Download The Kazhdan-Lusztig Cells in Certain Affine Weyl Groups by Jian-Yi Shi PDF

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By Jian-Yi Shi

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Then Proof: Cl Let b~__~group U 1<8G and H is an algebraic a subgroup. 1:e. In the preceding proposition the key was to generate a finite set which contains containing c*(uI G). C*(U 10;). In this case we generate a finite set Cl In order to do this. we '"ust examine the tensor in-. duced module for sums. Let 1-1 ,M 1 2 I ' •• ,Ms be k[H ]-modules (possibly some Hi I S are isomorphic - 39 - with some Mj's) and gl = 1 , gz , " ' , gn in H be a trallsversdl of G. We need a decomposition of into F * [G ]-invariant parts.

_-- = e/ker v. Ic I, ul G• is isomorphic with a direct summand of is algebraic. We conclude that V Step 8. V PE. is an VI G0 . is is a direct is algebraic, = PE. acts trivially on By step 7, V and - 47 - 1. C (E) therefore We now have shown that Z(G). El Thus (1)-(3) all P hold. 6)(d), vity of implies that every normal abelian subgroup of E structure of E the fonn primiti- P P is cyclic. l7). The extraspecial group Ip I > 8, of that result cannot occur if otherwise P would contain i3. noncyclic normal abelian subgroup.

4. n = 2. 16) we find that Obviously, V 41q+1 n V we is also is algebraic. 3). Z) is now obvious. Let of the isomorphism classes of irreducible generated by the rings t~d C(V ). vZ •... 6). C Z-module. Thus the elements of C be representatives t k[C ]-modules. Let C be the ring is a finitely genera- are integral. If V is irreducibly - 51 - generated then {vi E C so V is integral. 2). This application shows that the method, by itself, does not prove theor-erns. But it provides a schematic for constructing proofs.

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