In this chapter we turn to the analysis of Clifford algebras and modules, together with an analysis of the associated Dirac operators, on Riemannian manifolds more general than the open subsets of Euclidean space studied in the previous chapter. Concepts from differential geometry are needed from the outset, but in keeping with the spirit of making the material available to more classically trained analysts we have attempted to minimize the use of differential geometric machinery, possibly at the expense of clarity and elegance. In the first section, therefore, Dirac operators are introduced explicitly on a single coordinate patch of a manifold. This serves several useful purposes. It helps bring out quite simply the role that the curvature of the manifold plays in the expression for D2. For the ‘flat’ case studied earlier, (–D2) was the Euclidean Laplacian, and solutions of DF = 0 were automatically harmonic: the forerunner of the GCR property of operators. But the fundamental Bochner–Weitzenböck theorem expresses (–D2) in general as a second-order Laplacian together with a zero-order curvature operator. This idea is a basic one throughout the chapter. In the specific examples of the spinor Laplacian and the Hodge Laplacian, the curvature operator is explicitly calculated. By assuming that the coordinates form a normal coordinate system, we also express (–D2) asymptotically as a sum of an operator, which will play a fundamental role in the proof of the Atiyah–Singer index theorem, and a remainder operator.

Section 2 deals with the problem of passing from a local setting to a global setting.

Associated with any Euclidean space ℝi or Minkowski space ℝp,q is a universal Clifford algebra, denoted by and, respectively. Roughly speaking, a Clifford algebra is an associative algebra with unit into which a given Euclidean or Minkowski space may be embedded, in which the corresponding quadratic form may be expressed as the negative of a square. The real numbers ℝ, the complex numbers ℂ, and the quaternions ℝ are the simplest examples.

Our intent in this chapter is to give an elementary, coherent, and largely self-contained account of the theory of Clifford algebras. In sections 1 and 2 we present the definitions basic to all of our work. The balance of section 2 is devoted to three constructive proofs of the existence of universal Clifford algebras: two basis-free constructions using tensor algebras and exterior algebras, and a basis-dependent construction. The reader who is willing to accept the existence of Clifford algebras may wish to proceed directly to the statement of the major structural results in section 3. Sections 4, 5, and 6 explore the interconnections between Clifford algebras and orthogonal groups; the spin representation and spin groups will be studied in detail, with Spin(p, q) and Spin(p, q + 1) both being realized in using the notion of transformers. The reader who is primarily interested in the analytic applications of Clifford algebras may wish to proceed directly to the discussion of the Euclidean case in section 7. Section 8 is a discussion of spin groups as Lie groups. In section 9 we construct various realizations of Spin(p, q), p + q ≤ 6, whereby these groups are explicitly identified with classical Lie groups.

In this book we present a comprehensive introduction to the use of Clifford algebras and Dirac operators in harmonic analysis and analysis more generally. In the past 30 years, Clifford algebras and Dirac operators have played a key role in three of the most important areas of mathematical research during that time: the boundedness of the Cauchy integral on Lipschitz surfaces, the realization of discrete series representations of semi-simple Lie groups, and the celebrated Atiyah–Singer index theorem. Much as an analyst would like to understand and appreciate these developments, however, there are formidable technical barriers to doing so, particularly for more classically trained analysts, as we have found to our cost over the years. Thus our aim from the outset has been to meld into a coherent and reasonably self-contained whole a body of ideas from classical singular integral theory, representation theory and analysis on manifolds, with a view to making this material accessible to more classically trained analysts.

Now the starting point for much of classical harmonic analysis is the study of the boundary regularity of harmonic functions in domains in Euclidean space. Classical Hardy space theory explores the consequences of the improved boundary regularity obtained when consideration is restricted to analytic functions in the plane. On the other hand, for SL(2, ℝ), the starting-point for representation theory of semi-simple Lie groups, some important unitary representations become irreducible only on restriction to analytic functions.

Discussion of further applications of the theory of Dirac operators and Clifford algebras now begins. The style of exposition will change somewhat, with fewer details being given than before, so that greater demands are placed upon the reader. A wider variety of topics can then be covered. In this chapter we shall discuss the representation theory for the group Spin(V, Q), concentrating almost entirely on the case of the compact group Spin(n) which arises when (V, Q) is an n-dimensional (real) positive- or negative-definite quadratic space (see also chapter 5). In the non-compact case, the characterization of the irreducible unitary representations is only just being discovered. But in the compact case these representations are all finite-dimensional and their ‘parameterization’ has been known for many years, thanks to the effort of Cartan, Weyl et al. This will be given in section 2 after some routine preliminaries have been disposed of in section 1. For detailed applications to analysis, however, various explicit realizations of these representations are needed. Representations of O(n), hence SO(n), and of Spin(n), on spaces of harmonic polynomials on ℝn are well-known, and their role in such singular integrals as Riesz transforms is well-understood. Realizations of more general representations of O(n) on harmonic polynomials on the space ℝr × n of real matrices are far less widely known; their use in singular integrals has hardly begun. In sections 4 and 5 we give a fairly detailed account of some aspects of the theory of polynomials of matrix argument, using it as a vehicle for presenting some aspects of polynomial invariant theory.

Over the past few years, we ran a Seminar in Harmonic Analysis at the Mathematics Department of the University of Rome “La Sapienza”. In this seminar many of the talks given by staff members and visitors were concerned, directly or indirectly, with infinite trees or tree-like graphs, and their automorphism groups. Seminar notes were occasionally taken by one or both of us, and sometimes written up informally for distribution to newcomers to the seminar. After a while, we felt that it would be convenient to give a more coherent organization to these notes. Once this decision was taken it became apparent that, at the cost of some omission, the general aim of describing the group of automorphisms of a homogeneous tree and its irreducible unitary representations would provide a convenient focus which would include much of the material we had in mind. We felt that this approach would shed light on the connection between harmonic analysis on trees and harmonic analysis on hyperbolic spaces, by emphasizing the strict analogy between the group of automorphisms of the tree and real rank 1 semisimple Lie groups. This choice left out a lot of valuable material specifically concerning free groups and free products of finite groups. We felt however that the notes [F-T P2] and the memoir [F-T S2] could provide an introduction to these topics. We also decided not to treat the case of a semihomogeneous tree.