By Ronald S. Irving

ISBN-10: 0821824821

ISBN-13: 9780821824825

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By definition, 0wm(sw)* TTS(V(WO)*) = ra(5w)*. Thus the two Lemmas yield = ^5(5(^0^)*). w)*. 4. Let weW. Then 0wo6 = 6o0w. Proof. Both maps are 2Z[g 1 ' 2 ,g~ 1 ' 2 ] semi-linear, so it suffices to verify the equality on {l(y)* : y G 5 W } . 1. Thus it is also obvious for the elements 0(w) defined as compositions of 0 5 's. Proceeding inductively on £(w), we obtain the result for the summand 0W of0(w). • 40 RON S. 3. An involution of M*s In this subsection we study an important semi-linear involution on M*s.

4, and (5) follows from (4) as before. Thus, both (5) and (5 ; ) are valid. By the inductive hypothesis, for y < x we may replace 9sm{yY in (5) by its counterpart from (ii). Comparing the resulting formula with (5') yields (ii) for 0,m(x)*. 2. One can prove that (ii) implies (iii) directly by an argument analogous to that used to show that (ii) implies (iv). 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).

IRVING write 6(m(w)*) as a 7L[qll2, g_1 ^2]-linear combination of m(z)* 's via the TlziW'& and then rewrite each m(z)* as a 7L[qxl2, g""1^2]-linear combination of /(y)*'s via the Qy^'s. Comparing the coefficient of l(y)* in each of the two expressions yields the formula. 3. € W5 and z £ 5 W. 2 A. £' (1) = 1 if y < w and there exists a (not necessarily simple) reflection s £ W sucA that y = stF. 0*0 ^ y , « ( ! ) = ° otherwise. Proof. Choose w G SW and s £ B so that ws > w. (l) = KAl) ifys£ 5 W = ftyfW(i) + - Kysvi1) tt;,itt,(i) if ys>y ifys y, we have 7ZytW(l) = 0 unless y = w.

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