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\title{{\bf Tangential deformation quantization and polarized
symplectic groupoids}}
\author{Alan Weinstein\\Department of Mathematics\\
University of California\\
Berkeley, CA 94720 USA\\ \tt{alanw@math.berkeley.edu}
\thanks{Research
partially supported by NSF Grant DMS-96-25122.}}
\date{First finished draft, August 20, 1996. Final version appeared
as \\Weinstein, A., Tangential deformation quantization and polarized
symplectic groupoids, {\em Deformation Theory and Symplectic Geometry},
S. Gutt, J. Rawnsley, and D. Sternheimer, eds., {\em Mathematical Physics
Studies} {\bf 20}, Kluwer, Dordrecht, 1997, 301-314.}
\begin{document}
\maketitle
\begin{abstract}
We derive geometric analogues of theorems of Cahen-Gutt-Rawnsley and
Asin Lares concerning deformation quantizations of the dual of a Lie
algebra which are compatible with coadjoint orbits.
\end{abstract}
\tableofcontents
\section{Introduction}
The symmetrization isomorphism between the universal enveloping
algebra $U(\frakg)$ and the symmetric algebra $S(\frakg)$ can be used
\cite{be:remarks}
to produce a ``standard'' deformation quantization
\cite{bffls:deformation} for the Lie-Poisson
structure on the dual $\frakg^*$ of the Lie algebra $\frakg$. Since
the symplectic leaves of $\frakg^*$ are the coadjoint orbits, it is
interesting to know whether this, or any, deformation quantization of
$\frakg^*$ restricts to give deformation quantizations of the orbits.
Recent work of Cahen, Gutt, and Rawnsley \cite{ca-gu-ra:tangential} has shown
that the existence of any deformation quantization of $\frakg^*$ by
bidifferential operators compatible with the coadjoint orbit
decomposition implies that $\frakg$ must satsify a very strong
algebraic condition which excludes, for instance, all semisimple Lie
algebras. In addition, Asin Lares \cite{as:tangential} has shown that
the {\em standard} deformation quantization of $\frakg^*$ restricts to
a given coadjoint orbit $\calo$ if and only if the coadjoint isotropy
subalgebra $\frakg_{\mu}$ for any $\mu$ in $\calo$ is an ideal in
$\frakg$.
The proofs by Cahen-Gutt-Rawsley (CGR) and Asin Lares are very
computational and do not provide geometric connections between the
hypotheses and conclusions of the theorems.
The purpose of this note is to establish such connections by using the
idea of quantization via symplectic groupoids.
Although our results do not (yet?) lead to new proofs of the
theorems on deformation quantization, they are valuable for other reasons.
We have established another strong analogy between deformation
quantization and quantization by symplectic groupoids, adding to the
evidence in \cite{we:noncommutative} for a close relation between
these two kinds of quantization. In fact, we now propose to call
quantization by symplectic groupoids {\bf
geometric deformation quantization}, with the possible confusion with
``geometric quantization'' entirely intentional on our part, as we
shall explain below.
The main idea of our paper is to introduce a structure called a {\bf
strict geometric deformation quantization} and to show (Theorems
\ref{thm:asin} and \ref{thm:cgr} below) that the
precise conclusions of the Asin Lares and CGR theorems hold when
algebraic deformation quantizations are replaced by their geometric analogues.
In the CGR case, our conclusion is even stronger, and our work
suggests the following possible sharpening of the CGR result,
as well as a possible route to a more conceptual proof.
\begin{conj}
\label{conj:cgr}
$\frakg^*$ admits a deformation quantization by bidifferential
operators compatible with the coadjoint orbit decomposition if and
only if
there is a flat torsion-free affine connection on a neighborhood of
the identity in $G$ for which the induced connection on the vector
bundle $T^*G$ is compatible with the decomposition into the left (or
right) translates of coadjoint orbits.
\end{conj}
This is of course not the first instance where flat torsion free
connections have appeared in deformation quantization theory (e.g. in
quantizations of Moyal and Fedosov type), but their occurrence here
appears, at least on the surface, to be for quite different reasons.
We end this introduction with two questions.
1. It is known \cite{ar-ca-gu:deformations} that when $\frakg$ is
semisimple of compact type, there {\em is} an algebraic deformation
quantization of the polynomials on $\frakg^*$ which is compatible with
the coadjoint orbit decomposition. The operators defining this
quantization are {\em not} bidifferential. We also note that the idea
was raised already by Flato and Fronsdal and reported in
\cite{st:phase} that an orbit-compatible deformation quantization of
$\frakg^*$ might involve nonbidifferential, possibly {\em
pseudo}-differential operators. Is there something in our geometric
notion of deformation quantization which corresponds to
bidifferentiability?
2. Which Lie algebras admit a connection of the type described in
Conjecture \ref{conj:cgr}?
\noindent {\bf Acknowledgements}~~ I was stimulated to do the work described in
this paper by lectures of Simone Gutt and Santos Asin Lares at the
CIMPA Summer School on Quantization, Nice, July 1996, and by
conversations with Gutt, Asin Lares, Michel Cahen, and John Rawnsley
there and at the conference on Deformation Theory and Mathematical
Physics, Ascona, June 1996. I would like to think the sponsors and
organizers of both of those conferences for providing an excellent
working atmosphere.
\section{Geometric deformation quantization}
\label{sec:geometric}
The cotangent bundle $T^*P$ of a manifold $P$ is a ``geometric model''
for the space of functions (or half-densities) on $P$. In this model,
the multiplication of functions is represented by the operation of
addition in the fibres of $T^*P$. (For instance, the multiplication
of exponential functions corresponds to the addition of their
frequencies.)
Independently, Karasev \cite{ka:analogues},
Zakrzewski \cite{za:quantum}, and the author \cite
{co-da-we:groupoides}
\cite{we:noncommutative}
have developed a geometric
model for algebraic deformation quantization in which $T^*P$ with its
fibrewise addition operation is replaced by a general symplectic
groupoid whose base is $P$. Such a symplectic groupoid determines a
Poisson structure on $P$, just as an algebraic deformation
quantization does. Even though it is still unclear how to pass back
and forth between these
geometric objects and algebraic deformation quantizations,
we can consider them in their own right since
they have quite a rich structure themselves. For
precision, we make the following definitions.
The first definition corresponds to the case where we have a
1-parameter family of associative algebras deforming $\cinf (P)$, but
where the algebras are not necessarily isomorphic as vector spaces to
$\cinf (P)$. We will see in Section \ref{sec:explosion} below how
the deformation parameter appears.
\begin{dfn}
A {\bf geometric deformation quantization} of a Poisson manifold $P$
is a symplectic groupoid $\Gamma$ over $P$.
\end{dfn}
The next definition corresponds to the case where we have identified
all the deformed algebras with $\cinf (P)$ as vector spaces.
\begin{dfn}
A {\bf strict geometric deformation quantization} of a Poisson
manifold $P$ is a symplectic groupoid $\Gamma$ over $P$ together with
a symplectic isomorphism from $\Gamma$ to the cotangent bundle $T^*P$
which maps the identity section to the zero section and which
conjugates inversion in $\Gamma$ to multiplication by $-1$ in $T^*P$.
\end{dfn}
We call two manifolds {\bf globally transverse} if they are
transverse and intersect in a single point. The following proposition
is a simple globalization of Theorem 7.1 in \cite{we:symplectic}. We
omit the proof.
\begin{prop}
There is a one-to-one correspondence between strict geometric
deformation quantizations and pairs consisting of a geometric
deformation quantization $\Gamma$ and a real polarization of the symplectic
manifold $\Gamma$ with complete (in the natural flat affine connection)
simply connected leaves, each of which is globally transverse
to the zero section and is invariant under inversion.
\end{prop}
For some purposes, local quantizations are
sufficient.
\begin{dfn}
A {\bf [strict] local geometric deformation quantization} of a Poisson manifold
$P$ is a local symplectic groupoid over $P$ [together with a local
polarization which is transverse to the identity section].
\end{dfn}
A strict local geometric deformation quantization exists for any
Poisson manifold. Without the strictness condition, it is unique up
to isomorphism, but there are in general many ways in which the
polarization can relate to the groupoid structure. It is not clear
how these facts are connected to the existence and uniqueness of
algebraic deformation quantizations.
\section{Explosion of manifolds}
\label{sec:explosion}
To see how the objects defined above are really deformations, we
introduce the notion of ``explosion of manifolds''. (This term was
introduced in \cite{we:blowing}, but the idea appears already in
algebraic form in a 1966 paper by Gerstenhaber \cite{ge:deformation}. (See
\cite{we:blowing} for other references.)
\begin{dfn}
The {\bf explosion} $E(X,Y)$ of a differentiable manifold $X$ along a
submanifold $Y$ is, as a set, the product $X \times
\reals$ with the slice $X\times\{0\}$ replaced by
the normal bundle $N(X,Y)=T_Y X/TY$. A precise definition of the
differentiable structure on $E(X,Y)$ is given in \cite{we:blowing}.
\end{dfn}
If $(y,z)$ are local coordinates on $X$ with $Y$ defined by $z=0$,
then there is a corresponding set of local coordinates
$(y,z',\epsilon)$ on $E(X,Y)$, where $\epsilon$ is the
projection to $\reals$. Over the set where $\epsilon\neq 0$, the
projection from $E(X,Y)$ to $X$ is given in these coordinates by
$(y,z',\epsilon)\mapsto (y,\epsilon z)$. It follows that this
projection extends over $\epsilon = 0$ to a smooth map $p:E(X,Y)\to X$
when we map $N(X,Y)$ to $Y\subset X$ by the bundle projection.
If $X$ carries a symplectic structure $\omega$, we can pull $\omega$
back by the projection $p$ to a closed 2-form on $E(X,Y)$. We
consider this pullback instead as a family of 2-forms on the fibres of
the projection $\epsilon$. We denote this family by $p^{-1}\omega$.
If $Y$ is an isotropic submanifold of $X$, the quotient
$p^{-1}\omega/\epsilon$ is still a smooth family of closed forms. In
particular, if $Y$ is a lagrangian submanifold of $X$, then by using
local Darboux coordinates on $X$ which are adapted to $Y$ we can see
that $p^{-1}\omega/\epsilon$ is symplectic on every fibre of
$\epsilon$, and that the symplectic structure on the leaf $\epsilon=0$
is exactly that given by the usual identification of the normal bundle
of a lagrangian submanifold with its cotangent bundle. To
summarize:
\begin{prop}
If $Y$ is a lagrangian submanifold of the symplectic manifold $(X,\omega)$,
then the exploded manifold $E(X,Y)$ is a smooth family of symplectic
manifolds obtained from the product $X\times\reals$ carrying the form
$\omega / \epsilon$ on $X\times\{\epsilon\}$ by replacing the
slice $X\times \{0\}$ by the cotangent bundle $T^*Y$ with its
canonical symplectic structure.\footnote{The reader might want to
compare this result with Corollary 5.3 in \cite{we:blowing}, where a
manifold is exploded along a legendrian submanifold, the result being
a Poisson manifold in which the open dense subset $\epsilon\neq
0$ is a single symplectic leaf.}
\end{prop}
The case of most interest to us is that where $X$ is a (possibly
local) geometric deformation quantization of a Poisson manifold $P$ and $Y$ is
the identity section. In this case, the slice $\epsilon = 0$ in
$E(X,Y)$ is just
$T^*P$ with its canonical symplectic structure. Now suppose that the
deformation is strict. By choosing Darboux coordinates $(y,z)$ on $X$
in which $Y$ is defined by $z=0$ and the leaves of the polarization
are defined by $y=\mbox{constant}$, we get a family of polarizations
on the symplectic leaves of $E(X,Y)$ for which the one over
$\epsilon=0$ is just the polarization by the fibres of the cotangent
bundle.
We continue with a discussion of the effect of explosion on some other
geometric structures.
If $Y$ is a leaf of a foliation of $X$, then the foliations on the
fibres for $\epsilon\neq 0$ extend over $\epsilon=0$ to give a
foliation on all of $E(X,Y)$. (One sees this easily by using
coordinates $(y,z)$ in which the foliation is given by
$z=\mbox{constant}$.) The leaves of the induced foliation on the slice
$\epsilon = 0=N(X,Y)$ are the parallel sections for a flat linear
(``Bott'') connection on the normal bundle to the leaf. When the $X$ is
symplectic and the foliation is a polarization, then the induced
foliation on $T^*Y$ is lagrangian, so that the connection is not only flat
but torsion free as well. This is the flat affine structure on a leaf
of a polarization discovered by Libermann \cite{li:probleme} (also see
\cite{we:symplectic}).
Finally we investigate when happens to a Lie groupoid structure on $X$
when we explode along its identity section $Y$. The easiest case to
look at is that where $X$ is a group. Here, $N(X,Y)$ is the Lie
algebra, and $E(X,Y)$ is a smooth 1-parameter family of Lie groups,
with the group structure for $\epsilon=0$ being addition. For a
general Lie groupoid, it follows from our discussion of foliations
transverse to $Y$ that the source and target foliations of $X$ {\em
both} limit to the foliation by fibres in $N(X,Y)$, and then it is
easy to see that the limit of the groupoid multiplication is addition
in these fibres. In other words, $E(X,Y)$ is a 1-parameter family of
groupoids over $Y$ which are isomorphic to $X$ for $\epsilon \neq 0$, and for
$\epsilon = 0$ to the vector bundle $N(X,Y)$, which is just the Lie
algebroid of $X$. It can also be considered as a single groupoid over
$X\times \reals$.
Two special cases are worth noting here. First of all, if
$X=Y\times Y$ is the pair groupoid, then $N(X,Y)$ is the tangent
bundle $TY$, and $E(X,Y)$ is called the {\bf tangent groupoid} of
$Y$. We refer to \cite{co:noncommutative} for a discussion of the
central role played by this groupoid in index theory.
For our second example, we take $X$ to be a symplectic groupoid over a
Poisson manifold $Y$. In this case, $E(X,Y)$ becomes a 1-parameter
family of symplectic groupoids over $Y$ (or a single Poisson groupoid
over $Y\times\reals$) for which the fibres over $\epsilon\neq 0$ are
isomorphic to $X$ as groupoids, but with symplectic structure
multiplied by $1/\epsilon$, while the fibre over $\epsilon=0$ is the
vector bundle $T^*Y$ with its canonical symplectic structure. If $X$
is strict as a geometric deformation quantization of $Y$, than the
limit in $T^*Y$ of the polarization on $X$ is always the polarization
by fibres of the cotangent bundle. It is in this sense that our
geometric deformation quantizations are actually deformations.
\section{From geometric to algebraic deformations}
In this section we shall review the application (see
\cite{we:noncommutative}) of ideas from
geometric quantization and microlocal analysis for the construction of
noncommutative algebras from symplectic groupoids. In the next
sections, we will apply these ideas to obtain geometric analogues of
certain conditions which one might impose upon algebraic
deformations.
We being by appealing to the basic ``dictionary'' (see, for example
\cite{ba-we:lectures}) connecting
symplectic geometry and linear algebra (or analysis): thus, a
symplectic manifold $X$ should ``correspond'' to a complex vector space
$V(X)$, while a lagrangian submanifold $L$ of $X$ ``corresponds'' to an
element of $V(X)$. Taking the product of symplectic manifolds
produces the tensor product of their vector spaces, and multiplying a
symplectic structure by -1 (we denote the resulting symplectic
manifold by $\overline{X}$) produces the dual vector space. A
lagrangian submanifold $R$ of $\overline{X}\times Y$ (called a canonical
relation from $X$ to $Y$) then corresponds to a linear mapping $V(R)$ from
$V(X)$ to $V(Y)$. Quantization should be functorial in the sense that a
composition of canonical relations should go over to the composition
of the corresponding linear mappings.
Continuing with our hypothetical mode of argument, we note that the
multiplication in a symplectic groupoid $X$ is a canonical relation
from $X\times X$ to $X$, so it corresponds to a bilinear
multiplication on $V(X)$, whose associativity follows from that of the
groupoid if quantization is functorial. The identity section $Y$ of
$X$ corresponds to the unit element of the algebra $V(X)$, and
inversion in $X$ to an anti-automorphism. If we construct the
explosion $E(X,Y)$ of $X$ along $Y$, we get a 1-parameter deformation
of unital involutive algebras starting with the quantization of
$T^*Y$, which is just $\cinf (Y)$ with pointwise multiplication, unit
element 1, and complex conjugation.
To pass from the hypothetical to a more constructive mode, we
introduce ideas from geometric quantization and microlocal analysis.
According to geometric quantization, the way to construct $V(X)$ for a
symplectic manifold $X$ is to introduce a complex line bundle $F$ with
a connection whose curvature is the symplectic form on $X$, and then
to take the space of sections which are parallel in the direction of a
chosen polarization. To
quantize a lagrangian submanifold $L$ in $X$, one should first lift
$L$ to a parallel section $s$ of $F$ (which exists when $L$ satisfies
a quantization condition) and then take the parallel section of $F$
which agrees with $s$ along $L$. Of course, this procedure works
nicely only when $L$ is globally transverse to each leaf of the
polarization, or more generally when $L$ intersects
the leaves of the polarization with constant rank, in which case the
support of the quantization of $L$ should be the set of leaves which
intersect $L$.
The quantization of lagrangian submanifolds can be carried out more
systematically in the special case where the polarization is a
fibration by manifolds which are simply connected and are complete
with respect to their flat affine connection. If we assume the
existence of a ``base'' lagrangian submanifold $Y$ which is globally
transverse to each leaf of the polarization, then
$X$ is isomorphic to the cotangent bundle of $Y$ with the polarization
by fibres. In this case, lagrangian submanifolds of $T^*Y$ can be
quantized as asymptotic distributions on $Y$, i.e. distributions
depending on a parameter $\epsilon$. (See \cite{ba-we:lectures} or,
for more details, the last chapter of
\cite{gu-st:geometric}.) The asymptotic support of such a
distribution (i.e. the set where it is not rapidly decreasing as
$\epsilon\rightarrow 0$) is then contained in the projection of $L$
into $Y$.
When $X$ is a symplectic groupoid, applying the construction just
described to the graph of multiplication leads to an asymptotic
distribution on $Y\times Y \times Y$ which is a candidate for the
Schwartz kernel of a multiplication on $\cinf (Y)$. We discuss this
further in the next section.
\section{Deformations compatible with a submanifold}
In this section, we will define and analyze a property of geometric
deformation quantizations which is analogous to the compatibility of
algebraic deformations with a subset of the underlying manifold. We
begin with some algebraic and heuristic arguments, which
will lead us to a precise geometric condition.
If we have an associative multiplication $*$ on the space $\cinf(P)$
(which we may think of as a deformation of pointwise multiplication,
although this is not essential), the condition that it give rise by
restriction to a multiplication on the functions on a closed subset
$M\subset P$ is that the set of functions on $P$ which vanish on $P$
be a two-sided ideal in the algebra determined by $*$. If
$*_{\epsilon}$ depends on a parameter $\epsilon$, then we should
impose this condition for all $\epsilon$, or the consequent condition
on the coefficients of
all powers of $\epsilon$ in case the deformation is formal. When the
condition is satsified, we will say that the deformation is {\bf
compatible with} $M$.
Assume now that we have a (possibly local) strict geometric deformation
quantization $\Gamma$ of the Poisson manifold $P$, and identify $P$ with
the lagrangian submanifold of identity elements in $\Gamma$. We will
denote by $\pi$ the projection from $\Gamma$ to $P$ along the leaves of the
polarization.
Quantizing the canonical relation of groupoid multiplication in $\Gamma$
produces a product operation on $\cinf(P)$ which is defined by a
kernel distribution of the form $K(u',u'',u)~du'~du''$ on $P\times P\times P$;
i.e. the new multiplication of two functions $f$ and $g$ on $P$ is
given by an integral $$(f*g)(u)=\int_{P\times P}
f(u')g(u'')K(u',u'',u)~du'~ du''.$$ The asymptotic support of $K$ is
contained in the set of triples
$$(\pi(\gamma'),\pi(\gamma''),\pi(\gamma ' \gamma '')),$$ where
$(\gamma',\gamma'')$ ranges over all the composable pairs in $\Gamma\times
\Gamma$. Unless there are special cancellations in the quantization
construction, the asymptotic support will be in fact equal to this
set; we will take this condition as a working hypothesis.
\begin{lemma}
Let $M$ be a closed subset of $P$. If the set of functions which
vanish on $M$ forms an ideal with respect to the multiplication $*$
defined by the kernel $K$, then if $(u',u'',u)$ is in the support of
$K$ and $u$ is in $M$, $u'$ and $u''$ must also be
in
$M$.
\end{lemma}
\pf
Suppose that there is a triple $(u',u'',u)$ in the support of $K$
which does not satisfy the condition of the Lemma. Then we can find
bump functions $f$ and $g$ supported near $u'$ and $u''$ respectively
such that $(f*g)(u)$ is not zero.
\qed
The results above provide a heuristic justification for the following
definition.
\begin{dfn}
\label{dfn:compatible}
Let $\Gamma$ be a (possibly local) strict geometric deformation quantization of a
Poisson manifold $P$, and let $M$ be a subset of $P$. Let
$\pi:\Gamma\to P$ be the projection along the leaves of the
polarization We say
that the deformation is {\bf compatible with} $M$ if for any two
composible groupoid elements $\gamma'$ and $\gamma''$,
$\pi(\gamma' \gamma'') \in M $ implies that $\pi(\gamma')\in M$ and
$\pi(\gamma'')\in M$.
\end{dfn}
Notice that the compatibility condition on $M$ in the preceding
definition is really just a condition on the subset $\pi\inv (M)$ of
the groupoid $\Gamma$, namely that a product lies in $\pi\inv (M)$ if and
only if both of its factors do. We will call such a subset of a
groupoid {\bf
saturated}. (Equivalently, the
complement of the set is absorbing under multiplication.)
It turns out that saturated subsets have a simple
characterization. We recall some terminology. Let $\Gamma$ be a groupoid
over $P$ with source and target
maps $\beta$ and $\alpha$, so that a product $\gamma' \gamma''$ is
defined whenever $\beta(\gamma')=\alpha(\gamma'')$. Two elements
of $P$ are called equivalent if some element of $\Gamma$ has them as
its source and target. This is indeed an equivalence relation, and
the equivalence classes are called {\bf orbits} of $\Gamma$ in $P$.
As usual, we identify $P$ with the set of identity
elements in $\Gamma$, so that $\alpha$ and $\beta$ are retractions.
\begin{prop}
\label{prop:compatible}
Let $\Gamma$ be a groupoid over $P$ with source and target mappings
$\beta$ and $\alpha$. A
subset $C$ of $\Gamma$ is saturated if
and only if it is of the form $\alpha\inv (U)$, where $U$ is a union of
orbits in $P$ (in which case $C=\beta\inv(U)$ as well).
\end{prop}
\pf
Let $C=\alpha\inv (U)=\beta\inv (U)$, where $U$ is a union of
orbits in $P$. If $\gamma' \gamma'' \in C$, then $\alpha
(\gamma')=\alpha(\gamma' \gamma'' )$ and
$\beta(\gamma'')=\beta(\gamma' \gamma'')$ belong to $U$. This implies
that $\gamma'$ and $\gamma ''$ belong to $C$, so $C$ must be
saturated.
Conversely, suppose that $C$ is saturated. For
any $\gamma \in C$, the identity $\alpha(\gamma) \gamma = \gamma =
\gamma \beta(\gamma)$ implies that $C$ is closed under the source and
target mappings. Let $U$ be the image of $C$ under either of these
mappings, which also must equal the intersection of $C$ with $P$. Now
the identities $\alpha(\gamma)=\gamma \gamma \inv$ and $\beta(\gamma)=
\gamma\inv \gamma$ imply that $C$ is contained in, and hence equal to,
$\alpha\inv(U)$ and $\beta\inv(U)$. Finally, the fact that
$C=\alpha\inv (U)$ is
closed under $\beta$ implies that $U$ is a union of orbits.
\qed
The following Corollary follows immediately from Definition
\ref{dfn:compatible}, Proposition \ref{prop:compatible}, and the fact
that the mapping $\pi$ associated with a polarization is a
retraction.
\begin{cor}
\label{cor:compatible}
A (possibly local) strict geometric deformation quantization of the
Poisson manifold $P$ is compatible with a subset $M$ of $P$ if and
only if $M$
is a union of
symplectic leaves in $P$ and $\pi\inv(M)=\alpha\inv (M)=\beta\inv (M)$.
\end{cor}
\noindent
{\bf Remarks.} If $\Gamma$ is $\alpha$-connected, the orbits in $P$ are
just the symplectic leaves. Thus, $M$ must be a union of symplectic
leaves. The inverse image under $\alpha$ of a symplectic leaf is
coisotropic; hence, so is $C=\alpha\inv (M)=\beta\inv
(M)$. Since, for $u$ in $M$, $\pi\inv(u)$ is lagrangian
and contained in the coistropic submanifold $C$,
$\pi\inv(u)$ must contain the characteristic submanifold of $C$
which passes through $u$. If $C$ is a single orbit, this
characteristic submanifold is just the isotropy subgroup $\Gamma_u$ of $u$.
\bigskip
To close this section, we consider the case where a strict geometric
deformation quantization is
compatible with the decomposition of the Poisson manifold $P$ into its
symplectic leaves, i.e. the quantization is compatible with each leaf
separately. By analogy with the algebraic case, we will such a
quantization {\bf tangential}. For simplicity, we will assume that $\Gamma$ is
$\alpha$-connected, so that the leaves are the orbits of $\Gamma$. The
inverse images of the leaves under $\alpha$ or $\beta$ form a
decomposition of $\Gamma$ into coisotropic subgroupoids which we will call
the {\bf fundamental decomposition} of the groupoid $\Gamma$.
\begin{cor}
\label{cor:decomposition}
A (possibly local) strict deformation quantization $\Gamma$ of the
Poisson manifold $P$ is tangential if and only if the polarization of
$\Gamma$ is a
refinement of the fundamental decomposition.
\end{cor}
\section{The Lie-Poisson case}
We will now apply the results of the previous situation to the
situation where $P=\frakg ^*$ is the dual of the Lie algebra of the
Lie group $G$, in order to obtain geometric analogues of the theorems
of Asin Lares and Cahen-Gutt-Rawnsley on algebraic deformation
quantizations compatible with coadjoint orbits.
The cotangent bundle $\Gamma=T^*G$ is the underlying symplectic
manifold of a geometric deformation quantization of $\frakg^*$, with a
groupoid structure for which the source and target maps are the left
and right translations to the cotangent space $T_e^*G$ at the group
identity, this cotangent space being the manifold of identities in
$\Gamma$. (See \cite{co-da-we:groupoides} or
the exposition in \cite{va:lectures}.)
The cotangent bundle of course carries a natural polarization by its
fibres, but this does not constitute a strict geometric deformation
quantization because the leaves are not transverse to the groupoid
identity manifold. In fact, when the method of geometric quantization
is applied to this polarization, the resulting algebra is the
convolution algebra $A(G)$ of densities on the group $G$.
It seems to have been originally an idea of Berezin \cite{be:remarks}
that the group convolution algebra should be connected with the
functions on $\frakg^*$ by the composition of two operations. The first
of these operations is $\exp^*$, pullback by the exponential map of
$G$, and the second is the Fourier transform from densities on
the Lie algebra $\frakg$ to functions on $\frakg^*$. Berezin showed
that this correspondence applied to the subalgebra of $A(G)$
consisting of distribution densities supported at $e$ produces the standard
``symmetrization'' identification between the universal enveloping
algebra $U(\frakg)$ and the symmetric algebra $S(\frakg)$ of
polynomial functions on $\frakg^*$. He expressed the Schwartz kernel
of the new multiplication on $\frakg^*$ in terms of the
Campbell-Hausdorff mapping $\frakg\times\frakg \to \frakg$, and he
observed that the
commutator in $U(\frakg)$ is a perturbation of the
Lie-Poisson bracket on $\frakg^*$. (No deformation parameter appears
in Berezin's work.) The symmetrization
correspondence was later shown by Gutt \cite{gu:explicit} to give a formal
algebraic deformation quantization, while
Rieffel used the Fourier transform picture to show that one had a
strict deformation quantization in the sense of $C^*$-algebras.
It is well known in geometric quantization (see, for example
\cite{wo:geometric}) that the Fourier transform between functions (or
densities) on a vector space $V$ and its dual $V^*$ is represented
geometrically by the pairing between the spaces obtained from the
``horizontal'' and ``vertical'' polarizations of the symplectic manifold
$V \times V^*$. This
leads us to make the following construction. By the exponential
mapping (restricted if necessary to an open neighborhood of $0$ in
$\frakg$ on which it is an embedding, although we will seldom mention
this restriction explicitly from now on), we identify $T^*G$ with
$T^*\frakg$ and hence with $\frakg \times \frakg^*$. The latter
symplectic space has a ``horizontal'' polarization whose leaves are
globally transverse to $\{0\}\times \frakg^*$. Under our exponential
identification, this gives a polarization of $T^*G$ whose leaves are
globally transverse to $T_e^* G$, which we will call the {\bf exponential
polarization}. The symplectic groupoid $T^*G$ with this polarization
constitutes a (possibly local) strict geometric deformation
quantization of $\frakg^*$ which will may call the {\bf
exponential quantization}.
Our version of the algebraic theorem of Asin Lares
\cite{as:tangential} now goes as follows.
\begin{thm}
\label{thm:asin}
Let $\calo$ be a coadjoint orbit in $\frakg^*$. Then the following
three conditions are equivalent: (i) the exponential
quantization is compatible with $\calo$; (ii) $\calo$ is flat--i.e. it
is an open subset of an affine subspace of $\frakg^*$; (iii) the
isotropy algebra $\frakg _u$ of any $u\in \calo$ is an ideal in $\frakg$.
\end{thm}
\pf
We begin with a proof of the equivalence of (ii) and (iii), close to
the one for the nilpotent case which can be found in
\cite{co-gr:representations}. (See Theorem 3.2.3 in this book, which
also exhibits the importance of flat orbits in
representation theory.)
Recall that the tangent space to a
coadjoint orbit $\calo$ at a point $u$ is parallel to the
annihilator in $\frakg^*$ of the isotropy algebra $\frakg _u$. The
orbit is flat if and only if its tangent spaces are parallel to one
another, which is
the case just when the isotropy is independent of $u$. The isotropy
subgroups of the points of $\calo$ are the conjugates of that of $u$,
so the isotropy subalgebras are all the same when the isotropy
subgroup is normal, i.e. when $\frakg _u$ is an ideal.
The rest of the proof will based on the formula for the derivative of
the exponential mapping of a Lie group:
$$T_a \exp = T_{e}\ell_{\exp a} \circ \frac {1-e^{\ad_a}}{\ad_a},$$
which, following a suggstion of Duistermaat, we prefer to write in the
integral form
$$T_a \exp = T_{e}\ell_{\exp a} \circ \int_0^1 e^{-s \ad_a}~ds .$$
In both of these formulas, $\ell_g$ denotes left translation by the
group element $g$.
We will use this formula to compute the mapping $\pi:T^*G\to \frakg^*$
associated with the exponential polarization, using the identification of
$T^*G$ with $G \times \frakg^*$ given by left translations. Under
this identification, the mappings $\alpha$ and $\beta$ from $G \times
\frakg^*$ to $\frakg ^*$
are given by
the projection of $G \times \frakg^*$ and the coadjoint action
respectively.
For $g=\exp a \in G$ and $\mu\in \frakg^*$, we have
\begin{eqnarray}
\pi(g,\mu) & = & (T_g \ell_g^{-1})^* \mu \circ T_a \exp \nonumber \\
& = & \mu \circ \ T_g \ell_g^{-1}\circ T_e \ell_g
\circ \int_0^1 e^{-s \ad_a}~ds \nonumber \\
& = & \mu \circ \int_0^1 \Ad_{\exp (-sa)}~ds \nonumber \\
& = & \int_0^1 \Ad^*_{\exp (-sa)} \mu~ds.
\end{eqnarray}
Suppose now that the orbit $\calo$ is flat, and suppose that
$(g,\mu)\in \alpha^{-1}(\calo)$, i.e. $\mu \in \calo$. For $g$ near
$e$, $\pi(g,\mu)$ is a convex combination of elements of $\calo$ near
$\mu$ and hence must belong to the flat orbit $\calo$; thus we have
shown that $\alpha^{-1}(\calo)\subseteq \pi^{-1}(\calo)$. Since
$\alpha^{-1}(\calo)$ and $\pi^{-1}(\calo)$ are manifolds of the same
dimension, both containing $\calo$, they must be equal near $\calo$,
i.e. near the identity of $G$, so the quantization is compatible with
$\calo$.
Conversely, suppose that the quantization is compatible with $\calo$.
Fix $\mu\in \calo$ for the moment and consider the map
$\phi:g\mapsto\pi(g,\mu)$. Since $(g,\mu)=\in\alpha^{1}(\calo)$, by
compatibility we have $(g,\mu)\in \pi^{-1}(\calo)$, so $\phi(g,mu)$ is
an element of $\calo$ which is also in the convex hull of $\calo$. On
the other hand, it is easy to check (note that $\pi(\exp a,\mu)$ is an
average of points on a path from $\mu$ to $\Ad^*_{\exp(-a)}\mu$) that
$T_e\phi :\frakg\to\frakg^*$ is the map $b\mapsto -\frac{1}{2} \Ad^*_b
\mu$, whose image is $T_{\mu}\calo$. Thus $\phi$ maps a neighborhood
of $e$ in $G$ to a neighborhood of $\mu$ in $\calo$. It follows that
$\mu$ is locally convex near $\mu$, and since $\mu$ was arbitrary,
$\calo$ must be flat.
\qed
We turn now to the theorem of Cahen-Gutt-Rawnsley
\cite{ca-gu-ra:tangential}. Our geometric version is Corollary
\ref{cor:cgr} below, which will follow from the more general theorem
which follows, and which is the inspiration for Conjecture
\ref{conj:cgr}.
\begin{thm}
\label{thm:cgr}
$\frakg^*$ admits a local tangential strict geometric deformation
quantization if and only if a neighborhood of the identity in the
corresponding Lie group $G$ admits a flat torsion-free affine
connection for which the associated connection on $T^*G$ is compatible
with the decomposition into left (or right) translates of coadjoint orbits.
\end{thm}
\pf
Any symplectic groupoid for $\frakg^*$ is a neighborhood of the
identity section in $T^*G$. By Corollary \ref{cor:decomposition}, the
polarization of a tangential strict deformation quantization
is a refinement of the fundamental decomposition, which
is given by the left or right translates of coadjoint orbits. In
particular, the zero section of $T^*G$ corresponds to the coadjoint
orbit $\{0\}$, and so it must be a leaf of the polarization. If we
now explode $T^*G$ around this leaf, as in Section
\ref{sec:geometric}, we obtain a flat torsion free affine connection
on a neighborhood of the identity in $G$ for which the associated
connection on $T^*G$ is a refinement of the
exploded version of the fundamental decomposition. But since
multiplication by constants on $T^*G$ leaves the fundamental
decomposition invariant, the exploded decomposition is the same as the
original one. Thus we have a connection of the type described in
the statement of the theorem.
For the converse, we simply take the parallel sections for the given
connection as a polarization of the fundamental groupoid.
\qed
The conclusion of the following Corollary is precisely that of the
main theorem in \cite{ca-gu-ra:tangential}.
\begin{cor}
\label{cor:cgr}
If $\frakg^*$ admits a local tangential strict geometric deformation
quantization, and if $B$ is an $\ad^*$-invariant bilinear form
on $\frakg^*$, then the image of the associated linear map
$\tilde{B}:\frakg^*\to \frakg$ is a 2-step nilpotent ideal in
$\frakg$.
\end{cor}
\pf The fact that $\tilde{B}(\frakg)$ is an ideal is just a
consequence of $\ad^*$-invariance and has nothing to do with
quantization. (We give a proof below.) To prove that it is 2-step
nilpotent, we begin with the case where $B$ is nondegenerate. The
pullback $\overline{B}$ of $B$ to $T^*G$ by left or right translations is the
kinetic energy function of a bi-invariant pseudoriemannian metric on
$G$. Since $B$ is invariant, it is constant on coadjoint orbits;
hence $\overline{B}$ is constant on each of the submanifolds in the
fundamental decomposition. Since the connection produced in Theorem
\ref{thm:cgr} leaves the fundamental decomposition invariant, it
leaves $\overline{B}$ invariant. Being torsion free, it must be the
Levi-Civita connection for this metric. But we also know that this
connection is flat. Now there is a simple formula for the riemannian
curvature of a bi-invariant metric on a Lie group. (See for example
\cite{mi:morse}, p. 113; positive definiteness of the metric
is unnecessary.)
$$R(X,Y)Z = \frac{1}{4} [[X,Y],Z]$$
for left invariant vector fields $X$, $Y$, and $Z$.
The vanishing of $R$ means that all triple brackets are zero, which is
just the condition that $\frakg$ be 2-step nilpotent.
In the nondegenerate case, $\frakg = \tilde{B}(\frakg^*)$, and our
proof is complete. For the general case, we note first that, since
$\tilde{B}$ is $\mbox{ad}^*$-invariant, so is its kernel, which is the
annhilator of $\frakh=\tilde{B}(\frakg)$. It follows that
$\frakh$ is ad-invariant, and hence an ideal. Since the
connection preserves $\overline{B}$, it must preserve the left (which
are the same as the right) translates of $\frakh$, and therefore the
corresponding subgroup $H\subset G$ is totally geodesic. It follows
that the connection restricts to a flat, torsion free metric
connection on $H$, from which it follows as above that $\frakh$ is
2-step nilpotent.
\qed
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