Arithmetic and Analytic Theories of Quadratic Forms and ... · 1. Quadratic forms and associative...

Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups

http://dx.doi.org/10.1090/surv/109

Mathematical Surveys

and Monographs

Volume 109

Arithmetic and Analytic Theories of Quadratic Forms

and Clifford Groups

Goro Shimura

American Mathematical Society

EDITORIAL COMMITTEE Jer ry L. Bona Michael P. Loss Pe te r S. Landweber , Chair Tudor Stefan Ra t iu

J. T . Stafford

2000 Mathematics Subject Classification. P r imary 11D09, 11E08, 11E12, 11E25, 11E41, l l F x x , 15A66, 22E99.

For addi t ional information and upda tes on this book, visit w w w . a m s . o r g / b o o k p a g e s / s u r v - 1 0 9

Library of Congress Cataloging-in-Publication D a t a Shimura, Goro, 1930-

Arithmetic and analytic theories of quadratic forms and Clifford groups / Goro Shimura. p. cm. — (Mathematical surveys and monographs ; v. 109) Includes bibliographical references and index. ISBN 0-8218-3573-4 (alk. paper) 1. Forms, Quadratic. 2. Linear algebraic groups. 3. Number theory. I. Title. II. Mathe

matical surveys and monographs ; no. 109.

QA243.S49 2004 512.7,4-dc22 2003063826

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TABLE OF CONTENTS

Preface vii Notation and Terminology ix Introduction 1 Chapter I. Algebraic theory of quadratic forms, Clifford

algebras, and spin groups 9 1. Quadratic forms and associative algebras 9 2. Clifford algebras 15 3. Clifford groups and spin groups 20 4. Parabolic subgroups 28

Chapter II. Quadratic forms, Clifford algebras, and spin groups over a local or global field 37 5. Orders and ideals in an algebra 37 6. Quadratic forms over a local field 45 7. Lower-dimensional cases and the Hasse principle 52

Part I. Clifford groups over a local field 62 Part II. Formal Hecke algebras and formal Euler factors 72

9. Orthogonal, Clifford, and spin groups over a global field 80 Chapter III. Quadratic Diophantine equations 93

10. Quadratic Diophantine equations over a local field 93 11. Quadratic Diophantine equations over a global field 101 12. The class number of an orthogonal group and sums of

squares 113 13. Nonscalar quadratic Diophantine equations; Connection

with the mass formula; A historical perspective 126 Chapter IV. Groups and symmetric spaces over R 139

14. Clifford and spin groups over R; The case of signature (1, m) 139

15. The case of signature (2, m) 146 16. Orthogonal groups over R and symmetric spaces 154

Chapter V. Euler products and Eisenstein series on orthogonal groups 163 17. Automorphic forms and Euler products on an orthogonal

group 163

18. Eisenstein series on Ou 173 19. Eisenstein series on Ov 181 20. Arithmetic description of the pullback of an Eisenstein

series 187 21. Analytic continuation of Euler products and Eisenstein

series 196 Chapter VI. Euler products and Eisenstein series on

Clifford groups 205 22. Euler products on G+(V) 205 23. Eisenstein series on G(H, 2~1r]) 212 24. Eisenstein series of general types on a Clifford group 218 25. Euler products for holomorphic forms on a Clifford group 226 26. Proof of the last main theorem 234

Appendix 243 Al. Differential operators on a semisimple Lie group 243 A2. Eigenvalues of integral operators 250 A3. Structure of Clifford algebras over R 261 A4. An embedding of G1(F) into a symplectic group 265 A5. Spin representations and Lie algebras 268

References 272 Frequently used symbols 274 Index 275

PREFACE

The spirit of a three-year-old will be kept until 100 — A Japanese saying

There are two principal themes in the present book. The first one concerns a quadratic Diophantine equation of the form X^\-=1

(fijxixj — <7? where ip = (Pij) is a symmetric matrix with coefficients in Z and 0 / g G Z. For given ip and q a solution x = (x{) with Xi G Z is called primitive if the Xi have no nontrivial common divisor. When the equation is of the form x\ +x\ + x\ = q, Gauss showed in his Disquisitiones that the number of primitive solutions is an elementary factor times the class number of primitive binary quadratic forms of discriminant —q. His proof is roundabout; besides, though later researchers treated the cases of five and seven squares and obtained the formula for the number of primitive representations in terms of special values of Dirichlet L-functions, the connection with the class number was never explained satisfactorily. In the present book I show that the number of orbits of primitive solutions under a group of units is essentially the class number of an orthogonal group of degree n — 1, in such a way that the fact specialized to the case of three squares gives the result of Gauss in a more direct way.

The other principal theme is a certain Euler product, the idea of which originated in January 1961 when I presented a theory of Hecke operators for Sp(n, Q) in a seminar at the University of Tokyo. Taking the group of similitudes instead of Sp{n, Q), I found, when n — 2, an Euler product of degree 4, and also a congruence relation for each Euler p- fact or. The results were published in a short article two years later, and I turned my attention to other directions since then. Though I was always conscious of this problem, I was unable to gain a clear perspective until a few years ago, when I realized that I could treat it as a special case of the theory for Clifford groups. A large portion of this book is devoted to a full exposition of the theory including meromorphic continuation of Euler products and Eisenstein series on such groups, attaining my desire 42 years ago in a more general setting. At the same time we treat Euler products and Eisenstein series on orthogonal groups, which are actually special cases of those on Clifford groups. One of the new points of this work is that the gamma factors can be determined explicitly when the automorphic forms in question are nonholomorphic eigenfunctions of differential operators.

vii

V l l l PREFACE

Since Clifford groups are closely related to quadratic forms, there is nothing special about presenting these two themes in the same volume, but traditionally Clifford groups have been discussed merely as a tool for investigating quadratic forms. In the present book, however, we put the groups on center stage, and investigate them for their own sake, as this approach gives a better perspective in the theory of Euler products.

Though my chief intention is to present some new ideas on these two topics, I have also included expository material concerning arithmetic of quadratic forms such as the Hasse principle and the algebraic theory of Clifford algebras and spin groups, often with new methods, so that those portions of the book may serve as an introduction to some such subjects.

Princeton May 2003 Goro Shimura

N o t a t i o n a n d T e r m i n o l o g y

We denote by Z, Q, R , and C the ring of rational integers, the fields of rational numbers, real numbers, and complex numbers, respectively. We put

T={z<=C\\z\ = l}.

If p is a rational prime, Zp and Qp denote the ring of p-adic integers and the field of p-adic numbers, respectively. For an associative ring R with identity element we denote by Rx the group of all invertible elements of R, by Mn(R) the ring of all square matrices of size n with entries in R, and by l n the identity element of Mn{R); we put then GLn(R) = Mn{R)x, Rx2 = {a2 | a G Rx}, and

SLn(R) = { a G GLn(R) \ de t (a ) = 1 }

if R is commutative. We also denote by R™ the set of all (m x n)-matrices with entries in R; thus Mn(R) = R™, but we use Mn(R) when the ring-structure is in question. We put R171 = R™ for simplicity. For x G R™ and an ideal a of R we write x -< a if all the entries of x belong to a. (There is a variation of this; see §18.1.) The transpose, determinant, and trace of a matr ix x are denoted by lx, det(x) , and t r (x) ; we put

x = ( t x ) _ 1

if x is square and invertible. The zero element of R™ is denoted by 0™ or simply by 0. The size of a zero matr ix block writ ten simply 0 should be determined by the size of adjacent nonzero matrix blocks. If X\, . . . , xr are square matrices, diag[xi, . . . , xr] denotes the matr ix with x\, . . . , xr in the diagonal blocks and 0 in all other blocks. For a complex number or more generally for a complex matr ix a we denote by Re(a ) , Im(a ) , and a the real part , the imaginary part , and the complex conjugate of a. For complex hermitian matrices x and y we write x > y and y < x if x — y is positive definite, and x > y and y < x if x — y is nonnegative. For r G R we denote by [r] the largest integer < r.

Given a set A, the identity map of A onto itself is denoted by id^. To indicate tha t a union X = \JieIYi is disjoint, we write X = \_\ieI Y\. We understand tha t

Yli=a — 1 a n d Yli=a — 0 if ^ > / - For a finite set X we denote by # X or # ( X ) the number of elements in X. If H is a subgroup of a group G, we put [G : H] — #(G/H). However we use also the symbol [K : F] for the degree of an algebraic extension K of a field F. The distinction will be clear from the context.

The idele group of an algebraic number field K is denoted by K£, and for x G K^ we denote by \X\A, often writ ten simply \x\, the idele norm of x\ see §9.3. By a Hecke character \ of K, we mean a continuous T-valued character of K^ trivial on K x , and we denote by \* ^ n e ideal character associated with \- As for the notation concerning localization and adelization of algebraic groups, see §§9.1 and 9.3. We shall employ formal Dirichlet series of the form J^ a c(a)[a], where a runs over the integral ideals of the number field in question and [a] is a certain multiplicative symbol; see §17.10. We shall often put |a| = TV ( a ) - 1 for a fractional ideal a in both local and global cases.

ix

A P P E N D I X

A l . Differential operators on a semisimple Lie group

A l . l . For a C°° manifold M we denote by C°°(M) the set of all C-valued G°° functions on M. Since all the manifolds appearing in this section have real-analytic structure, we can speak of real-analytic functions on them, which we simply call analytic. We consider a symmetric space TC = G/K with a connected noncompact semisimple Lie group G with finite center and a maximal compact subgroup K of G. We take a Cart an decomposition go = o + Po, where go denotes the Lie algebra of G and to its subalgebra corresponding to K. We denote by g resp. t the complexification of go resp. to; also we denote by 11(g) resp. il(t) the universal enveloping algebra of g resp. t. We take a point o so that K = {g G G | g(o) = o} . Fixing a scalar-valued factor of automorphy J (a , z) = Ja(z) defined for a G G and z G H we define a one-dimensional representation p : K —» C x by p(k) = J{k, o) for every k G K. Since K is compact, we have \p(k)\ = 1 for every k G if. Then we denote by C°°(G, p) the set of all G°° functions f on G such that f(gk) — Pik'^fig) for every k G if and <? G G. For / G C°°(W) and a G G we define f\\pae C°°{H) and f G G°°(G) by

(Al.l) (f\\pa)(z) = Uz)-1f(az)1 F>(g) = (f\\pg)(o) (geG).

Then we easily see that / \-+ fp gives a bijection of C°°(H) onto the set C°°(G, p). Strictly speaking, as p does not necessarily determine J&(z), the notation /||pce is misleading, but we use it with the understanding that Ja is fixed once for all.

Now we let S)(p) denote the ring of all G°° differential operators W on H such that W(f\\pa) - {Wf)\\pa for every / G G°°(H) and every a G G. We also denote by X>(p) the ring of operators obtained from the elements D of 11(g) which map C°°(G, p) into itself.

Theorem A1.2. (i) There is a ring-isomorphism W ^> Wp of 1)(p) onto V(p) such that Wp(fp) = {WfY for every f G G°°(H).

(ii) S)(p) is commutative. (iii) Let ^ be the kernel of the natural homomorphism ofU(t) into C obtained

from —dp : t —> C. Then an element of 11(g) annihilates C°°(G, p) if and only if it belongs to ll(g)9Tp, and consequently 11(g)9tp is a two-sided ideal ofil(#).

(iv) Let U(g)K denote the set of all Ad(K)-invariant elements of 11(g), where Ad : G —• Aut(g) is the adjoint representation. Then every element of 11(g)K

243

244 APPENDIX

gives an element of V(p), and the map il($)K —>• *D(p) is surjective and has kernel H ( 0 ) K n i l ( g ) ^ p .

(v) Let C(Q) be the center of 11(g). Then the natural map of C(Q) into V(p) is surjective, provided G is classical.

The first assertion is included in [S02b, vol. IV, p. 739, Theorem D]; the remaining four assertions are included in [S90, Propositions 2.1 and 2.3, and Theorem 2.4]. This theorem for trivial p, excluding (v), is classical and well-known. Thus the point of the theorem is that it holds even for nontrivial one-dimensional p. As to (v) for trivial p, Helgason showed that it is true for classical G, and even determined for which exceptional groups it is valid.

Lemma A1.3. Let Z be the element ofS(p) such that Zp is the element of T*(p) obtained from the Casimir element of 11(g). Then Z is an elliptic operator, so that every element of C°° (H) that is an eigenfunction of Z is analytic.

PROOF. Let @(X, Y) be the Killing form of go- It is well-known that $ is positive definite on po, negative definite on to, and $(po, to) = 0. Therefore we can find It-bases {Xi)r

i=1 of po and {Yh}sh=1 of to such that <P(Xi, Xj) — Sij and #(5^, Ye) —

—Shi- Then C = YH=i Xf — YLh=i h ls the- Casimir element. Now we have YF — -dp(Y)F for every F G C°°(G, p) and every Y G t0, and hence £ * = 1 Y*F = XF with A = X^/Ui dp(Yh)2, which is independent of F. We can identify H with R r

through the map (U)rl=l *-+ exp ( £ I = i UXi)(o). Given / G C°°(H) = C°°(R r) ,

define an element F of C°°(G, p) by F ( e x p ( £ [ = 1 UX^k) = p(k)~1f{t1, . . . , tr). Then F = fp, and hence, by Theorem A1.2 (i), (ZfY = CF. Thus

[(Z + A)/](0) = [(C + X)F](eMEUUXl))t=0

= £ U ^ 2 ^ ( e x p ( E [ = 1 UXi))t=0 = T,l=i(d/dti)2m, . . . , U)t=o •

Let 7r be the projection map G —+ H defined by ir(g) = go. Then there is a differential operator L on H such that (Lf) o n = C(f o n) for every / G C°°(H). (This C is the operator Z for trivial p.) It is well-known that L is the Laplace-Belt rami operator on TL, and is elliptic. Taking p to be trivial, we see that the above equality is true with Lf in place of (Z + A)/, and hence (*) (Z/)(o)=[(L-A)/](o) for every feC°°{n). Now Z(f\\pa) = (Zf)\\pa for every aGG, so that (Zf)(az) = ^a(^)^[^a(^) _ 1 / ( a ^) ] • Taking z = o and putting w = ceo, we obtain

(Zf)(w) = Ja(o)Z[Ja(z)-1f(az)]z=o = Ja(o)[(L-\)(Ja(z)-1f(az))]z^.

Similarly, (Lf)(w) — L(f(az)) = . Comparing these equalities, we see that the principal part of the operator Z is the same as the principal part of L. This proves our lemma.

Lemma A1.4. Let M and N be real analytic manifolds and f(x, y) an analytic function of (x, y) G M x N. Then, for every compact subset C of M, the function g(y) — Jc f(x, y)dx of y G TV is analytic, where dx is a measure on M such that every open set is measurable and every compact set has finite measure.

Al. DIFFERENTIAL OPERATORS ON A SEMISIMPLE GROUP 245

PROOF. Fix a point q of N and analytic coordinates 2/1, . . . , yr on an open neighborhood W of q so that y\(q) — ••• = yr(o) — 0. For each p in a given compact set C we can find a neighborhood U of p and a neighborhood V of q contained in W so that / has a uniformly convergent expansion

(t) /(^,!/) = E „ ^ W - ! / ? r

for (x, y) e U xV with analytic functions an on [/, where n = ( n ^ = 1 . Since each an can be obtained as a partial derivative of / , it is an analytic function defined on the whole M. Since C is compact, we can find a finite set {(pi, Ui, Vi)}iei of such points pi of C and neighborhoods Ui and Vi so that C C [jieI Ui. Put Y = f]ieI Vi-Then Y is a neighborhood of q and the series of (f) is uniformly convergent on C xY. Therefore term-wise integration of (f) over C gives the desired result.

Given / G C°°(G, p) and g, h e G, we put

(A1.2) (Mhf)(g) = / p(k)f(gkh)dk. JK

Here and throughout this and the next sections we normalize the measure of K so that JK dk = 1. We easily see that M h / G C°°(G, p) and

(A1.3) MkMf = p{k£)~1Mhf for every k, £ e K.

Theorem Al .5 . Let f be an analytic element ofC°°(G, p). Then there exists a neighborhood V of the identity element e of G and a sequence {D^l}^=0 of elements D1^ ofV(p) defined for h G V such that

00

(Mhf)(g) = J2 (Dtf)(9) for (g, h) e V x V. ra=0

Moreover, the D^ are independent of f in the following sense: if V and E^ play the roles of V and D1^ for an analytic ff G C°°(G, p), then D^ = E^ for every h in a neighborhood of e contained in V D V.

PROOF. Take {Xi}\=1 as in the proof of Lemma A1.3. This is an orthonormal basis of po with respect to the Killing form, which is ^4d(if)-invariant. We put I I E L i ^ l l = { E L i ^ } 1 / 2 f o r UeR. Put N5 = {Yep0\\\Y\\<8} for 0<SeH. Then Ns is Ad(K)-stable. Clearly (#, Y) *-+ f(g • exp(F)) is analytic in (g, Y) G G x p. Therefore, choosing a suitable neighborhood U of e in G and a small 5, we have an expansion

f(g • exp(r)) = £ „ an(g)h(Y)^ • • • tr(Y)^

for (g,Y) G U x N2s with analytic functions an on J7, where n = ( n ^ = 1 and y ' = E[= i U{Y)Xi. For - 2 < s < 2 and Y G N6 we have

(*) / ( ^ • e x p ( 5 y ) ) = E n a n ( ^ ) ^ H t i ( F ) n i - - - t r ( F ) ^ ,

where \n\ = YH=zini- On ^n e °ther hand, a well-known principle guarantees that f (g - exp(sY)) = Ylm=o(sTn /rn^(Yrn f)(g) f° r small |s|. Comparing this with (*) we see that (Ymf)(g) = E\n\=man(g)h(Y)^ • -.tr(Y)n*. Since (*) is valid for s = 1 and (g, Y) G U x N$, we thus obtain

246 APPENDIX

oo 1

(**) /(s-exp(F)) = E ^ y m / ) ( 2 ) 771 = 0

for (g, Y) £ U x N$, and the convergence is locally uniform. We now fix X G N6 and put h = exp(X) and Yfe = Ad(k)X for A: G A\ Then

Yk G A^, and for g £ U we have

(Mhf){g)= I p{k)f(gk-exp{X))dk = f f(gk • exp(X)k-1)dk JK JK

oo x

f(g • exp(Yk))dk = / ] T — ( r r / ) ( < ? ) *

by (**). We can change J K J^ m t o ]C IK1 a n c^ s o

oo ,

7^0 m ! ^

Clearly £?^ belongs to 11(g)A, and so produces an element D1^ of T)(p) by Theorem A1.2 (iv). Take a small neighborhood W of e in A\ Then the map (/i, f) H-+ /i^ gives a bijection of exp(A^) x W onto a neighborhood of e in G. For such (h, £) put D™ = p(£)-1L>^. Then, by (A1.3) we obtain the formula for (Mhf)(g) of our theorem for a suitable V. The assertion concerning the independence of D1^ from / is clear from our construction of D1^.

T h e o r e m A 1 . 6 . Let f be a nonzero element ofC°°(G, p) that is an eigenfunc-tion ofV(p) in the sense that Df = \(D)f for every D G V(p) with X(D) G C. Then there is an analytic function p on G depending only on X such that

(A1.4) / p(k)f(gkh)dk = p(h)f(g) JK

for every (g, h) G G x G.

PROOF. Let / = pp with p G C°°(TL). Then p is an eigenfunction of the operator Z of Lemma A1.3, so tha t p is analytic by tha t lemma. Consequently / is analytic on G. Take V as in Theorem A 1.5. Then, for g, h G V we have

* oo

/ p(k)f(gkh)dk = (Mhf)(g) = £ KD^ftg) = »{h)f(g), j K m=0

where p(h) = Xlm=o M^™)- By Lemma A1.4 the first integral is analytic in (g, h), and hence we have JK p(k)f(gkh)dk = p(h)f(g) for every (g, h) G G x V. Take go G V so tha t f(go) ^ 0. Then p(h) = f(go)~1JKp(k)f(gokh)dk, and the last integral is analytic in h. Therefore p can be extended to an analytic function on the whole G, with which (A1.4) holds, as both sides of (A1.4) are analytic in (g, h) G G x G. Since p(h) = J2m=o M ^ m ) f ° r h £ V, the function p depends only on A.

Theorems A1.5 and Al .6 with trivial p are proven in Helgason [Hel, Ch. X, Theorem 7.2 and Corollary 7.4].

A 1.7. We now generalize the notion of spherical functions as follows. In the setting of §A1.1 we call a function / in COC)(G) a /^-spherical funct ion if it satisfies the following three conditions:


(A1.5a) /(e) = 1, where e is the identity element ofG; (A1.5b) f(kfxk) = pik'k)-1/^) for every k'\ k G K; (A1.5c) / is an eigenfunction of V(p) in the sense that Df = Xf(D)f with

Xf(D) G C for every D G V(p).

If p is trivial, this is the definition of a spherical function on G. We call a nonzero element / of C°°(G, p) an eigenfunction oiV(p) if / has property (Al.5c), and use the symbol Xf(D) always in that sense.

Proposition A1.8. Let f be a nonzero element of C°°(G). Then f is p-spherical if and only if

(A1.6) / p(k)f(xky)dk = f(x)f(y) for every x, y G G. JK

PROOF. Suppose / is p-spherical; then we have an analytic function \x satisfying (A 1.4). Putting y = e, we find that p = / , which proves the 'only if'-part. Conversely suppose (A1.6) holds. Substituting £km for k with I, m G K, we see that / satisfies (A1.5b). Take x to be an element xo such that f(xo) ^ 0 and take y = e. Then f(xo)f(e) = JK p(k)f(xok)dk = f{xo), so that /(e) = 1. Next, for D G V(p) we have f(x)(Df)(y) = JK p{k)(Df)(xky)dk. Put y = e. Then (Df)(e)f(x) = JK p(k)(Df)(xk)dk = (Df)(x), so that / is an eigenfunction. This proves our proposition. At the same time, we obtain (i) of the following lemma.

Lemma A1.9. (i) If cp is a p-spherical function, then X^D) = (Dcp)(e) for every D eV(p).

(ii) Given an eigenfunction f G C°°(G, p) ofV(p), define p by (A1.4). Then p is a p-spherical function and AM(D) = Xf(D) for every D G £>(/>).

(iii) If cp and i\) are p-spherical functions such that Xip(D) = X^(D) for every D G £>(p), then <p = t/;.

PROOF. TO prove (ii), take g0 G G so that f(go) ^ 0. Then f{go)p(x) = JK p(k)f(gokx)dk, and we can easily verify that p has properties (A1.5a, b). For D G V(p) we have

f(g0)(Dp)(x) = / p(k)(Df)(g0kx)dk JK

= Xf(D) [ p(k)f(g0kx)dk = Xf(D)f(g0)p(x), JK

so that Dp = Xf(D)p. This proves (ii). As for (iii), take any B G il(g) and observe that [Ad{k)Bf](x) = p(k)(Bf){xk) for every / G C°°(G, p) and every k G K. Moreover, if f {kx) = p(k~1) f {x) for every k G K, then (Bf)(kx) = p(k-1)(Bf)(x), so that p{k)(Bf)(k) = (Bf)(e). Now put D = fKAd(k)Bdk. Then D belongs to the set 11(g)K of Theorem A1.2 (iii), and so it defines an element Do of V(p). Thus, for any p-spherical ip we have (Doip)(e) = (D(p)(e) = JK [Ad(k)B(p](e)dk = JK p(k)(B(p)(k)dk = (B(p)(e). Therefore if <p and ^ are as in (iii), then by (i), (B(p)(e) = (Bip)(e) for every B G 11(g), which means that <p and ip have the same derivatives at the origin. Since they are analytic, they must coincide.

ALIO. We now take a decomposition of the form G = RBK with closed Lie subgroups R and B such that

248 APPENDIX

(A1.7) RBDK = BHK and bRb~l C R for every be B.

For example, we can take a parabolic subgroup P of G such tha t G = PK, and consider a decomposition P = RB with the unipotent radical RotP and a reductive factor B of P. An Iwasawa decomposition G = NAK is another example. We then put

(A1.8) £R(p) = { / e C°°{G, p) | f(rg) = f(g) for every r e R and 5 € G }.

Let pi be the restriction of p to 5 fl K Taking {£?, I? f) i^, B D R, pi} in place of {G, if, i2, p} , we can similarly define a subset £BDR{PI) of C°°(B, pi). Let to resp. bo be the Lie algebra of R resp. B. Pu t t ing r = to 0 R C and b — bo 0 R C, denote by it(r) resp. 11(b) the universal enveloping algebra of r resp. b. Then we denote by T>(pi) the ring of operators obtained from the elements of it(b) tha t map C°°{B1 pi) into itself. For every / G £R{P) the restriction of / to B belongs to £BDR{PI), and we easily see tha t the restriction gives a bijection of £R(P) onto £BnR{pi)- Indeed, if rbk = r\b\ki with r, r\ G R, b, b\ G B, and k, k\ G K, then kk^1 eRBnK = BHK by (A1.7), and hence r ~ V e RnB.

L e m m a A l . l l . Given f G £R(P), let p be the restriction of f to B. Suppose p is an eigenfunction ofV(pi) in the sense that Tp — P(T)p with (3(T) G C for every T G £>(pi). Then Yf = 7 ( F ) / with -y(Y) G C for every Y G V(p). Moreover, 7 ( F ) is completely determined by p, Y, and /3.

P R O O F . This was given in [S02b, vol. IV, p. 750, Theorem 1]. Here we reproduce the proof for the reader's convenience. We first prove:

(A1.9) If h G C°°(G) and h{rg) = h(g) for every r G R and g G G, then (Xh)(b) — 0 for every X G r and every b G B.

To prove this, we may assume tha t i G t o - Then, for b G B we have (Xh)(b) = {d/dt)h(b • e x p ( t X ) ) t = 0 = (d/dt)h(b • e x p ^ X ) ^ ) = (d/dt)h(b) = 0,

since 6 • e x p ( t X ) 6 - 1 G i?. This proves (A 1.9). Now extend — dp : £ —> C to a homomorphism a : ii(f!) - • C. Then W / = a ( W ) / if W G i l( t) and / G C 0 0 ^ , p). Let y G il(£|). Since $ = t + b + £, by a well-known principle, Y can be expressed as a finite sum of elements of the form UZW with U G U(t), Z G 11(b), and W G il(6). (The expression 3 = r -f b + 1 may not be a direct sum, but tha t causes no problem.) Therefore we can find a finite sum E = ]T ZW with such Z and W, such tha t Y — E G rll(g). Let / G £R{P). Then for any Q G 11(g) we can take Qf to be /i in (A1.9). Therefore [(V - E)f](b) = 0 for every be B. Thus

(Yf)(b) = (E / ) (6 ) = £ ( Z W 7 ) ( 6 ) = £ ( < * ( W W ) ( 6 ) = ( 2 » ( 6 ) ,

where T = ^ a ( l ^ ) Z , which is an element of 11(b). Take Y G D(p). Since / G £R{P)I w e easily see tha t y / G £R(P), and hence T(/? G £BHR(PI)- NOW / 1—> (p is a bijection of £ R ( P ) onto £B(IR{PI)- Therefore Tp G £BHR(PI) for every <p G £BC\R(PI), SO tha t T G D(pi). Thus T<p = P(T)p by our assumption, and hence y / coincides with (3(T)f on 5 . Since both Yf and /3(T)f belong to £R(P) and they coincide on B, we obtain Yf = /3(T)/ , which proves our lemma.

We note here a simple fact: Let 9Tp be the kernel of a , which is exactly 9tp of Theorem Al .2 (hi). Given Y G 11(g), there exists an element T of 11(b) such that


(ALIO) y -TGtU(f l )+U(f l )91 p .

Indeed, we have seen that Y - E G ril(g). Now E - T = ]T Z(W - a(W)), and W - a(W) G 9tp, and hence we obtain (ALIO).

A1.12. Given an eigenfunction / G C°°(G, p) of V(p), we have a set of eigenvalues {\f(D) | D G D(/?)}. By Lemma A1.9 (ii) there exists a p-spherical function p such that AM(D) = Xf(D) for every D G £>(/?)• Moreover, if /(e) = 1, then (A1.4) shows that p(x) = JK p(k)f(kx)dk. Now we can choose / as follows. We take an Iwasawa decomposition G = NAK, denote by do the Lie algebra of A, and take a continuous homomorphism ( : A —> C x , which is of course obtained by £(exp(X)) = exp (u(X)) for X G do with an R-linear map v : do —> C. Define gv G C°°(G, p) by gv(nak) = p(k)~1((a) for n e N, a e A, and A: G K. Take (JV, A) to be (i?, 5 ) of §A1.10. Then gv G £jy(p), and hence ^ is an eigenfunction of V(p) by Lemma ALII , and the eigenvalues are determined by v. Put

(ALU) <pv(x) = / p(k)gv(kx)dk (x G G).

Since gv(e) = 1, ^ is exactly the function p if we take gv to be / . Thus (pu is p-spherical and A^^(D) = X9u(D) for every D G 2}(p). Now all possible sets of eigenvalues of T>(p) can be obtained in this fashion because of the following theorem.

Theorem A1.13. Every p-spherical function is of the type (Al.ll) with a suitable v. Consequently in view of Lemma A 1.9 (ii), for every eigenfunction f G C°°(G, p) ofV(p), there exists an H-linear map v : do —> C such that Xf(D) = X9u(D) for every DeV(p).

PROOF. This is due to Harish-Chandra if p is trivial. The case of nontrivial p can be proved in the same manner by modifying the proof of [He2, p. 418, Theorem 4.3], or rather, by modifying the arguments of various auxiliary facts required in the proof. First of all, by Theorem A 1.2 (iv), V(p) can be obtained from 11(g)x, which is independent of p. However, the kernel 9Tp depends on p\ in fact, ii(g)^flp = il(g)E if p is trivial. Now the symbols D(G), D^(G) , and D(^4) in [He2] correspond to il(g), ii(g)K, and 11(d) here. Therefore we have to replace D(G)t by H(g)9tp, which is a two-sided ideal of 11(g) as noted in Theorem AL2 (iii). Then all arguments go through for nontrivial p. Without presenting all the details, we content ourselves with noting two examples of how the arguments must be modified: (I) Given Y G H(g), there exists a unique T G 11(d) such that Y -T G nii(fl) + ii(fl)9lp. This generalizes [He2, p. 302, Lemma 5.14]. The existence of T was given in (ALIO); we take (R, B) there to be (JV, A). To prove the uniqueness, let D G H(d) D [nii(g) + ii($)9Tp]. Then D = E + F with E G nll(s) and F G H(Q)^IP- Take an arbitrary if G C°°(A) and define an element / of C°°(G, p) by f(nak) = p{k)-1LP(a). By (A1.9) with R = JV we have Ef = 0; also, by Theorem A1.2 (iii), Ff = 0. Thus Df = 0. On the other hand, for a G A we have (Df)(a) = (Dip)(a), so that Dp = 0 for every p G C°°(A), so that D = 0. This proves the uniqueness. (II) For the proof of [He2,p. 305, (38)], we have to show that A(Sd-1(gi)t) C i l ^ + E e ^ M ^ W ) -Indeed, for X G t we have X = a(X)+X-a(X), and hence X(X) G C + 9 V Thus, for y G S ^ 1 ^ ) we have A(FX) - X(Y)X(X) G E e <d A(5c(fl)) and X(Y)X(X) G A(5'cf~1(g)) + 1 1 ( 9 ) ^ , which gives the desired fact.

250 APPENDIX

In fact, the result of Harish-Chandra gives a more precise result by considering the action of the Weyl group on the vector space of v. This can be generalized to the case of nontrivial p, but since we do not need it, we leave the details to the reader.

Proposition A1.14. Define 8sJ and E^(z, s; / , T) by (18.8) and (18.13) with f in the set S% of §17.13. Then 5sj is an eigenfunction of £>£ (for the notation see §17.12) for every s G C and E^ip(z, s; / , T) is an eigenfunction of T>% for Re(s) > (m + n - l ) /2 .

PROOF. Let SO% resp. SO$ denote the identity component of SO% resp. SO%. We consider the decomposition SOfi = RBK with

K = {a eSO% | a l a ; = l a ; } ,

B = {diag[a_1, e, a] |aGZ\ a , eeSO$}, A= {diag[oi, . . . , am] |0 < a*GR},

iJ = | x G P a | ^J = 1? ^x is unipotent upper triangular}.

Here P is as in §18.1; dx and ex are the d-block and e-block of x as in (18.14). Define functions p on Z* and g on Z» by p(ft) = 5(hl*)k/2f(hl*) for ft G SO£ and q{g) = 5{glu)k>258j(glu) for # G SOft. By (18.8) and (18.11) we have q(rg) = g(#) for r G P and g(diag[a-\ e, a]) = det(a)-2su(f\\ke)(l*) = det(a)~2sup(e). Since / G <S[, p is an eigenfunction of V of §17.12, and hence we easily see that det(a)_2sup(e) as a function on B is an eigenfunction of V(p\). Therefore, by Lemma Al . l l , 5sj is an eigenfunction of D£ for every s G C. The same is true for E^(z, s; / , T), since D(6sJ\\ka) = (D6sJ)\\ka for every a G T.

It should be noted that similar facts hold in the symplectic and unitary cases; see [S02b, vol. IV, p. 753, Theorem 3].

A2. Eigenvalues of integral operators

A2.1. With H = G/K, J a (c) , and p as in §A1.1 we now consider an analytic function P(z, w) on 7i x H such that

(A2.1) P(uz, aw) = Ja(z)Ja(w)P(z, w) for every a G G,

and also an analytic function Q(g, h) on G x G satisfying the following two conditions:

(A2.2a) Q(ag, ah) = Q(g, h) for every a G G, (A2.2b) Q(gk, hi) = p(k)~1p(()Q(g, h) for every fc, £ G A'.

Given P satisfying (A2.1), put Q(g, ft) = {J^(o) Jf l(o)}"1P(po, fto) for g, h e G. Then Q satifies (A2.2a, b). Conversely, given Q satisfying (A2.2a, b) , we obtain P satisfying (A2.1) by putting P(go, fto) = Jg{o)Jh{o)Q(g, ft). Notice that J(fc, o) = J(fe, o ) " 1 for fee A'.

Now, given Q satisfying (A2.2a, b), put R(g) — Q(g, 1). Then Q(g, ft) = R(h~lg) and

(A2.3) ii(fyfc) = p(k)-lp(€)-lR(g) for every fc, £ G AT.

A2. EIGENVALUES OF INTEGRAL OPERATORS 251

Conversely, given an analytic R satisfying (A2.3), put Q(g, h) = R(h~1g). Then Q satisfies (A2.2a, b) .

We can define a real-valued function r on H by

(A2.4) T(9O) = \J(g, o)\~\

since the right-hand side depends only on gK. We fix a Haar measure dg on G and also a G-invariant measure d^ on Ti so tha t

/ f(z)dz = / f(go)dg Jn JG

for every continuous function / on H.

T h e o r e m A 2 . 2 . Let cp be a nonzero element of C°°(7i) that is an eigenfunction of D(p) in the sense that Dcp = X(D)^p for every D G *3{p) with X(D) G C. Let P be an analytic function on H x Ti satisfying (A2.1). Then

(A2.5) / P(z, w)ip(w)r(w)dw = c • ip(z) Jn

with c G C, provided the integral is convergent. Moreover, c depends only on A and P.

P R O O F . Let / = (pp; define Q and R as above so tha t Q(g, h) = R(h~1g) and P(go, ho) = Jg(o)Jh(o)Q(g, h). Then we can easily verify tha t (A2.5) is equivalent to

(A2.6) / R(h-1g)f(h)dh = c.f(g) JG

when the integral is convergent. To prove it, let F(g) denote the left-hand side of (A2.6). Then, for every k G K we have

F(g) = f R(y-1)f(gy)dy = [ R(y-1k-1)f(gky)dy = / R{y'1)p{k)f(gky)dy. JG JG JG

Therefore

F(g)= I F(g)dk= [ [ R{y-l)p{k)f{gky)dydk JK JK JG

R(y~l) / P(k)f(gky)dkdy = f(g) / R(y-l)p(y)dy JG JK JG

by Theorem A1.6, where the function p is determined by A as in tha t theorem. As to the convergence: Our assertion is trivial if / is identically 0; so assume tha t f(g) ^ 0 for some g, and the integral of (A2.6) is convergent. Then the double integral JG j K is convergent, which implies tha t JG R(y~1)p(y)dy is convergent. Therefore we obtain (A2.6) with c = j G R(y~1)p(y)dy. This completes the proof.

R e m a r k A 2 . 3 . If p is trivial, (p is nonnegative on the whole H, and (A2.5) is convergent for z = ZQ with a point ^o such tha t <p(zo) > 0, then it is convergent for every z G H. Indeed, our reasoning combined with these assumptions shows tha t fG \R(y~1)p(y)\dy < oo. Then reversing our argument, we find tha t R(y~1)f(gky) is integrable as a function of (y, k) G G x K for every g G G. Therefore, for every fixed g G G, jGR{y~l)f{gky)dy is meaningful for almost all k G K. Since the last integral is independent of /c, we obtain the desired conclusion.

252 APPENDIX

A2.4. We can naturally ask whether Theorems A1.6 and A2.2 can be generalized to the case of an arbitrary representation p. Let us now show that if 7i is hermitian and classical, then there are formulas of type (A2.5) with higher-dimensional p, applicable to holomorphic f.

We first make the following observation. Let h(w, z) be a function on H x H antiholomorphic in w and holomorphic in z. Then ft is completely determined by h(z, z). In other words, ft = 0 on the whole TL x TL if h(z, z) = 0 for every z £ H. Indeed, it is sufficient to prove this by replacing TLxTLhyUxU with a neighborhood U of 0 in C n . Put p(z, z') = h(z — ~z!, z + z'). Then p is a holomorphic function of (z, z') G V x V with a sufficiently small neighborhood V of 0 in C n . Suppose ft(z, z) = 0 for every z e U; then p(x, iy) — h(x-\-iy, x + iy) — 0 for x, y e RnflV, so that p — 0 on V x V. Thus ft = 0 in a neighborhood of (0, 0) which proves the desired fact.

Now in each hermitian and classical case there is a complex Lie group Kc and Xc-valued (canonical) holomorphic factor of automorphy Aa(z) defined for (a, z) G G xTL. This Kc is a subgroup of GLm(C) x GLn(C) with some (m, n), and stable under (x, y) i—> (x, y)*, where (x, ?/)* = (x*, y*) and x* = tx. There is also a map E :TL-^ Kc such that

(A2.7a) Aa(z)*S(az)Aa(z) = E(z) for every (a, z) e G xTL.

The explicit forms of TL and Kc for each G are given in [S94, pp. 143-144]. We put E(z) = (£(z), ^(^)) with £ and 7/ given there; we define also Aa(z) by Aa(;z) = (A(a, z), /x(a, 2;)) with A and /x of [S94, (1.2)]; see also (2.2b, c) of the same paper. In each case the entries of E are polynomial functions in ~z and z, and therefore we can define a Kc-valued function E(w, z) of (w, z) G TL x TL antiholomorphic in w and holomorphic in z such that E(z, z) — E{z). This E(w, z) is unique for the reason explained at the beginning. The same principle combined with (A2.7a) implies that

(A2.7b) Aa(wYE(aw1 az)Aa(z) = E(w, z) for every (a, w, z) G G x TL x TL.

As we said, Kc C GLrn(C)xGLn(C); so we put det0(x, y) = det(y) for (x, y)e Kc; we also put

(A2.8) 6(w, z) = deto(S(i£;, *)), S(z) = det0(S(^)), ja(z) = det0(Aa(^)).

Then ja(w)ja(z)5(aw, az) = 5(w, z). If G = Sp(n, R), for example, then Kc — {(a, a) | a G GL n (C)}, and so we can

identify i^c with GLn(C). Then E(w, z) — (i/2)(w — z) and Aa(z) = caz + da, where (ca da) is the lower half of a. We shall discuss the unitary case in §A2.8. Now the formula with higher-dimensional p for holomorphic functions can be given as follows:

Theorem A2.5. Let p : Kc —» GfLiy(C) be a holomorphic representation. Then for s G C and a hoiomorphic map / : TL —» C^ we ftave

(A2.9) / |J(ti;, ^ r ^ ^ W V ^ ^ - ^ K z)*)f(w)dw = c(s)/(z)


provided the integral is convergent, where c(s) is a constant independent of p and / , that can be determined explicitly. (In the unitary and symplectic cases, c(s) coincides with ck(s) of (A2.21a) and (A2.21b) below if we put k = O.J

PROOF. We first observe tha t every holomorphic function on H is an eigenfunction of £)(pi) , if pi is a one-dimensional representation. Indeed, a set of generators of V{pi) was given in [S90]. There are two types, but each operator of one type is of the form Mz = 0DzEw; see [S90, Theorem 3.6 and Proposition 4.1]. As explained in the last paragraph of [S90, Section 7] and also in [S94, Proposition 2.2], such an Mz corresponds to an operator on TL of the form 0Dz^azEz (which is denoted by Lp), and Ez annihilates holomorphic functions. Therefore every holomorphic function on 7i is annihilated by all such nonconstant operators, and hence (A2.5) is applicable with a constant c independent of p.

Now put Pk(z, w) = \5{w, z)\~2s5(w, z)-kS(z)sS{w)s for k G Z Then Pk satisfies (A2.1) with Ja = ja{z)k; also 5(w)k gives r{w) with r of (A2.4). Thus (A2.5) with Pk as P implies

(A2.10) / 6(w, z)~k\5(w, z)\-2s5(z)s5(w)k+sh(w)dw = ck(s)h(z) Jn

for every holomorphic function h on H when the integral is convergent, where ck(s) is a constant independent of h. Given p and a holomorphic map / as in our theorem, put g(w) — p(E{w, zo)*)f(w) with a fixed ZQ £ H. Then g is holomorphic, and so (A2.10) with k — 0 and g as h gives

/ \S(w, z)\~2s5(z)s8(wyP(Z(w, zo)*)f(w)dw = c0(s)p(S(z1 z0y)f(z).

Put t ing z = ZQ, we obtain (A2.9) with c(s) = Co(s), which is independent of p and / . This proves our theorem, except for the last point concerning the explicit form of c(s).

Since we have seen tha t c(s) — co(s), the question is the explicit form of CQ(S) in (A2.10). Before discussing it, we first note tha t (A2.10) is essentially a special case of (A2.9). Indeed, take p(x, y) = det(y)k and substi tute s + k for s; then the left-hand side of (A2.9) becomes the left-hand side of (A2.10). Therefore (A2.10) with an arbitrary k follows from the case with k = 0, and we obtain

(A2 . l l ) Cfc(s)=co(s + fc).

We note also tha t [S97, Propositions A2.9 and A2. l l ] are exactly (A2.10) in the unitary and symplectic cases, though the determination of the constant ck(s) in terms of gamma functions as given in those propositions requires some nontrivial calculations. In §A2.8 below we shall determine ck{s) by a method simpler than tha t of [S97]. The method is applicable to other types of groups, which is what is meant by the statement tha t c(s) can be determined explicitly, though we leave the details of calculation in those cases to the reader.

A 2 . 6 . Returning to the setting of §A2.1 and Theorem A2.2, we call an element <p of C°°(H) an eigenfunction of^{p) if Dip = Xip(D)p for every D e ! 9 ( p ) . Now Theorem A2.2 implies tha t formula (A2.10) is valid even for such an eigenfunction p of 2)(p), not necessarily holomorphic, if we replace Ck(s) by a certain constant

254 APPENDIX

c£(s) depending on k, s, and the eigenvalues A^(D). We are going to determine this c£(s) explicitly in the orthogonal, symplectic, and unitary cases. (In the orthogonal case, we consider the space of Section 16 which may not be hermitian; also we take S(w, z) of (16.4d) which is different from the function given in [S94]. Still we have a formula of type (A2.10) in such a nonhermitian case.) Now we determine c^(s) in the following way. Given an arbitrary eigenfunction / of £>(p), we find an eigenfunction /o of £>(p) with the same eigenvalues as / for every element of 2)(p), and such that /o(nafc) = p(fc)_1C(a) for (n, a, k) G N x A x K with a homomorphism £ : A —> C x , as we discussed in §A1.12. The existence of such an /o is guaranteed by Theorem A 1.13. Clearly £ is parametrized by C r

with r = dim(.A). Therefore the eigenvalues are parametrized by C r , and we shall determine c£(s) as a function of fc, s and the parameters in C r in Theorems A2.10, A2.12, and A2.15.

A2.7. First, to state a few elementary integral formulas, let K denote R, C, or the division ring of Hamilton quaternions. For x G K™ we put x* — fx, where the bar is either complex conjugation or quaternion conjugation; we then put x = (#*) - 1 when x is square and invertible. For h = h* G GLn(K) we write ft > 0 if xhx* > 0 for every x G K^, ^ 0. We put

(A2.12) 1 = [K : R], K = n(n) = 1 + t,(n - l ) /2 . n - l

(A2.13) r ' ( s ) = T T " * " - 1 ) / 4 J J r ( s - (tk/2)),

(A2.14) 5 = Sn = {h G K£ I x = x*}, Q = {h G 5 | h > 0}.

For a: G 5 the map x 1—> ((û)ft=i? (ûO^j) §i v e s a bijection of 5 onto R n x jgn{n-i)/2t identifying K with R^ in an obvious way, we can identify S with Rn/C, and so we have a natural measure on 5. We restrict this measure to Q. Then we have

(A2.15) / e-tr{hy) det(y)a-*dy = r^(s) det(h)~s (heQ, Re(s)>K - 1), JQ

(A2.16) / det(ft + xx*)~sdx = d e t ( / i ) ( t m / 2 ) - ^ m n / 2 r ^ ( s - (tn/2))/r^{s)

(h G Q, Re(s) > n(m + n) - 1),

(A2.17) J det(/i - ia)-a det{h + ia)~pda = TvnK det(2h)K~a-f3

•rârtr^pr'râ + p-K) (heQ, Re(a + /?) > 2 / c - l ) ,

(A2.18) J det(ln + y)"^ det(y)^dy = ^ ' ^ ^ (Re(a)>K- l , Re{p-a)>K-l).

The first formula is well known; (A2.16) is given in [S97, Lemma A2.7, (A2.7.4)]. As for (A2.17), since —i(ih+a) = h — icr, the integral can be transformed to that of [S82, (1.31)]. In view of the rule [S82, (1.11)], we obtain (A2.17). The last formula is a generalized beta integral, and can be proved as follows. By (A2.15), r^(/3) times the left-hand side of (A2.18) equals


-tr(P(i+y)) det(pf-Kdp det{y)a-Kdy Q

-tr(p) det(p) fi — K e~tr{py)det{y)a-Kdydp Q

= r^a) / e- t r^det(p)^-^ = (a)^(/3-a), JQ

which proves (A2.18). Let U = Un be the set of all upper triangular matrices in GLn(K) whose diagonal

entries are all equal to 1. We define the measure on U by the bijection u \—> (uij)x<j of U onto K71^-1^2. Let g = diag[#i, • • • , gr] with 0 < g%G R. Then

(A2.19) de t [ l n + w * H ~ a d w = nL{n-1)2/2r(s - (n - 1)^ /2) n V^.^s)-

n^-n)t /2(i+^) (n- iw2- ( R e ( s ) > ( n - 1 ) ^ / 2 ) .

This is trivially true if n — 1. The general case can be proved by induction on 1 # 1

with x G K^_ x and v G Un-\\ put also £ = 0 v gn}. Then we can easily verify tha t

n as follows. Pu t u

diag[p2, •••

de t [ l n + u*gu] = (1 + #i) d e t [ l n _ i + v*£v + c • #*#],

where c = g1/(l + gx). Thus, by (A2.16) the integral of (A2.19) equals

M w o / / ^ /o^is — (n — 1)L/2)

Un-det[ln-1+v*ev]L/2 *dv.

Applying our induction to the last integral, we obtain (A2.19).

A 2 . 8 . In this subsection we discuss Cfc(s) for holomorphic / in the symplectic and unitary cases. First, we take G to be the group

" 0 0 -ilr

(A2.20) G = SU(ip) = {ae SLn(C)\a(pa* = c^}, <p •• 0 0 Hr 0

, and a measure

with n = 2r + t, r > 0, t > 0, and 0 < 6> = 6>* G GLt(C); we let G act on the Til space 3 of [S97, (6.1.4)]. Define the origin i of 3 by i = t

r

L r diu on 3 by [S97, (6.6.7)]. Next, take z — i, k = 0, and ft, to be the constant 1 in (A2.10), and consider the map t : *B —> 3 of [S97, Lemma A2.3]. Then we find tha t Co(5) equals, up to a constant, J* £(z)sd2: with *B and (5 of [S97, (A2.1.6) and (A2.1.1c)], and the value of the integral can be obtained from [S97, Lemma A2.7]. Employing (A2 . l l ) , we obtain

(A2.21a) ck(s) = 2 n r 7 r r ^ - r ) det{0)rr2(s + fc + r - n)r?(s + A;)"1,

as already stated in [S97, Proposition A2.9]. This procedure is simpler than the proof of tha t proposition, which is somewhat roundabout .

In the symplectic case, G = Sp(n,H), the space is S) = {z G C™ | lz — z, lm(z) > 0 } , 5(z, w) = det ((i/2)(w - z)), and £(2) = det ( lm(z)) . Now

256 APPENDIX

CQ(S) = / \6(w, z)-25(z)5(w)\sdw, dw = 5{w)~n-1 \{ {{i/2)dwhkdwhk}. Jf) h<k

Putting z = iln and w = x + iy with real x and y, we obtain

co(s) = 22na / / \dei(x + i{y+ln))\~2sdxdet(yy-2*dy, JQ JS

where K = {n + l ) /2 . By (A2.17),

/ | det (x + i(y + l n ) ) |"2sda: = 2n^+1~2sKnK det(y + l )"- 2 s r , l (2s - /c)r^(s)"2 .

Therefore our problem is reduced to JQdet(y + l ) K _ 2 s det(y)s~2Kdy, to which (A2.18) is applicable. By (A2.ll) a straightforward calculation shows that

(A2.21b) ck(s) = 2n^+1)7rn*rn1(s + k - n)r^{s + jk)"1,

which differs from the constant given in [S97, Proposition A2.ll] by a factor (2i)-nk2-2ns, because det(z-w) was employed there instead of 5(w, z) here.

A2.9. Let us now consider the case of eigenfunctions that are not necessarily holomorphic. We first take our group G to be the unitary group of (A2.20), and consider an Iwasawa decomposition G = NAK, where K = {a G G | a(i) = i} , A consists of all the diagonal matrices

(A2.22) a = diag[a^\ . . . , a"1, It, ai, . . . , ar]

with 0 < ai G R, and Ar consists of all the matrices of the form

(A2.23) n = u 0 0

b U 0

c iOb*u

u ueUr, beCr

t, c=(cr + 2-1ib0b*)u, a G 5 r ,

where [/r and 5 r are the symbols of §A2.7 in the case K — C. (We use n in order to distinguish it from the size n of our matrices.) These (u, b, a) give independent parameters of N. This form of N can be obtained by first taking a decomposition G — PK with a suitable parabolic subgroup P of G which can be obtained from P j of [S97, Proposition 6.4], and then showing P = NA.

Theorem A2.10 (Unitary case). Let 3 be as in [S97, (6.1.4)]; put

P(z, w) = S(w, z)-k\5(w, z)\-2s5(z)s6{w)s ((z, w) G 3 x 3)

with S of[S97, (6.3.11) and (6.6.8)], s G C, and k G Z; define D(p) as in §A1.1 with 3 = 7i and Ja(z) = ja{z)k, where ja(z) is the standard scalar factor ofautomorphy of [S97, (6.3.10)]. Let f0 be the function on 3 such that /£(na) = n [= i aT for

n E N and a G A as in (A2.22), where ^ G C. Then /o is an eigenfunction of D(p). Moreover, if f is an eigenfunction ofV(p) with the same eigenvalues as /o, then (A2.24) [ P(z, w)S(w)kf(w)dw = det(6)rck(s, {az})f(z) with

ck(s, {al})=2^7r^-l)r2(s + k)-1r2(s)-1

rls-r + i+——^\rls + r-n + l-t+——


provided the integral is convergent. For / = /0 , the integral is convergent in any right half plane on which co(s + (A;/2), {&{}) is finite. If f is an eigenfunction such that 5(w)k/2f(w) is bounded on the whole 3, then the integral is convergent for Re{s) > n - l - ( f e / 2 ) .

Before proving this, let us insert a remark. This includes the formula for holo-morphic / . Indeed, as explained in the proof of Theorem A2.5, all holomorphic functions belong to the same eigenvalues of 33 (p). Thus we can take / 0 = 1. Then /o(na) = ni=i a r f c> a n d therefore we obtain Cfc(s, {c^}) for holomorphic / by taking c^ = — k for every z, which gives exactly (A2.21a).

PROOF. AS we said in the proof of Theorem A2.5, P satisfies (A2.1). We may assume that 0 = l t , as the problem can be reduced to that case. Now, f^(nak) — p(k)~1fQ (a) = p(k)~x n r= i a?N a n d hence by Lemma Al . l l , f£ is an eigenfunction of 23(p); therefore by Theorem A2.2, (A2.24) holds with a constant ck (s, {cti}) when the integral is convergent. For the moment we assume the convergence, to which we will turn at the end. Thus our task is to find the explicit value of Cfc(s, {c*i})-Since /o(i) = <5(i) = 1, we have

ck(s, {a2}) = J 5(w, i)-k-sS^J)~SS(w)s^kfo(w)dw.

Recall the formula fGp(g)dg = fAfNfKp(ank)dadndk for an integrable continuous function p on G. Therefore, if q is an integrable continuous function on 3, then

/ q(w)dw = A / / q(ani)dadn «/3 J A JN

with a positive constant A that depends on the choice of various measures. Since /o(ani) = ja n(i) f c /o(an) and /£(an) = fSiana^a) = /o(a)> w e n a v e

ck(s, {a,}) = A / / <5(ani, i)~fc-s(5(ani, i) ~85(aidy+kja(i)kfg(a)dadn. J A JN

By [S97, (6.3.7), (6.3.10), and (6.3.13)], S(ani) = \j(an, i) |~2 - n [ = i ^ " 2 . By a straightforward calculation we find, for a of (A2.22) and n of (A2.23), that

2r£(ani, i) = n [= i <h2 ' d e t 0 + i(J) w i t h V = h2 + mx - 1 + 2~166*,

where h — diag[ai, . . . , ar\. Thus the last double integral fAJN can be written />oo /»oo r /*

2r(2s+fc) / . / TTaf*"1 / det(p + ia)-k-sdet(p-ia)-sd(u, a, 6)dai-- .da r ,

where ^ = c^ + 2s + k. By (A2.17) the last integral over N equals

B(s) [ det [h2 -f m T 1 + 2-1bb*]r~k~2sd(u1 6),

where B(s) is determined by that formula, and M is the space of (u, b). By (A2.16) the last integral equals

C(s) [ det(h Ju

2 ^^-l^r-k-28 du

258 APPENDIX

with C(s) determined by tha t formula. Since det(/i2 + uu x) — de t ( l + u*h2u), formula (A2.19) reduces our problem to

D(s)ll (1 + a1)n-k-l-2s afs+i-r)+k+a'-ldat

with D(s) determined by tha t formula. This equals D(s)E(s) with

r(s - r + i + (k + al)/2)r(s + r - n + 1 - i + (k - a , ) / 2 ) £(5)=2^n r ( 2 s + /c + l - n )

In this way we have shown tha t

(*) ck(s,{ai})=F(s)f[r(s-r + i + ^ ^ \ r ( a + r - n + l - i + ^

with a function F independent of the a^. This is valid if Re(s) is sufficiently large. To determine F, we take /o = 1, in which case ai — —k for every ?', as remarked just after the statement of the theorem. This is the case for holomorphic / , and therefore Cfc(s, {—k}) must coincide with the constant Cfc(s) given in (A2.11a). Therefore, comparing it with (*) for c^ = —h, we find an explicit form for F , which gives Cfc(s, {c^}) as stated in (A2.24).

To discuss the convergence, put s' = s + /c/2. Then our problem is the convergence of

(**) j \S(w, z)\-2s'S(Wy'g(w)dw, g(w) = \S(w)k^2f(w)\.

We first take / = / Q . Then ^(nai) = n [ = i a i e , and so g is of the same type as /o; thus the integral times (5(^)s is a special case of the integral of (A2.24) with k = 0. As we said in Remark A2.3, the integral is convergent if it is convergent at a point z = zo such tha t g(zo) > 0. The above calculation shows that the integral of (A2.24) is convergent at z — i for sufficiently large Re(s) , and consequently (**) is convergent for every z G 3 on the same domain of s'. Now, the principle of [S97, Lemma A1.5] (see also [S99c, Lemma 2.1]) shows tha t (**) is convergent for s' in any right half plane on which CQ(S'', {Re(c^)}) is holomorphic, and hence we obtain our last assertion about the convergence when f = f0.

Next, take / to be an eigenfunction such tha t 5k/2f is bounded. Then we replace g in (**) by the constant 1. For tha t choice of g the integral becomes co(s ' ,{0}) , which is finite for Re(«s/) > n — 1. Therefore the original integral of (A2.24) is convergent for Re(s) > n — 1 — (k/2). This completes the proof.

A 2 . l l . Let us next consider the orthogonal case by taking our setting to be tha t of §16.1 with r > 0. We define SO^ by (16.1), put G = SO^ for simplicity, and consider an Iwasawa decomposition G — NAK1 where K is C\ of Proposition 16.6(h), A consists of the matrices of (A2.22) with 0 < ai G R, and N consists of all the matrices of the form

(A2.25) n = u b c 0 lt 0 • t\m 0 0 u

ueUri 6 G R tr , c=(a-\-2~1be-tb)u, a G S'


where Ur is the group of §A2.7 in the case K = R and S' = {a G R£ | V = — a}. These (u, 6, a) give independent parameters of N. This form of N can be obtained by first taking the decomposition G = P*C\ as in Proposition 16.6 (ii), and by showing that P£ = AN. (Our groups G and K are not connected, but still we have G — NAK with connected N and A, and so we call it an Iwasawa decomposition; we can even replace them by O^ and C of (16.9), which causes no problems.)

Theorem A2.12 (Orthogonal case). Define the space Z and the point 1 by (16.3b) and (16.9). Let P{z, w) = \8(z)5(w)5(z, w)~2\s for (*, w) G Z x Z with 8 of (16Ad) and s G C; let 55 denote the ring of all G-invariant differential operators on Z, that is, 35(p) of§Al.l with trivial p, and let /o be the function on Z defined by /o(nal) = n[= i aT for n e N and a e A as in (A2.22), where a* G C. Then /o is an eigenfunction of®. Moreover, if f is an eigenfunction of 25 with the same eigenvalues as /o, then

(A2.26) / P(z, w)f(w)dw = det{6)r/2c(s, {oti})f{z) with

c(s, {a,}) = 2^/27r^-1)/2rr1(5)-1rr

1(5 + (i/2))_1

.n^(^a i+2

i"r)^(^-^+i+2

n"r"2

provided the integral is convergent. For / = / 0 , the integral is convergent in any right half plane on which this expression for c(s, {on}) is finite. If f is an eigenfunction bounded on the whole Z, then the integral is convergent for Re(s) > (n - 2)/2.

PROOF. This can be proved more or less in the same manner as for Theorem A2.10. However, we have to modify the argument nontrivially at two points. The first point is that a of (A2.25) belongs to the set Sf of alternating matrices, not to S of (A2.14). Therefore, instead of (A2.17) we need

(A2.27) / det(a + p)-*°da = 2W<*<* d e t ^ - D / 2 - 2 * ffi2 « " f " 1 ) / 2 > JS' "r (s) "r (5 + V 2 )

( R e ( 2 s ) > r - l ) , where 0 < p = tp G R£, and da — Yli<j d&ij- This is given in [S99c, (2.13), Case I]. Notice that (A2.27) is true even when r = 1, in which case Sf = {0}. Eventually we find that

(#) c(s, {„ i }) = F(S)f[r(s + ^ ^ ) r ( s - *±i+L±±zl)

with a function F independent of the a^, provided Re(s) is suffciently large. To obtain the explicit form of F, we use the formula

(A2.28) Jz \6(W)S(z)6(w, z)->\'dw = 2 " " V < « + ^ ^ ^ ^ y ^

valid for Re(s) > (n — 2)/2 and 9 = lt. Once this is known, we see that the right-hand side is c(s, {0}). Therefore, comparing it with (#) with a^ = 0, we find that

F(S) = 2rn/27rr(n-1)/2rr1(5)"1rr

1(5 +1/2)-1,

260 APPENDIX

which proves (A2.26). As for the convergence, what we said in the unitary case is valid in the present case. To prove (A2.28), we can take z = 1. Put w — t(z) with t of (16.15b) and z in B of (16.14c). Employing (16.16), we find that the integral over Z equals 2 r n / 2 J B det ( l g - y • *y) s"n /2dy. By [S97, Lemma A2.7] the last integral over B equals 7rqr/2r*(s - (q - l)/2)r*(s + 1/2) -1 . Thus we obtain (A2.28).

Corollary A2.13. Suppose r = 1 in the setting of Theorem A2.12; let f be a C°° function on Z such that Cf = a(a — 2~1t)f with a G C, where C is defined by (16.18). Then we have

(A2.29a) / \5{w)S(z)5(w, z)~2\sf{w)dw = det((9)1/2c(s, a)f(z) with

r(s-a)r(s + a-(t/2)) (A2.29b) c(s, a) - 2™/V n - 1 ) / 2 •

r(,)r( , +(i/2)) provided the integral of (A2.29a) is convergent, which is the case if S(w)Tf(w) is bounded on Z with r G R and Re(s) > Max{—r, r -f 2~1t}.

P R O O F . Take fQ(z) = S(z)(J in Theorem A2.12. This means that OL\ = -2a . Using the explicit form of C given in (16.18), we find that Cfo — a (a — 2~1t)/o-Therefore we obtain our formula as a special case of Theorem A2.12. If S(w)T f(w) is bounded on Z, then the convergence can be reduced to the case /o — $(w)~r, and so we obtain the condition as stated.

Put A = G(G — 2 _ 1 t ) ; then both A and c(s, a) are invariant under the transformation a i—• 2~lt — a. Thus, c(s, a) is a function of A.

A2.14. Let us now take G = NAK with G = Sp(n, R), K = SO(l2n) H G and A, AT determined as follows: A consists of the diagonal matrices

(A2.30) a = diag[a^\ . . . , a"1, ai , . . . , an]

with 0 < ai G R, and AT consists of all the matrices of the form u bu 0 u

where Un and Sn are as in §A2.7 with K = R. We let G act on 9) = {z G C™ | tz = z, Im(z) > 0} as usual, and put 5(w, z) — det [[i/2){w — z)), <5(z) = det (lm(z)), and jQ(^) = det(caz + da) for 2; G f) and a G G, where [ca da] is the lower half of a. We define a measure dz on 5} by dz = 8(z)~n~1 Yih<k {(i/2)dzhkd~Zhk}'

Theorem A2.15 (Symplectic case). Put

P(z, w) = 6(w, z)~k\5(w, z)\-2sS(z)sS(w)s ((z, w) G S) x $))

with s G C and k G Z; define £>(p) as in §A1.1 with S) = H and Ja(z) = ja(z)k, Let /o be the function on ft such that /0

p(na) = fllLi aT for n E N and a £ A as in (A2.30), where cti G C. Then /o is an eigenfunction ofD(p). Moreover, if f is an eigenfunction of 2) (p) with the same eigenvalues as /o, then

/ P(z, w)5{w)kf(w)dw = ^ ( s , {ai})/(z) with

(A2.31) n = ueUni be Sn,

A3. STRUCTURE OF CLIFFORD ALGEBRAS OVER R 261

Cfc(s, {at}) = 2^+1)7r-2rn1(5)-1r?l(5 + ^ ) - 1

n ( k + ai+i-n\ f k - a% - i - n\

i = A 8 + ^ )r{S+ 2 J' where n = (n -f- l ) /2, provided the integral is convergent. For / = /0 , the integral is convergent in any right half plane on which co(s + (k/2), {c^}) is finite. If f is an eigenfunction such that 5(w)k/2f(w) is bounded on the whole fj, then the integral is convergent for Re(s) > n — (k/2).

PROOF. This can be proved in exactly the same fashion as for Theorem A2.10. For the same reason as explained after Theorem 2.10, the case of holomorphic / is included as a special case with cti = —k for every i.

In this section we treated the orthogonal, symplectic, and unitary groups of the above types, but our methods are applicable to other types of groups such as Sp(n, C) and SO(n, C). The necessary integral formulas are given in [S99c, Section 7], in addition to what we quoted from [S82] and [S97].

A3. Structure of Clifford algebras over R

A3.1. In this section we study the structure of A(V, <p) and A+(V, if) when the basic field is R. We begin with some elementary facts on involutions of an algebra. Let A be an algebra with identity element over a field F of characteristic ^ 2, and <T an F-linear involution of A. For a vector space X over F denote by dim/? X the dimension of X over F, and put

(A3.1) d+(<r) = dimF {xeA \xa = x), d-(a) = dimF {xeA | xa = -x}.

Then d+{a) + d_(a) = [A : F}.

Lemma A3.2. With A and o as above, let xp = sx°'s~l with an element s of Ax such that sa = es, e = ±1. Then p is an F-linear involution of A. Moreover, d±(cr) = d±(p) if e — 1 and d±(cr) = d^(p) if e = —1.

PROOF. The first assertion can be verified in a straightforward way. Next, for x G A we have xp = ±x if and only if (xs)a = ±sxs, and hence we obtain the equalities about d±.

Lemma A3.3. (i) If A = Mn(F) and xa = lx for xeA, then d±(a) = n(n ± l ) /2 .

(ii) If A — Mn(B) with a quaternion algebra B over F and xG — lxl for xeA, where i is the main involution of B and {xij)L — (x^), then d±(cr) = n(2n =F 1).

These are easy exercises.

A3.4. We now take (V, ip) with nondegenerate <p. Throughout the rest of this section we put n = dim(y). We consider the canonical involution * of A(V); we can thus define d+(*). Since A+(V) is stable under *, we can also define d+ on A+(V), which we denote by d+(*). If {ei}^=1 is an orthogonal basis of V over F, then (ei • • • er)* = (-l)r^r-1^2e1 • • • er. Thus

(A3.2a) ^(*)= E ( i ) + S 0<k<n/4 ^ ' 0<fc<(n- l ) /4

Ufc+i)'

262 APPENDIX

(A3.4) dPM

(A3.2b) d°(*) = J2 (ll)-

To compute these values, we start with the equality

2 J2 ( 2k ) ^ = (1 + X)n + (1 " X)U' 0<k<n/2 ^ '

Substituting ix for x with z = A / ^ T and adding the new formula to the original one, we obtain

(A3.3) 4 J2 (lb) xAk = C1 + x ) n + i1 ~ x ) n + (X + ' ^ ) n + C1 " ^ ) n -0<k<n/4 ^ '

Put t ing x = 1, we obtain the value of (A3.2b). Since (1 + i)2r = (2i)r and {l-i)2r = ( - 2 i ) r , we have

{ (_^n/42(n+2)/2 [f n = Q ( m ( ) d ^

0 if n = 2 (mod 4), ( - l ) ^ 2 - 1 ) / ^ 4 " 1 ) / 2 if n = 1 (mod 2).

Therefore, by (A3.2b) and (A3.3) we obtain

2n~2 + ( - l ) n / 4 2 ( n - 2 > / 2 if n = 0 (mod 4), 2 n ~ 2 if 72 = 2 (mod 4), 2n~2 + ( _ i ) ( - 2 - i ) / 8 2 ^ - 3 ^ 2 if n = 1 (mod 2).

Similarly, taking (1 + x ) n — (1 — x)n instead of (1 + x)n + (1 — x)n, we obtain (I + x)n - (1 - x)n - i(l + ix)n + i(l - ix)n = 4 ^ ( n

Adding this to (A3.3) and put t ing x = 1, we find tha t ( 271'1 + (_l)(n-D/42(n-l)/2 j f /? = x ( m o d 4^

2 n ~ 1 if 7i = 3 (mod 4),

2 n - i + 2(n-2)/2 if n = 0 or 2 (mod 8),

( 2 7 1- 1 - 2 ( n ~ 2 ) / 2 if n = 4 or 6 (mod 8).

A 3 . 5 . Let A be an F-algebra with Q or R as F , and a an F-linear involution of A. We call a pos i t i ve if TT(xxa) > 0 for every x G A, 0, where Tr is the reduced trace map A —> F . We can show tha t an algebra with a positive involution is semisimple, but we do not need this result, since we deal only with semisimple algebras. Let A — © t G / Ai with simple F-algebras A% and let a be a positive involution of A. If 0 ^ x G Ai and x a G Aj with j ^ i, then x x a = 0, a contradiction, as Tr(xi;(T) > 0. Thus each Ai is stable under cr. In this way any problem about a positive involution of a semisimple algebra can be reduced to tha t of a simple algebra.

Now take F to be R . Then there are three types of simple algebras over R : M n ( R ) , M n ( C ) , and M n ( H ) , where H denotes the Hamilton quaternions. Let a be a positive involution of one of these. If the algebra is A/ n (C) , a cannot be the identity map on C, since Tr(x2) < 0 for some x G C. Thus a must be complex conjugation on C.

(A3.5) d+(*) = {

A3. STRUCTURE OF CLIFFORD ALGEBRAS OVER R 263

To treat all cases uniformly, we let K denote any of these three division rings R, C, and H, and denote by x the complex conjugate or the quaternion conjugate of x G K; we have x = x if K = R. Extend this to Mn(K) by putting x = (xtj) for x = (xij). Then we easily see that x •—» fx is a positive involution of i l /n(K).

Now let a be an involution of Jl/n(K); we assume that a gives complex conjugation on C if K = C. Let us now show that xa = a • % _ 1 for every x G Mn(K) with an element a of GLn(K) such that *a = ea with e = ±1 if K = R or H and e = 1 if K = C. Indeed, given c\ we see that x i—• ^xY is an automorphism of Mn(K) over its center, and hence is an inner automorphism. Thus xa = a • ^ a " 1

for every # G il/n(K) with an element a of GLn(K). Since (x(J)(J = a:, we see that taa-1 commutes with every element of il /n(K), so that fftQ_1 = c with c G R if K is R or H; taa~1 = c with c G C if K = C. From the relation la = ca we obtain cc = 1. Thus c = ±1 if K = R or H. If K = C, we can put c — b/b with b G C x . Replacing a by ba< we may assume that *a = a, which proves the expected fact.

Lemma A3.6. Let xa = a - lxa~l with a G GLn(K) and e as above. Then G is positive if and only if e = 1 and a is positive or negative definite.

PROOF. TO prove the direct part, suppose e = 1 and a is positive definite. Then we can find a positive definite 8 = *5 such that a = S2. Given x G M n (K) , ^ 0, put z = S^xS. Then Tr(xxa) = T r ( 5 " 1 ^ 2 • 'xJ" 1 ) = Tr(^ • *z) > 0, so that a is positive. If a is negative definite, we obtain the same conclusion by taking S2 = -a.

We prove the converse part only in the case K = H, as the other two cases are similar and easier. Let 7 = pa • */? with (3 G GLn(H). For y = p~1x/3 with x G i\/n(H) we have j5yyaj3~l = #7 • *X7_1, so that cr is positive if and only if the involution x H-> 7 • ^ary-1 is positive. Therefore the nature of a does not change by replacing a by pa • */?. Now suppose la = —a. Let i, j , A; be the standard quaternion units. It is well known that there exists p G GLn(H) such that Pa-lp = j l n ; see [S99c, Lemma 3.2], for example. Thus we may assume that a = jln. Then for x = iln we have xxa = — l n , so that TY(xxa) < 0. Therefore ta = a if cr is positive. Suppose a is not definite; then t:uau = 0 for some u G H™, ^ 0. Put y = U'lu. Then y = 0 and Tr(ycv • tya~1) = Tr(u • ^ a w • tua~1) = 0, and so a is not positive. This completes the proof.

Lemma A3.7. Let A he a semisimple algebra over R with a positive involution a. Then there exists an H-linear isomorphism f of A onto 0 i G / B^ with Bi of the form M n (K), such that f{xCT) = {lai)i€i if f{x) = (a^)^G/.

PROOF. We have seen that every simple component of A is stable under cr, and hence we may assume that A is simple and isomorphic to Mn(K). By Lemma A3.6 we have xa = a • lxa~l with a positive definite hermitian element a, changing a for — a if necessary. We can find a hermitian element p such that p2 = a. Put f(x) = p~1xp. Then f{xG) = tf(x), which proves our lemma.

Lemma A3.8. For (V, tp) with nondegenerate if defined over R the following assertions hold:

264 APPENDIX

(i) The canonical involution of A(V) gives a positive involution on A+(V) if (f is positive or negative definite.

(ii) If <p is positive resp. negative definite, then the map x H-» x* resp. x i—> (V)* is a positive involution of A(V).

P R O O F . In view of Lemma 3.8, it is sufficient to prove (i) when cp is positive definite. The structure of A+(V) described in Theorem 2.8 shows that if we denote by T+(a) the trace of the image of a G A+(V) under the regular representation over R, then T+(a) = cTr(a) with 0 < c G Q x . Therefore it is sufficient to prove that T+(xx*) > 0 for 0 / x G A+(V). Our proof is by induction on n. The one-dimensional case is trivial. Suppose n > 1; let V = [7 + R e with an element e such that e2 - 1 and U = (Re) 1 . Then A+{V) = A+(U) © A~{U)e. For 7 G A~(U)e, right multiplication by 7 sends A+{U) resp. ^4~({7)e into A~(U)e resp. T4+(17). Therefore T+(7) = 0 for every 7 G A"(L7)e. Let x = a + 6e with a G -A+(£7) and 6G A" (C7). Then xx* = (a + 6e)(a* + e6*) = aa* + 66* + (6a* -a6*)e and 6a*-a6* G A~(C7), so that T+(xx*) = T+(aa* +66*). Take an element g of 7/ such that g2 = 1. Then 6# G +(77) and 66* = (bg)(bg)*. Since A+(V) = A+(/7)©v4+(^)^e, a regular representation of A+(V) restricted to A+(U) is twice a regular representation of A+{U). Therefore by induction, T+(aa*) > 0 and T+(66*) = T((bg)(bg)*) > 0. If x ^ 0, then either a ^ O or 6 7 0, so that T+(xx*) > 0. This proves (i).

As for A{V), we first assume that p is positive definite and take an element h of V such that h2 = 1. Then A~{V) = A+{V)h and A(V) = A+(V) 0 A+(V)/i. Let T denote the trace of a regular representation of A(V). Then T(a) = 2T+(a) for a G ^ + ( ^ ) and T(c) = 0 for c G A"(V). Let x = a + 6/1 with a, 6 G ^ + ( V ) . Then xx* = aa* + 66* + d with d = a/16* + bha*. Since d G A~(V), we have T(xx*) = 2T+(aa* + 66*), from which we obtain the desired conclusion. Suppose if is negative definite; take h so that h2 = —1. Then for x — a + bh with a, 6 G A+(V) we have x(x')* = aa* + 66* + e with e = 6/ia* — a/16*, and obtain the same conclusion. This completes the proof.

A3.9. Let us now determine A+(V) when F = R and p is positive or negative definite. By Lemma 3.8, we may assume that p is positive definite.

(I) Suppose n is odd; put n = 2p + 1 and m = 2 P _ 1 . By Theorem 2.8, A+(V) is a central simple algebra over R, which is isomorphic to M2m(R) or M m (H) . In the former case, combining Lemma A3.3 with Lemma A3.6, we have d+(*) = m(2m + 1); similarly, we have d+(*) = m(2m — 1) in the latter case. Comparing this with (A3.4), we obtain

(A3.6) A+{V) M 2 m (R) if ra = ±1 (mod* Mm (H) if n = ±3 (mod

(II) Suppose n is even; put n = 2p, m = 2P~2, and 2; = h\- — hn with an orthogonal basis {/ii}™=1 of V as in Lemma 2.7; we take h2 = 1 for every i. Then ^2 = ( - I ) P . By Theorem 2.8, A+(V) is isomorphic to M2 m(C) if p is odd. If p is even, A+(V) is the direct sum of two central simple algebras over R, which are isomorphic to M2m(R) or M m (H) . By Lemmas A3.3 and A3.6 we can compute d+(*) in each case. Comparing the result with (A3.4), we obtain

A4. AN EMBEDDING OF G ^ V ) INTO A SYMPLECTIC GROUP 265

(A3.7) A+(V) ^ I ' Af2m(R) © M2 m(R)

M2 m(C) . M ^ H l f f i M ^ H )

if n = 0 (mod 8), if n = 2 (mod 4). if n = 4 (mod 8).

(A3.8) A(V) =* {

A3.10. Let us now determine the structure of A(V) when p is positive or negative definite. We first assume that n is odd. Take an orthogonal basis {hi}f=1 of V such that hf = e where e = 1 or —1 according as p is positive or negative definite; put z = hi - — hn and £ = R + Rz. By Theorem 2.8, £ is the center of A(V), A(V) = A+(V) (g)R £, and z2 - £n(_1)n(n-i)/2> T h u g £ ig i s o m o r p h i c t o c

or R 0 R depending on (s, n). Therefore from (A3.6) we can easily derive [ M2 m(R) 0 M2 m(R) if ra = e (mod 8),

M2 m(C) if n= -£ (mod 4). Mm(H) 0 Mm (H) if n = be (mod 8).

where m = 2 ( n _ 3 ^ 2 . Let us next assume that n is even and put p = n/2 and ra = 2 P _ 1 . Then, with

5 as above, we have f M2 m(R) if p = 0 or s (mod 4),

<A 3 '9> ^ ' " { M ^ H , i ( p = 2 o , - £ ( m o d 4 ) . Indeed, A(V) in this case is central simple over R by Theorem 2.8 (i), and hence isomorphic to M 2 m(R) or M m (H) . Suppose £ = 1; then by Lemmas A3.3 and A3.7, d+(*) is ra(2ra + 1) of ra(2ra — 1) accordingly. Comparing this with (A3.5), we obtain (A3.9) when e = 1.

Next suppose e — — 1; put xCT = (a/)* for x G ^4(V). Since x a = — x* for x G J 4 ~ ( V ) , we easily see that

(A3.10) d+(*) + d+(a) = 2d!j_(*) + 2 n " 1 .

Combining this with (A3.4) and (A3.5), we obtain d+(a). Now cr is positive by Lemma A3.8. Therefore, by the same type of argument as in the case e = 1, we obtain (A3.9) for e = - 1 .

A3.11. The structure of A(V) and A+(V) with p of an arbitrary signature can be reduced to the case of definite p as follows. We take a Witt decomposition

r (A3.ll) V = X + J](Re2 + Rfi)

2 = 1

as in (1.2a) with a subspace X on which </? is definite. By Theorem 2.6, A(V) is isomorphic to MS(A(X)) with s = 2 r; also A+(V) is isomorphic to MS(A+(X)) with s = 2 r i f l ^ {0} and to M s / 2 (R) 0 M s / 2 (R) if X = {0}. Since A{X) and A+(X) are determined by (A3.6), (A3.7), (A3.8), and (A3.9), this settles our question about A(V) and A+(V) with <p of an arbitrary signature.

A4. An embedding of Gl{V) into a symplectic group

A4.1. We discussed in §15.14 an isomorphism of 5p(2, R) with Gl{V, p) when F = R and (p is of signature (2, 3), and also mentioned an embedding of GX{V) into Sp(/x, R) in a more general case. Let us now explain these in detail. We

266 APPENDIX

take our setting to be that of §15.1 with n > 5. Thus V = U + R^ + R K and U = T + Re. Put xa = e~1x*£ for x G A(U). This is an involution of A{U). Let us now prove that a gives a positive involution of A+(U). Given x G A+(U). we can put x = a + be with a G A+(T) and b G / l _ ( r ) . Denote by tr the trace of a regular representation of A+(U). Then xx*7 = (a + be)(a* — £&*) = aa* — bb* + c£ with c G A~(T), so that tr^xx*7) = tv(aa* — bb*) for the same reason as in the proof of Lemma A3.8. We can find an element 77 of T such that if = — 1. Then bi] G A+(T) and {bi])(br))* = -66*, so that t^ra*7) - tr (aa* + ( M ( M * ) > °> s i n c e

a 1—• a* is a positive involution of A+(T) by Lemma A3.8 (i). Next suppose that n — A = ±1 (mod 8). Then for every subspace A" of T of

codimension 1, by (A3.6), A+(X) is isomorphic to M^/2(R) with p = 2 ( n _ 3 ) / 2 . Since X is a core subspace of U, we see that A+(U) is isomorphic to M/1(R) as explained in §A3.11. By Lemma A3.7 we can identify A+(U) with i l / ^R) so that xa = lx for every x G .A+(£/). Put /7c = U 0 R C and extend a C-linearly to A(UC). Then we can identify A+{UC) with M^(C) so that ;ra = *x for every x G T4+(£/C)- We easily see that (eu,)<T = eu for every u G /7c-

Define E by (2.14) with £ as g, and f) by (15.4a). In §15.14 we have seen that E(Gl(V)) C Sp(/.i, R) . We are going to discuss the action of Gl(V) on 9) defined in §15.6 in connection with the action of Sp(/j,< R) on the space

(A4.1) H = {w G A/M(C) I *w = w< lm(w) > 0}.

For £ G 5p(^, R) and w G W we define £(w) G 7Y and a GL^(C)-valued factor of automorphy X^(w) as ususal; to be precise, we have

(A4.2) £ 1 A*(«0-

Let us now show that the actions are consistent in the sense that

(A4.3) £$ C W, (A4.4) ze9) and £ = ~(a) with a e G ^ V )

= * ? ( « ) = £•"(=) and X^sz) = Xa(z).

Let £ = E(a) with a € G1(V) and z £ Sj. Then from (15.6) we obtain

(A4.5) £

Take in particular c

(A4.6)

£ 0 0 1

ie. Then

1

#(a ) 1

• a(ie) 1

e-a(c) 1

Aa(fe).

Aa(s).

By Theorem 15.7 (6) every point of 55 is of the form a{ie) with some a G G 1 (y) . Comparing (A4.6) with (A4.2) for w = i l^ , we find that e • a(z^) = £(il^) G W, which proves (A4.3). Thus ez G 7Y. Therefore, comparing (A4.5) with (A4.2) for w = ez, we obtain (A4.4). If n = 5, then E^G1^)) = 5p(2, R) as proven in §15.14, and so we have eS) = H, from which we obtain (15.19a).

As for scalar factors of automorphy, from Lemma 3.18 we obtain

(A4.7) det (Xc(ez))=iy(Xa(z)Y

A4. AN EMBEDDING OF G1^') INTO A SYMPLECTIC GROUP 267

where m = 2^^~2. We can embed Gl(V) into a symplectic group over R and prove analogues of

(A4.3) and (A4.4) even when n — 4 ^ ±1 (mod 8). We can also find such an embedding over a number field. We will not treat these topics here, as they require lengthy preliminaries.

A4.2. If n = 5, there is a simple way of obtaining a quaternion unitary group, and in particular 5p(2, F) , as Gl{V) for Lp not necessarily isotropic. This can be done purely algebraically as follows. We take (V, </?) given by V = F ® B with a quaternion algebra B over F and y?[(a, b)] = a2 + dbb1 for (a, b) G F ® B, where d is a fixed element of F x and t is the main involution of B. Define a map q : V -+ M2{B) and p : V -> M2{B) x M2(B) by

db' (A4.8)

(A4.9) p(s) - (g(x), -q(x)) (x G V).

We have clearly p(x)2 = ip[x] for every x e V. The elements q(x) for all a? € V generate M2(B). Indeed, we have seen in §7.4, (B) that the elements q(0, b) for all b G B generate M2(B). By the definition of A(V), there exists an F-linear ring-homomorphism of A(V) into M2(B) x M2(B) such that f(x) = p(x) for every x G V. Now / maps ^ (V^) into {(y, y) | y G A/2(^)}- Since -A+(V) is a simple algebra by Theorem 2.8 (ii), / is either 0 or an injection. Since / ^ 0 and dim (KA+(V)) — 24, we have f(A+(V)) = {(y, y) | y G A/2(B)}. From (A4.9) we see that f(A(V)) contains an element of the form (z, —z) with z G Bx, and hence we see that both (y, y) and (y, —y) belong to /(A(V)) for every y e B. Therefore we have /(A(V)) = M2(B) x M2{B). In view of the dimension of /(A(V)) we can thus put

(A4.10) A(V) = M2(B) x A/2(B), A+(V) = {(y, y) | y G A/2(J5)}.

A4.3. For a p q v s

pL rL

ql sL , and put also a* G M2{B) put V =

5 • taLS~1 with ^ = diag[d, 1], where d is the element fixed in §A4.2. Then the canonical involution of A(V) can be given by (a, /?)* = (a*, /?*), as this is so for (a,/?) Gp(V). Put

(A4.ll) G = {crG GL2{B) \ aa* G F x }.

Let us now show that G+(V) is isomorphic to G, or more precisely,

(A4.12) G+(V) = {(a, a) | a G G} and i/((a, a))

To see this, we first observe that aaT

(A4.13) q{V) = {x G A/2(B) | x* = x, tr(x) = 0},

where tr : M2(B) —» F is the reduced trace. Every element of A+(V)X is of the form (a, a) with a G GL2{B). This belongs to G+(V) if and only if a~lq(V)a = q(V). By (A4.13), this is so if and only if (a~1xa)* = a~lxa for every x G q{V), which is so if and only if aa* commutes with every x G q(V). Since q(V) generates AI2(B), we obtain (A4.12).

268 APPENDIX

We can state (A4.12) in the following way. We simply identify G+(V) with G in an obvious way, and define the map r : G+(V) —• SO^(V) by

(A4.14) q(x)r(a) = q(a~1xa) (x G V, a G G).

Then the spin group is given by

(A4.15) Gl(V) = {ae GL2(B) \ aa* = l } ,

which is a quaternion unitary group. Take in particular B = M2(F) and d = 1. Then yL = j • ^ j - 1 for y G M2(F)

as in §1.10. Viewing a with j = a b c d G M2(B) with a, 6, c, d G

1 0 M2(F) as an element of M^{F) with no rearrangement, we see that a* = Jo otJ^ , where Jo = diag[j, j] and *a is the transpose of a as a matrix of size 4. Thus

(A4.16) G+(V) = {aeGL4(F)\aJ0-ta = u(a)J0 with ^ ( ce )GF x } .

This is the group of similitudes of Jo, and Gl(V) becomes the symplectic group defined with the alternating matrix J0.

A5. Spin representations and Lie algebras

A5.1. We take (V, </?) over F, and assume for the moment that F is algebraically closed; we put n = dim(V) as before. We are interested in the representation of G1(F) obtained from its injection into A+(V). Since F is algebraically closed, the core dimension of (V, ^ is 0 or 1. Clearly G+(V) = FxGl(V), and hence T(G1(V)) =SO^.

First suppose n = 2r + 1 with 0 < r G Z; then, taking dim(X) = 1 in Theorem 2.6, we see that A+(V) is isomorphic to Mrn(F) with m = 2 r . Next suppose n = 2r G 2Z. Then by the same theorem with AT = {0}, A+(V) is isomorphic to Mm/2(F) x M m / 2 ( i ? ) . Thus the injection Gl(V) -> A+(V)X defines an injective homomorphism

x (GLm(F) if ra = 2r + l, (A5.1) „ : G (V) - > | GL^{F) x G ^ / 2 ( F ) .f n = ^

where m = 2r. This a; is called the spin representation of G?1(V). If n = 2r, we put u(a) — (u;i(a), ^ ( a ) ) with two homomorphisms CJI, uo2 : G1(V) —• GLm/2(F). To state our theorem on a;, we fix a Witt decomposition V = Z + E^i(Fez + F / , ) with dim(Z) < 1.

Theorem A5.2. (i) The spin representation is irreducible in the sense that uo(Gl{V)) spans Mm(F) or Mm/2(F) x Mrn/2(F) according as n is even or odd.

(ii) Given ai, . . . , a r G C x , there exists an element a of G1(V) with the following two properties: (1) eir(a) = a~2e^ and far (a) = a 2 ^ for 1 < i < r and zr(a) — z for every z G Z\ (2) if n = 2r + 1, then UJ{OL) is a semisimple matrix whose characteristic roots are a^1 • • • a£

rr with all elements (ei)\=1 G { ± l } r ;

if n = 2r, then the two matrices uJk{®) are semisimple, and the characteristic roots

A5. SPIN REPRESENTATIONS AND LIE ALGEBRAS 269

of uJk(a) are given by {a\x • • • a%r | (£i)ri=1 G Ek} with subsets E\, E2 of { ± l } r such

that {±l}r = E1U E2l and fe)Li £ Ei lf ( ~ ^ ) L i ^ E2. PROOF. Assertion (i) means that ou(Gl(V)) spans A+(V) over F. Both (i) and

(ii) are obvious if n = 1. Assuming n > 2, put e = ei, / = / 1 , and J7 = ^ + E L 2 ( F ^ + F ^ ) ; t h e n ^ = f / + i ? e + F / - P u t 7 a = a e / + « _ 1 / e with a G F x . Then 7 a G G 1 ^ ) , er ( 7 a ) = a~2e, and / r ( 7 a ) - a 2 / by (3.12a). If n = 2, the elements 7 a for all a £ Fx span A+(V). Thus both (i) and (ii) hold when n = 2. We therefore assume that n > 2 and prove our theorem by induction on n. Define & by (2.8) and E by (2.14) with an element g G U such that g2 = 1. We take g G Z if n £ 2Z and # = er + / r if n G 2Z. Then ~"(A+(V)) = M2(A+(J7)) by (2.15). From (2.8), (2.9), and (2.14) we obtain

(A5.2) S(7a) = 0 H(#(ae -1 / ) )

0

Both 7 a and g(ae + a */) belong to Gl(V). Clearly the matrices of (A5.2) span the space of all (2 x 2)-matrices over F. Also, by (2.11), E(/3) = diag[#_1/?<7, /3] for every (3 G A+(U) and g^A+^g = A+(U). By induction, G\U) spans A+(U) over F. Therefore E(G1(V)) spans £T(yl+(V)) over F. This proves (i). To prove (ii), first suppose n = 2r; then r > 2 and we have an isomorphism £ : -A+(/7) —* MS(F) x MS(F) with 5 = 2 r~2 . Put (((3) = (Ci(/?), C2(/?)) for /? G A+(U) with two

maps C/c : A+{U) -+ MS(F). Given a G ^ + ( V ) , put S (a ) ' a b Then we

may assume that uok '• A+iV) —» M2S{F) is given by cjfc(a) By

C d

Ck(a) \k(b) _(k(c) (k(d)_

induction, Gl(U) has an element j3 such that e^r(/^) = a~2ei and far((3) = a2/^ for i > 2, and 0fc(/2) n a s characteristic roots as described in (ii). Put a = 7ai/?-Then a G G1(V) and O0k(ct) = diagfaiO^g/^g-1), l Cfe (/?)]• We can easily verify that a has the required properties of (ii). Since the case of odd n can be proved in a similar and simpler way, we obtain our theorem.

Remark. The element a of the above theorem is uniquely determined by ai, . . . , ar if the a$ are generic, as can easily be seen. The uniqueness fails, however, for some special a . Take, for example, n = 2 and a\ = y —L Then —a has the same properties as a.

A5.3. Let us now describe the Lie algebra of Gx{y) as a subset of A+(V), and discuss its relationship with the Lie algebra of SO^. We let F denote either R or C, and consider the Lie algebras over F. We fix (V, if) over F. We have two exponential maps:

exp : End(l/, F) —> GL(V, F), exp : A(V) —> A{V)X,

defined as usual by exp(x) = S m = o x m / m ' * Then the Lie algebra of GL(V, F) resp. A(V)X is End(V, F) resp. A(V). The structure of A(V) as a Lie algebra is defined with respect to

[x, y] — xy — yx for x, y G A(V).

Let g^ resp. g1 denote the Lie algebra of SO^ resp. G1(V). By a standard principle we have

270 APPENDIX

(A5.3a) tf> = {£ G End(V) | exp(^) G 5 0 ^ for every t G R } ,

(A5.3b) g1 = {£ G A+(F) | exp(^) G G 1 ^ ) for every * G R } .

These two Lie algebras are isomorphic, as r gives a local isomorphism of Gl(V) to SO^. We can easily verify that

(A5.4) g^ = {£ G End(V) | </?(i>£, w) = -<p(v, w£) for every v, w G V},

which can be written g^ = {£ e End(F) | £<£o = —<£o * *£}> if we identify End(V) and O^ with Mn(F) and O(^o) with a symmetric matrix cpo as we did in §1.6. Taking cpo in the diagonal form, we can easily show that dim(g^) = n(n — l ) /2 . Let us next prove

(A5.5) g1 = {£ G A+(V) | T = - £ [K f] C T/}.

Indeed, if £ G g1, then exp(££) exp(££*) = exp(^) exp(*£)* = 1 and exp{—t£)v •exp(££) G V for every t G R and every t> G V. Then we easily see that g1 is contained in the right-hand side of (A5.5). Conversely, suppose £ G ^4+(V^) and [V, f] C V. Define i% G End(V) by vR$ = [v, f] for v G V. By induction we can

easily verify that vRg = E ^ o l " 1 ) * ( ™ j £ X m ~ N s o t h a t exp(-*£)*> exp(*f) =

v • exp(«%) G V. Thus exp(*f) G G+(F). If f* = - £ , then exp(*f)exp(*0* = h

so that exp(££) G G^V) . This proves (A5.5).

Theorem A5.4. If {xi}f=1 is an orthogonal basis of V, then we have g1 = J2i<j FxiXj. Moreover, if dr denotes the isomorphism of g1 onto g^ obtained from the map r : Gl{V) —• SO^(V), then v • dr(£) = [v, f] for every v eV and every

P R O O F . We have seen that r(exp(££)) =exp(fi?^), from which we obtain dr(^) = R^. This proves the second assertion. Next, for x, y, v G V we have xyv — vxy — x(yv + vy) — (xv + vx)y G V; also, if xy = —yx, then (xy)* = —xy, and hence XiXj G g1 if i 7 j . Since g1 is isomorphic to g^, we have dim(g1) = n(n — l ) /2 . Therefore g1 = Xw<j Fxixj> This completes the proof.

A5.5. Let us now take F = R and investigate the Cartan decomposition go = £o + Po of §A1.1 for SO^ and Gl(V). If 99 is definite, then the groups are compact, and the question is trivial. Therefore we assume that ip is not definite, and take a decomposition V = X 0 Y with nontrivial subspaces X and V, on which cp is positive and negative definite, respectively. By Theorem 14.2 (iv), G1(X)G1(Y) is a maximal compact subgroup of G 1 ^ ) , and T(G1(X)G1(Y)) = SO^(X) x SO*{Y), which is a maximal compact subgroup of SOQ(V).

Define an element 9 of 0^{V) by (x + y)0 = x — y for x G A" and y G Y. Then /? 1—> 9~1/36 is an automorphism of SOQ(V), and

(A5.6) SO^(X) x SO*(Y) = {/? G SO£(F) | fl"1/^ = /?}.

Take the Cartan decomposition $v = t* + p^ with respect to this automorphism. Then V*> corresponds to SO*(X) x SO*(Y). If we take ip = diag[lz, - l m ] , X = J2i=i ^eii a n d ^ — S j = i R^j with the standard basis {£;}f=1 of R^, then

A5. SPIN REPRESENTATIONS AND LIE ALGEBRAS 271

(A5.7a) F = {diag[a, d] | - ' a = a G R{, -*d = d G R™},

(A5.7b) p1^ - 0 6 'ft 0

6GR!

Now, by Lemma 3.8 the map v •-» v9 for v G V can be extended to an automorphism ftH^ of A(V) such that r(ae) — 6~1r{a)9 for every a G G(V). Let us now prove

(A5.8) G1{X)G1{Y) = {ae G^V) \ ae = a}.

We easily see that Y = 7 for 7 G A+(X) U A + ( r ) , and so G1(X)G1(Y) is contained in the right-hand side of (A5.8). Suppose a G Gl(V) and ae = a; then r(a) belongs to the right-hand side of (A5.6), and hence r(a) = T(J) with 7 G G1(X)G1{Y). Thus a - ±7 G G^AOGH^), which proves (A5.8).

Let g1 = t1 + p1 be the Cartan decomposition of g1 with respect to the automorphism a i-> a61 of Gl(V). For £ G g1 and t G R we have exp (f • d9(£)) = exp(t£)e = exp(f^), so that d0{£) = £e for every ^egK Thus

(A5.9) P1 = { € e g 1 | ^ = - e } -Take an orthogonal bases {xt} of A" and {yh} of Y. By Theorem A5.4, g1 is spanned by the elements XjXj for i ^ j , ynyk for ft ^ ft, and x ^ . Therefore we obtain

(A5.10) p1 is spanned over R by tiie elements xy for all x G X and all y G Y.

A5.6. Finally let us discuss the 6-dimensional case with F = R and <p = 16

as an easy example. By Lemma A3.8, the map X H / is a positive involution of A+(V); also A+(V) ^ i\/4(C) by (A3.7). By Lemma A3.7 there is an isomorphism / ofA+{V) onto M4(C) such that f(x*) = tJ(x). Therefore we can identify A+(V) with i\/4(C) so that x* = fx. Let us now prove

(A5.ll) G\V) = {xe 5L4(C) I xx* = l } ,

that is, Gl{V) can be identified with 517(4). First, by Theorem 14.2, G 1 ^ ) is connected and compact, and hence det (Gl{V)) = 1. Therefore G1(V) is contained in the right-hand side of (A5.ll). The same principle as in (A5.3a, b) shows that the Lie algebra of 517(4) is

{ £ € M 4 ( C ) | r = - £ tr(O = 0}.

This has dimension 15, which is dim(g1). It is well known that 517(4) is connected, and therefore we obtain (A5.ll).

Let us next consider A+(Vc) for Vc = V 0 R C with the above </? extended naturally to VQ. Then T4 + (VC) = A+(V) 0 R C, which can be identified with A/4(C) x i\/4(C) through the map £ 0 a i-> (af, a?) for £ € ^ + ( V ) = A/4(C) and a G C. Since (£ 0 a)* = ^ ® o , we see that (a, /?)* = (*/?, *a) for (a, /i) G A+(Vc) = M±(C) x M4(C). Now the Lie algebra of 5L4(C) is of dimension 15 over C. Therefore the same type of reasoning as above shows that

(A5.12) GHVc) = {(«, 'a-1) \ a G 5L 4 (C)} .

Thus G1(Vc) can be identified with 5L4(C), and the spin representation u can be given by uj(a) = (a, ta~1) for a G 5L4(C).

R E F E R E N C E S

[And] A. N. Andrianov, Euler products corresponding to Siegel modular forms of genus 2, Uspekhi Mat. Nauk, 29:3 (1974), 43-110 (Russian Math. Surveys, 29:3 (1974), 45-116.

[Ano] Anonymous, Correspondence, Ann. of Math. 69 (1959), 247-251. [B] P. Bachmann, Die Arithmetik der Quadratischen Formen, Leipzig, Teubner,

1898. [C] C. Chevalley, Sur la theorie du corps de classes dans les corps finis et les

corps locaux, J. Fac. Sci. Univ. Tokyo 2 (1933) 365-476. [D] M. Deuring, Algebren, Springer, 1935, 2nd ed. 1968. [DD] L. Dirichlet, Vorlesungen iiber Zahlentheorie, supplement by R. Dedekind,

Braunschweig, 1894. [El] M. Eichler, Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher

Algebren iiber algebraischen Zahlkorpern und ihre L-Reihen, J. Reine u. Angew. Math. 179 (1938), 227-251.

[E2] M. Eichler, Die Ahnlichkeitsklassen indefiniter Gitter, Math. Z. 55 (1952), 216-252.

[E3] M. Eichler, Quadratische Formen und orthogonale Gruppen, Springer 1952, 2 n d e d . 1974.

[G] C. F. Gauss, Disquisitiones Arithmeticae, 1801, English translation by A.A. Clarke, Yale Univ. Press, 1966.

[Ha] Harish-Chandra, Automorphic forms on semisimple Lie groups, Lecture notes in Math. 62, Springer, 1968.

[Hel] S. Helgason, Differential geometry and symmetric spaces, Academic Press, 1962.

[He2] S. Helgason, Groups and geometric analysis, Academic Press, 1984. [K] M. Kneser, Klassenzahlen indefiniter quadratischer Formen in drei oder mehr

Veranderlichen, Arch. d. Math. 7 (1956), 323-332. [S63] G. Shimura, On modular correspondences for Sp(n, Z) and their con

gruence relations, Proc. of the National Academy of Sciences, 49 (1963), 824-828 (= Collected Papers, vol. I, 464-468).

[S82] G. Shimura, Confluent hypergeometric functions on tube domains, Math-ematische Annalen, 260 (1982), 269-302 (= Collected Papers, vol. I l l , 388-421).

[S90] G. Shimura, Invariant differential operators on hermitian symmetric spaces, Annals of Mathematics, 132 (1990), 237-272 (= Collected Papers, vol. IV, 68-103).

[S94] G. Shimura, Differential operators, holomorphic projection, and singular forms, Duke Mathematical Journal, 76 (1994), 141-173. ( = Collected Papers, vol. IV, 351-383).

[S97] G. Shimura, Euler Products and Eisenstein series, CBMS Regional Conference Series in Mathematics, No. 93, Amer. Math. S o c , 1997.

[S99a] G. Shimura, An exact mass formula for orthogonal groups, Duke Math. J. 97 (1999), 1-66 (= Collected Papers, vol. IV, 509-573).

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[S99c] G. Shimura, Generalized Bessel functions on symmetric spaces, Journal fiir die reine und angewandte Mathematik, 509 (1999), 35-66. (= Collected Papers, vol. IV, 609-636).

[S00] G. Shimura, Arithmeticity in the theory of automorphic forms, Mathematical Surveys and Monographs, vol. 82, Amer. Math. Soc. 2000.

[S02a] G. Shimura, The representation of integers as sums of squares, Amer. J. Math. 124 (2002), 1059-1081.

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(1944), 577-622 (= Gesammelte Abhandlungen, II, 421-466).

FREQUENTLY USED SYMBOLS

a . . . 80

A( ) ,A + ( M ~ ( ) . . . 16 a9() . . . 176 \[a}q . . . 178 d(v?) . . . 12 D[ , ] . . . 181 3), D_ . . . 146 dil( ) . . . 44, 167

c T O . . . 168 eA, ea, eh, ev . . . 183 e£, e™,e£\e™ . . . 183 e() . . . 176 rym . . . 12 rj{ ) . . . 141, 148 J c . . . 165

T% . . . 169

G( ),G+( ),G"( ) . . . 20

G'( ) . . . 22

GH ) •••23 G°( ) . . . 139 g . . . 37

r{L) ... 103 r n o ... 185 r » . . . 254 h . . . 80 5} . . . 148 i . . . 229

il3 . . . 182 «( ) . . . 71, 75, 210 0 ( ) . - - 3 2 Lc( , ) . . . 170 L[g, b] . . . 103 A0( ) . . . 175 Mk ••• 230 i / ( ) . . . 23 0 . . . 243 O^ . . . 9

oo ••• 12

1 . . . 156, 164 p( ) . . . 159, 175 &{) . . . 18 JR( O • • • 73 Sk . . . 230

S£ . . . 170

SO* . . . 9

S0 . . . 85

cr( ) . . . 23

T . . . ix

T(s, f, \ ) .. .170, 208 r( ) . . . 20 v . . . 80 Z, Z£, Z(r, 0) . . . 154 Z* . . . 163 Z(s, f, \ ) . . . 170, 208

274

INDEX

algebra anisotropic automorphic form binary form canonical automorphism canonical involution capacity ideal Center of a Clifford algebra claSS of a lattice class of a symmetric matrix ClaSS n u m b e r of an algebraic group class number of an order Clifford algebra Clifford group Conduc to r of an order congruence subgroup 165, core dimension core subspace cusp form 170, denominator ideal discriminant algebra discriminant field d i s c r i m i n a n t of a quadratic form d i s c r i m i n a n t of an order

13 9

169,230 120

16 16

75,210 19 81 91 81

116 16 20

106 205,229

11 11

, 206,230 44 12 12 12

106 eigenfunction of differential operators 247 elementary divisor even Clifford algebra even Clifford group exceptional prime genUS of a lattice genUS of a symmetric matrix geilUS of maximal lattices global field Hasse principle Hecke algebra Hecke character Hecke eigenform 170, involution isotropic Laplace-Belt rami operator lattice local field

44 17 20 83 81 91 83 37 60 73 ix

207,232 13 9

144,158 37 37

main involution mass maximal lattice maximal order Order in an algebra origin orthogonal basis orthogonal complement orthogonal group parabolic subgroup (<£>, L)-primitive positive involution primitive matrix primitive solutions proper ideal quaternion algebra reduced expression reduced norm reduced trace simple algebra special orthogonal group spherical function spin group spin representation spinor norm split Witt decomposition strong approximation sum of 3 squares sum of 5, 7, 9 squares symmetry totally isotropic two-sided ideal weak approximation weak Witt decomposition Witt decomposition Witt's theorem

15 133 45 37 37

149,156 12 9 9

29 95

262 44

1 107 14 44 14 14 13 9

246 23

268 24 11 81

118,119 123,135

21 9

37 85 11 11 10

275

Titles in This Series

109 Goro Shimura, Arithmetic and analytic theories of quadratic forms and Clifford groups, 2004

108 Michael Farber, Topology of closed one-forms, 2004 107 Jens Carsten Jantzen, Representations of algebraic groups, 2003 106 Hiroyuki Yoshida, Absolute CM-periods, 2003 105 Charalambos D . Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces with

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Recurrence sequences, 2003 103 Octav Cornea, Gregory Lupton, John Oprea, and Daniel Tanre,

Lusternik-Schnirelmann category, 2003 102 Linda Rass and John Radcliffe, Spatial deterministic epidemics, 2003 101 Eli Glasner, Ergodic theory via joinings, 2003 100 Peter Duren and Alexander Schuster, Bergman Spaces, 2004 99 Phil ip S. Hirschhorn, Model categories and their localizations, 2003 98 Victor Guil lemin, Viktor Ginzburg, and Yael Karshon, Moment maps,

cobordisms, and Hamiltonian group actions, 2002 97 V. A. Vassiliev, Applied Picard-Lefschetz theory, 2002 96 Martin Markl, Steve Shnider, and J im Stasheff, Operads in algebra, topology and

physics, 2002 95 Seiichi Kamada, Braid and knot theory in dimension four, 2002 94 Mara D . Neuse l and Larry Smith, Invariant theory of finite groups, 2002 93 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 2:

Model operators and systems, 2002 92 Nikolai K. Nikolski, Operators, functions, and systems: An easy reading. Volume 1:

Hardy, Hankel, and Toeplitz, 2002 91 Richard Montgomery , A tour of subriemannian geometries, their geodesies and

applications, 2002 90 Christian Gerard and Izabella Laba, Multiparticle quantum scattering in constant

magnetic fields, 2002 89 Michel Ledoux, The concentration of measure phenomenon, 2001 88 Edward Frenkel and David Ben-Zvi , Vertex algebras and algebraic curves, 2001 87 Bruno Poizat, Stable groups, 2001 86 Stanley N . Burris, Number theoretic density and logical limit laws, 2001 85 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Spectral problems associated with

corner singularities of solutions to elliptic equations, 2001 84 Laszlo Fuchs and Luigi Salce, Modules over non-Noetherian domains, 2001 83 Sigurdur Helgason, Groups and geometric analysis: Integral geometry, invariant

differential operators, and spherical functions, 2000 82 Goro Shimura, Arithmeticity in the theory of automorphic forms, 2000 81 Michael E. Taylor, Tools for PDE: Pseudodifferential operators, paradifferential

operators, and layer potentials, 2000 80 Lindsay N . Childs, Taming wild extensions: Hopf algebras and local Galois module

theory, 2000 79 Joseph A. Cima and Wil l iam T. Ross , The backward shift on the Hardy space, 2000 78 Boris A. Kupershmidt , KP or mKP: Noncommutative mathematics of Lagrangian,

Hamiltonian, and integrable systems, 2000 77 Fumio Hiai and Denes Petz , The semicircle law, free random variables and entropy,

2000

TITLES IN THIS SERIES

76 Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmiiller theory. 2000 75 Greg Hjorth, Classification and orbit equivalence relations, 2000 74 Daniel W . Stroock, An introduction to the analysis of paths on a Riemannian manifold,

2000 73 John Locker, Spectral theory of non-self-adjoint two-point differential operators, 2000 72 Gerald Teschl, Jacobi operators and completely integrable nonlinear lattices, 1999 71 Lajos Pukanszky, Characters of connected Lie groups, 1999 70 Carmen Chicone and Yuri Latushkin, Evolution semigroups in dynamical systems

and differential equations, 1999 69 C. T. C. Wall (A. A. Ranicki, Editor) , Surgery on compact manifolds, second edition,

1999 68 David A. Cox and Sheldon Katz , Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N . Wallach, Continuous cohomology, discrete subgroups, and

representations of reductive groups, second edition, 2000 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra,

1999 64 R e n e A. Carmona and Boris Rozovskii , Editors, Stochastic partial differential

equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W . Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic

algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Wil l iams, Morita equivalence and continuous-trace

C*-algebras. 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B . Frenkel, and Alexander A. Kirillov, Jr., Lectures on

representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives. 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum

groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in

analysis, 1997 53 Andreas Kriegl and Peter W . Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V . G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in

domains with point singularities, 1997 51 Jan Maly and Wil l iam P. Ziemer, Fine regularity of solutions of elliptic partial

differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential

equations, 1997 48 Paul-Jean Cahen and Jean-Luc Chabert , Integer-valued polynomials. 1997 47 A. D . Elmendorf, I. Kriz, M. A. Mandell , and J. P. May (with an appendix by

M. Cole) , Rings, modules, and algebras in stable homotopy theory, 1997

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