By P. Eymard, J. Faraut, G. Schiffmann, R. Takahashi
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Extra resources for Analyse harmonique sur les groupes de Lie: seminaire, Nancy-Strasbourg
W In this section The G-module M Small PIM's (L) is essential. 2) can always be d e c o m p o s e d into a direct sum of G-submodules classes involved. , located in the lowest alcove, the G-summand b e l o n g i n g to the linkage class of ~ + (p-l)@ will turn out to be just QX' I = (~ - 6)o in A (with perhaps tion on p). 1 Orbits of weights If ~ ~ X, we denote by W fixed (but arbitrary) LEMMA. all ~ ~ ~+. its s t a b i l i z e r in W. , P Assume that p does not divide f. Then: . <~,e> < p for (a) If ~ is a weight of M and ~ ~ ~ (mod pX), then ~ = B.
O i ( 1 ) . , oi(t) = T j ( 1 ) . . Tj(t). where we can apply the following enveloping (A 2) YlrY2r+SYl s (B 2) YlrY2r+SYlr+2SY2 s (G 2 ) = ~ the condition: "'" ~ in W, we Because W is a Coxeter group, to verify this in rank 2, cf. Bourbaki in the universal into a the sum of all Z~ as I runs over a linkage Given two reduced expressions it suffices just defined (*) T~ = 0 We assert that the maps T i (i < i < ~) also satisfy have Ti(1) ... Ti(t) identities [~, IV, w of Verma Prop. [~, (5)(6)(7)] algebra of [C: = v Sv r+Sv r -i -2 -i = Y2sYlr+2sY2r+SYl r YlrY2r+s'~12r+3sY2r+2sYlr+3SY2S Y2SYl r+ 3SY2r+2SYl2r+3sY2r+SYl r Here r,s c sides become .
The t h e o r e m implies that For G = SL(3,K), dim Ql is typically e i t h e r 6p 3 or 12p 3. The proof of the t h e o r e m is rather lengthy, so we shall be content to sketch briefly the steps i n v o l v e d and most complicated, (cf. also Verma ). The first, step is to show that Ql §247eIZl for some integer U eI. In turn, since Z l §247Ed M (sum over linkage class of l), we have U cl~ = eld ~. 3), so fl = el/d I is a constant depending only on the linkage class. Now we consider the dimension of the block of u a s s o c i a t e d w i t h I.