This 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.