Cells in affine Weyl groups. II.

*(English)*Zbl 0625.20032This is a continuation of part I [Adv. Stud. Pure Math. 6, 255-287 (1985; Zbl 0569.20032)]. The main theme of the previous paper was the study of a numerical function \(w\mapsto a(w)\) on a Coxeter group W which in the case of Weyl groups is closely related to the Gelfand-Kirillov dimension of certain modules over the corresponding enveloping algebra. There, the function was defined purely in terms of multiplication in the Hecke algebra and several properties of it were proved for Weyl groups.

In this paper, the author proves them for a larger class of Coxeter groups including the affine Weyl groups. One of the main themes of the paper is provided by certain distinguished involutions of W, one in the left cell. A. Joseph has shown that for each left cell of a Weyl group, the function \(y\mapsto \ell(y)-2\delta(y)\) \((\ell=\) length, \(\delta=\) degree of the polynomial \(P_{e,w})\) reaches its minimum at a unique element of that left cell [cf. J. Algebra 73, 295-326 (1981; Zbl 0482.17002); D. Kazhdan and the author, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. The author shows that this minimal value is \(a(y)\) for any y in the left cell, from this Joseph’s conjecture follows easily and also gives the analogous result for affine Weyl groups. An important role in the author’s proof is played by a ring J which may be regarded as an asymptotic version of the Hecke algebra H and he proves the comparison theorem between H and J.

In this paper, the author proves them for a larger class of Coxeter groups including the affine Weyl groups. One of the main themes of the paper is provided by certain distinguished involutions of W, one in the left cell. A. Joseph has shown that for each left cell of a Weyl group, the function \(y\mapsto \ell(y)-2\delta(y)\) \((\ell=\) length, \(\delta=\) degree of the polynomial \(P_{e,w})\) reaches its minimum at a unique element of that left cell [cf. J. Algebra 73, 295-326 (1981; Zbl 0482.17002); D. Kazhdan and the author, Invent. Math. 53, 165-184 (1979; Zbl 0499.20035)]. The author shows that this minimal value is \(a(y)\) for any y in the left cell, from this Joseph’s conjecture follows easily and also gives the analogous result for affine Weyl groups. An important role in the author’s proof is played by a ring J which may be regarded as an asymptotic version of the Hecke algebra H and he proves the comparison theorem between H and J.

Reviewer: E.Abe

##### MSC:

20G05 | Representation theory for linear algebraic groups |

17B35 | Universal enveloping (super)algebras |

16E10 | Homological dimension in associative algebras |

20F40 | Associated Lie structures for groups |

14F10 | Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials |

##### Keywords:

Gelfand-Kirillov dimension; enveloping algebra; Hecke algebra; Coxeter groups; affine Weyl groups; involutions; left cell
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##### References:

[1] | Joseph, A, Goldie rank in the enveloping algebra of a semisimple Lie algebra III, J. algebra, 73, 295-326, (1981) · Zbl 0482.17002 |

[2] | Kazhdan, D; Lusztig, G, Representations of Coxeter groups and Hecke algebras, Invent. math., 53, 165-184, (1979) · Zbl 0499.20035 |

[3] | Kazhdan, D; Lusztig, G, Schubert varieties and PoincarĂ© duality, (), 185-203 |

[4] | Lusztig, G, Characters of reductive groups over a finite field, () · Zbl 0930.20041 |

[5] | Lusztig, G, Cells in affine Weyl groups, () · Zbl 0884.20026 |

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