Dirac operators in representation theory.

*(English)*Zbl 1103.22008
Mathematics: Theory & Applications. Basel: Birkhäuser (ISBN 0-8176-3218-2/hbk). x, 199 p. (2006).

Let \(G\) be a connected real reductive Lie group with a maximal compact subgroup \(K\) and a corresponding Cartan decomposition \(\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{p}_0\) where \(\mathfrak{g}_0\) and \(\mathfrak{k}_0\) are the Lie algebras of \(G\) and \(K\), respectively, and let \(\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}\) be the complexified Cartan decomposition. The Killing form on \(\mathfrak{g}\), which is nondegenerate on \(\mathfrak{p}\), defines the complex Clifford algebra \(C(\mathfrak{p})\) as an associative algebra with unit. Given an orthonormal basis \(Z_i\) of \(\mathfrak{p}\), Vogan defined an algebraic version of the Dirac operators as \(D=\sum_i Z_i\otimes Z_i\in U(\mathfrak{g})\otimes C(\mathfrak{p})\) where \(U(\mathfrak{g})\) is the universal enveloping algebra of \(\mathfrak{g}\) with center \(Z(\mathfrak{g})\). Here \(D\) is independent of the choice of the basis \(Z_i\) and \(K\)-invariant (for the adjoint action of \(K\) on both factors).

If \(X\) is a \((\mathfrak{g},K)\)-module and \(S\) is a space of spinors (a simple \(C(\mathfrak{p})\)-module), then \(D\) acts on \(X\otimes S\). The Dirac cohomology is defined to be \(H_D(X)={\roman {Ker}}\,D/{\roman{Ker}}\,D\cap{\roman{Im}}\,D\). Let \(\mathfrak{k}_\Delta\) denote a diagonal embedding of \(\mathfrak{k}\) into \(U(\mathfrak{g})\otimes C(\mathfrak{p})\). Vogan’s conjecture proved in J.-S. Huang and P. Pandzic [J. Am. Math. Soc. 15, 185-202 (2002; Zbl 0980.22013)] consists in the following: For any \(z\in Z(\mathfrak{g})\) there is a unique \(\zeta(z)\in Z(\mathfrak{k}_\Delta)\) and there are \(K\)-invariant elements \(a,b\in U(\mathfrak{g})\otimes C(\mathfrak{p})\) such that \(z\otimes 1=\zeta(z)+Da+bD\). Moreover, \(\zeta:Z(\mathfrak{g})\to Z(\mathfrak{k}_\Delta)\) is an algebra homomorphism having an explicit description in terms of Harish-Chandra isomorphisms. This allowed the authors to identify the infinitesimal character of an irreducible \((\mathfrak{g},K)\)-module that has nonzero Dirac cohomology, which, in turn, enabled them to simplify the proofs of a few classical theorems and to sharpen some of them.

Contents: Preface. 1. Lie Groups, Lie Algebras and Representations.

1.1. Lie groups and algebras. 1.2. Finite-dimensional representations. 1.3. Infinite-dimensional representations. 1.4. Infinitesimal characters. 1.5. Tensor products of representations.

2. Clifford Algebras and Spinors.

2.1. Real Clifford algebras. 2.2. Complex Clifford algebras and spin modules. 2.3. Spin representations of Lie groups and algebras.

3. Dirac Operators in the Algebraic Setting.

3.1. Dirac operators. 3.2. Dirac cohomology and Vogan’s conjecture. 3.3. A differential on \((U(\mathfrak{g})\otimes C(\mathfrak{p}))^K\). 3.4. The homomorphism \(\zeta\). 3.5. An extension of Parthasarathy’s Dirac inequality.

4. A Generalized Bott-Borel-Weil Theorem.

4.1. Kostant cubic Dirac operators. 4.2. Dirac cohomology of finite-dimensional representations. 4.3. Characters. 4.4. A generalized Weyl character formula. 4.5. A generalized Bott-Borel-Weil theorem.

5. Cohomological Induction.

5.1. Overview. 5.2. Some generalities about adjoint functors. 5.3. Homological algebra of Harish-Chandra modules. 5.4. Zuckerman functors. 5.5. Bernstein functors.

6. Properties of Cohomologically Induced Modules.

6.1. Duality theorems. 6.2. Infinitesimal character, \(K\)-types and vanishing. 6.3 Irreducibility and unitarity. 6.4. \(A_{\mathfrak{q}}(\lambda)\) modules. 6.5. Unitary modules with strongly regular infinitesimal character.

7. Discrete Series.

7.1. \(L^2\)-index theorem. 7.2. Existence of discrete series. 7.3. Global characters. 7.4. Exhaustion of discrete series.

8. Dimensions of Spaces of Automorphic Forms.

8.1. Hirzebruch proportionality principle. 8.2. Dimensions of spaces of automorphic forms. 8.3. Dirac cohomology and \((\mathfrak{g},K)\)-cohomology. 8.4. Cohomology of discrete subgroups.

9. Dirac Operators and Nilpotent Lie Algebra Cohomology.

9.1. \(\mathfrak{u}\)-homology and \(\bar{\mathfrak{u}}\)-cohomology differentials. 9.2. Hodge decomposition in the finite-dimensional case. 9.3. Hodge decomposition for \(\mathfrak{p}^-\)-cohomology in the unitary case. 9.4. Calculating Dirac cohomology in stages. 9.5. Hodge decomposition for \(\bar{\mathfrak{u}}\)-cohomology in the unitary case. 9.6. Homological properties of Dirac cohomology.

10. Dirac Cohomology for Lie Superalgebras.

10.1. Lie superalgebras of Riemannian type. 10.2. Dirac operators for \((\mathfrak{g},\mathfrak{g}_0)\). 10.3. Analog of Vogan’s conjecture. 10.4. Dirac cohomology for Lie superalgebras.

References. Index.

If \(X\) is a \((\mathfrak{g},K)\)-module and \(S\) is a space of spinors (a simple \(C(\mathfrak{p})\)-module), then \(D\) acts on \(X\otimes S\). The Dirac cohomology is defined to be \(H_D(X)={\roman {Ker}}\,D/{\roman{Ker}}\,D\cap{\roman{Im}}\,D\). Let \(\mathfrak{k}_\Delta\) denote a diagonal embedding of \(\mathfrak{k}\) into \(U(\mathfrak{g})\otimes C(\mathfrak{p})\). Vogan’s conjecture proved in J.-S. Huang and P. Pandzic [J. Am. Math. Soc. 15, 185-202 (2002; Zbl 0980.22013)] consists in the following: For any \(z\in Z(\mathfrak{g})\) there is a unique \(\zeta(z)\in Z(\mathfrak{k}_\Delta)\) and there are \(K\)-invariant elements \(a,b\in U(\mathfrak{g})\otimes C(\mathfrak{p})\) such that \(z\otimes 1=\zeta(z)+Da+bD\). Moreover, \(\zeta:Z(\mathfrak{g})\to Z(\mathfrak{k}_\Delta)\) is an algebra homomorphism having an explicit description in terms of Harish-Chandra isomorphisms. This allowed the authors to identify the infinitesimal character of an irreducible \((\mathfrak{g},K)\)-module that has nonzero Dirac cohomology, which, in turn, enabled them to simplify the proofs of a few classical theorems and to sharpen some of them.

Contents: Preface. 1. Lie Groups, Lie Algebras and Representations.

1.1. Lie groups and algebras. 1.2. Finite-dimensional representations. 1.3. Infinite-dimensional representations. 1.4. Infinitesimal characters. 1.5. Tensor products of representations.

2. Clifford Algebras and Spinors.

2.1. Real Clifford algebras. 2.2. Complex Clifford algebras and spin modules. 2.3. Spin representations of Lie groups and algebras.

3. Dirac Operators in the Algebraic Setting.

3.1. Dirac operators. 3.2. Dirac cohomology and Vogan’s conjecture. 3.3. A differential on \((U(\mathfrak{g})\otimes C(\mathfrak{p}))^K\). 3.4. The homomorphism \(\zeta\). 3.5. An extension of Parthasarathy’s Dirac inequality.

4. A Generalized Bott-Borel-Weil Theorem.

4.1. Kostant cubic Dirac operators. 4.2. Dirac cohomology of finite-dimensional representations. 4.3. Characters. 4.4. A generalized Weyl character formula. 4.5. A generalized Bott-Borel-Weil theorem.

5. Cohomological Induction.

5.1. Overview. 5.2. Some generalities about adjoint functors. 5.3. Homological algebra of Harish-Chandra modules. 5.4. Zuckerman functors. 5.5. Bernstein functors.

6. Properties of Cohomologically Induced Modules.

6.1. Duality theorems. 6.2. Infinitesimal character, \(K\)-types and vanishing. 6.3 Irreducibility and unitarity. 6.4. \(A_{\mathfrak{q}}(\lambda)\) modules. 6.5. Unitary modules with strongly regular infinitesimal character.

7. Discrete Series.

7.1. \(L^2\)-index theorem. 7.2. Existence of discrete series. 7.3. Global characters. 7.4. Exhaustion of discrete series.

8. Dimensions of Spaces of Automorphic Forms.

8.1. Hirzebruch proportionality principle. 8.2. Dimensions of spaces of automorphic forms. 8.3. Dirac cohomology and \((\mathfrak{g},K)\)-cohomology. 8.4. Cohomology of discrete subgroups.

9. Dirac Operators and Nilpotent Lie Algebra Cohomology.

9.1. \(\mathfrak{u}\)-homology and \(\bar{\mathfrak{u}}\)-cohomology differentials. 9.2. Hodge decomposition in the finite-dimensional case. 9.3. Hodge decomposition for \(\mathfrak{p}^-\)-cohomology in the unitary case. 9.4. Calculating Dirac cohomology in stages. 9.5. Hodge decomposition for \(\bar{\mathfrak{u}}\)-cohomology in the unitary case. 9.6. Homological properties of Dirac cohomology.

10. Dirac Cohomology for Lie Superalgebras.

10.1. Lie superalgebras of Riemannian type. 10.2. Dirac operators for \((\mathfrak{g},\mathfrak{g}_0)\). 10.3. Analog of Vogan’s conjecture. 10.4. Dirac cohomology for Lie superalgebras.

References. Index.

Reviewer: Yuri I. Karlovich (Cuernavaca)

##### MSC:

22E47 | Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) |

22-02 | Research exposition (monographs, survey articles) pertaining to topological groups |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

22E46 | Semisimple Lie groups and their representations |

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

81T70 | Quantization in field theory; cohomological methods |