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
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