The Weil representation, Maslov index and Theta series

342 Pages·1980·9.027 MB·English

by  Gérard Lion, Michèle Vergne (auth.)

Progress 111I'/lathematics 6
Editedby
J. Coatesand
S.Helgason
Gerard Lion
Michele Vergne
TheWeil .
representation,
Maslov index and
Theta series
SpringerScience+BusinessMedia,LLC
MicheleVergne
DepartmentofMathematics
MassachusettsInstituteofTechnology
Cambridge.MA02139
USA
GerardLion
UniverstedeParis
XU.E.R.deSciencesEconomiques
92001Nanterre
France
LibraryofCongressCataloginginPublicationData
Lion.Gerard.1949-
TheWellrepresentation.Maslovindex,andthetaseries.
(Progressinmathematics;6)
Includesbibliographicalreferences.
1.Symplecticgroups.2.Representationsofgroups.3.
Series.Theta.4.Forms,Modular.5.Liftingtheory.I.Vergne,
Micheel,jointII.Title.III.Title:Maslovindex,andthetaseries.IV
Series:Progressinmathematics(Cambridge);6.
OA171.L765 512'.22 80-15942
CIP-KurztitelaufnahmederDeutschenBibliothek
Uon,Gerard:
TheWei!representation,Maslovindexandtheta
series/GerardLion:MicheleVergne.- Boston,
Basel,Stuttgart:Birkhauser.1980.
(Progressinmathematics:6)
ISBN 978-0-8176-3007-2 ISBN 978-1-4684-9154-8 (eBook)
DOI 10.1007/978-1-4684-9154-8
NE:Vergne,Michele:
Allrightsreserved.Nopart01thispublicationmaybereproduced.
storedInaretrievalsystem,ortransmitted,Inanyformorbyany
means.electronic.mechanical,onotocoovmo.recordingorornerwrse.
Withoutpriorpermissionofthecopyrightowper
©SpringerScience+BusinessMediaNewYork.!.980
OriginallypublishedbyBirkhauserBostonin1980.
>-30<t;orl1> 3:l1>i3...0 '<
0....,
3:...'""'0'"<l1>...(JQ::>l1>
Preface.
In these notes, the Shale-Weil representation of the
symplectic group is discussed, as well as some of its applications
to number theory.
The monograph is composeu of two parts:
In Part I, written by Gerard Lion and Michele Vergne, we
introduce the Shale-WeiI representation and establish a relation
between its cocycle and the Maslov index.
In Part II, written by Michele Vergne, applications of
q-series to liftings of modular forms are given.
Although the results of the first part enlightens the
exposition of the classical transformation properties of
A-functions, a reader mainly interested by these applications
to liftings could read directly the second part with an eventual
glance to earlier paragraphs. The two parts have separate
introductions and bibliographical notes.
The authors
Table of Contents
Part I: The Shale-Weil representation and the Maslov index,
by G€rar d Lion and Michele Vergne.
1.0. 2:ntroduction.
1.1. Symplectic vector spaces and the Heisenberg Lie
algebra.
1.2. The Heisenberg group and the Schrodinger
representation.
1.3. The Weyl transformand the Stone-Von Neumann
theorem.
1.4. Fourier transforms and intertwining operators.
1.5. Maslov index.
1.6. The cocycle of the Shale-Weil representation and
the Maslov index.
1.7. Oriented Lagrangian planes and the metaplectic
group.
1.8. The universal covering group of SL(2,R).
1.9. The universal covering group of the symplectic
group.
A. Appendix: Ageneralization of Maslov index to
local fields.
B. Bibliographical notes.
Part II. Theta-series and applications, by Michele Vergne.
2.0. Introduction.
2.1. Lattices and representations of the Heisenberg group.
2.2. The multiplier of ~.
2·3· Modular forms on the upper half plane.
2.4. Modular forms of weight 1/2.
2.5. The Shale-Weil representation associated to a
quadratic form.
2.6. ~-series associated to quadratic forms.
2.7. The Shimura correspondence.
2.8. Zagier modular forms and the Doi-Naganuma
correspondence.
2.9. Cohen correspondence.
B. Bibliographical notes.
Part I: The Shale-Well representation
and the Maslov Index.
by
Gerard Lion and Michele Vergne
4
1.0. Introduction:
Relations between symplectic geometry, Maslov index,
representations of the Heisenberg group and the Shale-Weil
representation of the symplectic group are discussed in this
chapter. We first give in 1.1 the basic definitions and
properties of symplectic vector spaces, Lagrangian subspaces,
the Heisenberg Lie algebra and the action of the symplectic
group of those objects.
The Schr8dinger representation of the Heisenberg group N
associated to a Lagrangian plane £ is constructed in 1.2.
We prove in 1.3 the Stone-von Neumann theoremwhich asserts
that all unitary representations of N whose restriction to the
center of N acts by the same non-trivial character are essentiall:
the same: two such irreducible representations are equivalent.
Although this uniqueness theorem underlies the construction
of the Shale-Weil representation R, wewill give however a
direct construction of R, independent of the proof of this
theorem. Thus results of Section 1.3 will not be needed sub
sequently.
The Schr8dinger representations W£ and W£. of the
Heisenberg group associated to the Lagrangian planes £ and £.
are equivalent: we give in 1.4 a canonical choice of an operator
f£.,£ such that:
Fe;,
~" £
w£(n) = £ w£.(n) for every n € N.
(This operator in appropriate coordinates is a partial Fourier
4
4
transform.)
Before going further, we have to introduce, in 1.5, the
Maslov index of a triple of Lagrangian planes: Let (V,B) be
a symplectic vector space, £1' £2 and £3 three Lagrangian
planes, then
Q123(Xl$X2$X3) = B(xl,x2) + B(x2,x3) + B(x3,xl)
is a quadratic form on £1 @£2 $ £3' which can be diagonalized
with p times the eigenvalue 1 and q times the eigenvalue
-1. Amodified definition of the Maslov index ~(£1'£2'£3) due
to M. Kashiwara is ~(£1'£2'£3) =sign Q =p - q. We prove
123
in 1.5 that this Maslov index verifies a fundamental chain
property.
The symplectic group G acts on Lagrangian planes, on the
Heisenberg Lie algebra n and on the Heisenberg Lie group N.
If W£ is the Schr8dinger representation of N associated to
£, the representation W'(n) =w£(g.n) is equivalent to Wi:
The natural action A(g) of the symplectic group on functions
on N transforms the representation W£ of N into the
representation n ~ Wg'hn(g·n). The Fourier operator ~hn,g'hn
intertwines the representation W £ with Wi' Thus the
g.
canonical unitary operator R£(g) = ~,g.£ A(g) satisfies the
fundamental relation:
W£(g.n) = R£(g) w£(n) R£(g)-l, for every n € N.
We prove in (1.6) that