Affine Hecke algebras and their graded version.

*(English)*Zbl 0715.22020Affine Hecke algebras \(H_{v_ 0}\) arise naturally in the representation theory of semisimple p-adic groups. Representations of \(H_{v_ 0}\) correspond to representations with Iwahori-fixed vectors, which are important to understand for applications to number theory.

A classification of simple modules for \(H_{v_ 0}\) is obtained by D. Kazhdan, G. Lusztig [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] in the special case that the c(s) in the defining relations \((T_ s+1)(T_ s-v_ 0^{2c(s)})=0\) are independent of s. The approach, based on equivariant K-theory, is not suited to the general case.

The author [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026)], defined a graded analogue \(\bar H_{r_ 0}\) of \(H_{v_ 0}\), for which the representation theory in the general case may be studied using equivariant homology and intersection cohomology. In the present paper, the author connects the representation theory of the \(H_{v_ 0}\) to that of \(\bar H_{r_ 0}\). The center \({\mathcal Z}\) of \(H_{v_ 0}\) is determined, generalizing unpublished work of Bernstein from the special case. A simple \(H_{v_ 0}\)-module M determines a unique maximal ideal T of \({\mathcal Z}\) such that \(TM=0\). The completion \(\hat {\mathcal Z}\) of the center is taken with respect to the T-adic topology. Then the completion of the affine Hecke algebra is defined by \(\hat H_{v_ 0}=H_{v_ 0}\otimes_{{\mathcal Z}}\hat {\mathcal Z}.\)

Graded algebras \(\bar H_{r_ 0}\) are defined via powers of a maximal ideal of a certain commutative subalgebra \({\mathcal O}\) of \(H_{v_ 0}\). The centers are determined and again completions are defined. The first reduction theorem is that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of \(n\times n\) matrices over the completion of a smaller affine Hecke algebra. The second reduction is that a certain natural homomorphism from an affine Hecke algebra into a suitable completion of its graded version becomes an isomorphism when the first algebra is completed. The classification of simple \(H_{v_ 0}\)-modules is then reduced essentially to the classification of simple \(\bar H_{r_ 0}\)-modules, in the case the parameter \(v_ 0\in {\mathbb{C}}^ x\) has infinite order.

A classification of simple modules for \(H_{v_ 0}\) is obtained by D. Kazhdan, G. Lusztig [Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] in the special case that the c(s) in the defining relations \((T_ s+1)(T_ s-v_ 0^{2c(s)})=0\) are independent of s. The approach, based on equivariant K-theory, is not suited to the general case.

The author [Publ. Math., Inst. Hautes Étud. Sci. 67, 145-202 (1988; Zbl 0699.22026)], defined a graded analogue \(\bar H_{r_ 0}\) of \(H_{v_ 0}\), for which the representation theory in the general case may be studied using equivariant homology and intersection cohomology. In the present paper, the author connects the representation theory of the \(H_{v_ 0}\) to that of \(\bar H_{r_ 0}\). The center \({\mathcal Z}\) of \(H_{v_ 0}\) is determined, generalizing unpublished work of Bernstein from the special case. A simple \(H_{v_ 0}\)-module M determines a unique maximal ideal T of \({\mathcal Z}\) such that \(TM=0\). The completion \(\hat {\mathcal Z}\) of the center is taken with respect to the T-adic topology. Then the completion of the affine Hecke algebra is defined by \(\hat H_{v_ 0}=H_{v_ 0}\otimes_{{\mathcal Z}}\hat {\mathcal Z}.\)

Graded algebras \(\bar H_{r_ 0}\) are defined via powers of a maximal ideal of a certain commutative subalgebra \({\mathcal O}\) of \(H_{v_ 0}\). The centers are determined and again completions are defined. The first reduction theorem is that the completion of an affine Hecke algebra with respect to a maximal ideal of the center is isomorphic to the ring of \(n\times n\) matrices over the completion of a smaller affine Hecke algebra. The second reduction is that a certain natural homomorphism from an affine Hecke algebra into a suitable completion of its graded version becomes an isomorphism when the first algebra is completed. The classification of simple \(H_{v_ 0}\)-modules is then reduced essentially to the classification of simple \(\bar H_{r_ 0}\)-modules, in the case the parameter \(v_ 0\in {\mathbb{C}}^ x\) has infinite order.

Reviewer: C.D.Keys

##### MSC:

22E50 | Representations of Lie and linear algebraic groups over local fields |

22E35 | Analysis on \(p\)-adic Lie groups |

16W50 | Graded rings and modules (associative rings and algebras) |

20G05 | Representation theory for linear algebraic groups |

14G20 | Local ground fields in algebraic geometry |

11F85 | \(p\)-adic theory, local fields |

14F43 | Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) |

##### Keywords:

representation theory of semisimple p-adic groups; simple modules; equivariant homology; intersection cohomology; affine Hecke algebra; Graded algebras; completions
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\textit{G. Lusztig}, J. Am. Math. Soc. 2, No. 3, 599--635 (1989; Zbl 0715.22020)

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##### References:

[1] | David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153 – 215. · Zbl 0613.22004 |

[2] | Robert W. Kilmoyer, Principal series representations of finite Chevalley groups, J. Algebra 51 (1978), no. 1, 300 – 319. · Zbl 0389.20008 |

[3] | George Lusztig, Singularities, character formulas, and a \?-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208 – 229. · Zbl 0561.22013 |

[4] | George Lusztig, Some examples of square integrable representations of semisimple \?-adic groups, Trans. Amer. Math. Soc. 277 (1983), no. 2, 623 – 653. · Zbl 0526.22015 |

[5] | George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145 – 202. · Zbl 0699.22026 |

[6] | I. G. Macdonald, Spherical functions on a group of \?-adic type, Ramanujan Institute, Centre for Advanced Study in Mathematics,University of Madras, Madras, 1971. Publications of the Ramanujan Institute, No. 2. · Zbl 0302.43018 |

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