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**Classification of unipotent representations of simple \(p\)-adic groups.**
*(English)*
Zbl 0872.20041

Let \(\mathfrak G_0\) be the group of \(\mathbb{F}_q\)-rational points of a reductive connected algebraic group defined over a finite field \(\mathbb{F}_q\). Earlier [Characters of reductive groups over a finite field (Ann. Math. Stud. 107, 1984; Zbl 0556.20033)] the author classified the unipotent representations of \(\mathfrak G_0\) independently of \(q\). Let \(G\) be a simply connected almost simple algebraic group over \(\mathbb{C}\) with Lie algebra \(\mathfrak g\). The set of pairs \((\mathcal C,\mathcal F)\) for \(\mathcal C\) a nilpotent \(G\)-orbit in \(\mathfrak g\) and \(\mathcal F\) an irreducible \(G\)-equivariant local system on \(\mathcal C\) (up to isomorphism) is similar to the set of unipotent representations of \(\mathfrak G_0\). At one extreme, pairs \((\mathcal C,\mathcal F)\) from the Springer correspondence [T. A. Springer, Invent. Math. 36, 173-207 (1976; Zbl 0374.20054)] are in bijective correspondence with the irreducible representations of the Weyl group. At the other extreme, there are the rare cuspidal pairs that the author has previously studied [Invent. Math. 75, 205-272 (1984; Zbl 0547.20032)]. A major theme of the paper under review is that these two classifications coalesce in the representation theory of \(\mathfrak G\), the group of rational points of a split adjoint simple algebraic group over a nonarchimedean local field.

An irreducible admissible representation of \(\mathfrak G\) is unipotent if its restriction to some parahoric subgroup contains a subspace on which the parahoric subgroup acts through a unipotent cuspidal representation of the “reductive quotient” of that parahoric subgroup (a reductive group over a finite field). (Note that replacement of \(\mathfrak G\) by \(\mathfrak G_0\) and parahoric by parabolic yields the notion of unipotent representation for \(\mathfrak G_0\).) An earlier conjecture of the author [Trans. Am. Math. Soc. 227, 623-653 (1983; Zbl 0526.22015)] is that the set of isomorphism classes of unipotent representations of \(\mathfrak G\) is in one-to-one correspondence with the set of triples \((s,y,{\mathcal V})\) modulo the natural action of \(G\). Here \(s\) is a semisimple element of \(G\), \(y\) is a nilpotent element of the Lie algebra \(\mathfrak g\) of \(\mathfrak G\) such that \(\text{Ad}(s)y=qy\) for \(q\) the number of elements of the residue field of the local field, and \(\mathcal V\) is an irreducible representation of the group of components of the simultaneous centralizer of \(s\) and \(y\) in \(G\), on which the center of \(G\) acts trivially. The present paper proves the conjecture by means of a program the author set out at the 1990 International Congress of Mathematicians [Proc. Int. Congr. Math., Tokyo 1990, 155-174 (1991; Zbl 0749.14010)]. The author already carried out several steps of his program [most recently in CMS Conf. Proc. 16, 217-275 (1995; Zbl 0841.22013)]. The first step in completing the program and establishing the conjecture in the current paper is the observation that the endomorphism algebra of a representation induced by a unipotent cuspidal representation of the reductive quotient of a parahoric subgroup is an affine Hecke algebra with presentation similar to that of N. Iwahori, H. Matsumoto [Publ. Math., Inst. Hautes Étud. Sci. 25, 5-48 (1965; Zbl 0228.20015)] but with possibly unequal parameters. Recent work of A. Moy and G. Prasad [Invent. Math. 116, No. 1-3, 393-408 (1994; Zbl 0804.22008)] and L. Morris [Invent. Math. 114, No. 1, 233-274 (1993; Zbl 0854.22022)] showed that these unipotent representations are parametrizable by the simple modules of the above endomorphism algebras.

The author shows further that given a subgroup of \(G\) that is the centralizer of some semisimple element and a unipotent class of the subgroup that carries a local cuspidal system, it is possible to construct explicitly an affine Hecke algebra (usually with unequal parameters). Those Hecke algebras arising from the geometry of \(G\) are the same as those that arise from the representation theory of \(\mathfrak G\). That reduces the problem to classification of the simple modules of the affine Hecke algebras that correspond to cuspidal local systems.

The author linearizes the affine Hecke algebras with respect to various points in the spectrum of the center to obtain graded Hecke algebras that he has previously studied [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026) and J. Am. Math. Soc. 2, No. 3, 599-635 (1989; Zbl 0715.22020)]. The representation theory of those algebras yields the representation theory of the affine Hecke algebras in a way analogous to that in which the representation theory of a Lie group is recoverable from that of its Lie algebra. This thus reduces the problem at hand to classification of the simple modules of the linearized algebras. It turns out that the author’s earlier 1995 paper, which used equivariant theory and perverse sheaves, provides the necessary representation theory of those linearized algebras.

The paper also classifies the unipotent representations of inner forms of \(\mathfrak G\), and the author conjectures that his methods should also apply to any form of \(\mathfrak G\) that becomes split after an unramified extension of the ground field.

An irreducible admissible representation of \(\mathfrak G\) is unipotent if its restriction to some parahoric subgroup contains a subspace on which the parahoric subgroup acts through a unipotent cuspidal representation of the “reductive quotient” of that parahoric subgroup (a reductive group over a finite field). (Note that replacement of \(\mathfrak G\) by \(\mathfrak G_0\) and parahoric by parabolic yields the notion of unipotent representation for \(\mathfrak G_0\).) An earlier conjecture of the author [Trans. Am. Math. Soc. 227, 623-653 (1983; Zbl 0526.22015)] is that the set of isomorphism classes of unipotent representations of \(\mathfrak G\) is in one-to-one correspondence with the set of triples \((s,y,{\mathcal V})\) modulo the natural action of \(G\). Here \(s\) is a semisimple element of \(G\), \(y\) is a nilpotent element of the Lie algebra \(\mathfrak g\) of \(\mathfrak G\) such that \(\text{Ad}(s)y=qy\) for \(q\) the number of elements of the residue field of the local field, and \(\mathcal V\) is an irreducible representation of the group of components of the simultaneous centralizer of \(s\) and \(y\) in \(G\), on which the center of \(G\) acts trivially. The present paper proves the conjecture by means of a program the author set out at the 1990 International Congress of Mathematicians [Proc. Int. Congr. Math., Tokyo 1990, 155-174 (1991; Zbl 0749.14010)]. The author already carried out several steps of his program [most recently in CMS Conf. Proc. 16, 217-275 (1995; Zbl 0841.22013)]. The first step in completing the program and establishing the conjecture in the current paper is the observation that the endomorphism algebra of a representation induced by a unipotent cuspidal representation of the reductive quotient of a parahoric subgroup is an affine Hecke algebra with presentation similar to that of N. Iwahori, H. Matsumoto [Publ. Math., Inst. Hautes Étud. Sci. 25, 5-48 (1965; Zbl 0228.20015)] but with possibly unequal parameters. Recent work of A. Moy and G. Prasad [Invent. Math. 116, No. 1-3, 393-408 (1994; Zbl 0804.22008)] and L. Morris [Invent. Math. 114, No. 1, 233-274 (1993; Zbl 0854.22022)] showed that these unipotent representations are parametrizable by the simple modules of the above endomorphism algebras.

The author shows further that given a subgroup of \(G\) that is the centralizer of some semisimple element and a unipotent class of the subgroup that carries a local cuspidal system, it is possible to construct explicitly an affine Hecke algebra (usually with unequal parameters). Those Hecke algebras arising from the geometry of \(G\) are the same as those that arise from the representation theory of \(\mathfrak G\). That reduces the problem to classification of the simple modules of the affine Hecke algebras that correspond to cuspidal local systems.

The author linearizes the affine Hecke algebras with respect to various points in the spectrum of the center to obtain graded Hecke algebras that he has previously studied [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026) and J. Am. Math. Soc. 2, No. 3, 599-635 (1989; Zbl 0715.22020)]. The representation theory of those algebras yields the representation theory of the affine Hecke algebras in a way analogous to that in which the representation theory of a Lie group is recoverable from that of its Lie algebra. This thus reduces the problem at hand to classification of the simple modules of the linearized algebras. It turns out that the author’s earlier 1995 paper, which used equivariant theory and perverse sheaves, provides the necessary representation theory of those linearized algebras.

The paper also classifies the unipotent representations of inner forms of \(\mathfrak G\), and the author conjectures that his methods should also apply to any form of \(\mathfrak G\) that becomes split after an unramified extension of the ground field.

Reviewer: J.F.Hurley (Storrs)

### MSC:

20G05 | Representation theory for linear algebraic groups |

20G40 | Linear algebraic groups over finite fields |

20G25 | Linear algebraic groups over local fields and their integers |

17B20 | Simple, semisimple, reductive (super)algebras |

17B45 | Lie algebras of linear algebraic groups |

14L15 | Group schemes |

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