## 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
Full Text:

### References:

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