INTRODUCTION

7

in the same manner in the orthogonal case. However, in the case of Clifford groups,

the analysis of a pullback becomes more complicated, and so we have thought it is

more practical to discuss the orthogonal case first as preliminaries to the case of

Clifford groups.

There is one essential difference of the present theory from those in [S97] and

[S00]: here we deal with nonholomorphic eigenfunctions of differential operators,

while holomorphic forms were discussed in those books, and one crucial point was

the explicit forms of eigenvalues of an integral operator, which was necessary for

the determination of gamma factors. In the present case we have to know such

explicit eigenvalues for nonholomorphic functions, for which there is no previously

known result. Therefore we devote two sections in the Appendix for the purpose

of developing a general theory and calculating explicitly such eigenvalues in the

nonholomorphic case.

Thus the main portion of our theory concerns nonholomorphic functions, which

is inevitable, as our symmetric space has a complex structure only in a very special

case. In the last part of the book, however, we shall treat holomorphic forms, but

before discussing it, let us first explain how the whole book is organized.

III . The main subjects of Chapter I are Clifford algebras, Clifford groups and

spin groups over an arbitrary field F. The basic properties of these are well known,

but we give a detailed exposition with some new methods, among which we mention

the isomorphism

(10) ¥ : A(V)^ M2(A(U)),

which can be explicitly given if V = U 0 Ft © FK with elements i and K such

that — (f[i\ = P[K] = 1 and (p(u, K) — 0. We almost constantly use this & in order

to simplify our problems, often in an essential way.

In Chapter II we consider quadratic forms and Clifford groups over a global or

a nonarchimedean local field. The main new points are the construction of the

subgroup Jv of G + ( V % in (6) and the determination of an explicit rational form

for (8).

The theory of Diophantine equations explained in I will be developed in Chapter

III. A short history of the theory of the equation ip[h] — q will be given in §13.15.

Notable investigations from Lagrange to Siegel will be discussed in connection with

our theory.

Chapter IV concerns the case F = R. We are particularly interested in the

symmetric spaces associated with our groups. In Section 16 we consider the case of

an arbitrary signature. If the signature of if is (1, n — 1) or (2, n — 2), then we can

present the symmetric space and the group action algebraically within the Clifford

algebra, which will be the topics of Sections 14 and 15. When V — U 0 Rx © Rft

with L and n as above, the space is given as a half space H contained in U + R^

or a tube domain S) contained in U 0 R C, according as the signature is (1, m) or

(2, m). Then each a G G+(V) acts on the space as a generalized fractional linear

transformation obtained from \P(a) with the above \P; see (14.9) and (15.6) for their

explicit forms.

In Chapters V and VI we discuss Euler products and Eisenstein series about

which we already explained in II . Let us now discuss the question of holomorphic