A Filtered Category OS and Applications by Ronald S. Irving

By Ronald S. Irving

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1 and we carry over the notation introduced there. In particular, SW = Ws\W. Notice that SW depends on v. u) : w £ W}. The definition of O de- RONALD S. IRVING 54 pends on i/. v. A T h e set of 5-dominant weights in the W-orbit of 5 : w G W } . Such weights are necessarily S-integral. We will denote t h e corresponding generalized Verma modules and simple modules by M(w) and L(w). W i t h this convention, L(e) = M(e). Let Os be the full subcategory of t he category O consisting of modules all of whose composition : w G 5 W } .

Let x,y £ 5 W with y < x. Let s G B satisfy xs > x and either ys < y orys ^ s W. Then i/(y, x) — vs(y,x). Proof. By setting q — 1 in the definition of Py )W(q), we obtain (1) /(«,) = £(-l)«")-^>>P y , ro (l)m(y). y Assume the Py}W's coincide with the Kazhdan-Lusztig polynomials. ) Let x be as in the RONALD S. IRVING 22 Lemma and let w = xs. Recall that 9J(x) = 2l(x) + l(w) + J2 "»(*- *)'(2)x x-^)p y ,,(l)m(j/) y y

Fw) modulo 2. But z = lw, so we need only verify that £(sw) — £{w) is odd. This follows because 5, being a conjugate of a simple reflection, has odd length. • III. THE FILTERED CATEGORY Os 5. 1. The category and its Grothendieck group For the remainder of this paper, let us fix a complex, semisimple Lie algebra g, a Cartan subalgebra rj, and a Borel subalgebra b containing f). Let R be the root system and W(R) the Weyl group associated to f). To each a G R is associated a reflection sQ in W(R).

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