##
**Finite groups of Lie type. Conjugacy classes and complex characters.**
*(English)*
Zbl 0567.20023

Pure and Applied Mathematics. A Wiley-Interscience Publication. Chichester-New York etc.: John Wiley and Sons. XII, 544 p. £42.50 (1985).

As the title indicates, the book under review is concerned with the conjugacy classes and the representation theory of finite groups of Lie type. Let G be a connected reductive algebraic group defined over a finite field, F a Frobenius morphism of G, and \(G^ F\) the finite group of F-fixed points. The finite groups obtained in this way are the groups of Lie type. These groups inherit some of the structure of the algebraic group G, and in particular they have a split BN-pair with Weyl group W, where B is an F-fixed Borel subgroup containing an F-fixed maximal torus T (such a torus is said to be maximally split), \(N=N(T)\) and \(W=N/T\). The book therefore begins in Chapter 1 with an exposition (without proofs) of the theory of algebraic groups over an algebraically closed field. The BN-pair axioms are given and some consequences are derived in Chapter 2, including the pattern of intersections of parabolic subgroups. Chapter 3 continues the basic material by describing the classification of \(G^ F\)-conjugacy classes of maximal tori in G; these are in bijection with the F-conjugacy classes of W.

The representation theory of finite groups of Lie type entered a new phase in the early seventies with the development of what is now known as Harish-Chandra theory by Harish-Chandra and T. A. Springer [see Cusp forms in finite groups, in Lect. Notes Math. 131, C1-C24 (1970; Zbl 0263.20024)]. An even more exciting breakthrough was achieved by the paper of P. Deligne and G. Lusztig in 1976 [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. Let T be any F-fixed maximal torus of the algebraic group G as above. Then for any character \(\theta\) of \(T^ F\) (i.e. a homomorphism into \(\bar Q^*_{\ell}\), for a prime \(\ell\) not equal to the characteristic of the finite field) Deligne and Lusztig (loc. cit.) defined a virtual representation, or equivalently, a generalized character \(R^ G_ T(\theta)\) of \(G^ F\) over \(\bar Q_{\ell}\) by using the \(\ell\)-adic cohomology of certain subvarieties of the flag variety. These characters have the expected properties of orthogonality, and indeed \(R^ G_ T(\theta)\) and \(R^ G_{T'}(\theta ')\) have no irreducible constituents in common unless the pairs (T,\(\theta)\), (T’,\(\theta\) ’) are related by ”geometric conjugacy”. If \(G^*\) is a group in duality with G, the geometric conjugacy classes of pairs (T,\(\theta)\) are in bijection with the F-stable semisimple conjugacy classes of \(G^{*F}\). The concepts of geometric conjugacy and duality are explained in Chapter 4. If \(\theta\) is in ”general position”, \(R^ G_ T(\theta)\) is irreducible up to sign. Moreover, if \(x=su\) is the Jordan decomposition of \(x\in G^ F\), the value of \(R^ G_ T(\theta)\) at x can be described in terms of \(\theta\) and of the values of the analogous generalized characters for \(C^ F_ G(s)\) at unipotent elements. Assuming the basic properties of \(\ell\)-adic cohomology (which are stated in an appendix) the author derives these properties of the Deligne-Lusztig characters in Chapter 7. In Chapter 8 he describes (when the center of G is connected) certain rational linear combinations of the \(R^ G_ T(\theta)\) which turn out to be irreducible characters and which he calls semisimple and regular characters. There is precisely one semisimple and one regular character in each geometric conjugacy class, and the regular characters are those appearing in the Gelfand-Graev representation, which is also described in this chapter.

The Harish-Chandra theory states that the irreducible representations of G fall into families in the following way: if \(\rho\) is an irreducible character of \(G^ F\) then \(\rho\) is a constituent of an induced character \(Ind^{G^ F}_{P^ F}(\tau)\) where \(\tau\) is the pullback of a parabolic subgroup \(P^ F\) of a ”cuspidal” character of a Levi factor \(L^ F\) of \(P^ F\), and the pair (L,\(\tau)\) is unique up to \(G^ F\)-conjugacy. A connection of this theory with the Deligne-Lusztig theory is given by the fact that if \(\theta\) is a character of \(T^ F\) in general position, then the irreducible (up to sign) character \(R^ G_ T(\theta)\) is cuspidal if and only if T is anisotropic, i.e. not contained in any proper parabolic subgroup of G. These results are described in Chapter 9. Thus the Harish-Chandra theory leads to the problems of classifying the cuspidal characters and of decomposing the characters induced from cuspidal characters of parabolic subgroups. R. B. Howlett and G. I. Lehrer [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)] described the structure of the endomorphism algebras of such induced representations and this theory is described in Chapter 10.

The Deligne-Lusztig theory, on the other hand, leads naturally to the problems of decomposing the \(R^ G_ T(\theta)\) when \(\theta\) is not in general position, and of classifying the cuspidal constituents. As a first step one can take \(\theta =1\), and then the constituents of the \(R^ G_ T(1)\) are the unipotent characters. In a series of papers starting from 1977 Lusztig, working case by case, classified the unipotent characters, identified the cuspidal ones, and decomposed the \(R^ G_ T(1)\) (in some cases, for large q). For the classical groups this was completed in 1982. Finally in a brilliant tour de force [Characters of reductive groups over a finite field. Ann. Math. Stud. 107 (1984; Zbl 0556.20033)] he gave a decomposition of all the \(R^ G_ T(\theta)\) when G has a connected center. For this work he used the intersection cohomology theory defined by Goresky and MacPherson. A consequence of this work is that the irreducible characters of \(G^ F\) are in bijection with \(G^{*F}\)-conjugacy classes of pairs (s,\(\phi)\) where s is a semisimple element in \(G^{*F}\) and \(\phi\) is a unipotent character of \(C_{G^*}(s)\), thus yielding a ”Jordan decomposition” of characters.

Lusztig’s classification shows that the unipotent characters fall into families in a remarkable way. When G is F-split and \(\phi\) is an irreducible character of W let \(R_ w\) be \(R^ G_{T_ w}(1)\), where \(T_ w\) is a torus obtained from a maximally split torus by twisting by \(w\in W\), and let \(R_{\phi}=(1/| w|)\sum_{w}\phi (w)R_ w\). (There is a similar definition in the twisted case.) Then it is sufficient to decompose the \(R_{\phi}\), and Lusztig’s results show that the constituents of the \(R_{\phi}\) are all in the same family. The decomposition is described by a ”Fourier transform” matrix. These results are described without proofs in Chapters 12 and 13. Other topics described in these two chapters include cells of Weyl groups, the Springer correspondence between unipotent classes and Weyl group representations, and the Jordan decomposition of irreducible characters due to Lusztig mentioned earlier. Chapter 11 discusses representations of Coxeter groups as a preparation for Chapter 12.

As for conjugacy classes, a standard argument reduces the problem of classifying the conjugacy classes to that of classifying the semisimple and the unipotent classes. In good characteristics, Springer defined a G- equivariant bijection between the nilpotent variety in the Lie algebra of G and the unipotent variety in G. Thus we can look at the orbit of G on the nilpotent variety in the Lie algebra. Springer and Steinberg gave a classification of these G-orbits by weighted Dynkin diagrams by using the Jacobson-Morozov theorem when the characteristic is sufficiently large. The author and Bela gave another approach using results of Richardson. Both these theories are described in Chapter 5. The case of small primes is also described without proofs, and this is a useful addition to the literature. Finally the semisimple classes are also described here, using a geometric approach due to Deriziotis.

The author has given in Chapters 5-10 a careful, detailed treatment of the unipotent conjugacy classes and of the representation theory including the paper of Deligne-Lusztig and the results of Howlett-Lehrer. The tables and other information in Chapter 13 and the extensive bibliography will also be useful to workers in the field; for example all the ”generic degrees” for exceptional groups are available for the first time in one place. In the later chapters 11-12 he has given an account of the deep results of Lusztig on unipotent characters and the decomposition of the \(R^ G_ T(1)\), but in the author’s own words, the exposition here is very sketchy. In the reviewer’s opinion what is missing are indications of the proofs and of the ideas involved in this deep work. It is also disappointing that no mention is made of recent work on the computation of Green functions which play an important role in the character tables of the groups. However, in sum the book is a most valuable source of information and is excellent preparation for the book of Lusztig (loc. cit.) in which the whole story is told.

The representation theory of finite groups of Lie type entered a new phase in the early seventies with the development of what is now known as Harish-Chandra theory by Harish-Chandra and T. A. Springer [see Cusp forms in finite groups, in Lect. Notes Math. 131, C1-C24 (1970; Zbl 0263.20024)]. An even more exciting breakthrough was achieved by the paper of P. Deligne and G. Lusztig in 1976 [Ann. Math., II. Ser. 103, 103-161 (1976; Zbl 0336.20029)]. Let T be any F-fixed maximal torus of the algebraic group G as above. Then for any character \(\theta\) of \(T^ F\) (i.e. a homomorphism into \(\bar Q^*_{\ell}\), for a prime \(\ell\) not equal to the characteristic of the finite field) Deligne and Lusztig (loc. cit.) defined a virtual representation, or equivalently, a generalized character \(R^ G_ T(\theta)\) of \(G^ F\) over \(\bar Q_{\ell}\) by using the \(\ell\)-adic cohomology of certain subvarieties of the flag variety. These characters have the expected properties of orthogonality, and indeed \(R^ G_ T(\theta)\) and \(R^ G_{T'}(\theta ')\) have no irreducible constituents in common unless the pairs (T,\(\theta)\), (T’,\(\theta\) ’) are related by ”geometric conjugacy”. If \(G^*\) is a group in duality with G, the geometric conjugacy classes of pairs (T,\(\theta)\) are in bijection with the F-stable semisimple conjugacy classes of \(G^{*F}\). The concepts of geometric conjugacy and duality are explained in Chapter 4. If \(\theta\) is in ”general position”, \(R^ G_ T(\theta)\) is irreducible up to sign. Moreover, if \(x=su\) is the Jordan decomposition of \(x\in G^ F\), the value of \(R^ G_ T(\theta)\) at x can be described in terms of \(\theta\) and of the values of the analogous generalized characters for \(C^ F_ G(s)\) at unipotent elements. Assuming the basic properties of \(\ell\)-adic cohomology (which are stated in an appendix) the author derives these properties of the Deligne-Lusztig characters in Chapter 7. In Chapter 8 he describes (when the center of G is connected) certain rational linear combinations of the \(R^ G_ T(\theta)\) which turn out to be irreducible characters and which he calls semisimple and regular characters. There is precisely one semisimple and one regular character in each geometric conjugacy class, and the regular characters are those appearing in the Gelfand-Graev representation, which is also described in this chapter.

The Harish-Chandra theory states that the irreducible representations of G fall into families in the following way: if \(\rho\) is an irreducible character of \(G^ F\) then \(\rho\) is a constituent of an induced character \(Ind^{G^ F}_{P^ F}(\tau)\) where \(\tau\) is the pullback of a parabolic subgroup \(P^ F\) of a ”cuspidal” character of a Levi factor \(L^ F\) of \(P^ F\), and the pair (L,\(\tau)\) is unique up to \(G^ F\)-conjugacy. A connection of this theory with the Deligne-Lusztig theory is given by the fact that if \(\theta\) is a character of \(T^ F\) in general position, then the irreducible (up to sign) character \(R^ G_ T(\theta)\) is cuspidal if and only if T is anisotropic, i.e. not contained in any proper parabolic subgroup of G. These results are described in Chapter 9. Thus the Harish-Chandra theory leads to the problems of classifying the cuspidal characters and of decomposing the characters induced from cuspidal characters of parabolic subgroups. R. B. Howlett and G. I. Lehrer [Invent. Math. 58, 37-64 (1980; Zbl 0435.20023)] described the structure of the endomorphism algebras of such induced representations and this theory is described in Chapter 10.

The Deligne-Lusztig theory, on the other hand, leads naturally to the problems of decomposing the \(R^ G_ T(\theta)\) when \(\theta\) is not in general position, and of classifying the cuspidal constituents. As a first step one can take \(\theta =1\), and then the constituents of the \(R^ G_ T(1)\) are the unipotent characters. In a series of papers starting from 1977 Lusztig, working case by case, classified the unipotent characters, identified the cuspidal ones, and decomposed the \(R^ G_ T(1)\) (in some cases, for large q). For the classical groups this was completed in 1982. Finally in a brilliant tour de force [Characters of reductive groups over a finite field. Ann. Math. Stud. 107 (1984; Zbl 0556.20033)] he gave a decomposition of all the \(R^ G_ T(\theta)\) when G has a connected center. For this work he used the intersection cohomology theory defined by Goresky and MacPherson. A consequence of this work is that the irreducible characters of \(G^ F\) are in bijection with \(G^{*F}\)-conjugacy classes of pairs (s,\(\phi)\) where s is a semisimple element in \(G^{*F}\) and \(\phi\) is a unipotent character of \(C_{G^*}(s)\), thus yielding a ”Jordan decomposition” of characters.

Lusztig’s classification shows that the unipotent characters fall into families in a remarkable way. When G is F-split and \(\phi\) is an irreducible character of W let \(R_ w\) be \(R^ G_{T_ w}(1)\), where \(T_ w\) is a torus obtained from a maximally split torus by twisting by \(w\in W\), and let \(R_{\phi}=(1/| w|)\sum_{w}\phi (w)R_ w\). (There is a similar definition in the twisted case.) Then it is sufficient to decompose the \(R_{\phi}\), and Lusztig’s results show that the constituents of the \(R_{\phi}\) are all in the same family. The decomposition is described by a ”Fourier transform” matrix. These results are described without proofs in Chapters 12 and 13. Other topics described in these two chapters include cells of Weyl groups, the Springer correspondence between unipotent classes and Weyl group representations, and the Jordan decomposition of irreducible characters due to Lusztig mentioned earlier. Chapter 11 discusses representations of Coxeter groups as a preparation for Chapter 12.

As for conjugacy classes, a standard argument reduces the problem of classifying the conjugacy classes to that of classifying the semisimple and the unipotent classes. In good characteristics, Springer defined a G- equivariant bijection between the nilpotent variety in the Lie algebra of G and the unipotent variety in G. Thus we can look at the orbit of G on the nilpotent variety in the Lie algebra. Springer and Steinberg gave a classification of these G-orbits by weighted Dynkin diagrams by using the Jacobson-Morozov theorem when the characteristic is sufficiently large. The author and Bela gave another approach using results of Richardson. Both these theories are described in Chapter 5. The case of small primes is also described without proofs, and this is a useful addition to the literature. Finally the semisimple classes are also described here, using a geometric approach due to Deriziotis.

The author has given in Chapters 5-10 a careful, detailed treatment of the unipotent conjugacy classes and of the representation theory including the paper of Deligne-Lusztig and the results of Howlett-Lehrer. The tables and other information in Chapter 13 and the extensive bibliography will also be useful to workers in the field; for example all the ”generic degrees” for exceptional groups are available for the first time in one place. In the later chapters 11-12 he has given an account of the deep results of Lusztig on unipotent characters and the decomposition of the \(R^ G_ T(1)\), but in the author’s own words, the exposition here is very sketchy. In the reviewer’s opinion what is missing are indications of the proofs and of the ideas involved in this deep work. It is also disappointing that no mention is made of recent work on the computation of Green functions which play an important role in the character tables of the groups. However, in sum the book is a most valuable source of information and is excellent preparation for the book of Lusztig (loc. cit.) in which the whole story is told.

Reviewer: B.Srinivasan

### MSC:

20G05 | Representation theory for linear algebraic groups |

20-02 | Research exposition (monographs, survey articles) pertaining to group theory |

20G10 | Cohomology theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

14L35 | Classical groups (algebro-geometric aspects) |

17B35 | Universal enveloping (super)algebras |