
g
′
g
= E
r(g
′
)
.
If G is merely a local groupoid then (6) is replaced by the isomorphisms
U
g
: C
∞
(U
g
, r
∗
(E)) → C
∞
(U
g
−1
, r
∗
(E))(7)
defined for the ope n subsets U
g
⊂ G
d(g)
and U
g
−1 ⊂ G
r(g)
defined in (5).
Let B ⊂ R
n
be an open subset. Define the space S
m
(B × R
n
) of symbols on the
bundle B × R
n
→ B as in [14] to be the set of smooth functions a : B × R
n
→ C
such that
|∂
α
y
∂
β
ξ
a(y, ξ)| ≤ C
K,α,β
(1 + |ξ|)
m−|β|
(8)
for any compact set K ⊂ B and any multiindices α a nd β. An e le ment of one of
our spaces S
m
should properly be said to have “order less than or equal to m”;
however, by abuse of language we will say that it has “order m”.
A symbol a ∈ S
m
(B × R
n
) is called classical if it has an asymptotic expa nsion
as an infinite sum of homo geneous symbols a ∼
P
∞
k=0
a
m−k
, a
l
homogeneous of
degree l: a
l
(y, tξ) = t
l
a
l
(y, ξ) if kξk ≥ 1 and t ≥ 1. (“Asymptotic expansion”
is used here in the sense that a −
P
N−1
k=0
a
m−k
belongs to S
m−N
(B × R
n
).) The
space of classical symbols will be denoted by S
m
cl
(B × R
n
). We shall be working
exclusively with classical sy mbols in this paper.
This definition immediately extends to give spaces S
m
cl
(E; F ) of symbols on E
with values in F , where π : E → B and F → B are smooth euclidian vector bundles.
These spaces, which are independent of the metrics used in their definition, are
sometimes denoted S
m
cl
(E; π
∗
(F )). Taking E = B × R
n
and F = C one recovers
S
m
cl
(B × R
n
) = S
m
cl
(B × R
n
; C).
A pseudodifferential operator P on B is a linear map P : C
∞
c
(B) → C
∞
(B)
that is locally of the form P = a(y, D
y
) plus a regularizing operator, where for any

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 7
complex valued symbol a on T
∗
W = W × R
n
, W an open subset of R
n
, one defines
a(y, D
y
) : C
∞
c
(W ) → C
∞
(W ) by
a(y, D
y
)u(y) = (2π)
−n
Z
R
n
e
iy·ξ
a(y, ξ)ˆu(ξ)dξ .(9)
Recall that an operator T : C
∞
c
(U) → C
∞
(V ) is called regularizing if and only if it
has a smooth distribution (or Schwartz) kernel. This happens if and only if T is
pseudodifferential of order −∞.
The class of a in S
m
cl
(T
∗
W )/S
m−1
cl
(T
∗
W ) does not depe nd on any choices; the
collection of all these classe s, for all coordinate neighborhoods W , patches to gether
to define a class σ
m
(P ) ∈ S
m
cl
(T
∗
W )/S
m−1
cl
(T
∗
W ) which is called the principal sym-
bol of P . If the ope rator P acts on sections of a vector bundle E, then the principal
symbol σ
m
(P ) will belong to S
m
cl
(T
∗
B; End(E))/S
m−1
cl
(T
∗
B; End(E)). See [14] for
more details on a ll these constructions.
We shall sometimes refer to pseudodifferential operators acting o n a s mooth
manifold as ordinary pseudodifferential operators, in or der to distinguish them from
pseudodifferential operators on groupoids, a c lass of operator s which we now define
(and which are really families of ordinary pseudodifferential o perators).
Throughout this paper, we shall denote by (P
x
, x ∈ G
(0)
) a family of order m
pseudodifferential operators P
x
, acting on the spaces C
∞
c
(G
x
, r
∗
(E)) for some vector
bundle E over G
(0)
. Operators betwe e n sections of two different vector bundles E
1
and E
2
are obtained by considering E = E
1
⊕ E
2
.
Definition 6. A family (P
x
, x ∈ G
(0)
) as above is called differentiable if for any
open set V ⊂ G, diffeomorphic through a fi ber preserving diffeomorphism to d(V ) ×
W , for some open subset W ⊂ R
n
, and any φ ∈ C
∞
c
(V ), we can find a ∈ S
m
cl
(d(V )×
T
∗
V ; End(E)) such that φP
x
φ corresponds to a(x, y, D
y
) under the diffeomorphism
G
x
∩ V ≃ W , for each x ∈ d(V ).
A fiber preserving diffeomorphism is a diffeomorphism ψ : d(V ) × W → V
satisfying d(ψ(x, w)) = x. Thus we require tha t the operators P
x
be given in local
coordinates by symbols a
x
that depend smoothly on all variables , in particular on
x ∈ G
(0)
.
Definition 7. An order m invariant pseudodifferential operator P on an almost
differentiable groupoid G, acting on sections of the vector bundle E, is a differ-
entiable family (P
x
, x ∈ G
(0)
) of order m classical pseudodifferential operators P
x
acting on C
∞
c
(G
x
, r
∗
(E)) and satisfying
P
r(g)
U
g
= U
g
P
d(g)
(invariance)(10)
for any g ∈ G
(1)
, where U
g
is as in (6).
Replacing the coefficient bundle E by E ⊗ D
λ
and using the is omorphism Ω
λ
d
≃
r
∗
(D
λ
), we obtain operators acting on sections of density bundles. Note that P
can generally not be considered as a single pseudodiffere ntial operator on G
(1)
.
This is because a family of pseudodifferential operators on a smooth manifold M,
parametrized by a smooth manifold B, is not a pseudodifferential operator on the
product M × B, although it ac ts naturally on C
∞
c
(M × B). (See [2] or [14], page
94.)
Recall [13] that distributions o n a manifold Y with coefficients in the bundle E
0
are continuous linear maps C
∞
c
(Y, E
′
0
⊗ Ω) → C, where E
′
0
is the dual bundle to E
0

8 V. NISTOR, A. WE INSTEIN, AND PING XU
and Ω = Ω(Y ) is the space of 1-densities on Y . The collection o f all distributions on
Y with coefficients in the (finite dimensiona l complex vector) bundle E
0
is denoted
C
−∞
(Y ; E
0
).
If P = (P
x
, x ∈ G
(0)
) is a family of pseudodifferential operators acting on G
x
denote by k
x
the distribution kernel of P
x
k
x
∈ C
−∞
(G
x
× G
x
; r
∗
1
(E) ⊗ r
∗
2
(E)
′
⊗ Ω
2
).(11)
Here Ω
2
is the pull-back of the bundle of vertical densities Ω
d
on G
x
to G
x
× G
x
via the second projection. These distribution kernels are obtained using Schwartz’
kernel theorem. We define the support of the operator P to be
supp(P ) = ∪
x
supp(k
x
).(12)
The support of P is contained in the closed subset {(g, g
′
), d(g) = d(g
′
)} of the
product G
(1)
× G
(1)
. In particula r (id × ι)(supp(P )) ⊂ G
(2)
. If all op e rators P
x
are
of order −∞, then each kernel k
x
is a smooth section. Actually we have more
Lemma 1. The collection of all distribution kernels k
x
of a differentiable family
P = (P
x
, x ∈ G
(0)
) of order −∞ operators defines a smooth section k of r
∗
1
(E) ⊗
r
∗
2
(E)
′
⊗ Ω
2
on {(g, g
′
), d(g) = d(g
′
)}.
Proof. Indeed if ψ : d(V ) × W → V is a fib e r preserving diffeomor phism as in
Definition 6, then it fo llows from the definition that k is smooth on d(V )×W ×W ⊂
{(g, g
′
), d(g) = d(g
′
)}. Since in this way we obtain an atlas of {(g, g
′
), d(g) = d(g
′
)},
we obtain that k is smooth as claimed.
Definition 8. The family P = (P
x
, x ∈ G
(0)
) is properly supported if p
−1
i
(K) ∩
supp(P ) is a compact set for any compact subset K ⊂ G, where p
1
, p
2
: G × G → G
are the two projections. The family P is called compactly su pported if its support
supp(P ) is compact; and, finally, P is called uniformly supported if its reduced
support supp
µ
(P ) = µ
1
(supp(P )) is a compact subset of G
(1)
, where µ
1
(g
′
, g) =
g
′
g
−1
.
It immediately follows from the definition that a uniformly supported operator
is also properly supported, and that a compactly supported operator is uniformly
supported. If the family P = (P
x
, x ∈ G
(0)
) is properly supported then each P
x
is
properly supported, but the converse is no t true.
Recall that the composition of two ordinar y pseudodifferential operators is de-
fined if one of them is properly supported. It follows that we can define the compo-
sition P Q of two properly supported families of operators P = (P
x
, x ∈ G
(0)
) and
Q = (Q
x
, x ∈ G
(0)
) on G
(1)
by pointwise compos itio n P Q = (P
x
Q
x
, x ∈ G
(0)
). The
action on sec tions of r
∗
(E) is also defined pointwise as follows. For any smooth
section f ∈ C
∞
(G, r
∗
(E)) denote by f
x
the restriction f |
G
x
. If each f
x
has compact
support and P = (P
x
, x ∈ G
(0)
) is a family of ordinary pseudodifferential operators,
then we define P f by (P f )
x
= P
x
(f
x
).
Lemma 2. (i) If f ∈ C
∞
c
(G, r
∗
(E)) and P = (P
x
, x ∈ G
(0)
) is a differentiable
family of ordinary pseudodifferential operators then P f ∈ C
∞
(G, r
∗
(E)). If P is
also properly supported then P f ∈ C
∞
c
(G, r
∗
(E)).
(ii) The composition P Q = (P
x
Q
x
, x ∈ G
(0)
) of two properly supported differen-
tiable families of operators P = (P
x
, x ∈ G
(0)
) and Q = (Q
x
, x ∈ G
(0)
) is a properly
supported differentiable family.

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 9
Proof. If P consists of regularizing operators then
P f (g) =
Z
G
x
k
x
(g, h)f(h) , where x = d(g).
Lemma 1 implies that the formula above for P f involves only the integration of
smooth (uniformly in g) compactly supported sections, and hence we can exchange
integration and derivation to obtain the smoothness of P f. This proves (i) in ca se
P consists of regularizing operators. The proof o f (ii) if both P and Q consist of
regularizing operators follows the same reasoning.
We prove now (i) for P arbitra ry. Fix g ∈ G
x
and V a neig hborhood of g fiber
preserving diffeomorphic to d(V )×W for some open convex subset W in R
n
, 0 ∈ W ,
such that (x, 0) maps to g. Replacing P
x
by P
x
−R
x
for a smooth regularizing family
R
x
we can assume that the distribution kernels k
x
of P
x
satisfy
p
−1
1
(d(V ) × W/4) ∩
∪ supp(k
x
) ⊂ (d(V ) × W/4) × (d(V ) × W/2).
The smoothness of P f, res pectively of P Q if Q consists of r e gularizing operators,
reduces in this way to a computation in local coordinates. This completes the proof
of (i) in general, and of (ii) if Q is regularizing.
For arbitrary Q we can replace Q, in view of what has already b e en prove d, with
Q − R, wher e R is a regularizing family. In this way we may as sume that
p
−1
1
(d(V ) × W/2) ∩
∪ supp(k
′
x
) ⊂ (d(V ) × W/2) × (d(V ) × 3W/4)
where k
′
x
are the distribution kernels of Q
x
. The support estimates above for P
and Q show that the P
y
Q
y
for y ∈ d(V ) are the compositions of smooth families of
pseudodifferential operators acting on W ⊂ R
n
. The result is then known.
The smaller class of uniformly supported operators is also closed under compo-
sition.
Lemma 3. The composition P Q = (P
x
Q
x
, x ∈ G
(0)
) of two uniformly supported
families of operators P = (P
x
, x ∈ G
(0)
) and Q = (Q
x
, x ∈ G
(0)
) is uniformly
supported.
Proof. The reduced suppor t supp
µ
(P Q) (see (12)) of the composition P Q satisfies
supp
µ
(P Q) ⊂ µ

supp
µ
(P ) × supp
µ
(Q)

where µ is the composition of arrows. Since supp
µ
(P ) and supp
µ
(Q) are compact,
the equation above completes the proof of the lemma.
Let G be an almost differentiable groupoid. The space of order m, invariant, uni-
formly supported pseudodifferential operators on G, acting on sections of the vector
bundle E will b e denoted by Ψ
m
(G; E). We denote Ψ
∞
(G; E) = ∪
m∈Z
Ψ
m
(G; E)
and Ψ
−∞
(G; E) = ∩
m∈Z
Ψ
m
(G; E). Thus an operator P ∈ Ψ
m
(G; E) is actually a
differentiable family P = (P
x
, x ∈ G
(0)
) of ordinary pseudodifferential operators.
Theorem 1. The set Ψ
∞
(G; E) of uniformly supported invariant pseudodifferent ial
operators on an almost differentiable groupoid G is a filtered algebra, i.e.
Ψ
m
(G; E)Ψ
m
′
(G; E) ⊂ Ψ
m+m
′
(G; E).
In particular Ψ
−∞
(G; E) is a two-sided ideal.

10 V. NISTOR, A. WE INSTEIN, AND PING XU
Proof. Let P = (P
x
, x ∈ G
(0)
) and Q = (Q
x
, x ∈ G
(0)
) be two invariant uniformly
supported pseudodifferential oper ators on G, of order m and m
′
respectively. Their
composition P Q = (P
x
Q
x
), is a uniformly supported operator of order m + m
′
, in
view of L emma 3. It is also a differentiable family due to Lemma 2. We now check
the invar iance co ndition. Let g be an arbitra ry a rrow and U
g
: C
∞
c
(G
x
, r
∗
(E)) →
C
∞
c
(G
y
, r
∗
(E)), x = d(g) and y = r(g), be as in the definition above. Then
(P Q)
y
U
g
= P
y
Q
y
U
g
= P
y
U
g
Q
x
= U
g
P
x
Q
x
= U
g
(P Q)
x
.
This proves the theorem.
Properly supported invariant differentiable fa milies of pseudodifferential opera-
tors also form a filtered algebra, deno ted Ψ
∞
prop
(G; E). While it is clear that in order
for our class of pseudodifferential operators to form an algebra we need some con-
dition on the support of their distribution kernels, exactly what support condition
to impose is a matter of choice. We prefer the uniform support condition because
it leads to a better control at infinity of the family of op e rators P = (P
x
, x ∈ G
(0)
)
and allows us to identify the re gularizing ideal (i.e. the ideal of order −∞ operators)
with the gro upoid convo lution algebra of G. The choice of uniform support will also
ensure that Ψ
m
(G; E) behaves functorially with res pect to open embeddings. The
compact support condition enjoys the same pr operties but is usually too restrictive.
The issue of support will be discussed again in examples.
The definition of the principal symbol extends easily to Ψ
m
(G; E). Deno te by
π : A
∗
(G) → M, (M = G
(0)
) the projection. If P = (P
x
, x ∈ G
(0)
) ∈ Ψ
m
(G; E) is
an order m pseudodiffere ntial differential operator on G, then the principal symbol
σ
m
(P ) of P will be represented by sections of the bundle E nd(π
∗
E) and will be
defined to satisfy
σ
m
(P )(ξ) = σ
m
(P
x
)(ξ) ∈ End(E
x
) if ξ ∈ A
∗
x
(G) = T
∗
x
G
x
(13)
(the equation above is mod S
m−1
cl
(A
∗
x
(G); End(E))). This equation will obviously
uniquely determine a linear map
σ
m
: Ψ
m
(G) → S
m
cl
(A
∗
(G); End(E))/S
m−1
cl
(A
∗
(G); End(E)).
provided we can show that for any P = (P
x
, x ∈ G
(0)
) ther e exists a symbol
a ∈ S
m
cl
(A
∗
(G); End(E)) whose restriction to A
∗
x
(G) is a representative of the prin-
cipal symbol of P
x
in that fiber for each x. We thus need to choose for each P
x
a representative a
x
∈ S
m
cl
(A
∗
x
(G); End(E)) o f σ
m
(P
x
) such that the family a
x
is
smooth and invariant. Assume first that E is the tr iv ial line bundle and proceed
as in [14] Section 18.1, espec ially Equatio n (18.1.27) and below.
Choose a connection ∇ on the vector bundle A(G) → G
(0)
and consider the
pull-back vector bundle r
∗
(A) → G of A(G) → G
(0)
endowed with the pull-back
connection
e
∇ = r
∗
∇. Its restriction on any fiber G
x
defines a linear connection in
the usual sense, which is denoted by ∇
x
. These connections are right invariant in
the sense that
R
∗
g
∇
x
= ∇
y
, ∀g ∈ G such that r(g) = x and d(g) = y.(14)
Using such an invariant connection, we may define the exponential map of a
Lie algebroid, which generalizes the usua l exponential map of a manifold with a
connection and the exponential map of a Lie algebra as follows. For any x ∈ G
(0)
,

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 11
define a map exp
x
: A
x
→ G as the compo sition of the maps :
A
x
i
−→ T
x
G
x
˜exp
x
−−−−→ G
where i is the natural inclusion and ˜exp
x
= exp
∇
x
is the usual exponential map
at x ∈ G
x
on the manifold G
x
. By varying the point x, we obtain a ma p exp
∇
defined in a neighbor hood of the zero section, called the exponential map of the Lie
algebroid
2
. Clearly, exp
∇
is a local diffeomorphism
A(G) ⊃ V
0
∋ v −→ exp
∇
(v) = y ∈ V ⊂ G(15)
mapping an open neighborho od V
0
of the zero section in A(G) diffeomorphically to
a neighborhood V of G
(0)
in G, and sending the zer o section onto the set of units.
Choose a cut-off function φ ∈ C
∞
(G) with support in V and equal to 1 in a smaller
neighborhood of G
(0)
in G. If y ∈ V , x = d(y) and ξ ∈ A
∗
x
(G) let v ∈ V
0
be the
unique vector v ∈ A
x
(G) such that y = exp
∇
(v) and denote e
ξ
(y) = φ(y)e
iv·ξ
which
extends then to all y ∈ G due to the cut-off function φ. Define the (∇, φ)–complete
symbol σ
∇,φ
(P ) by
σ
∇,φ
(P )(ξ) = (P
x
e
ξ
)(x), ∀ξ ∈ T
∗
x
G
x
= A
∗
x
(G).(16)
Lemma 4. If P = (P
x
, x ∈ G
(0)
) is an operator in Ψ
m
(G) then the function
σ
∇,φ
(P ) defined above is differentiable and defines a symbol in S
m
cl
(A
∗
(G)). More-
over if (∇
1
, φ
1
) is another pair consisting of an invariant connection ∇
1
and a cut-
off function φ
1
then σ
∇,φ
(P )−σ
∇,φ
1
(P ) is in S
−∞
cl
(A
∗
(G)) and σ
∇,φ
(P )−σ
∇
1
,φ
1
(P )
is in S
m−1
cl
(A
∗
(G)).
Proof. For each ξ ∈ A
∗
x
the function e
ξ
is smooth with compact support on G
x
so P
x
e
ξ
is defined. Equation (18.1.27) of [14] shows that a(ξ) = σ
∇,φ
(P )(ξ) is
the restriction of the complete symbo l of P
x
φ to T
∗
x
G
x
if the complete symbol is
defined in the normal coordinate system at x ∈ G
x
(given by the exponential map).
The normal coordinate system defines, using a local trivialization of A(G), a fiber
preserving diffeomorphism ψ : d(V ) × W → V for some open subset W of R
n
(i.e.
satisfying d(ψ(x, w)) = x). From the definition of the smoothness of the family P
x
(Definition 6) it follows that the complete symbol of P φ is in S
m
cl
(d(V ) × T
∗
W ) if
the support of φ is chosen to be in V . This proves that σ
∇,φ
(P ) is in S
m
cl
(A
∗
(G)).
The rest follows in exactly the same way.
The lemma above justifies the following definition of the principal symbol as the
class of σ
∇,φ
(P ) modulo terms of lower order (for the triv ial line bundle E = C).
This definition will be, in view of the same lemma, independent on the choice of
∇ or φ and will satisfy Equation (13). If E is not trivial one can still define a
complete symbol σ
∇,∇
′
,φ
(P ), depending also o n a second connection ∇
′
on the
bundle E, which is used to trivialize r
∗
(E) on V ⊂ G (assuming also that V
0
is
convex). Alternative ly, we can use Propo sition 3 below.
Proposition 1. Let ∇ and φ be as above. The choice of a connection ∇
′
on E
defines a complete symbol map σ
∇,∇
′
,φ
: Ψ
m
(G; E) → S
m
cl
(A
∗
(G)). The principal
symbol σ
m
: Ψ
m
(G; E) → S
m
cl
(A
∗
(G))/S
m−1
cl
(A
∗
(G)), defined by
σ
m
(P ) = σ
∇,∇
′
,φ
(P ) + S
m−1
cl
(A
∗
(G))(17)
2
See [17] for an alternative definition of the exponential map. One should not confuse this
map with the exponential map from Γ(A) to the bisections of the groupoid as defined in [16].

12 V. NISTOR, A. WE INSTEIN, AND PING XU
does not depend on the choice of the connections ∇, ∇
′
or the cut-off function φ.
Proof. The (∇, ∇
′
, φ)–complete symbol σ
∇,∇
′
,φ
(P ) is defined as follows. Let w be a
vector in E
x
. Using the connection ∇
′
we can define a section ˜w of r
∗
(E) on G
x
∩V
by pa rallel transport along the geodesics of ∇ star ting at x, and which coincides
with w at x. Then denote e
ξ,w
= e
ξ
˜w and let
σ
∇,∇
′
,φ
(P )(ξ)w = (P
x
e
ξ,w
)(x) ∈ E
x
, ∀ξ ∈ T
∗
x
G
x
= A
∗
x
(G).(18)
The rest of the proof proceeds along the lines of the proof of Lemma 4.
Note that the principal symbol of P determines the principal symbo ls of the indi-
vidual opera tors P
x
by the invar iance with respect to right translations. Precisely,
we have σ
m
(P
x
) = r
∗
(σ(P ))|
T
∗
G
x
.
The following re sult extends some very well known properties of the calculus of
pseudodifferential operators on smooth manifolds. We shall prove the surjectivity
of the principal symbol in section 5.
Proposition 2. (i) The principal symbol map
σ
m
: Ψ
m
(G; E) → S
m
cl
(G; End(E))/S
m−1
cl
(G; End(E))
has kernel Ψ
m−1
(G; E) and satisfies Equation (13).
(ii) The composition P Q of two operators P, Q ∈ Ψ
∞
(G; E), of orders m and,
respectively, m
′
, satisfies σ
m+m
′
(P Q) = σ
m
(P )σ
m
′
(Q).
Proof. (i) The ope rator P = (P
x
, x ∈ G
(0)
) ∈ Ψ
m
(G; E) is in the kernel of σ
m
if
and o nly if all symbols σ
m
(P
x
) vanish. This implies P
x
∈ Ψ
m−1
(G; E) for all x
and hence P = (P
x
, x ∈ G
(0)
) ∈ Ψ
m−1
(G; E). As already observed for E a trivial
line bundle, the fa ct that Equation (13) is s atisfied was contained in the proof of
Lemma 4. The general case is similar or can be proved using P roposition 3.
The second s tatement is known for pseudodifferential operators on smooth man-
ifolds [14]; this accounts for the second equality sign in the next equation. We
obtain using Equation (13) that
σ
m+m
′
(P Q)(v) = σ
m+m
′
(P
x
Q
x
)(v) = σ
m
(P
x
)σ
m
′
(Q
x
)(v) = σ
m
(P
x
)(v)σ
m
′
(Q
x
)(v)
where v ∈ A
∗
x
(G).
Although for the most of this paper we shall be concerned with groupoids, the
definition of Ψ
m
(G; E) easily extends to local groupoids. Indeed it suffices to modify
the invaria nce condition in Definition 7, using the notation in Equation (7), as
follows. We assume that for any g ∈ G
(1)
and any smooth compactly suppor ted
function φ on U
g
there exists a regularizing operator R
g,φ
such that
U
g
(φ)P
r(g)
U
g
f − U
g
(φP
d(g)
f) = R
g,φ
f(19)
for any function f ∈ C
∞
c
(U
g
). We thus replace the strict invariance of the original
definition by ‘invariance up to regularizing operators’.
We denote by Ψ
m
lo c
(G; E) the set of differentiable properly supported families
P = (P
x
, x ∈ G
(0)
) of order m pseudodifferential ope rators satisfying the c ondition
(19) above. Note that if we regard an almost differentiable gro upoid G as a loc al
groupoid then Ψ
∞
(G; E) ⊂ Ψ
m
lo c
(G; E). The inclusion is gener ally a str ic t one,
though, because Equation (19) gives no condition for order −∞ operators, and s o
Ψ
−∞
lo c
(G; E) consists of arbitrary smooth families P = (P
x
, x ∈ G
(0)
) of re gularizing

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 13
operators. This ideal is too big to reflect the structure of G. The “sy mbolic” part
remains however the same:
Ψ
m
prop
(G; E)/Ψ
−∞
prop
(G; E) ≃ Ψ
m
lo c
(G; E)/Ψ
−∞
lo c
(G; E).
If the sets U
g
are all connected (in which ca se the local g roupoid G is sa id to be
d–connected) an easier condition to use than (19) is
[X, P ] ∈ Ψ
−∞
lo c
(G; E)(20)
for all r–vertical left–invariant vector fields X on G
(1)
. With this, the following
analog o f Theorem 1, becomes str aightforward.
Theorem 2. Assume that G is a d–connected almost differentiable groupoid. Then
the space Ψ
∞
lo c
(G) is a filtered algebra, with Ψ
−∞
(G; E) as residual ideal.
Proof. The only thing to check is that Ψ
∞
lo c
(G; E) is close d under composition. The
composition of two differentiable, properly suppor ted families P, Q ∈ Ψ
∞
lo c
(G; E) is
again differentiable and properly s upported, as it has already been proved. The
infinitesimal invariance condition [X, P Q] = [X, P ]Q + P [X, Q] ∈ Ψ
−∞
lo c
(G; E) (20)
follows from the fact that Ψ
−∞
lo c
(G; E) is an ideal of Ψ
∞
lo c
(G; E).
3. Differential operators and quantization
In this section, we examine the differential operators in Ψ
∞
(G; E), if G is a
global groupoid, or in Ψ
∞
lo c
(G; E) if G is a local groupoid. We also show how a
simple algebraic construction applied to G and to the algebras Ψ
∞
lo c
(G; E) leads to
a concrete construction of a deformation quantization of the Lie-Poisson structure
on the dual of a Lie algebroid.
In this section, G will be an almost differentiable local gr oupoid, This gener ality
is necessary in order to integrate arbitrary Lie alg e broids. Nevertheless, when A is
the Lie algebroid of an almost differentiable global groupoid G (that is not just a
local groupoid) then all results we shall prove for Ψ
∞
lo c
(G; E) in this section extend
immediately to Ψ
∞
(G; E), although we shall not mention this each time.
Lemma 5. Let P = (P
x
, x ∈ G
(0)
) be an operator in Ψ
∞
lo c
(G; E). If P
x
is a mu lti-
plication operator for all x, then there exists a smooth endomorphism s of E such
that P
x
(g) = s(r(g)) for all g ∈ G
x
. Conversely, every smooth section s of End(E)
defines a multiplication operator in Ψ
0
lo c
(G; E).
Proof. By assumption P
x
(g) is in End(E
r(g)
). The invariance relation shows that
P
x
(g) depends only on r(g). This defines the section s of End(E) such that P
x
(g) =
s(r(g)). To show that s is smooth, we let φ be a smooth section of E over G
(0)
and
let
˜
φ(g) = φ(r(g)). By assumption P
˜
φ is smooth and hence sφ = P
˜
φ|
G
(0)
is also
smooth. Since φ is arbitrary this implies the smoothness of s.
Conversely, if s is a smooth endo morphism of E, then if we let P
x
(g) = s(r(x))
we obtain a multiplication operator in Ψ
0
lo c
(G; E).
The following proposition will allow us to assume that E is a trivial bundle,
which is sometimes useful in applications.
Proposition 3. Let E be a vector bundle on G
(0)
embedded into a trivial hermitian
bundle, E ⊂ C
N
. Denote by e
0
the projection onto E regarded as a matrix of
multiplication operators in M
N
(Ψ
0
lo c
(G)), the algebra of N × N matrices with values
in Ψ
0
lo c
(G). Then Ψ
∞
lo c
(G; E) ≃ e
0
M
N
(Ψ
∞
lo c
(G))e
0
as filtered algebras.

14 V. NISTOR, A. WE INSTEIN, AND PING XU
Proof. The multiplication operator e
0
defines an element of Ψ
0
lo c
(G; E) by the
lemma above; hence it acts on all spaces C
∞
c
(G
x
, C
N
). Then
C
∞
c
(G
x
, r
∗
(E)) = e
0
C
∞
c
(G
x
, C
N
)
and every pseudodifferential operator P
x
on C
∞
c
(G
x
, r
∗
(E)) extends in this way to an
operator on C
∞
c
(G
x
, C
N
). This gives an inclusion Ψ
∞
lo c
(G; E) ⊂ e
0
M
N
(Ψ
∞
lo c
(G))e
0
.
Conversely if P
x
is a pseudodifferential operator on C
∞
c
(G
x
, C
N
) then e
0
P
x
e
0
is a
pseudodifferential operator on C
∞
c
(G
x
, C
N
). This gives the opposite inclusion.
The following proposition shows the intimate connection between A(G), the Lie
algebroid of G, and Ψ
∞
lo c
(G) (Ψ
∞
(G) if G is global). It is morally an equivalent
definition of the Lie alg e broid associated to an almost differentiable local groupoid.
Proposition 4. Let G be an almost differentiable local groupoid.
(i) The algebra C
∞
(G
(0)
) is the algebra of multiplication operators in Ψ
0
lo c
(G).
(ii) The space of sections of the Lie algebroid A(G) can be identified with the
space of order 1 differential operators in Ψ
1
lo c
(G) without constant term.
(iii) The Lie algebroid structure of A(G) is induced by the commutator operations
[ , ] : Ψ
1
lo c
(G) × Ψ
1
lo c
(G) → Ψ
1
lo c
(G) and [ , ] : Ψ
1
lo c
(G) × Ψ
0
lo c
(G) → Ψ
0
lo c
(G).
Proof. The first part is a particular case of Lemma 5, only easier. Or der 1 dif-
ferential operato rs without constant term are vector fields, right invaria nt by the
definition of Ψ
1
lo c
(G), so they can be identified with the sections of the Lie algebroid
A(G) of G. This proves (ii). In order to check (iii) recall that, if we regard vector
fields on G as linear maps C
∞
(G) → C
∞
(G), then the Lie bracket coincides with
the commutator of linear maps. More over the commutator [X, f] of a vector field
X and of a multiplication map f is [X, f] = X(f), again regarded as a linea r map.
Then (iii) follows in view of the discussion above.
The Lie algebroid A = A(G) turns out to determine the str uctur e of the algebra of
invariant tangential differential operators on G, denoted Diff(G). We shall see that
the subalgebra Diff(G) ⊂ Ψ
∞
lo c
(G) is a concre te model of the universal enveloping
algebra of the L ie algebroid A [15, 35], a concept whose definition we now recall.
Given a Lie algebroid A → M with anchor ρ, we can make the C
∞
(M)-module
direct sum C
∞
(M) ⊕ Γ(A) into a Lie algebra over C by defining
[f + X, g + Y ] = (ρ(X)g − ρ(Y )f) + [X, Y ].
Let U = U(C
∞
(M) ⊕ Γ(A)) be its universal enveloping algebra. For any f ∈
C
∞
(M) and X ∈ Γ(A), denote by f
′
and X
′
their ca nonical image in U . Denote
by I the two-sided ideal of U generated by all elements of the form (fg)
′
− f
′
g
′
and
(fX)
′
− f
′
X
′
. Define
U(A) = U/I.(21)
U(A) is called the universal enveloping algebra of the Lie algebroid A. When A
is a Lie algebra, this definition r e duces to the usual universal enveloping algebr a.
We shall see, for example, that for the tangent bundle T M this is the algebra of
differential operators on M .
The maps f → f
′
and X → X
′
considered above descend to linear embeddings
i
1
: C
∞
(M) → U (A), and i
2
: Γ(A) → U(A); the first map i
1
is an algebra

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 15
morphism. These maps have the following properties:
i
1
(f)i
2
(X) = i
2
(fX), [i
2
(X), i
1
(f)] = i
1
(ρ(X)f )), [i
2
(X), i
2
(Y )] = i
2
([X, Y ]).
(22)
In fact, U(A) is universal among triples (B, φ
1
, φ
2
) having these properties (see [15]
for a proof o f this easy fact).
In particular, if M is the space of units of an almost differentiable groupoid
G with Lie algebroid A(G), the natural morphisms φ
1
: C
∞
(M) → Diff(G) and
φ
2
: Γ(A) → Diff(G) o bta ined from Proposition 4 extend to a unique algebra
morphism τ : U(A) → Diff(G). (Recall that we denoted by Diff(G) the algebra o f
right invariant tangential differential operators on G.) Denote by U
n
(A) ⊂ U (A)
the space generated by C
∞
(M) and the images of X
1
⊗ X
2
⊗ . . . ⊗ X
k
∈ U =
U(C
∞
(M) ⊕ Γ(A)), for k ≤ n, under the canonical projection U → U(A) = U/I.
Then
U
0
(A) ⊂ U
1
(A) ⊂ · · · ⊂ U
n
(A) ⊂ · · ·(23)
is a filtration of U (A). The r e lations (22) show that, a s in the Lie algebra case, the
graded algebra ⊕U
n
(A)/U
n−1
(A) is commutative . Similarly, Diff(G) is naturally
filtered by degree.
Lemma 6. The map τ : U (A) → Diff(G) m aps U
n
(A) onto the space Diff
n
(G) of
operators of order ≤ n.
Proof. Let D ∈ Diff(G) be an invar iant tangetial differential operator of order
≤ n. By right invariance D is completely determined by the res trictions (Du)|
G
(0)
,
u ∈ C
∞
c
(G). Since D acts on the fibers of d we can write
(Du)|
G
(0)
=
n
X
i=1
D
i
u
where D
i
is a superposition of derivations D
i
u = X
(i)
1
X
(i)
2
· · · X
(i)
k
i
u defined using
the tangential derivations X
(i)
j
∈ Γ(A). By definition it follows that D is the sum
of τ(X
(i)
1
X
(i)
2
· · · X
(i)
k
i
).
Denote by Symm(A) the symmetric tensor product of the bundle A, that is
Symm(A) =
∞
M
n=0
S
n
(A)
where S
n
(A) is the symmetric quotient of the bundle A
⊗n
, and is iso morphic to
the subspace of symmetric tensors, if S
0
(A) is C
∞
(M) by co nvention. The spa c e
Γ(Symm(A)) of smooth sections of Symm(A) identifies with the space of smooth
functions on A
∗
polynomial in each fiber. The complete symbol map σ
∇,φ
(D) of
an invariant differential operator Diff(G) ⊂ Ψ
∞
lo c
(G) (defined in Equation (16)) does
not depend on the cut-off function φ and will be a polynomial in ξ, denoted simply
by σ
∇
(D).
Using the algebra morphism τ : U (A) → Diff(G) obtained from the universality
property of U (A), we have the following Poincar´e-Birkhoff-Witt type theorem for
Lie algebr oids. Note that both U(A) and Γ(Symm(A)) have natura l filtrations (see
(23)).

16 V. NISTOR, A. WE INSTEIN, AND PING XU
Theorem 3 (Poincar´e-Birkhoff-Witt). The composite map
U(A) ∋ D → σ
∇
(τ(D)) ∈ Γ(Symm(A))
is an isomorphism of filtered vector spaces. In particular τ : U(A) → Diff(G) is an
algebra isomorphism.
Proof. It follows from definitions that the map σ = σ
∇
◦ τ considered in the state-
ment maps U
n
(A) to ⊕
n
k=0
Γ(S
k
(A)) and hence it pres e rves the filtra tion. By abuse
of notation we shall still denote by σ the induced map U
n
(A)/U
n−1
(A) → Γ(S
n
(A)).
It is enough to prove that the map of graded spaces
σ :
M
U
n
(A)/U
n−1
(A) →
M
Γ(S
n
(A)) = Γ(Symm(A))
is an isomorphism. By Lemma 6 this map is onto. We now prove that it is one-to-
one.
The inclusion of C
∞
(M) in U (A) makes U(A) a C
∞
(M)–bimodule. The filtra-
tion U
n
(A) of U (A) cons ists of C
∞
(M)–bimodules. Moreover, since the graded
algebra ⊕U
n
(A)/U
n−1
(A) is commutative, the quotient U
n
(A)/U
n−1
(A) co ns ists
of central elements for this action (i.e. the left and right C
∞
(M)–module structure
coincide). It follows from the definition that the subspace Γ(A)
⊗n
of the universal
enveloping alge bra U = U(C
∞
(M) ⊕ Γ(A)) maps o nto U
n
(A)/U
n−1
(A). The pre-
vious discussion shows that this map descends to a map from the tensor product
Γ(A) ⊗
C
∞
(M)
· · · ⊗
C
∞
(M)
Γ(A) of C
∞
(M)-modules. By the commutativity of the
graded algebra of U(A) this further descends to a C
∞
(M)–linear surjective map
q : Γ(S
n
(A)) → U
n
(A)/U
n−1
(A).
The composition σ ◦ q : Γ(Symm(A)) −→ Γ(Symm(A)) is multiplicative since
both q and σ are multiplicative. Moreover σ ◦ q is the identity when restricted to
C
∞
(M) (the order 0 elements) and Γ(A) (the elements of order 1). Since these form
a system of generators of the commutative algebra Γ(Symm(A)) it follows that σ ◦q
is the identity. This completes the pr oof.
Remark The Poincar´e-Birkhoff-Witt theorem was proved in the algebraic con-
text by Rinehart [35] for (L, R)-algebras (an algebraic version of Lie algebroids). It
essentially stated that the associated graded algebra grU(A) = ⊕
n
U
n+1
(A)/U
n
(A)
is isomorphic to s ymmetric algebra S(Γ(A)) = Γ(Symm(A)). The role of the con-
nection ∇ o n A → M is to esta blish an explicit isomorphism σ
∇
◦ τ between U (A)
and Γ(Symm(A)).
We will now use the results of this and the pr e vious section to obtain an explicit
deformation quantization of A
∗
. In order to do that we nee d to establish the relation
between commutators and the Poisson bracket in our calculus.
For any x ∈ G
(0)
, T
∗
G
x
is a symplectic manifold, so T
∗
d
G
def
= ∪
x∈G
(0)
T
∗
G
x
is a
regular Poisson manifold with the leafwise symplectic str uctures. Now the Poisson
structure on A
∗
can be considered as being induced from that on T
∗
d
G. More pre-
cisely, let Φ : T
∗
d
G → A
∗
be the natural projection induced by the right translation,
used to define a map Φ
∗
: C
∞
(A
∗
(G)) → C
∞
(T
∗
d
G). We then have:
Lemma 7. The map Φ is a Poisson map.
Of course this lemma is really the definition of the Poisson structure on A
∗
. The
point is to show that the subspace Φ
∗
(C
∞
(A
∗
(G))) of C
∞
(T
∗
d
G) is closed under the
Poisson bracket.

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 17
Proof. It is enough to check that
Φ
∗
({f, g}) = {Φ
∗
(f), Φ
∗
(g)}(24)
where f and g are two smooth function on A
∗
with polynomial restrictions on each
fiber of A
∗
, that is for f and g in Γ(Symm(A)). Since the Poisson bracket is a
derivation in each var iable it is further enough to check this for f constant or f
linear in e ach fiber. If both f and g are constant in each fiber then both sides of
Equation (24) vanish. If f and g are of degree one in each fib er then they correspond
to sections X and Y of A, and their Poisson bracket will identify to [X, Y ] (so in
particular will also be of degree one in each fiber and this justifies the na me of
Lie-Poisson structure for this Poisson structure). For this situation the relatio n
(24) follows from the identification of Γ(A) with d–vertical right inva riant vector
fields on G and the fact that T
∗
d
G is a Lie-Poisson manifold itself. The remaining
case is treated similarly.
We shall use the fo llowing general fact on the principal symbols of commutators.
Proposition 5. When E is the trivial line bundle, the commutator [P, Q] satisfies
σ
m+m
′
−1
([P, Q]) = {σ
m
(P ), σ
m
′
(Q)}, where { , } is the Poisson structure on A
∗
(G).
Proof. The map Φ
∗
: C
∞
(A
∗
(G)) → C
∞
(T
∗
d
G) is a Poisson map according to Lemma
7. Since Φ
∗
(σ
m
(P )) = σ
m
(P
x
) on T
∗
G
x
the re sult follows from
Φ
∗
(σ
m+m
′
−1
([P, Q])) = σ
m+m
′
−1
([P
x
, Q
x
]) = {σ
m
(P
x
), σ
m
′
(Q
x
)}
= {Φ
∗
(σ
m
(P )), Φ
∗
(σ
m
′
(Q))} = Φ
∗
({σ
m
(P ), σ
m
′
(Q)}).
Since Φ
∗
is one-to-one this proves the last statement.
We now use the results above to construct deformatio n quantizations.
Deform the L ie bracket str uctur e on the Lie algebroid A on M to obtain a new
algebroid, the adiabatic algebroid A
t
associated to A, defined over M × [0, ∞) as
follows. As a bundle A
t
is the lift of the bundle A to M × [0, ∞). Regard the
sections X of A
t
as functions X : [0, ∞) → Γ(A), t → X
t
. Then the algebroid
structure is obtained by letting
[X, Y ]
t
= t[X
t
, Y
t
] and
ρ(X)
t
= tρ(X
t
)
so that ρ(X)f is the function whose restriction to {t} × M is tρ(X
t
)(f
t
) where
for any f ∈ C
∞
(M × [0, ∞)) we denote by f
t
∈ C
∞
(M) the restriction of f to
{t} × M ≡ M.
Observe that C
∞
([0, ∞)) ⊂ C
∞
(M ×[0, ∞)) is ac ted upon trivially by Γ(A
t
) and
hence will define a central subalgebra of the universal enveloping algebra U(A
t
) of
the adiabatic Lie algebroid A
t
. Denote by t ∈ C
∞
([0, ∞)) the identity function.
Theorem 4. The inverse limit proj lim U (A
t
)/t
n
U(A
t
) is a deformation quanti-
zation of Γ(Symm(A)), the algebra of polynomial functions on A
∗
. Therefore, it
induces a ∗-product on the Lie-Poisson space A
∗
in the sense of [3].
Proof. It follows from the PBW theorem for Lie algebroids (Theorem 3) that the
inverse limit proj lim U(A
t
)/t
n
U(A
t
) is isomor phic to Γ(Symm(A))[[t]] as a C[[t]]
module via the complete symbol map σ
∇
= σ
∇,φ
defined in Equation (16). Denote
by { , }
′
the Poisson bracket on A
∗
t
and identify C
∞
(A
∗
) with the subset of functions

18 V. NISTOR, A. WE INSTEIN, AND PING XU
on A
∗
t
that do not depend on t. Then {f, g}
′
= t{f, g} if f, g are smooth functions
on A
∗
t
and { , } is the Poisson bracket on A
∗
.
For any polynomial function f on A
∗
denote by q(f) ∈ U(A
t
) the element
with complete symbol σ
∇
(q(f))(ξ, t) = f (ξ) obtained, as an application of the
isomorphism in the PBW theorem for A
t
. (We treat τ as the identity, which
justifies replacing σ
∇
◦ τ with σ
∇
.) The proof will be complete if we check the
following quantization relatio n
q(f)q(g) − q(g)q(f ) = tq({f, g}) + t
2
h, h ∈ U (A
t
).(25)
It is a c tua lly enough to do s o for f and g among a set of generators of the alge bra
Γ(Symm(A)). Choose the set of generators to be the union o f C
∞
(M) and Γ(A).
Then Equation (25) will o bviously be satisfied for f and g in this generating set
(with no t
2
–term) in view of the definition o f the Lie bracket on A
t
.
Remark When M is a L ie group G and ∇ is the right invariant trivial connec-
tion making all right invariant vector fields parallel, this construction, restricted to
right invariant differential operato rs, reduces to the symmetrization correspondence
between U(g) and S(g) studied by Berezin [4] and Gutt [12]. See also Rieffel’s paper
[34]. On the other hand, when the Lie algebroid A is the ta ngent bundle Lie alge-
broid T P , this construction gives rise to a quantization for the canonical symplectic
structure on cotangent bundle T
∗
M.
The quantization of the Lie-Poisson structure on A
∗
was investigated by Lands-
man in terms of Jordan-Lie algebras [17]. His quantization axioms are closer to
those of Rieffel’s strict deformation quantization. It was conjectured in [17] that
the quantization o f A
∗
is related to the groupoid C
∗
-algebra o f the corresponding
groupoid G, and the transitive case was proved in [18].
4. Examples
As anticipated in the introduction, we recover many previously defined classes of
operators as pseudodifferential operators on gr oupoids. We begin by showing that
pseudodifferential operators on a manifold, in the classica l sense, are obtained as a
particular case of our cons truction. In this section we will consider only operators
with coefficients in the trivial line bundle E = C. We include the description of the
Lie algebroids associated to each example.
Denote by Ψ
m
prop
(M) the space of properly supported pseudodifferential opera-
tors on a smooth manifold M, and by Ψ
m
comp
(M) the subspace o f operators with
compactly supported Schwartz kernel, re garded as a distribution on M × M .
Example 1. L et M be a smooth manifold and G = M × M be the pair groupoid:
G
(1)
= M × M, G
(0)
= M , d(x, y) = y, r(x, y) = x, (x, y)(y, z) = (x, z). According
to the definition, a pseudodifferential operator P ∈ Ψ
m
(G) is a uniformly supported
invariant family of pseudodifferential operators P = (P
x
, x ∈ M ) on M × {x}. T he
action by right translation with g = (x, y) identifies M × {x} with M × {y}. After
we identify all fibers with M , the invariance condition r e ads P
x
= P
y
for a ll x, y
in M. This shows tha t the fa mily P = (P
x
)
x∈M
is co nstant, and hence reduces
to one operator P
0
on M. The family P = (P
x
)
x∈M
is uniformly supported if
and only if the distribution kernel of P
0
is compactly supported. The family P is
properly supported if and only if P
0
is properly supported. If M is not compact
then P = (P
x
)
x∈M
will not be compactly supported unless it vanis hes. We obtain
Ψ
m
(G) = Ψ
m
comp
(M).

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 19
In this case, the Lie algebroid A(G) is the tangent bundle T M.
Example 2. If G has only o ne unit, i.e. if G = G, a Lie group, then Ψ
m
(G) ≃
Ψ
m
prop
(G)
G
, the algebra of properly supported pseudodifferential operators on G,
invariant with respe c t to right translations. In this example, every invariant prop-
erly supp orted operator is also uniformly supported. Again, there are no nontrivial
compactly supported operators unless G is compact.
In this example A(G) is the Lie algebra of G.
We continue with some more elaborate examples.
Example 3. If G is the ho lonomy group oid
3
of a fo liation F on a smooth manifold
M, then Ψ
∞
(G) is the algebra of pseudodifferential operators along the leaves of
the foliation [6, 8, 27, 39]. Suppose fo r simplicity that the foliation is given by a
(right) locally free action of a Lie group G on a manifold M, and that the isotropy
representation of G
x
, the stabilizer of x, on N
x
, the normal space to the orbit
through x, is faithful. This is equivalent to the condition tha t the holonomy of
the leaf pass ing thro ugh x be isomorphic to the discrete group G
x
. Then the
holonomy groupoid of this foliation is the trans formation groupoid G
(1)
= X × G,
G
(0)
= X, d(x, g) = xg, r(x, g) = x and (x, g)(xg, g
′
) = (x, gg
′
). The algebra
Ψ
∞
(G; E) consists of families of pseudodifferential operators on G parametrized by
X, invariant with res pect to the diagonal action of G and with support contained
in a set of the form {(x, kg, xk, g)} ⊂ (X × G)
2
where g ∈ G is arbitrary but x ∈ X
and k ∈ G belong to compact sets that depend on the family P .
The Lie alg e broid A = A(G) is the integrable subbundle of T M corres ponding
to the foliation F.
Example 4. L et G b e the fundamental groupoid of a compact smooth manifold M
with fundamental group π
1
(M) = Γ. Recall that if we denote by
˜
M a universal
covering of M and let Γ act by covering transformatio ns, then G
(0)
=
˜
M/Γ = M,
G
(1)
= (
˜
M ×
˜
M)/Γ and d and r are the two projections. Each fiber G
x
can be
identified with
˜
M, uniquely up to the action of an element in Γ. L e t P = (P
x
, x ∈
M) be an invariant, uniformly supported, pseudodifferential operator on G. Then
each P
x
, x ∈ M is a pseudodifferential opera tor o n
˜
M. The invariance condition
applied to the elements g such that x = d(g) = r(g) implies that each operator P
x
is
invariant with respec t to the a c tion of Γ. This means that we can identify P
x
with an
operator on
˜
M and that the resulting operator does not depend on the identification
of G
x
with
˜
M. Then the invariance condition applied to an arbitrary arrow g ∈ G
(1)
gives that all operators P
x
acting on
˜
M coincide. We obtain Ψ
m
(G) ≃ Ψ
m
prop
(
˜
M)
Γ
,
the algebra of properly supported Γ-invariant pseudodifferential op e rators on the
universal covering
˜
M of M . An alterna tive definition of this algebra using crossed
products is given in [28]. See also [6].
The Lie algebroid is T M , as in the first example.
Example 5. L et Γ be a discrete group acting from the right by diffeomorphisms on
a smooth compact manifold M . Define G as follows, G
(0)
= M, G
(1)
= M × M × Γ
with d(x, y, γ) = yγ, r(x, y, γ) = x and (x, y, γ)(yγ, y
′
, γ
′
) = (x, y
′
γ
−1
, γγ
′
). Then
3
The holonomy groupoids of some foliations are non-Hausdorff manifolds. We believe that our
constructions will extend to this case with the use of the technique in [7] (page 564), where the
groupoid al gebra is generated by continuous functions supported on Hausdorff open sets.

20 V. NISTOR, A. WE INSTEIN, AND PING XU
Ψ
∞
(G) is the algebra genera ted by Γ and Ψ
∞
prop
(M) acting on C
∞
(M)⊗C[Γ], where
Γ acts diagonally, Ψ
∞
prop
(M) acts on the first variable, a nd C[Γ ] denotes the set of
finite sums of elements in Γ with complex coefficients. This algebra coincides with
the crossed product algebra Ψ
∞
prop
(M) ⋊ Γ = {
P
n
i=0
P
i
g
i
, P
i
∈ Ψ
∞
prop
(M), g
i
∈ Γ}.
The regularizing algebra Ψ
−∞
(G) is isomorphic to Ψ
−∞
prop
(M)⋊Γ ≃ Ψ
−∞
prop
(M)⊗C[Γ].
If we drop the condition that M b e co mpact we obtain Ψ
∞
(G) ≃ Ψ
∞
comp
(M) ⋊ Γ.
In general if a discrete group Γ acts on a groupoid G
0
then
Ψ
∞
prop
(G
0
⋊ Γ) ≃ Ψ
∞
prop
(G
0
) ⋊ Γ.
This construction does not change the Lie algebroid.
In the following example we realize the algebra of families of operators in Ψ
m
(G)
parametrized by a compact s pace B as the algebra of pseudodifferential operators
on the product groupoid G × B. This example shows that our class of operators on
groupoids is closed under formation of families of o perators.
Example 6. If B is a compact manifold with corner s, define G × B by (G × B)
(0)
=
G
(0)
× B, (G × B)
(1)
= G
(1)
× B with the structural maps preserving the B-
component. Then Ψ
m
(G × B) contains Ψ
m
(G) ⊗ C
∞
(B) as a dense subset in the
sense that Ψ
m−1
(G) ⊗ C
∞
(B) = Ψ
m
(G) ⊗ C
∞
(B) ∩ Ψ
m−1
(G × B) and Ψ
m
(G) ⊗
C
∞
(B)/Ψ
m−1
(G) ⊗ C
∞
(B) is dense in Ψ
m
(G × B)/Ψ
m−1
(G × B) in the correspond-
ing Frechet topology (defined by the isomorphism of Theorem 8). It follows that
Ψ
m
(G × B) consists of smooth families of operators in Ψ
m
(G) parametrized by
B, see [2], page 122 and after, where families of pseudodifferential operators are
discussed.
We obtain that A(G × B) is to pull back of A to G
(0)
× B.
The following example generalizes the tangent groupoid of Connes; here we
closely follow [6, II,5]. The groupoid defined below also a ppears in [36] and is
related to the notion of explosion of manifolds.
Example 7. T he adiabatic groupoid G
adb
associated to G is defined as follows. The
space of units is G
(0)
adb
= [0, ∞) × G
(0)
with the product manifold structure. The
set of arrows G
(1)
adb
is defined to be the disjoint union A(G) ∪ (0, ∞) × G
(1)
, and
d(t, g) = (t, d(g)), r(t, g) = (t, r(g)) if t > 0, d(v) = r(v) = (0, x) if v ∈ T
x
G
x
. The
composition is µ(γ, γ
′
) = (t, gg
′
) if γ = (t, g) and γ
′
= (t, g
′
) for t > 0 (necessarily
the same t!) and µ(v, v
′
) = v + v
′
if v, v
′
∈ T
x
G
x
.
The smooth structure on the set of arrows is the product structure for t > 0.
In order to define a coordinate chart a t a point v ∈ T
x
G
x
choose first a coordinate
system ψ : U = U
1
× U
2
→ G
(1)
, U
1
⊂ R
p
and U
2
⊂ R
n
being open sets containing
the origin, U
2
convex, with the following properties: ψ(0, 0) = x ∈ G
(0)
⊂ G
(1)
,
d(ψ(s, y
1
)) = d(ψ(s, y
2
)) = φ(s) and ψ(U )∩G
(0)
= ψ(U
1
×{0}). Here φ : U
1
→ G
(0)
is a coordinate chart of x in G
(0)
. We identify, using the differential D
2
ψ of the
map ψ, the vector space {s} × R
n
and the tangent space T
φ(s)
G
φ(s)
= A
φ(s)
(G).
We obtain then coordinate charts ψ
ǫ
: [0, ǫ) × U
1
× ǫ
−1
U
2
→ G
(1)
, ψ
ǫ
(0, s, y) =
(0, (D
2
ψ)(s, y)) ∈ T
φ(s)
G
φ(s)
= A
φ(s)
(G) and ψ
ǫ
(t, s, y) = (t, ψ(s, ty)) ∈ (0, 1) × G
(1)
.
For ǫ very small the range of ψ
ǫ
will contain v.
For G = M ×M a s in the first ex ample the groupoid G
adb
is the tangent groupoid
defined by Connes, and the a lgebra of pseudodifferential operators is the alge bra of
asymptotic pseudodifferential operators [3 8]. In general an operator P in Ψ
m
(G
adb
)

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 21
will restrict to an adiabatic family P = (P
t,x
, t > 0, x ∈ G
(0)
) which will have an
“adiabatic limit” at t = 0 given by the operator P at t = 0.
The Lie algebroid of G
adb
is the adia batic Lie algebroid associated to A(G),
A(G
adb
) = A(G)
t
(using the notation of Theorem 4). This gives a procedure for
integrating adia batic Lie algebroids. Using pseudodifferential operators on the adi-
abatic groupoid we obtain an explicit quantization of symbols on A
∗
generalizing
Theorem 4. The proof procee ds e xactly in the same way.
Theorem 5. The inverse limit pr oj lim Ψ
∞
(G
adb
)/t
n
Ψ
∞
(G
adb
) is a deformation
quantization of the commutative algebra S
∞
cl
(A
∗
(G)) of classical symbols.
The space S
∞
cl
(A
∗
(G)) appearing in the statement of the theorem above is the
union of all symbo l spaces S
m
cl
(A
∗
(G)) and is a commutative algebra under pointwise
multiplication.
Of course Theorem 4 provides us with a ∗-product whose multiplication is given
by differential opera tors, and hence this ∗-product extends to all smooth functions
(even to functions defined on open subsets). The usefulness of the theorem above
is that it gives in pr inciple a nonperturbative (i.e. not just formal) deformation
quantization, close in spirit to that of strict deformation quantization introduced
by Rieffel [33]. The algebras A
t
[33] turn out to be groupoid algebras.
Example 8. T his example provides a treatment in o ur settings of the b- and c-calculi
defined by Melrose [21, 22, 26, 25] on a manifold with bo unda ry M.
Define first a groupoid G
φ
(M) associated to M and an increasing diffeomorphism
φ : R → (0, ∞) as fo llows. If M = [0, ∞) the action by transla tion of R on
itself extends to an action on M fixing 0, not smooth in general, defined using
the isomorphism φ. Define G
φ
(M) to be the transformation groupoid associated
this action of R on M. If M = [0, 1) then G
φ
(M) is defined to be the reduction
G
φ
(M) = G
φ
([0, ∞)) ∩ d
−1
(M) ∩ r
−1
(M) of G
φ
([0, ∞)) to [0, 1).
Suppose next that M = ∂M × [0, 1). We then define G
φ
(M) = G
φ
([0, 1)) ×
(∂M × ∂M) where ∂M × ∂M is the equivalence groupo id of ∂M considered in the
first example. For an arbitrary manifold with boundary M write M = M
0
∪ U
where U = M \ ∂M and M
0
is diffeomorphic to ∂M × [0, 1). (Our construction
will depend on this diffeomorphism.) Then we define G
(0)
φ
(M) = M and G
(1)
φ
(M) =
G
(1)
φ
(M
0
) ∪ (U × U) with the induced operations.
If φ(t) = e
t
or φ(t) = −t
−1
(for t << 0) then G = G
φ
will be an almost differ-
entiable groupoid and we obtain Ψ
m
(G) ⊂ Ψ
b
(M) in the first case and Ψ
m
(G) ⊂
Ψ
c
(M) in the second case. The first gro upoid does not depend on any choices.
5. Distribution kernels
In this section we characterize the reduced (or convolution) distribution ker-
nels of operato rs in Ψ
m
(G; E) following [21] (see also [1 4]) as compactly supported
distributions on G, conormal to the set of units G
(0)
.
Denote by END
G
(E) the bundle Hom(d
∗
(E), r
∗
(E)) = r
∗
(E) ⊗ d
∗
(E)
′
on G
(1)
,
where V
′
denotes as usual the dual of the vector bundle V . Using the relations
d ◦ ι = r and r ◦ ι = d we see that END
G
(E) satisfies
ι
∗
(END
G
(E)) ≃ d
∗
(E) ⊗ r
∗
(E)
′
≃ END
G
(E)
′
.(26)

22 V. NISTOR, A. WE INSTEIN, AND PING XU
We define a convolution product on the space C
∞
c
(G
(1)
, END
G
(E) ⊗ d
∗
(D)) of
compactly supported smooth sections of the bundle END
G
(E) ⊗ d
∗
(D)) by the
formula
f
1
∗ f
2
(g) =
Z
{(h
1
,h
2
),h
1
h
2
=g}
f
1
(h
1
)f
2
(h
2
) .(27)
The multiplication on the right hand side is the composition of homomorphisms
giving a linear map
Hom(E
d(h
1
)
, E
r(h
1
)
) ⊗ Hom(E
d(h
2
)
, E
r(h
2
)
) ⊗ D
d(h
1
)
⊗ D
d(h
2
)
−→
Hom(E
d(g)
, E
r(g)
) ⊗ D
d(h
1
)
⊗ D
d(h
2
)
f
1
(h
1
) ⊗ f
2
(h
2
) −→ f
1
(h
1
)f
2
(h
2
),
defined since d(h
1
) = r(h
2
). To see that the integration is defined we parametrize
the set {(h
1
, h
2
), h
1
h
2
= g} as {(gh
−1
, h), h ∈ G
d(g)
} which shows that this set is
a smooth manifold, and notice that we can invariantly define the integration with
respect to h taking advantage of the 1-density factor D
d(h
1
)
= D
r(h)
= (Ω
d
)
h
. If we
choose a hermitian metric on D
−1/2
⊗ E, we obtain a conjugate–linear involution
(making C
∞
c
(G
(1)
, END
G
(E) ⊗ d
∗
(D)) into a ∗-algebra).
Consider an operator P = (P
x
, x ∈ G
(0)
) ∈ Ψ
−∞
(G; E) and let k
x
be the distri-
bution kernel of P
x
, a smoo th section k
x
∈ C
∞
(G
x
× G
x
; r
∗
1
(E) ⊗ r
∗
2
(E)
′
⊗ Ω
2
), using
the notation Ω
2
= p
∗
2
(Ω
d
) = r
∗
2
(D) of (11). We define the reduced distribution
kernel k
P
of the smoothing operator P by
k
P
(g) = k
d(g)
(g, d(g)) ∈ E
r(g)
⊗ E
′
d(g)
⊗ D
d(g)
.(28)
This definition will be later extended to all of Ψ
∞
(G; E).
The following theorem is one of the main reasons we consider uniformly sup-
ported operators.
Theorem 6. The reduced kernel map P → k
P
(28) defines an isomorphism of the
residual ideal Ψ
−∞
(G; E) with the convolution algebra C
∞
c
(G
(1)
, END
G
(E)⊗d
∗
(D)).
Proof. Let P and k
x
be as a bove. We know from Lemma 1 tha t the collec tion of
all sections k
x
defines a smooth se c tion of r
∗
1
(E) × r
∗
2
(E)
′
⊗ Ω
2
over the manifold
{(g
1
, g
2
), d(g
1
) = d(g
2
)}. The r e lation P
r(g)
U
g
= U
g
P
d(g)
gives the invariance re-
lation k
r(g)
(h
′
, h) = k
d(g)
(h
′
g, hg) ∈ E
r(h
′
)
⊗ E
′
r(h)
⊗ D
r(h)
for all arrows g ∈ G
(1)
,
and h, h
′
∈ G
r(g)
. It follows that k
d(h)
(h
′
, h) = k
r(h)
(h
′
h
−1
, r(h)) = k
P
(h
′
h
−1
).
The section k
P
is well defined, smooth and co mpletely determines all kernels k
x
and hence also the op e rator P . Moreover the section k
P
has compact support be-
cause supp(k
P
) = supp
µ
(P ) = µ ◦ (id × ι)

∪
x
supp(k
x
)

and the reduced suppor t
supp
µ
(P ) of P is compact s ince P is uniformly supported. The distribution kernel
k
P Q
x
of the product P
x
Q
x
of two operators P
x
, Q
x
∈ Ψ
−∞
(G
x
) is
k
P Q
x
(g, g
′′
) =
Z
G
x
k
P
x
(g, g
′
)k
Q
x
(g
′
, g
′′
)dg
′
where k
P
x
and k
Q
x
are the distribution kernels of P
x
and, respectively, Q
x
. From
this, taking into account the definitions o f k
P Q
, k
P
and k
Q
, we obtain
k
P Q
(g) =
Z
G
x
k
P
(gg
′
−1
)k
Q
(g
′
)dg
′
.

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 23
This means that k
P Q
= k
P
∗ k
Q
and hence the map P → k
P
establishes the desired
isomorphism.
We will now use duality to extend the definition of the r educed distribution kernel
to any o perator P ∈ Ψ
∞
(G; E). Let L = Ω
G
(0)
be the line bundle of 1 -densities on
G
(0)
and D = Ω
d
|
G
(0)
be the bundle of vertical 1-dens ities as above. Define
Ψ
−∞
(G; E)
L
= Ψ
−∞
(G; E) ⊗
C
∞
(G
(0)
)
C
∞
(G
(0)
, D
−1
⊗ L) ≃
Ψ
−∞
(G; E) ⊗
C
∞
(G
(0)
)
C
∞
(G
(0)
, D
−1
) ⊗
C
∞
(G
(0)
)
C
∞
(G
(0)
, L)
where the tensor products ar e defined using the inclusion C
∞
(G
(0)
) ⊂ Ψ
∞
(G; E).
We note that the bundle L plays an important role in connection w ith the modular
class of a groupoid [11], since it carrie s a natural representation of the groupoid.
The relation k
P f
(g) = k
P
(g)f(d(g)) for f ∈ C
∞
(G
(0)
) and P ∈ Ψ
−∞
(G; E) give
using Theorem 6 the isomorphism
Ψ
−∞
(G; E)
L
≃ C
∞
c
(G
(1)
, END
G
(E) ⊗ d
∗
(L)).(29)
The space Ψ
−∞
(G; E)
L
comes equipped with a natural linear functional T s uch
that if P
0
∈ Ψ
−∞
(G; E), ξ ∈ C
∞
(G
(0)
, D
−1
) and ν ∈ C
∞
(G
(0)
, L) then
T(P
0
⊗ ξ ⊗ ν) =
Z
G
(0)
tr(k
P
0
(x)ξ(x))dν(x)
defined by integra ting the function tr(k
P
0
(x)ξ(x)) with r e spect to the 1-density (i.e.
measure) ν. An operator P ∈ Ψ
m
(G; E) defines a continuous linear functional (i.e.
distribution) k
ι
P
: Ψ
−∞
(G; E)
L
→ C by the formula k
ι
P
(P
0
⊗ξ⊗ν) = T(P P
0
⊗ξ⊗ν).
It is easy to see using Equation (26) that the map f →
˜
f = f ◦ ι, ι(g) = g
−1
,
establishes isomor phisms
Φ : C
∞
c
(G
(1)
, END
G
(E) ⊗ d
∗
(L))
ι
∗
−→ C
∞
c
(G
(1)
, END
G
(E)
′
⊗ r
∗
(L)) ≃(30)
C
∞
c
(G
(1)
, (END
G
(E) ⊗ d
∗
(L))
′
⊗ d
∗
(L) ⊗ r
∗
(L)) ≃
C
∞
c
(G
(1)
, (END
G
(E) ⊗ d
∗
(L))
′
⊗ Ω
G
)
whose compo sition we denote by Φ, so that Φ(P
0
⊗ ξ ⊗ ν) = (k
P
0
ξν) ◦ ι = ι
∗
(k
P
0
ξν).
We obtain in this way from k
ι
P
a distribution k
P
∈ C
−∞
(G
(1)
; END
G
(E) ⊗ d
∗
(D))
defined by the fo rmula
hk
P
, f i = k
ι
P
(Φ
−1
(f)).(31)
An other way of writing the formula above is
hk
P
, ι
∗
(k
P
0
ξν)i = T(P P
0
⊗ ξ ⊗ ν) =
Z
G
(0)
tr(k
P P
0
(x)ξ(x))dν(x).(32)
Proposition 6. If P ∈ Ψ
−∞
(G; E) is a regularizing operator then the kernels k
P
defined in Equations (28) and (31) coincide.
Proof. To make a distinction for the purpose of this proof, denote by k
dist
P
the
distribution defined by (31). Let ν be a smooth section of L, ξ a smooth section of

24 V. NISTOR, A. WE INSTEIN, AND PING XU
D
−1
and P, P
0
∈ Ψ
−∞
(G; E). Using Equation (32) we obtain
hk
dist
P
, ι
∗
(k
P
0
ξν)i = T(P P
0
⊗ ξ ⊗ ν) =
Z
G
(0)
tr(k
P P
0
(x)ξ(x))dν(x) =
Z
G
(0)

Z
G
x
tr

k
P
(h
−1
)k
P
0
(h)ξ(x)


dν(x) = hk
P
◦ ι, k
P
0
ξνi = hk
P
, ι
∗
(k
P
0
ξν)i.
Definition 9. The distribution k
P
∈ C
−∞
(G
(1)
; END
G
(E)⊗d
∗
(D)), defined for any
operator P ∈ Ψ
m
(G; E) by Equation (31) will be called the reduced (or convolution)
distribution kernel of P , or simply the reduced kernel of P , and will be denoted k
P
.
We now r e late the action o f Ψ
∞
(G; E) by multiplication on Ψ
−∞
(G; E), respec-
tively on Ψ
−∞
(G; E)
L
, to that on C
∞
c
(G, r(E)).
Ψ
−∞
(G; E) ≃ C
∞
c
(G
(1)
; r
∗
(E)) ⊗
C
∞
(G
(0)
)
Γ(E
′
) ⊗
C
∞
(G
(0)
)
Γ(D)(33)
Ψ
−∞
(G; E)
L
≃ C
∞
c
(G
(1)
; r
∗
(E)) ⊗
C
∞
(G
(0)
)
Γ(E
′
) ⊗
C
∞
(G
(0)
)
Γ(L)(34)
such that the left action by multiplication of Ψ
∞
(G; E) on Ψ
−∞
(G; E) becomes
P (f ⊗η⊗ξ) = P f ⊗η⊗ξ where η is a smooth section of E
′
and ξ is a smooth section
of D or L. Moreover the kernel of P
0
= f ⊗ η ⊗ ξ is k
P
0
(g) = f(g) ⊗ η(d(g))ξ(d(g)).
Thus in order to define the distr ibution k
P
, for arbitrary P , it is enough to compute
T(P f
0
⊗ η ⊗ ν) where ν is a density.
Fix a unit x and choose a coordinate chart φ : U
0
→ U ⊂ G
(0)
where U
0
is
an open subset of R
k
containing 0, k = dim G
(0)
and φ(0) = x. By decreasing
U
0
if necessary we can assume that the tangent space T G
(0)
is trivialized over U.
Consider the diffeomorphism exp
∇
: V
0
→ V ⊂ G
(1)
associated to a right invariant
connection ∇ as in (14) and (15) where V
0
⊂ A(G) is an open neighborho od of the
zero section. It maps the zero section of A(G) to G
(0)
. Choose a connection on E
which lifts to an invariant connection on r
∗
(E). By decreasing V if necessary and
using the invariant connection on r
∗
(E) we obtain canonical trivializations of r
∗
(E)
on each fiber V ∩ G
x
. Denote by θ
h
: E
r(h)
⊗ E
′
d(h)
→ End(E
x
) the iso morphism
induced by the connection ∇
′
(defined using parallel transport along the geodesics
of ∇) where x = d(h) and h is in V . Decreasing further V and U we can assume
that E is trivialized over d
−1
(U) ∩ V and that φ a nd exp
∇
give a fiber preserving
diffeomorphism ψ : U
0
× W → d
−1
(U) ∩ V where W ⊂ R
n
is an open set, identified
with an open neighborhood of the zero section in T
x
G
(0)
. The diffeomorphism ψ
we have just co ns tructed satisfies d(ψ(s, y)) = φ(s). The maps ψ and θ
h
yield
isomorphisms
C
−∞
(d
−1
(U) ∩ V, END
G
(E) ⊗ d
∗
(D)) ≃(35)
C
−∞
(U
0
× W, ψ
∗
(END
G
(E) ⊗ d
∗
(D))) ≃ C
−∞
(U
0
× W, E
x
⊗ E
′
x
)
whose compos itio n is denoted Θ
ψ
.
Next theorem describes the reduced distribution kernels k
P
of operator s P in
Ψ
m
(G; E). We use the notation introduced above.
Theorem 7. For any operator P = (P
x
, x ∈ G
(0)
) ∈ Ψ
m
(G; E) the reduced distri-
bution kernel k
P
satisfies:

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 25
(i) If ψ : U
0
× W → V
1
⊂ V , W ⊂ R
n
open, is a diffeomorphism satisfying
ψ(s, 0) = d(ψ(s, y)), then there exists a symbol a
P
∈ S
m
cl
(U
0
× R
n
; End(E
x
)), such
that k
P
◦ ι = Θ
−1
ψ
(k) on V
1
, where k is the distribution
k(s, y) = (2π)
−n
Z
R
n
e
−iy·ζ
a
P
(s, ζ)dζ ∈ End(E
x
) ,
the integral being an oscillatory integral. Moreover, after suitable identifications,
a
P
is a representative of the principal symbol of P .
(ii) The singu lar support of k
P
is contained in G
(0)
.
(iii) The support of k
P
is compact, more precisely supp(k
P
) = supp
µ
(P ).
(iv) For every distribution k ∈ C
−∞
(G; E
0
), satisfying the three conditions above,
there exists P ∈ Ψ
m
(G; E) such that k = k
P
.
Note that k
P
◦ ι, ι
∗
(k
P
) and k
ι
P
all denote the same distribution.
Proof. Write φ(s) for ψ(s, 0) = d(ψ(s, y)). According to Definitions 6 and 7, there
exists a class ical symbol a ∈ S
m
cl
(U
0
× T
∗
W ; End(E
x
)) such that P
φ(s)
= a(s, y, D
y
)
(modulo regularizing operators) o n G
φ(s)
∩ V
1
≃ W .
Let P
0
∈ Ψ
−∞
(G; E), ξ ∈ C
∞
(G
(0)
, D
−1
) and ν ∈ C
∞
(G
(0)
, L). Assume, using
the isomorphisms (33) and (34), that P
0
= f
0
⊗ η where η ∈ Γ(E
′
) = C
∞
(G
(0)
, E
′
)
and f
0
∈ C
∞
(G, r
∗
(E) ⊗ Ω
d
), so that f
0
ξ is a section of C
∞
c
(G, r
∗
(E)). Then we
have
tr(k
P P
0
(x)ξ(x)) = η(P
x
(f
0
ξ|
G
x
)(x)).
Suppose f
0
is supported in V
1
, and denote by f
s
the section of E
x
that correspo nds
to f
0
ξ|
G
φ(s)
under the diffeomorphism G
φ(s)
∩ V
1
≃ {s} × W = W induced by ψ.
We then have
hι
∗
(k
P
), k
P
0
ξνi = hk
P
, ι
∗
(k
P
0
ξν)i = T(P P
0
⊗ ξ ⊗ ν) =
Z
G
(0)
tr(k
P P
0
(x)ξ(x))dν(x) =
Z
G
(0)
∩U
η

P
x
(f
0
ξ|
G
x
)(x)

dν(x) =
Z
U
0
η

Z
R
n
Z
W
e
−iy·ζ
a(s, 0, ζ)f
s
(y)dydζ

dν(s) =
Z
U
0
Z
R
n
Z
W
e
−iy·ζ
tr

a(s, 0, ζ)f
s
(y) ⊗ η

dydζdν(s) =
Z
U
0
×W
tr

f
s
(y) ⊗ η
Z
R
n
e
−iy·ζ
a(s, 0, ζ)dζ

dydν(s)
where the first integral is really a pairing between the distribution k obtained from
(i) for a
P
(s, ζ) = a(s, 0 , ζ), and the smooth section f
s
⊗ η ⊗ ν. Since End(E
x
)
is canonica lly its own dual this shows that the distribution k
P
is the conormal
distribution to G
(0)
given by (i).
To prove (iii) and (iv) observe that k
x
is the restriction to G
x
× G
x
of the distri-
bution µ
∗
1
(k
P
), where µ
1
(h
′
, h) = h
′
h
−1
and the distribution µ
∗
1
(k
P
) is defined by
hµ
∗
1
(k
P
), f i = hk
P
(g),
R
h
1
h
2
=g
f(h
1
, h
2
)i. Then we can define P
x
by its distribution
kernel k
x
. From (i) it follows that k
x
is conormal to the diagonal and hence P
x
is
a pseudodifferential operator.
In order to check (ii) fix g 6∈ G
(0)
and let ϕ be a smooth cut-off function, ϕ = 1
in a neighborhood of G
(0)
, ϕ = 0 in a neighborhood of g. Consider again the
distribution µ
∗
1
((1−ϕ)k
P
) = (1−ϕ◦µ
1
)µ
∗
1
(k
P
). Its restriction to G
x
is (1−ϕ◦µ
1
)k
x

26 V. NISTOR, A. WE INSTEIN, AND PING XU
which is smooth since the singular support of k
x
(= the distribution kernel of P
x
)
is contained in the diagonal of G
x
× G
x
, and 1 − ϕ ◦ µ
1
vanishes there. It follows
that µ
∗
1
((1 − ϕ)k
P
) is smooth and hence (1 − ϕ)k
P
is also smooth.
Corollary 1. The distribution k
P
is conormal at G
(0)
and smooth everywhere else.
In particular the wave-front set of k
P
is a subset of t he annihilator of T G
(0)
:
W F (k
P
) ⊂ (T G/T G
(0)
)
∗
⊂ T
∗
G|
G
(0)
.
Proof. This is a standard consequence of (i) and (ii) above, see [14], section 12.2.
We remark that (T G/T G
(0)
)
∗
is naturally identified with A
∗
(G). Denote by
S
m
c
(A
∗
(G); End(E)) ⊂ S
m
cl
(A
∗
(G); End(E)) the space of clas sical symbols with sup-
port in a set of the form π
−1
(K) where π : A
∗
(G) → G
(0)
is the projection and
K ⊂ G
(0)
is a compact subset.
Corollary 2. Let V be a neighborhood of G
(0)
in G. Then any P ∈ Ψ
m
(G; E) can
be written as P = P
1
+ P
2
where P
1
has reduced support supp
µ
(P
1
) contained in V
and P
2
∈ Ψ
−∞
(G; E).
Proof. Let φ be a smooth cut-off function, equal to 1 in a neighborhood of G
(0)
and
with support in V . Define P
2
∈ Ψ
−∞
(G; E) by k
P
2
= k
P
(1 − φ). This is possible
using Theorem 6 since by (ii) of the theorem above k
P
(1−φ) is a smooth compactly
supported section of an appropriate bundle. Then P
1
= P − P
2
and P
2
satisfy the
requirements of the statement.
Theorem 8. The principal symbol map σ
m
in Equation (17) is onto; hence it
establishes an isomorphism
Ψ
m
(G; E)/Ψ
m−1
(G; E) ≃ S
m
c
(A
∗
(G); End(E))/S
m−1
c
(A
∗
(G); End(E))
for any m.
Proof. We only need to prove that σ
m
is onto. If follows from the proof of Theorem
7 that σ
m
(P ) is the class of the symbol a
P
appearing in equation in (i). Given
a symbo l a ∈ S
m
c
(A
∗
(G); End(E)) the equation in (i) defines a distribution k
0
in
a small neighborhood of G
(0)
in G. Using a smo oth cut-o ff function we obtain a
distribution k on G that coincides with k in a neighborhood of G
(0)
and is smooth
outside G
(0)
. From (iv) we conclude that there exists an operator P with k
P
= k
which will then nec essarily satisfy σ
m
(P ) = a + S
m−1
c
(A
∗
(G); End(E)).
6. The action on sections of E
In this section we define a natural action of Ψ
m
(G; E) on sections of G
(0)
, thus
generalizing the action of classical pseudodiffere ntial operators on functions.
Let φ be a smooth section of E over G
(0)
. Define
˜
φ ∈ C
∞
(G
(1)
, r
∗
(E)) ,
˜
φ(g) = φ(r(g)).(36)
Lemma 8. If P = (P
x
, x ∈ G
(0)
) belongs to Ψ
∞
(G; E), then for any section φ in
C
∞
(G
(0)
, E) there exists a unique section ψ ∈ C
∞
(G
(0)
, E) such that P
˜
φ =
˜
ψ.

PSEUDODIFFERENTIAL OPERATORS ON GROUPOIDS 27
Proof. Observe first that given a section γ of r
∗
(E) over G
(1)
we can find a s e c tion
φ of E ove r G
(0)
such that f =
˜
φ if and only if f(g
′
g) = f(g
′
) for all g and g
′
, i.e.
if and only if
U
g
f
x
= f
y
, for all g, x, y such that x = d(g) and y = r(g).(37)
We then have
U
g
˜
φ
x
=
˜
φ
y
⇒ P
y
U
g
˜
φ
x
= P
y
˜
φ
y
⇒ U
g
P
x
˜
φ
x
= P
y
˜
φ
y
⇒ U
g
(P
˜
φ)
x
= (P
˜
φ)
y
and hence P
˜
φ satisfies (37). Thus we can find a section ψ of E over G
(0)
such that
P
˜
φ =
˜
ψ. Note that P
x
˜
φ
x
is defined since P
x
is properly supported.
The uniqueness of the section ψ follows from the fact that the map φ →
˜
φ is
one-to-one, and the smoothness of ψ follows from Lemma 2.
The representation given by the following theorem reduces to the trivial repre-
sentation in the case a group (se e also comments bellow).
Theorem 9. There exists a canonical representation π
0
of the algebra Ψ
∞
(G; E)
on C
∞
(G
(0)
, E) given by π
0
(P )φ = ψ where, using the notation of the previous
lemma, ψ is the unique section satisfying
˜
ψ = P
˜
φ. Moreover π
0
(P ) maps compactly
supported sections to compactly supported sections.
Proof. The fact that π
0
is well defined follows from the uniqueness part of the
previous lemma. It is clearly a representation. We only need to check that π
0
(P )
maps compactly supported sections to compactly supported sections. L e t L
1
⊂ G
(0)
be the support of φ, L
2
= supp(P ). Then the support of π
0
(P ) is contained in
L
2
L
1
.
Assume that E is a trivial line bundle. Then C
∞
(M) and Γ(A) act naturally on
C
∞
(M) and this action satisfies the re lations (22) which means that it gives rise
to a representation of U (A) = Diff(G) on C
∞
(M). Then π
0
is an extension of this
representation. If G = G is a group, then π
0
extends the trivial representation. In
order to generalize this fact to arbitrary representation of G we need the fo llowing
definition.
Definition 10. An equivariant bundle (V, ρ) on G
(0)
is a differentiable vector bun-
dle E together with a bundle isomorphism ρ : d
∗
(V ) −→ r
∗
(V ) satisfying ρ(gh) =
ρ(g)ρ(h).
An equivariant bundle is also called a representation of G. Given an equi-
variant bundle (V, ρ), we c an define a re presentation π
ρ
of the groupoid algebra
C
∞
c
(G, d
∗
(D)) on C
∞
c
(G
(0)
, V ) by the formula
(π
ρ
(f)φ)(x) =
Z
G
x
f(h
−1
)ρ(h
−1
)φ(r(h)).(38)
Note that the integration is defined and gives an element of V
x
since f(h
−1
)φ(r(h))
is in C
∞
c
(r
∗
(V ) ⊗ Ω
d
) and hence that f (h
−1
)ρ(h
−1
)φ(r(h)) is a smooth compac tly
supported section of d
∗
(V ) ⊗ Ω
d
.
The following proposition has no obvious analog in the classical theory b e c ause
the pair groupoid has no nontrivial representations. If one move s one step up
and considers the fundamental groupoid, nontrivial representations exist, and the
following lemma says that geometric operators (i.e. the ones that lift to the universal
covering s pace) act on sections of fla t bundles. A representation of a groupoid thus
resembles a flat bundle.

28 V. NISTOR, A. WE INSTEIN, AND PING XU
Proposition 7. Let (V, ρ) be an equivariant bundle and E an arbitrary bundle on
G
(0)
. There exists a natural morphism T
ρ
: Ψ
∞
(G; E) → Ψ
∞
(G; V ⊗ E) and hence
there exist a canonical action π
ρ
= π
0
◦ T
ρ
of Ψ
∞
(G) on C
∞
(G
(0)
, E ⊗ V ) and
C
∞
c
(G
(0)
, E ⊗ V ) which extends the representation defined in (38).
Proof. Let
W
ρ,x
: C
∞
c
(G
x
; r
∗
(E)) ⊗ V
x
= C
∞
c
(G
x
; r
∗
(E) ⊗ d
∗
(V )) → C
∞
c
(G
x
; r
∗
(E ⊗ V ))
be the iso morphism defined by ρ as in the definition 10. It is easy to see tha t this
gives an isomorphism W
ρ
: C
∞
c
(G; r
∗
(E) ⊗ d
∗
(V )) → C
∞
c
(G; r
∗
(E ⊗ V )). Define a n
operator on C
∞
c
(G
x
; r
∗
(E ⊗ V )) by the formula
(T
ρ
(P ))
x
= W
ρ,x
(P
x
⊗ id
V
x
)W
−1
ρ,x
.
The relation W
ρ,x
(U
g
⊗ ρ(g)) = U
g
W
ρ,x
shows that the family (T
ρ
(P ))
x
, x ∈ G
(0)
satisfies the invariance condition (T
ρ
(P ))
x
U
g
= U
g
(T
ρ
(P ))
y
, for d(g) = x and
r(g) = y. The uniform support condition is satisfied since supp(T
ρ
(P )) = supp
µ
(P ).
It follows that the family (T
ρ
(P ))
x
defines an operator T
ρ
(P ) in Ψ
∞
(G; V ⊗E). The
multiplicativity condition T
ρ
(P Q) = T
ρ
(P )T
ρ
(Q) follows from definition and hence
T
ρ
is a morphism.
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Department of Mathematics, Pennsylvania State University
E-mail address: nistor@math.psu.edu
Department of Mathematics, University of California, Berkeley
E-mail address: alanw@math.berkeley.edu
Department of Mathematics, Pennsylvania State University
E-mail address: ping@math.psu.edu