##
**Characters of reductive groups over a finite field.**
*(English)*
Zbl 0556.20033

Annals of Mathematics Studies, 107. Princeton, New Jersey: Princeton University Press. XXI, 384 p. hbk: $ 65.00; pbk: $ 25.50 (1984).

Reductive groups over finite fields are classified by root systems in a lattice with an action of the Frobenius; a way to get these data from the connected reductive algebraic group \(G\) is to take the projective variety \({\mathcal B}\) of Borel subgroups of \(G\), the set of isomorphism classes of G- equivariant line bundles over \({\mathcal B}\) forms a lattice \(X\); the subgroups containing a given \(B\in {\mathcal B}\) are not conjugate under \(G\), and the orbits of the minimal ones by conjugation under \(G\) form a finite set, each one giving \({\mathcal B}\) as the total space of a \({\mathbb{P}}_ 1\)-fibration over it, hence by taking the tangent bundle along the projections we get elements of \(X\): this is the basis of the root system. Now, each orbit of \(G\) in \({\mathcal B}\times {\mathcal B}\) defines an automorhism of X using the two projections on \({\mathcal B}\), and these orbits form the Weyl group \(W\) of \(G\). The Galois group of the algebraic closure of \(F_ q\) is naturally \({\bar {\mathbb{Z}}}\), generated topologically by the Frobenius F, and the set \(H^ 1({\bar {\mathbb{Z}}},W)\) classifies the maximal tori defined over \(F_ q\) in \(G\).

For each \(w\in W\), let \({\mathcal B}_ w\) be the set of Borel subgroups \(B\in {\mathcal B}\) for which \({}^ FB\) is in position w with respect to B; with a maximal tori \(T\subset B\) defined over \(F\), P. Deligne and G. Lusztig constructed a variety \({\mathcal B}^ T_ w\) projecting over \({\mathcal B}_ w\) with fibers \(T(F_ q)\) and compatible action of \(G(F_ q)\) so the alternate sum of the \(\ell\)-adic cohomology groups give a virtual representation of \(G(F_ q)\) commuting with the action of \(T(F_ q)\): this leads to the representations \(R^ T_{\theta}\) for the characters \(\theta\) of \(T(F_ q)\) in \({\bar {\mathbb{Q}}}^ x_{\ell}\) [Ann. Math., II. Ser. 103, 103-171 (1976; Zbl 0336.20029)]. Up to equivalence, all the irreducible representations of \(G(F_ q)\) occur in these \(R^ T_{\theta}\), when T and \(\theta\) vary. What was not given in this fundamental article, is an explicit formula for the multiplicities of the irreducible components in the \(R^ T_{\theta}\)’s. The book answers this question, completely in the case \(G\) has a connected center (since, Lusztig obtained the general case).

One of the main tools is the étale intersection cohomology of P. Deligne, A. A. Beilinson and J. Bernstein [Astérisque 100 (1982; Zbl 0536.14011)], applied to the closures of the varieties \({\mathcal B}_ w\), the Schubert cells. Another one is a deep understanding of the Weyl groups and their Hecke algebras; some properties on them are obtained through the theory of primitive ideals of enveloping algebras of complex reductive Lie algebras; the book uses systematically the results obtained by its author and by D. Kazhdan and its author in the theory of Weyl and Coxeter groups. He shows how the classification of irreducible representations reduces to the classification of unipotent representations of the “endoscopic” groups, where the solution comes from the Hecke algebra of the corresponding Weyl group. Also, the author, using the Springer correspondence [T. A. Springer, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], gives a parametrisation of the irreducible representations in terms of the special conjugacy classes of the dual group of \(G\).

For each \(w\in W\), let \({\mathcal B}_ w\) be the set of Borel subgroups \(B\in {\mathcal B}\) for which \({}^ FB\) is in position w with respect to B; with a maximal tori \(T\subset B\) defined over \(F\), P. Deligne and G. Lusztig constructed a variety \({\mathcal B}^ T_ w\) projecting over \({\mathcal B}_ w\) with fibers \(T(F_ q)\) and compatible action of \(G(F_ q)\) so the alternate sum of the \(\ell\)-adic cohomology groups give a virtual representation of \(G(F_ q)\) commuting with the action of \(T(F_ q)\): this leads to the representations \(R^ T_{\theta}\) for the characters \(\theta\) of \(T(F_ q)\) in \({\bar {\mathbb{Q}}}^ x_{\ell}\) [Ann. Math., II. Ser. 103, 103-171 (1976; Zbl 0336.20029)]. Up to equivalence, all the irreducible representations of \(G(F_ q)\) occur in these \(R^ T_{\theta}\), when T and \(\theta\) vary. What was not given in this fundamental article, is an explicit formula for the multiplicities of the irreducible components in the \(R^ T_{\theta}\)’s. The book answers this question, completely in the case \(G\) has a connected center (since, Lusztig obtained the general case).

One of the main tools is the étale intersection cohomology of P. Deligne, A. A. Beilinson and J. Bernstein [Astérisque 100 (1982; Zbl 0536.14011)], applied to the closures of the varieties \({\mathcal B}_ w\), the Schubert cells. Another one is a deep understanding of the Weyl groups and their Hecke algebras; some properties on them are obtained through the theory of primitive ideals of enveloping algebras of complex reductive Lie algebras; the book uses systematically the results obtained by its author and by D. Kazhdan and its author in the theory of Weyl and Coxeter groups. He shows how the classification of irreducible representations reduces to the classification of unipotent representations of the “endoscopic” groups, where the solution comes from the Hecke algebra of the corresponding Weyl group. Also, the author, using the Springer correspondence [T. A. Springer, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)], gives a parametrisation of the irreducible representations in terms of the special conjugacy classes of the dual group of \(G\).

Reviewer: P.Gérardin

### MSC:

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

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

20G10 | Cohomology theory for linear algebraic groups |

17B35 | Universal enveloping (super)algebras |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14L30 | Group actions on varieties or schemes (quotients) |