By James E. Humphreys (auth.)

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3]) to compute the formal character of the sum of these ~ , using q as an a b b r e v i a t i o n for the d e n o m i n a t o r Z e(T)e(T~), e being the alternating Ts c h a r a c t e r of W: Z . ch(o~ + (p-l)6) geW ~ E Z = c(T)e(T(o~ + p~))/q = ~cW ~ TeW Z Z e(T)e(TO~e(pT6)/q = oEW ~ TeW ( e(op))( ~cW Z Z e(T)e(pT6)/q). TeW The first factor is w h a t we have denoted s(~), while the second factor is just the formal character ch((p-l)~) of the Steinberg module. 4) that dim Q~ = aldXpm. 1), along with shows that a I = [W:W ], and from this all assertions of the t h e o r e m follow immediately.

Y p-i to m + yields a n o n z e r o vector m- (of w e i g h t 6), m a p p e d ~i am by f onto a m u l t i p l e of v-. In turn, applying X p-i ... X p-i to m++ m yields a nonzero vector m (of w e i g h t (p-l)~) m a p p e d by f onto a ++ multiple of v +. Since m is k i l l e d by each X (cf. the statement . 1(a)), fore generates Z(p_l)6 it is actually a maximal vector. a submodule of M which It there- is a h o m o m o r p h i c image of = St; but this submodule is m a p p e d by f onto St, so we are done.

O follows: o i -go o o (i,i) oI (0,i) o2 (i,0) OlO 2 (i,-i) o2o I (-i,i) 00 (0,0) In this case the r e s u l t i n g m a t r i x identity matrix; For type A 2 these weights are as (Do(-e ~ T))O,TeW is simply the o it is in p a r t i c u l a r unipotent, and the column sums of its inverse give the desired numbers bo, here all equal to i. In general, H u l s u r k a r shows that the matrix in question is unipotent (which leads h i m to suggest a new partial ordering of W). shows that column sums in the inverse matrix which top alcove must yield b o = i, as m e n t i o n e d earlier.