An Apparent Dependence of the Apex and Velocity of Solar by Perrine C. D.

By Perrine C. D.

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Then Fix ~ = ~ - (ni+l)e i + is l i n k e d not hard else to ~ in A. +l + Y. v is a m a x i m a l v e c t o r of w e i g h t ~ (or i n I. + i = p, ~ = A), w h e r e we denote Y~. by Yi" This 1 determines taken a w-module in r e v e r s e homomorphism yields f : Z~ § a homomorphism g : Z~ § The same procedure , because 28 I = ~-(<~,~i > + l)e i. The composite gof = 0, in view of the fact that forcing Yi <~'~i>+l Yi +l <~,~i > = -ni-2, = Y~i = 0 ' Similarly ' fog = 0. 1(e) in which first the case just sketched, k and ~ are linked by a simple reflection readily ~ : Ker f is seen to coincide with Im g, and similarly Ker g = Im f.

I The Steinberg module For each ~ ~ Xp, R 1 denotes top (and bottom) tive KF-module multiple unless composition the PIM of KF having M 1 as its unique factor. Recall is a direct sum of PIM's. I) that each projec- Moreover, of pm (the exact power of p dividing ~ = (p-l)6, only irreducible ef. 2), IF]). the Steinberg module KF-module which dim RI is a Since dim MI < pm St = M(p_l)~ is the could possibly be projective. In fact: PROPOSITION. Proof. requires of F: St is a projective The only proof of this statement known to the author a highly nontrivial There exists ef.

P-2)6 + ~ (f) a composition for the "Borel subgroup" correspond bijeetively diagonal to factor of highest weight In his thesis, lower bound on dim R I by observing is that dim R I ~ Upon restriction one can force R I = QI in the regular of dimensions. 3). ) subgroup. The conclusion ~I is the linear character of the subgroup of F corresponding to I. Notice that the W-orbit of ~I has the same cardinality as the W-orbit of I in X/(p-I)X, is the same as the eardinality of the W-orbit of I + ~ in X/pX = A.

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