By Ronald S. Irving

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**Example text**

We obtain that the expressions of (1) and (2) are equal. Suppose now that ys < y and let x = ys. We wish to show that p(x)* = Osp(y)* — desired equality as J2 i *>y,**>* (3) j2^iw)~t(x)),2^A^m(wy w Vsiy, z)p{z)*. We may rewrite the two sides of this and RON S. IRVING 36 (4) J2qWw)-tW)/2QU^MwT- £ w * ^(y,^)E^ (u;) "^ ))/2 2^^)^H*w z>y,z*>z The coefficient of m(w)* in (3) is (5) «W"-)-^))/2e£,„(«). ) q-WqVW-WWQ^iq) if ws > w, qWqM*>)-*(y))/*Q$tW(q) + qMwaWvW2Q$%wa(q) i f ws 0 if ws £ Multiplying (5) and (6) by q(l(x)~l(w))/2 < w> S W.

One starts with 0S on m{wY instead of l(w)*, and obtains recursive formulas for Qy7w(q) which show that they coincide with the inverse KazhdanLusztig polynomials QyiW(q)- In a similar manner, one also obtains that (iii) implies (ii). 4. 1 has provided a characterization of the Verma basis in terms of the simple basis. In turn, one can characterize the simple basis in terms of the Verma basis. It depends on the existence of a semi-linear involution of M*. Definitions. (1) Let 6 : Ai* —• Ai* be the unique 2Z-linear map with 6(qU(wy) = q-Uiw)* for all w and i.

For s £ B, the following equation holds: wsoTs = (q1/20s-l)o7rs. Proof. The proof is a direct calculation on the Verma basis of MQ. ) Let w G W, with w — ww. 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.