Einhüllende Algebren halbeinfacher Lie-Algebren.

*(English)*Zbl 0541.17001
Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 3. Berlin etc.: Springer-Verlag. 298 S. DM 118.00 (1983).

Ten years ago, J. Dixmier’s book ”Algèbres enveloppantes” (Paris, 1974; Zbl 0308.17007, English ed. 1977; Zbl 0339.17007) established the algebraic study of the enveloping algebra \(U({\mathfrak g})\) of a finite- dimensional complex Lie algebra \({\mathfrak g}\) as a mathematical subject in its own right, which is very useful and of great interest from the point of view of various different other subjects, such as the representation theory of Lie groups and algebras, the theory of Noetherian rings, the theory of rings and modules of differential operators, or the theory of algebraic groups and the geometry of their homogeneous spaces, conjugacy classes, Schubert varieties. At that time, a fairly complete understanding of the case \({\mathfrak g}\) solvable had been achieved, and was about simultaneously presented also in a book by P. Gabriel, R. Rentschler, and the reviewer [Lect. Notes Math. 357 (1973; Zbl 0293.17005)\(=[BGR]]\). These two presentations of the solvable case are ”still valid today” (as the author states in the introduction), while this is not at all true for the chapters (7-9 resp. 10) of Dixmier’s book dedicated to the semisimple resp. the general case. Although an impressive amount of beautiful results about the semisimple case - pioneer work of Harish-Chandra, Kostant, Gelfand, Duflo, and others - was already known at that time, it became clear since then that this was merely a prelude to a period of exceptionally fast and fruitful developments in the past ten years, which led to an unexpectedly deep and rich theory about enveloping algebras of semisimple Lie algebras. However, it seems that only a relatively small circle of experts has been aware of the beautiful tree that grew out meanwhile of the germs popularized by Dixmier’s book.

The merit of the present book is that it makes for the first time an important portion of this theory - the ring-theoretical and the representation-theoretical aspects of the ”tree” - accessible to a larger circle of interested mathematicians. It will also be extremely useful as a basis and reference work for further research in the field. On the 300 pages of this book, the author gives a highly concentrated and condensed report of the results of many significant recent research papers - especially from the late seventies - which never appeared in a book before. The core of the book consists of a complete development of A. Joseph’s work about the classification of the primitive ideals in terms of Weyl group representations, by ring theoretic methods (Ore localizations, Goldie ranks, Gelfand-Kirillov dimension, etc.), in addition to the necessary reductions to (and theory of) highest weight modules and Harish-Chandra modules, due to M. Duflo, D. Vogan, J. Bernstein, I. M. Gelfand, S. I. Gelfand, and others (including the author and the reviewer).

Let me next give a rough outline of the contents of the book. The first three chapters are essentially of preparatorial nature. Here the author collects some necessary generalities or technical requisites about enveloping algebras (Kapitel 1) resp. semisimple Lie algebras (Kapitel 2) resp. centralizers in \(U({\mathfrak g})\) (Kapitel 3), starting with a few lines on Harish-Chandra’s description of the center \(Z({\mathfrak g})\) of \(U({\mathfrak g})\). The author’s general policy is to refer - wherever reasonably possible - to the books of Dixmier, or [BGR], or his own ”Moduln mit einem höchsten Gewicht” [Lect. Notes Math. 750 (1979; Zbl 0426.17001)\(=[MHG]]\), for proofs and more details. This applies also to his summary of the theory of highest weight modules in the next chapter (Kapitel 4), which is crucial and fundamental for most of the material presented in this book. Among the more advanced points treated here are e.g. 1) projective objects in the BGG (Bernstein-Gelfand-Gelfand) category \({\mathcal O}\); 2) translation functors including the ideas of Speh- Vogan about ”translation through the walls” and ”coherent continuation”; 3) Ext-groups in \({\mathcal O}\) including results of Delorme; 4) work of Gabber and Joseph describing the radical of a Verma module (4.18), and concerning the BGG resolution of simple highest weight modules. It has to be noted that this summary does not include the perhaps most important results of the subject, the ”KLBBBK character formula”, that is to say a certain formula (in terms of combinatorics of the Weyl group, or alternatively in terms of geometry of Schubert varieties) for the formal character of a simple highest weight module conjectured by Kazhdan- Lusztig (1979), and proved by Beilinson-Bernstein, and independently by Brylinski-Kashiwara (1981). It is a consequence of the basic conception of this book - namely to almost totally avoid the geometrical and the differential operator aspects and methods of the subject - that this result is only briefly stated towards the end of the book (Kapitel 16), necessarily in the combinatorial version, and without even comments on the proof.

The central topic of the book, the classification of primitive ideals in \(U({\mathfrak g})\), is taken up in the chapter on ”annihilators of simple highest weight modules” (Kapitel 5). The set \({\mathfrak X}\) of all such annihilators coincides with the set of all primitive ideals of \(U({\mathfrak g})\), by Duflo’s fundamental theorem, which is proved, however, only later (in Kapitel 7). This set \({\mathfrak X}\) is partitioned into subsets \({\mathfrak X}_{\lambda}\) corresponding to a specified character \(\chi_{\lambda}\) of the center \(Z({\mathfrak g})\) (Kapitel 3), and and the elements of each subset \({\mathfrak X}_{\lambda}\) are indexed by the elements of the Weyl group W (Kapitel 4); more precisely \[ {\mathfrak X}_{\lambda}=\{J_{w.\lambda}:=Ann L(w.\lambda)| \quad w\in W\}, \] where \(L(\mu)\) denotes the simple module of highest weight \(\mu\), and \(\lambda\) is chosen fixed (e.g. ”dominant”) in its W-orbit. It remains to be determined when two elements w,\(y\in W\) give the same primitive ideal \(J_{w.\lambda}=J_{y.\lambda}\). Thus the goal of classification comes down to the specification of a certain equivalence relation on W (depending on \(\lambda)\). For the special case \({\mathfrak g}={\mathfrak sl}_ n\), where W is the symmetric group \(S_ n\), Joseph has found a very elegant explicit combinatorial description of this equivalence relation, in terms of the Robinson-Schensted algorithm, which associates to each permutation a standard tableau. This appears as the highlight in this chapter (Satz 5.25). For this purpose (and many others in subsequent chapters), various general techniques are developed before, including 1) a ”translation principle” for relating different subsets \({\mathfrak X}_{\lambda}\) to each other (the original version, due to the author and the reviewer, being slightly refined here using work of Vogan), 2) Vogan’s concept of ”generalized \(\tau\)-invariant” to distinguish different elements of \({\mathfrak X}_{\lambda}\) from each other, and 3) Joseph’s concept of ”characteristic variety” (totally unrelated to the concept of the same name in the theory of D-modules) serving to verify coincidences of certain primitive ideals.

Then (in Kapitel 6) the author presents an algebraic approach to the theory of Harish-Chandra bimodules (i.e. \({\mathfrak g}\times {\mathfrak g}\)- modules with a locally finite diagonal action of finite multiplicities), following Bernstein and S. Gelfand. The main point here is an equivalence of this category with a certain subcategory of the category \({\mathcal O}\) (Kapitel 4), which was independently obtained by Enright, and Joseph. Since \(U({\mathfrak g})/J\) is a Harish-Chandra bimodule in an obvious way, for each primitive ideal J, this equivalence can be evaluated for the study of primitive ideals, which is done in Kapitel 7. A first easy consequence is the following statement, known as ”Dixmier’s problem” for a Verma module, say \(M(\lambda)\) with dominant regular integral highest weight \(\lambda\) : The map \(I\mapsto IM(\lambda)\) is a bijection of the set of all ideals of \(U({\mathfrak g})\) annihilating \(L(\lambda)\) to the set of all submodules of \(M(\lambda)\). Now Duflo’s theorem drops out as an immediate corollary. Further major points in this chapter are 1) the determination of the annihilators of simple Harish-Chandra bimodules (Duflo-Joseph), and 2) a characterization of the order relation on \({\mathfrak X}_{\lambda}\) (by inclusion) in terms of tensor products of simple highest weight modules with finite-dimensional modules (Vogan, 7.13). This latter result allows to compute (even the order relation on) \({\mathfrak X}_{\lambda}\) from the formal characters of the simple highest weight modules. Consequently, by combining the work of Duflo, Joseph, Vogan about primitive ideals with the ”KLBBBK character formula” mentioned previously in this review, the classification of primitive ideals is ultimately reduced to certain explicit (though often hopelessly complicated) combinatorics of the Weyl group. The deduction, however, is highly non-trivial and involves further new ideas. But there is fortunately a beautiful theory of Joseph, relating the classification of primitive ideals to irreducible Weyl group representations via his ”Goldie rank polynomials”, which e.g. serves Barbasch-Vogan to organize this deduction efficiently. As the ”KLBBBK formula” itself, this is only briefly reviewed without proofs and details in Kapitel 16 of the book.

On the other hand, this theory of Joseph may be developed independently of the ”KLBBBK formula”, and gives also additional, deeper insight into the classification of primitive ideals. A complete development of this theory - with full proofs and in much detail - is the dominating purpose in the second half of the book. This development culminates in Kapitel 14, entitled ”Goldie rank polynomials and Weyl group representations”, where the main results of the theory are formulated and proved. For this purpose, however, the author has to prepare a lot of further prerequisites in the preceding chapters. Fortunately, several of them are of some independent interest, as for instance the study of Gelfand- Kirillov dimensions and multiplicities (i.e. degree and highest coefficient of certain Hilbert-Samuel polynomials) in Kapitel 8, 9, and 10, the study of Ore set localizations in Kapitel 11, and the study of Goldie ranks of primitive ideals in Kapitel 12. Kapitel 13 on skew polynomial extensions, and Kapitel 15 on the ”generalized Gelfand- Kirillov conjecture” are dealing with ring-theoretic descriptions of the quotient \(U({\mathfrak g})/J\) by a primitive ideal J, or rather with the structure of its complete ring of fractions, which is - by Goldie’s theorem - a matrix ring over some skew field. The highlight of these chapters is Joseph’s proof for \({\mathfrak g}={\mathfrak sl}_ n\) of the experts’ general hope that this skew field is always isomorphic to the field of fractions of a Weyl algebra.

The last two chapters (16 and 17) are ”Ergebnisberichte” in the sense of the word (”im engeren Sinne”), as the author puts it, meaning a report of results without proofs. Concerning the fundamental results about ”Kazhdan-Lusztig polynomials and special Weyl group representations” reported in Kapitel 16, refer to my previous comments above. Let me add that Joseph’s classification theory of primitive ideals in terms of irreducible Weyl group representations is completed only here by the results (verified case by case, using the Kazhdan-Lusztig conjecture) of Barbasch-Vogan that 1) Joseph’s Weyl group representation coincides with Springer’s Weyl group representation attached to a certain nilpotent orbit in \({\mathfrak g}\), and 2) the list of all representations occuring here for integral central characters coincides with Lusztig’s so-called ”special” representations. While the last point still seems a bit mysterious today, the first point has meanwhile found a very natural, geometric explanation, after this book was published.

It is a characteristic feature of this book that it almost totally avoids the various geometrical aspects of the subject. The final Kapitel 17, on ”associated varieties”, is essentially the only exception. The ”associated variety” of a primitive ideal J in \(U({\mathfrak g})\) is defined as the zero set in \({\mathfrak g}\) of the associated graded ideal of J. This variety turns out to be always the closure of a single nilpotent orbit in \({\mathfrak g}\). This was suggested already a long time ago by the reviewer, but a full proof was completed only recently (by work of Joseph, Hotta, Brylinski, and the reviewer), when the book was already in press. The author presents here useful expositions of the proofs for the cases \({\mathfrak g}={\mathfrak sl}_ n\) resp. \({\mathfrak g}\) classical, given by Joseph resp. Brylinski and the reviewer. - Let me call attention also to an additional, more sophisticated relation of primitive ideals to conjugacy classes (coadjoint orbits): The author’s and the reviewer’s work on Dixmier’s orbit method, which establishes an injective map of the coadjoint orbit space of \({\mathfrak g}={\mathfrak sl}_ n\) into the space of completely prime primitive ideals, is only mentioned in passing in the book, although recent work of C. Moeglin, proving the bijectivity of this map, and of D. Vogan (modifying the orbit method, to deal with strange phenomena occuring if \({\mathfrak g}\) is not \({\mathfrak sl}_ n)\), seem to indicate that this aspect still deserves some attention.

The book concludes with an extensive, very useful bibliography (mainly of very recent papers), a useful index of notations, and a subject index (which could be more complete). Let me conclude this review with one more remark about recent developments: There exists by now a more geometrical understanding of Joseph’s classification theory for primitive ideals, in terms of characteristic (cohomology) classes on the flag variety, from which the relation to nilpotent orbits and Springer’s theory of Weyl group representations emerges quite naturally (recent work of Brylinski, MacPherson, and the reviewer). This approach, however, does make use of sheaves of modules of differential operators, and intersection homology theory, i.e. the main ingredients for the proof of the Kazhdan-Lusztig conjecture, avoided by the author in the present book.

The merit of the present book is that it makes for the first time an important portion of this theory - the ring-theoretical and the representation-theoretical aspects of the ”tree” - accessible to a larger circle of interested mathematicians. It will also be extremely useful as a basis and reference work for further research in the field. On the 300 pages of this book, the author gives a highly concentrated and condensed report of the results of many significant recent research papers - especially from the late seventies - which never appeared in a book before. The core of the book consists of a complete development of A. Joseph’s work about the classification of the primitive ideals in terms of Weyl group representations, by ring theoretic methods (Ore localizations, Goldie ranks, Gelfand-Kirillov dimension, etc.), in addition to the necessary reductions to (and theory of) highest weight modules and Harish-Chandra modules, due to M. Duflo, D. Vogan, J. Bernstein, I. M. Gelfand, S. I. Gelfand, and others (including the author and the reviewer).

Let me next give a rough outline of the contents of the book. The first three chapters are essentially of preparatorial nature. Here the author collects some necessary generalities or technical requisites about enveloping algebras (Kapitel 1) resp. semisimple Lie algebras (Kapitel 2) resp. centralizers in \(U({\mathfrak g})\) (Kapitel 3), starting with a few lines on Harish-Chandra’s description of the center \(Z({\mathfrak g})\) of \(U({\mathfrak g})\). The author’s general policy is to refer - wherever reasonably possible - to the books of Dixmier, or [BGR], or his own ”Moduln mit einem höchsten Gewicht” [Lect. Notes Math. 750 (1979; Zbl 0426.17001)\(=[MHG]]\), for proofs and more details. This applies also to his summary of the theory of highest weight modules in the next chapter (Kapitel 4), which is crucial and fundamental for most of the material presented in this book. Among the more advanced points treated here are e.g. 1) projective objects in the BGG (Bernstein-Gelfand-Gelfand) category \({\mathcal O}\); 2) translation functors including the ideas of Speh- Vogan about ”translation through the walls” and ”coherent continuation”; 3) Ext-groups in \({\mathcal O}\) including results of Delorme; 4) work of Gabber and Joseph describing the radical of a Verma module (4.18), and concerning the BGG resolution of simple highest weight modules. It has to be noted that this summary does not include the perhaps most important results of the subject, the ”KLBBBK character formula”, that is to say a certain formula (in terms of combinatorics of the Weyl group, or alternatively in terms of geometry of Schubert varieties) for the formal character of a simple highest weight module conjectured by Kazhdan- Lusztig (1979), and proved by Beilinson-Bernstein, and independently by Brylinski-Kashiwara (1981). It is a consequence of the basic conception of this book - namely to almost totally avoid the geometrical and the differential operator aspects and methods of the subject - that this result is only briefly stated towards the end of the book (Kapitel 16), necessarily in the combinatorial version, and without even comments on the proof.

The central topic of the book, the classification of primitive ideals in \(U({\mathfrak g})\), is taken up in the chapter on ”annihilators of simple highest weight modules” (Kapitel 5). The set \({\mathfrak X}\) of all such annihilators coincides with the set of all primitive ideals of \(U({\mathfrak g})\), by Duflo’s fundamental theorem, which is proved, however, only later (in Kapitel 7). This set \({\mathfrak X}\) is partitioned into subsets \({\mathfrak X}_{\lambda}\) corresponding to a specified character \(\chi_{\lambda}\) of the center \(Z({\mathfrak g})\) (Kapitel 3), and and the elements of each subset \({\mathfrak X}_{\lambda}\) are indexed by the elements of the Weyl group W (Kapitel 4); more precisely \[ {\mathfrak X}_{\lambda}=\{J_{w.\lambda}:=Ann L(w.\lambda)| \quad w\in W\}, \] where \(L(\mu)\) denotes the simple module of highest weight \(\mu\), and \(\lambda\) is chosen fixed (e.g. ”dominant”) in its W-orbit. It remains to be determined when two elements w,\(y\in W\) give the same primitive ideal \(J_{w.\lambda}=J_{y.\lambda}\). Thus the goal of classification comes down to the specification of a certain equivalence relation on W (depending on \(\lambda)\). For the special case \({\mathfrak g}={\mathfrak sl}_ n\), where W is the symmetric group \(S_ n\), Joseph has found a very elegant explicit combinatorial description of this equivalence relation, in terms of the Robinson-Schensted algorithm, which associates to each permutation a standard tableau. This appears as the highlight in this chapter (Satz 5.25). For this purpose (and many others in subsequent chapters), various general techniques are developed before, including 1) a ”translation principle” for relating different subsets \({\mathfrak X}_{\lambda}\) to each other (the original version, due to the author and the reviewer, being slightly refined here using work of Vogan), 2) Vogan’s concept of ”generalized \(\tau\)-invariant” to distinguish different elements of \({\mathfrak X}_{\lambda}\) from each other, and 3) Joseph’s concept of ”characteristic variety” (totally unrelated to the concept of the same name in the theory of D-modules) serving to verify coincidences of certain primitive ideals.

Then (in Kapitel 6) the author presents an algebraic approach to the theory of Harish-Chandra bimodules (i.e. \({\mathfrak g}\times {\mathfrak g}\)- modules with a locally finite diagonal action of finite multiplicities), following Bernstein and S. Gelfand. The main point here is an equivalence of this category with a certain subcategory of the category \({\mathcal O}\) (Kapitel 4), which was independently obtained by Enright, and Joseph. Since \(U({\mathfrak g})/J\) is a Harish-Chandra bimodule in an obvious way, for each primitive ideal J, this equivalence can be evaluated for the study of primitive ideals, which is done in Kapitel 7. A first easy consequence is the following statement, known as ”Dixmier’s problem” for a Verma module, say \(M(\lambda)\) with dominant regular integral highest weight \(\lambda\) : The map \(I\mapsto IM(\lambda)\) is a bijection of the set of all ideals of \(U({\mathfrak g})\) annihilating \(L(\lambda)\) to the set of all submodules of \(M(\lambda)\). Now Duflo’s theorem drops out as an immediate corollary. Further major points in this chapter are 1) the determination of the annihilators of simple Harish-Chandra bimodules (Duflo-Joseph), and 2) a characterization of the order relation on \({\mathfrak X}_{\lambda}\) (by inclusion) in terms of tensor products of simple highest weight modules with finite-dimensional modules (Vogan, 7.13). This latter result allows to compute (even the order relation on) \({\mathfrak X}_{\lambda}\) from the formal characters of the simple highest weight modules. Consequently, by combining the work of Duflo, Joseph, Vogan about primitive ideals with the ”KLBBBK character formula” mentioned previously in this review, the classification of primitive ideals is ultimately reduced to certain explicit (though often hopelessly complicated) combinatorics of the Weyl group. The deduction, however, is highly non-trivial and involves further new ideas. But there is fortunately a beautiful theory of Joseph, relating the classification of primitive ideals to irreducible Weyl group representations via his ”Goldie rank polynomials”, which e.g. serves Barbasch-Vogan to organize this deduction efficiently. As the ”KLBBBK formula” itself, this is only briefly reviewed without proofs and details in Kapitel 16 of the book.

On the other hand, this theory of Joseph may be developed independently of the ”KLBBBK formula”, and gives also additional, deeper insight into the classification of primitive ideals. A complete development of this theory - with full proofs and in much detail - is the dominating purpose in the second half of the book. This development culminates in Kapitel 14, entitled ”Goldie rank polynomials and Weyl group representations”, where the main results of the theory are formulated and proved. For this purpose, however, the author has to prepare a lot of further prerequisites in the preceding chapters. Fortunately, several of them are of some independent interest, as for instance the study of Gelfand- Kirillov dimensions and multiplicities (i.e. degree and highest coefficient of certain Hilbert-Samuel polynomials) in Kapitel 8, 9, and 10, the study of Ore set localizations in Kapitel 11, and the study of Goldie ranks of primitive ideals in Kapitel 12. Kapitel 13 on skew polynomial extensions, and Kapitel 15 on the ”generalized Gelfand- Kirillov conjecture” are dealing with ring-theoretic descriptions of the quotient \(U({\mathfrak g})/J\) by a primitive ideal J, or rather with the structure of its complete ring of fractions, which is - by Goldie’s theorem - a matrix ring over some skew field. The highlight of these chapters is Joseph’s proof for \({\mathfrak g}={\mathfrak sl}_ n\) of the experts’ general hope that this skew field is always isomorphic to the field of fractions of a Weyl algebra.

The last two chapters (16 and 17) are ”Ergebnisberichte” in the sense of the word (”im engeren Sinne”), as the author puts it, meaning a report of results without proofs. Concerning the fundamental results about ”Kazhdan-Lusztig polynomials and special Weyl group representations” reported in Kapitel 16, refer to my previous comments above. Let me add that Joseph’s classification theory of primitive ideals in terms of irreducible Weyl group representations is completed only here by the results (verified case by case, using the Kazhdan-Lusztig conjecture) of Barbasch-Vogan that 1) Joseph’s Weyl group representation coincides with Springer’s Weyl group representation attached to a certain nilpotent orbit in \({\mathfrak g}\), and 2) the list of all representations occuring here for integral central characters coincides with Lusztig’s so-called ”special” representations. While the last point still seems a bit mysterious today, the first point has meanwhile found a very natural, geometric explanation, after this book was published.

It is a characteristic feature of this book that it almost totally avoids the various geometrical aspects of the subject. The final Kapitel 17, on ”associated varieties”, is essentially the only exception. The ”associated variety” of a primitive ideal J in \(U({\mathfrak g})\) is defined as the zero set in \({\mathfrak g}\) of the associated graded ideal of J. This variety turns out to be always the closure of a single nilpotent orbit in \({\mathfrak g}\). This was suggested already a long time ago by the reviewer, but a full proof was completed only recently (by work of Joseph, Hotta, Brylinski, and the reviewer), when the book was already in press. The author presents here useful expositions of the proofs for the cases \({\mathfrak g}={\mathfrak sl}_ n\) resp. \({\mathfrak g}\) classical, given by Joseph resp. Brylinski and the reviewer. - Let me call attention also to an additional, more sophisticated relation of primitive ideals to conjugacy classes (coadjoint orbits): The author’s and the reviewer’s work on Dixmier’s orbit method, which establishes an injective map of the coadjoint orbit space of \({\mathfrak g}={\mathfrak sl}_ n\) into the space of completely prime primitive ideals, is only mentioned in passing in the book, although recent work of C. Moeglin, proving the bijectivity of this map, and of D. Vogan (modifying the orbit method, to deal with strange phenomena occuring if \({\mathfrak g}\) is not \({\mathfrak sl}_ n)\), seem to indicate that this aspect still deserves some attention.

The book concludes with an extensive, very useful bibliography (mainly of very recent papers), a useful index of notations, and a subject index (which could be more complete). Let me conclude this review with one more remark about recent developments: There exists by now a more geometrical understanding of Joseph’s classification theory for primitive ideals, in terms of characteristic (cohomology) classes on the flag variety, from which the relation to nilpotent orbits and Springer’s theory of Weyl group representations emerges quite naturally (recent work of Brylinski, MacPherson, and the reviewer). This approach, however, does make use of sheaves of modules of differential operators, and intersection homology theory, i.e. the main ingredients for the proof of the Kazhdan-Lusztig conjecture, avoided by the author in the present book.

Reviewer: W.Borho

##### MSC:

17B35 | Universal enveloping (super)algebras |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

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

22E46 | Semisimple Lie groups and their representations |

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

20G05 | Representation theory for linear algebraic groups |

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