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By Ronald S. Irving

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A FILTERED CATEGORY Os AND APPLICATIONS 45 (2) Combining the Theorems with the Corollary we may rewrite the Sy)W's in terms of inverse Kazhdan-Lusztig polynomials. 3. We will see there that certain elements of the projective basis are self-dual. By the Theorem, we will be able to deduce that they are in the self-dual basis as well. 4. 1. 2. By setting q = 1, we pass from A4* to M.. Let us call M be the specialized Hecke module. All that we have done for M* in this chapter carries over to M via specialization.

2. (1) For each w G 5 W , there exists a unique element d^ in M*s such that (i) 6(d*w) = d*w,and (n) where Sy}W(q) is a polynomial 12 in 7L[q ' ] such that SyiW(q) = 0 unless y < w, such that SWiUJ(q) = 1, and such that the highest degree occurring in Syw(q) is at most (£(w)-£(y)-l)/2ify w, the polynomials Sy>w satisfy the following recursion relation: SyiWS=T- £ M »(z | W ),«"'>-'W+ 1 >/ 2 S y ,, l x Z8<*,Zy qSVyW +SySfW ifys < y 0 ifys£sW.

1. (i) For w G 5VV, we have js(d(w)*) lently, d(w)* = = p(js(w))*- Equiva- js(p(js(w))*). (ii) In particular, d(w)* is well-defined. (Hi) The set {d(w)* : w G 5 W } is a basis ofM*s. (iv) For w G 5 W, we have y (v) Given w G 5 W and s G 5 , we have 9$d(w)* = (qll2 + q-^2)d(wY = d(ws)* + /^ ifws < w no(y,w)d(y)* ifws > w y yw)d(y)* ifws<£sw y y

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