Table Of ContentLecture Notes ni
Mathematics
Edited yb .A Dold dna .B nnamkcE
436
Louis Auslander
Richard Tolimieri
Abelian Harmonic Analysis,
Theta Functions and Function
Algebras on a Nilmanifold
galreV-regnirpS
Berlin. Heidelberg • New York 1975
Prof. Louis Auslander
The Graduate School and University Center
City University of New York
33 West, 42 Street
New York, N.Y. 10036/USA
Prof. Richard Tolimieri
Dept. of Mathematics
niversity U of Con necticut
Storrs, CT 06250/USA
Library of Congress Cataloging in Publication Data
Auslander, Louis.
Abelian harmonic analysis~ theta _~unctions, and
functionaaln alysis on a nilmanifold.
(Lecture notes in mathematics 436) ;
Bibliography: p.
Includes index.
io Lie groups~ 2o Manifolds (Mathematics) o3 Har-
monic analysis° 4° Functions, Thetao ,I Toli~ieri~
Richard, 1940- joint author. IIo Title° III. Se-
ries: Lecture notes in mathematics (Berlin) 436° ;
QA3.L28 no.436 ~A387] 510'o8s 0~55o'215..~
74-3236~ ._
AMS Subject Classifications (1970): 22EXX, 22E25, 43-XX, 43A85
ISBN 3-540-07134-2 Springer-Verlag Berlin • Heidelberg • New York
ISBN 0-387-07134-2 Springer-Verlag New York • Heidelberg • Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
reprinting, re-use of illustrations, broadcasting, reproduction by photo-
copying machine or similar means, and storage ni data banks.
Under § 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
© by Springer-Verlag Berlin • Heidelberg 1975. Printed ni Germany.
Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
PREFACE
These notes are concerned with the inter-relationship between abelian harmonic
analysis, theta functions and functional analysis on a certain nilmanifold. Some
of the results in these notes are new and some are old. However, our approach,
because it puts a certain nilmanifold and its function theory at center stage,
often leads to new proofs of standard results. For example, we view theta functions
as the analogue on a nilmanifold of the spherical functions on the sphere, where
the Heisenberg group plays the role of the orthogonal group. Thus the classical
theta identities will follow from basic group theoretic results.
Historically, there are many names that can be associated with the topics
treated in these notes. Because of the informal nature of these notes we have not
made any effort at giving complete biographical references for results and have
given only references to the sources we ourselves have used. If we have overlooked
someone's work or state a result without reference that someone knows to be his,
we apologize in advance. However, we would be less than honest if we did not admit
the great influence of the ideas of J.Brezin, G.W.Mackey and A.Weil on our work.
Indeed, after so many years of talking with Brezin many of the ideas or germs of
ideas in these notes may be his . In addition, we should also mention the work of
Cartier which stands somewhere in the middle ground between the work of Weil and
that presented in these notes.
It may be advisable at this point to explain to the reader how the material
in these notes has been labelled. The reader will find some material labelled
Chapter and some material labelled Appendix. The material with sections labelled
Chapter is more complete and well rounded. The material labelled Appendix is of a
more tentative nature and does not yet seem to have taken on a definitive form.
The last appendix has its own list of references and the other appendices and
chapters have a single list of references located on page 74.
TABLE OF CONTENTS
Cha~ter .I
Fourier transform and the nilmanifold f\N . . . . . . . . . . . . . . . . . . . . . I
Chapter II.
Functions on ~N and theta functions . . . . . . . . . . . . . . . . . . . . . . . . 51
Chapter III.
Elementary transformation theory . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Appendix to Chapter III.
Cohomology and theta functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Chapter IV.
Theta functions and distinguished subspaces . . . . . . . . . . . . . . . . . . . . 44
Appendix B.
The arithmetic of distinguished subspaces . . . . . . . . . . . . . . . . . . . . . 68
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
Appendix C.
Fourier analysis on the Heisenberg manifold . . . . . . . . . . . . . . . . . . . . 75
References for Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
CHA~ER I
THE FOURIER TRANSFOP~\~ AND THE NIL~NIFOLD F\N.
The deep relationship between abelian harmonic analysis, Jacobi theta fun-
ctions, and the function theory on a certain nilmanifold that is the subject of
these notes can be well motivated by examining a non-traditional proof of the
Plancherel theorem for the reals. Hints of the proof we will present can be
found in the proof of the general Plsncherel theorem given by A. Weil in [8].
We will begin by giving a formal treatment that ignores all convergence questions,
Let ~ denote the reals and let f(t) be a function on ~ . We define the
functions fn(~), 0 ~ ~ < I, n E Z, where Z denotes the integers, by the formula
fn(~) = f(n + ~) 0 g ~ < ,l n E Z .
Now let T denote the reals modulo 1 and let 2 = T T × T denote the 2-torus.
We, of course, may view fn(~) as a function on T for each value of n E Z.
Define ®f by the formula
®f = E fm(~) exp(2~im~) , 0 g ~ < 1
= m -~
2
and view @f as a formal sum on T Here (~,~) is being considered as a co-
ordinate system for 2 T , Let
UA®f = ~ fm(~) exp(-2~im~)
m= -~
Since 0 ~ ~ < I and 0 ~ ~ < ,I the function exp(-2~i~) is well defined,
although discontinuous, on 2 T . Thus we may define
oc
MuAef : exp(-2~i~)fm(~) exp(-2~im~)
and finally define
1
A M U ®(f) exp(-2~in~)d~
0
n E Z . Letting t = ~ + n, we have ~(-2~t) = ~(-2~(~+n)) .
A
We will now formally compute f(-2~t).
i
~(-2~(~+n)) = f ~ exp(-2~i~)fm(~) exp(-2~im~) exp(-2TTin~)d~
0 m= oc-
f exp(-2~i~)fm(~) exp(-2~imE) exp(-2~in~)d~ .
m= -~ 0
Setting s = ~ + ,m we have, because n and m are in ,Z that
~ -( )tTT2 f(s) exp (- 2wist)ds
Let us now see what all this formalism means when f E L2OR).
We begin our discussion by showing that ® is an isomorphism of L2OR) onto
L2(T2). First, note that for f E L2OR)
-~
m= gr) 2~)
where I[ [H [ denotes the norm and the subscript the Hilbert space in which the fun-
ction lies. Next observe that
fm (~) exp 2~im~ and fn(~) exp 2Nin~ , n ~ m
are orthogonal in L2(T2). Hence
2(T) L2~)
T(221lfSl] 2) = ~ llfm 2(T2) m= = z IIr m
L m= o¢_ Q¢-
This shows that ®: L2(~) *- L2(T 2) is an isometry.
We will now verify that ® is a surjection. Let F(~,~) E L2(T 2) and let
Z a exp(2~in~) exp(2Nim~)
nm
be the Fourier series for F(E,~). Then
0c
) 2cr2 = z lmnal 2
]IF n m=-~
and ~N exp 2Nim~ converges in 2 L to fn(~) with
n=-N anm
r 2
h fn(~) H = ~ lanm 12
L2(T) m= -~
Now let f(x) = fn({), where x = n + {, be considered as a function on ~.
We note that if $F = f then
2 II*FI1L2 = Ilftln2
because
: 0o : =2 moa
llflI 2(IR) = T n= ~o_ llfn({)ll 2(T) n,m L (T z)
Thus ~: L2(T 2) ~ L2(]R) is an isometry, It is a formal computation to show
that ~ = ®-I and so ® is a surjection. We have thus shown that ® is a
unitary operator.
Now let F 6 L2(T 2) . Then we may view F as a doubly periodic function on
~I 2 or F(x,y) is such that
F(x+n,y+m) = F(x,y) , n,m E Z (x,y) 2 E .
We define A U F(x,y) = F(y,-x) . Clearly
[IyIIL2(T2 > : 2T(2LIIFII
and as A U is invertable A U is a unitary operator.
Now let F E L2(T 2) , 1 g p , and let
(N[F)({,~) = exp(-2~i~)F(~,~) , 0 g ~,~ < 1 .
So again
and so M being invertable is a unitary operator of L2(T 2) .
This proves that
®-lM A U ®: L20R) -- L20R)
is a unitary operator. We will let ~ = ®-IM A ® U .
Let the Fourier expansion of A M U ®f = b exp 2~im~ exp 2~in~
where convergence is in the 2 L norm. Then
~(-2~({+n)) = C=-~ bmn exp 2~im~ ,
where convergence is in the 2 L norm on T.
We next note that if gn and
g E L2(T) , n = 1,2 ..... and if A and A is the m Fourier coefficient of
mn m
gn and g, respectively, then if gn ~ g in L2(T) we have lim A = A
n °~- mn m
This shows that for f E L2~), we have
N
~(-2~t) = lim f f(s) exp(-2~ist)ds
N -= -N
2
where the limit is in the L norm. Thus
: L2gR) -. L2(~)
is a unitary operator. This last statement is called Plancherel's theorem. Since
UAF(0,0) = F(0,0) and exp(-2~i~)(0,0)= 1 , we have
A
Z f(m) = Z ~(2r~m)
m: -m m: -~
when both M UA ®f and ®~ are continuous at (0,0). This is called the
Poisson summation formula.
The proof of the Plancherel theorem we have just presented is elementary,
but seems very artificial. Let us now present a more geometric treatment of the
above proof that will lead us to an explaination of why it works.
Consider the group N, often called the 3-dimensional Heisenberg group, of
all matrices of the form
where x,y,z E m . We will denote the elements of N by triples (x,y,z) and
then multiplication is given by
(x,y,z)(a,b,c) = (x+a,y+b~z+c+ya)
In N let F denote the discrete subgroup consisting of all triples
(ml,m2,m3), m~ E Z, i = 1,2,3 . Then the homogeneous space F\N consisting of
cosets of the form Fn, n 6 N, is a compact manifold, called a nilmanifold.
Further N has a left and right invariant Haar measure. Hence there exists a
unique probability measure ~ on F~N which is invariant under the action of
5
on F\N given by Fn *- Fnn 0 , n,n 0 6 N . We next note that the set
D = [(x,y,z) ~ NIO < x,y,z < i}
is a fundamental domain for F\N and that ~ is given in this coordinate system
by the usual Lebesgue measure.
Let us now consider L2(F~N) and for f £ L2(F\N) and nl,n 2 6 N, we define
R(nl)(f(Fn2)) = f(Fn2n I) .
Then the mapping R: n ~ R(n) is a unitary representation of N with kernel
C = [(0,O,m) E Nlm E Z} . This gives us a representation of N/C . The image
of I(0,0,z) E N, z E~h in N/C is then a central compact subgroup and so
L2(F\N) breaks up into a Hilbert space direct sum
L2(F\~) = ® Z H
n6Z n
where F E H if and only if
n
217inz
R(0,0,z)F = e F
If Pn is the orthogonal projection of L2(F~N) onto Hn, then we may write Pn
explicitly in the form
1
-2~int
Pn(F)(x,y,z) = f F(x,y,z+t)e dt , n 6 Z
0
It is clear that the spaces H n, n E Z , are invariant under the representation
R of N .
We will now look a bit harder at H and produce an orthonormal basis for
n
H Because D is a fundamental domain for F~N and the measure defining
n
L2(F\N) agrees with the Lebesgue measure on D, one easily verifies that
exp(2~ifflX) exp(2~iff2y) exp(2~i~3z) , ffl,ff2,ff3 E Z
0 ~ x,y,z < i form an orthonormal basis of .)N¢kF(2L In terms of this basis
of L2(I~N), H n has orthonormal basis all basis vectors with ~3 = n . Thus
the mapping
: L2(T 2) ~- H
n n
defined on an orthonormal basis of L2(T 2) by
dn(eXp(2~ifflX) exp(2~ia2Y)) = exp(2~ialx)(ex p 2~i~2Y)(exp(2~inz)
extends to a unitary operator.
The space I H will play the role of L2(T 2) in our geometric interpreta-
tion of our proof of the Plancherel theorem, It remains to find a geometric
meaning for the operators ~A and ® . We will begin with the operator h~JA -
Let L(N) denote the Lie algebra of N. Then L(N) consists of all matrices
of the form
a,b,c E]R
with the usual bracket relation for making these matrices into a Lie algebra. Let
li0il i l l i
X = 0 Y = 0 Z = 0
0 0 0
Then X,Y,Z is a basis of L(N) and Z spans the center of L(N). Further
[Y,X] = YX- XY = Z
Now let ~* be the linear mapping of L(N) given by
~*(Y) = X , ~*(X) = -Y and a*(Z) = Z .
Then
[a (y),£ (x)] = [ x,-Y] = [Y,x] = z
6"
and so is an automorphism of L(N) and so, since N is simply connected,
G* determines an automorphism G of N . We will by direct computation now
compute ~ . First note that
(x,y,z) = (x,O,O)(O,y,O)(O,O,z)
If exp( ) denotes the exponential map of L(N) into N then exp(xX) = (x,O,0),
etc. Hence
