##
**Clifford algebras and Dirac operators in harmonic analysis.**
*(English)*
Zbl 0733.43001

Cambridge Studies in Advanced Mathematics, 26. Cambridge (UK): Cambridge University Press. vi, 334 p. £37.50; $ 75.00 (1991).

The book under review is written by analysts and for analysts; therefore it is quite different from other publications on Clifford algebras and Dirac operators by authors who are geometers or topologists.

In chapter 1, the algebraic theory of Clifford algebras and spin groups is expounded on 80 pages. The Clifford algebra of the Euclidean space \(\mathbb{R}^n\) is denoted by \(\mathfrak A_n\). For an \(\mathfrak A_n\)-module \({\mathfrak H}\) and an open subset \(\Omega\) of \(\mathbb{R}^n\), the standard Dirac operator is then a differential operator \(D\) from \(\mathcal C^\infty(\Omega,\mathfrak H)\), the space of smooth functions on \(\Omega\) with values in \(\mathfrak H\), to itself.

Chapter 2 studies Clifford analytic functions, that is, solutions of \(Df=0\). Since \(D^2=-\Delta\), they are harmonic. Parallel to the classical theory of holomorphic functions of a complex variable, a theory of Clifford analytic functions is developed, including results about subharmonicity and a theory of Hardy spaces on the upper half-space and on Lipschitz domains.

The aim of chapter 3 is to give explicit constructions of certain irreducible representations of the orthogonal groups. After some generalities, the representation of \(O(n)\) on the space \(\mathcal H_m(\mathbb{R}^n)\) of harmonic homogeneous polynomials of degree \(m\) on \(\mathbb{R}^n\) is studied. Its highest weight is determined, and \(\mathcal H_m(\mathbb{R}^n)\) is decomposed into irreducible representations of \(O(n-1)\), using Gegenbauer polynomials. A larger class of irreducible representations of \(O(n)\) is then obtained by generalizing to harmonic polynomials of a matrix argument.

In chapter 4, the Dirac operators are generalized to the notion of operators of Dirac type (still on open subsets of Euclidean space). Important differential operators arising from geometry are seen to be of Dirac type; others are constructed from irreducible representations of Spin(n).

The final chapter 5 treats Dirac operators on manifolds and closes with a proof, due to Getzler, of the local index theorem for Dirac operators.

The book contains some serious mistakes and a number of inaccuracies: On p. 147 there is the incredible statement that \(\mathrm{Spin}(n)\) is topologically isomorphic to a sphere. On p. 13, no proof is given for the fact that a Clifford algebra is \(\mathbb{Z}/2\)-graded; since a more general definition of a Clifford algebra is adopted than by most authorities, this assertion is not obvious. On p. 268, the authors define what it means for a manifold to admit a spin structure; nothing is said about when two spin structures are equivalent. Then, simply as examples without comment or proof, but marred by a misprint, the numbers of inequivalent spin structures on spheres and real projective spaces are given. As a final sample, in the very basic definition (1.4) on p. 204, the function \(\partial_ AF\) is defined to be (A\(\circ \nabla)F\) instead of \(A\circ (\nabla F).\)

There are many sections in the book which are written in an informal style and in which the authors try to tell what is going on. This is very laudable, in principle; but sometimes greater care would have been required to obtain formulations which are really intelligible and free of jargon.

Two important works are missing in the list of references: M. F. Atiyah, R. Bott and A. Shapiro [Clifford modules. Topology 3, Suppl. 1, 3–38 (1964; Zbl 0146.19001)]; H. B. Lawson and M.- L. Michelsohn [Spin geometry. Princeton: Princeton University Press (1989; Zbl 0688.57001)].

In chapter 1, the algebraic theory of Clifford algebras and spin groups is expounded on 80 pages. The Clifford algebra of the Euclidean space \(\mathbb{R}^n\) is denoted by \(\mathfrak A_n\). For an \(\mathfrak A_n\)-module \({\mathfrak H}\) and an open subset \(\Omega\) of \(\mathbb{R}^n\), the standard Dirac operator is then a differential operator \(D\) from \(\mathcal C^\infty(\Omega,\mathfrak H)\), the space of smooth functions on \(\Omega\) with values in \(\mathfrak H\), to itself.

Chapter 2 studies Clifford analytic functions, that is, solutions of \(Df=0\). Since \(D^2=-\Delta\), they are harmonic. Parallel to the classical theory of holomorphic functions of a complex variable, a theory of Clifford analytic functions is developed, including results about subharmonicity and a theory of Hardy spaces on the upper half-space and on Lipschitz domains.

The aim of chapter 3 is to give explicit constructions of certain irreducible representations of the orthogonal groups. After some generalities, the representation of \(O(n)\) on the space \(\mathcal H_m(\mathbb{R}^n)\) of harmonic homogeneous polynomials of degree \(m\) on \(\mathbb{R}^n\) is studied. Its highest weight is determined, and \(\mathcal H_m(\mathbb{R}^n)\) is decomposed into irreducible representations of \(O(n-1)\), using Gegenbauer polynomials. A larger class of irreducible representations of \(O(n)\) is then obtained by generalizing to harmonic polynomials of a matrix argument.

In chapter 4, the Dirac operators are generalized to the notion of operators of Dirac type (still on open subsets of Euclidean space). Important differential operators arising from geometry are seen to be of Dirac type; others are constructed from irreducible representations of Spin(n).

The final chapter 5 treats Dirac operators on manifolds and closes with a proof, due to Getzler, of the local index theorem for Dirac operators.

The book contains some serious mistakes and a number of inaccuracies: On p. 147 there is the incredible statement that \(\mathrm{Spin}(n)\) is topologically isomorphic to a sphere. On p. 13, no proof is given for the fact that a Clifford algebra is \(\mathbb{Z}/2\)-graded; since a more general definition of a Clifford algebra is adopted than by most authorities, this assertion is not obvious. On p. 268, the authors define what it means for a manifold to admit a spin structure; nothing is said about when two spin structures are equivalent. Then, simply as examples without comment or proof, but marred by a misprint, the numbers of inequivalent spin structures on spheres and real projective spaces are given. As a final sample, in the very basic definition (1.4) on p. 204, the function \(\partial_ AF\) is defined to be (A\(\circ \nabla)F\) instead of \(A\circ (\nabla F).\)

There are many sections in the book which are written in an informal style and in which the authors try to tell what is going on. This is very laudable, in principle; but sometimes greater care would have been required to obtain formulations which are really intelligible and free of jargon.

Two important works are missing in the list of references: M. F. Atiyah, R. Bott and A. Shapiro [Clifford modules. Topology 3, Suppl. 1, 3–38 (1964; Zbl 0146.19001)]; H. B. Lawson and M.- L. Michelsohn [Spin geometry. Princeton: Princeton University Press (1989; Zbl 0688.57001)].

Reviewer: Wilhelm Singhof (Düsseldorf)

### MSC:

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

43A80 | Analysis on other specific Lie groups |

15A66 | Clifford algebras, spinors |

58J20 | Index theory and related fixed-point theorems on manifolds |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

30G30 | Other generalizations of analytic functions (including abstract-valued functions) |

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

32A35 | \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables |

57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |