A Filtered Category OS and Applications by Ronald S. Irving

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By Ronald S. Irving

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Definition. We call the basis {d(w)* : w G 5 W } the self-dual basis of M*s. 44 RON S. 2. 2). 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.

For such a u>, we have fj,s(x,w) = 38 RON S. 8. Thus we may rewrite the first formula as Osq(wsx)* = q(wsxs)* + ^ ns(x,w)q(wsw)*. 5, 0S and ITS commute. * 11/ w>x,w$>wtw^^W This proves the Corollary. 2. u;)*. As we will see later in the section, it is desirable to define 0W more generally for all w G W. For this reason, it is best to define 0W by working with the Hecke module MQ for W. We will then need to verify that for suitable w, the element 0W has the desired action on M*s. Assume then that we are working with A4Q.

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). (1) Under this action, M*s is a well-defined Ti-module. (2) The map ws is an H-module homomorphism. • The endomorphism 05 on Mj$ or M*s has the same action as the element q-1/2(Ts + 1). Thus we may view 9S as the element q-1,2(Ts 4- 1) of H. 2 as follows.

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