# A Filtered Category OS and Applications by Ronald S. Irving

By Ronald S. Irving

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2) Let fi"(x,z) be the coefficient in SXyZ of 9 W*M(*)-i)/2. Given w in 5 W and s in B with ws > 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. 3. For w G 5 W , we have d(w)* = d^. 2. R e m a r k s . 2 reveals the close analogy between the d(w)*'s and the l(w)*'s. 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.

Ifws £ SW, then there are two cases to consider: either ws > w and ws > w, or ws < w and ws < w. In either case, the desired equality is easily verified. If ws fc 5 W , then in W we have ws > w. 2 of [De 1]). Thus ws = (ws,)'w1 with "w the minimal length coset representative of ws in 5 W . We again find two cases to consider: either ws' > w;, in which case ws > w, or ws' < w_, in which case ws < w. The given factorization of ws allows one again to verify the desired equality directly. 3. For each s G B, let Ts act on M*s as (q1/29s - 1).

Projective covers exist in 0{b)\ we denote by Q(/i) the projective cover of L(ii). Assume for the remainder of the paper that a subset S of B (W(R)) is fixed. The subgroup Ws of W(R) generated by S is called a parabolic subgroup; it is the Weyl group of the root subsystem Rs of R spanned by roots of the reflections in S. There is a unique parabolic subalgebra ps of g which contains b and has Rs as root system. fi 5^ fi for all w £ VV5. There is a canonical choice of reductive part of ps, the subalgebra ms consisting of the sum of f) and the root spaces of g associated to roots of Rs.