Nilpotent orbits in representation theory.

*(English)*Zbl 1169.14319
Anker, Jean-Philippe (ed.) et al., Lie theory. Lie algebras and representations. Boston, MA: BirkhĂ¤user (ISBN 0-8176-3373-1/hbk). Progress in Mathematics 228, 1-211 (2004).

This article, based on a series of lectures given at the European School of Group Theory in Odense, provides a detailed treatment of the theory of nilpotent orbits in Lie algebras of reductive algebraic groups over algebraically closed fields. It continues the series of comprehensive surveys on the theory of orbits (conjugacy classes) in the adjoint representations that was started by T. A. Springer and R. Steinberg, Lect. Notes Math. 131 (1970; Zbl 0249.20024)]. The article is very well written and contains a lot of useful information.

The work contains many examples from classical groups with elegant use of linear algebra and varieties with group actions, thereby providing an excellent introduction to the field. One of the main points is that the characteristic of the ground field \(K\) is arbitrary, and the author attempts to present the result in a characteristic free fashion whenever possible. The article is divided into 13 sections and has an extensive bibliography. Most of the results, especially for classical Lie algebras, are presented with complete proofs.

In a sense, the article under review starts at the point where the above-mentioned article of Springer and Steinberg stops. Section 1 contains a thorough treatment of nilpotent orbits in the classical Lie algebras, i.e., for \(\mathfrak{sl}(V)\), \(\mathfrak{so}(V)\), and \(\mathfrak{sp}(V)\). All results are given with complete proofs. This includes the classification of nilpotent orbits in terms of partitions and pointing out a canonical representative in each orbit. It is then the general strategy of the author to discuss all new concepts first in the classical cases, using partitions and explicit matrix models.

Section 2 contains some general results. The most important of them is the finiteness of the number of nilpotent orbits, which is proved by Richardson’s method. It is in this section that the author introduces the notion of good and bad primes, which plays an important role in what follows. After these preparations, Section 3 gives a description of centralisers of nilpotent elements in the classical Lie algebras. A method for determining the dimension and the reductive part of the centraliser using the corresponding partition or its dual is given. Later, in Section 5, the structure of centralisers is discussed in the general context. A classification of nilpotent orbits based on the Bala-Carter approach is presented in Section 4. This section contains the usual material on distinguished parabolic subgroups, Levi subgroups, and distinguished nilpotent orbits.

Sections 6 and 7 provide general results on the nilpotent cone \(\mathcal N\subset\mathfrak g\). This includes irreducibility and dimension of \(\mathcal N\), a description of the ideal of \(\mathcal N\) in \(S({\mathfrak g}^*)\), properties of the quotient morphism \({\mathfrak g}\to{\mathfrak g}/G\), the structure of the algebra \(S({\mathfrak g}^*)^G\), and construction of Springer’s resolution \(p\colon\tilde{\mathcal N}\to\mathcal N\).

The material in Sections 1–7 is more or less standard and can be found in several textbooks on nilpotent orbits, algebraic groups, and invariant theory [see, e.g., D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie algebras, Van Nostrand Reinhold, New York (1993; Zbl 0972.17008); W. M. McGovern, in: Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action. Berlin: Springer. Encycl. Math. Sci. 131(2), 159–238 (2002; Zbl 1036.17007); R. Steinberg, Conjugacy classes in algebraic groups. Lect. Notes Math. 366 (1974; Zbl 0281.20037); E. B. Vinberg, V. V. Gorbatsevich and A. L. Onishchik, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 41 (1990; Zbl 0733.22003)]. The last six sections discuss more advanced topics. The titles of these sections already give a good understanding of the contents: 8. Functions on orbits and orbit closures; 9. Associated varieties; 10. Springer’s fibers and Steinberg’s triples; 11. Paving Springer’s fibers; 12. \(l\)-adic and perverse stuff; 13. Springer’s representations.

Section 8 provides a detailed description of the graded ring of functions, including McGovern’s theorem and, as a particular case, the situation for minimal and Richardson orbits. The normality of the orbit closures is also discussed. In Section 9, the author shows how to associate to each simple \(\mathfrak g\)-module a nilpotent orbit and discusses various features of this correspondence. Other topics discussed here are: Verma modules, primitive ideals in \(U({\mathfrak g})\), left and two-sided cells in the Weyl group. To this end, necessary basic notions of noncommutative theory are introduced. Sections 10 and 11 are devoted to the study of fine properties of the fibres of Springer’s resolution. Then this is used in Sections 12 and 13 for presenting Springer’s uniform construction of Weyl group representations via the cohomology of fibres of \(p:\tilde{\mathcal N}\to\mathcal N\).

For the entire collection see [Zbl 1054.17001].

The work contains many examples from classical groups with elegant use of linear algebra and varieties with group actions, thereby providing an excellent introduction to the field. One of the main points is that the characteristic of the ground field \(K\) is arbitrary, and the author attempts to present the result in a characteristic free fashion whenever possible. The article is divided into 13 sections and has an extensive bibliography. Most of the results, especially for classical Lie algebras, are presented with complete proofs.

In a sense, the article under review starts at the point where the above-mentioned article of Springer and Steinberg stops. Section 1 contains a thorough treatment of nilpotent orbits in the classical Lie algebras, i.e., for \(\mathfrak{sl}(V)\), \(\mathfrak{so}(V)\), and \(\mathfrak{sp}(V)\). All results are given with complete proofs. This includes the classification of nilpotent orbits in terms of partitions and pointing out a canonical representative in each orbit. It is then the general strategy of the author to discuss all new concepts first in the classical cases, using partitions and explicit matrix models.

Section 2 contains some general results. The most important of them is the finiteness of the number of nilpotent orbits, which is proved by Richardson’s method. It is in this section that the author introduces the notion of good and bad primes, which plays an important role in what follows. After these preparations, Section 3 gives a description of centralisers of nilpotent elements in the classical Lie algebras. A method for determining the dimension and the reductive part of the centraliser using the corresponding partition or its dual is given. Later, in Section 5, the structure of centralisers is discussed in the general context. A classification of nilpotent orbits based on the Bala-Carter approach is presented in Section 4. This section contains the usual material on distinguished parabolic subgroups, Levi subgroups, and distinguished nilpotent orbits.

Sections 6 and 7 provide general results on the nilpotent cone \(\mathcal N\subset\mathfrak g\). This includes irreducibility and dimension of \(\mathcal N\), a description of the ideal of \(\mathcal N\) in \(S({\mathfrak g}^*)\), properties of the quotient morphism \({\mathfrak g}\to{\mathfrak g}/G\), the structure of the algebra \(S({\mathfrak g}^*)^G\), and construction of Springer’s resolution \(p\colon\tilde{\mathcal N}\to\mathcal N\).

The material in Sections 1–7 is more or less standard and can be found in several textbooks on nilpotent orbits, algebraic groups, and invariant theory [see, e.g., D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie algebras, Van Nostrand Reinhold, New York (1993; Zbl 0972.17008); W. M. McGovern, in: Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action. Berlin: Springer. Encycl. Math. Sci. 131(2), 159–238 (2002; Zbl 1036.17007); R. Steinberg, Conjugacy classes in algebraic groups. Lect. Notes Math. 366 (1974; Zbl 0281.20037); E. B. Vinberg, V. V. Gorbatsevich and A. L. Onishchik, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 41 (1990; Zbl 0733.22003)]. The last six sections discuss more advanced topics. The titles of these sections already give a good understanding of the contents: 8. Functions on orbits and orbit closures; 9. Associated varieties; 10. Springer’s fibers and Steinberg’s triples; 11. Paving Springer’s fibers; 12. \(l\)-adic and perverse stuff; 13. Springer’s representations.

Section 8 provides a detailed description of the graded ring of functions, including McGovern’s theorem and, as a particular case, the situation for minimal and Richardson orbits. The normality of the orbit closures is also discussed. In Section 9, the author shows how to associate to each simple \(\mathfrak g\)-module a nilpotent orbit and discusses various features of this correspondence. Other topics discussed here are: Verma modules, primitive ideals in \(U({\mathfrak g})\), left and two-sided cells in the Weyl group. To this end, necessary basic notions of noncommutative theory are introduced. Sections 10 and 11 are devoted to the study of fine properties of the fibres of Springer’s resolution. Then this is used in Sections 12 and 13 for presenting Springer’s uniform construction of Weyl group representations via the cohomology of fibres of \(p:\tilde{\mathcal N}\to\mathcal N\).

For the entire collection see [Zbl 1054.17001].

Reviewer: Dmitri Panyushev (Bonn) (MR0352279)