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By David C. Vella

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F:E Let lpJ ~* E ' |B p -» E ( P } $ -(p-D/> and Ei>:STj(p) - -(p-l)p be the evaluation maps. Then the maps *pn are defined as respective 8 Ev) . LB p ((f) B J Consider the following diagram, where fi2 = -(P-1)e2(p)• J BHLjl U^1( P ) (p) . l L "j ® P2 ~ ~ J I BflLj I - -* - . ®- p 2 — »«° > f-(P) l(p)| 0 STj(p) E^' * 8 -lo-iw-r-ill -(P-Dejlio) 1 • ST J( P) ^ • fl(P> 8 - 0 Ev E (P) 3 -(p-1)/) The triangle commutes by definition of ^°, the top row is obtained by restricting the middle row to Lj.

25. 2, •*. ,an_1) Mj(-X) p . Let X = prw1 with r £ 0 (p = char k) . Then is irreducible, isomorphic to S(X ). Proof. 6, Myf-XHp Lli / AA ( N ) = 0 where N = Ker{Ev\ ! is irreducible if and only if * ). We have MT(-X) = M T (-«-) ( p > which S(X ) i l l has dimension n, and similarly S(X ) = S(w n )* p ' has dimension n + 1. This shows N is one dimensional, necessarily a character of Pj. Since the high weight X* of S(X ) is not a weight of MI(-X) we have N = X = pru) . 7. 7 of [1] we need only check this for large r.

L+w3 1 2u>2 0 [ Indeed (At l e a s t 2) L 0 ? (2 2CJ-+u>2 2 i,A(Mi(_w))) 0 4o>^ 2w- +u>2 i(L ) i 2u) 2 0 w l+w3 2o) 2 2w M 0 2u)2+u>2 4u>i itMjt-u)!! C 0 w 2u>2+W2 . 12 since lpI Mj(-a) = _ M | B f ° r those weights. This explains the top 3 rows. 25 when I = (02,02). o) 1 - W 2 J | P Comparing e v F2o) .

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