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An inversion formula for the Kazhdan-Lusztig polynomials for affine Weyl groups. (English) Zbl 0598.20044

From the author’s introduction: ”Let W be a Coxeter group. Kazhdan and Lusztig defined for each y,w\(\in W\) certain polynomials \(P_{y,w}\) over Z... Elementary properties of the polynomials \(P_{y,w}\) show that there exist polynomials \(Q_{x,z}\), x,z\(\in W\) such that the inversion formula \[ \sum_{z}(-1)^{\ell (z)-\ell (y)} P_{y,z}Q_{z,w}=\delta_{y,w} \] holds for all y,w\(\in W\). Here \(\ell\) denotes the length function on W. In the case where W is finite it is shown by Kazhdan and Lusztig that \(Q_{z,w}=P_{w_ 0w,w_ 0z}\) where \(w_ 0\) is the longest element in W. In the case where W is an affine Weyl group Lusztig defines certain Q-polynomials which ”generically” equal the \(Q_{z,w}'s\). In this paper we modify Lusztig’s construction and thereby obtain certain Q’- polynomials which coincide exactly with the \(Q_{z,w}'s.''\)
There are shown some connections with the Lusztig conjecture (this conjecture is the analogue of the Kazhdan-Lusztig conjecture for prime characteristic p; recall that the Kazhdan-Lusztig conjecture has now been proved). There are also formulated some conjectures.
Reviewer: G.A.Margulis

MSC:

20G05 Representation theory for linear algebraic groups
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