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**Example text**

N be a Cartan matrix, and choose a set of relatively prime integers d l , . . , dn such that (diaij) is symmetric. Fix also an indeterminate v, and set vi = v d~, i = 1 , . . , n . If m C N we write [m] = (v m - v - m ) / ( V - - V - I ) and [m]! = [ m ] [ m - 1]... [2][1]. For t C N, we set [t] - [m][m - 1]-.. [m - t + 1]/[t]!. , and [~]i for the corresponding expressions with v replaced by vi. 1 The quantum group attached to (aij) is the Q(r)-algebra U with generators Ei, Fi, Ki, K/--I, i = 1 , .

If m C N we write [m] = (v m - v - m ) / ( V - - V - I ) and [m]! = [ m ] [ m - 1]... [2][1]. For t C N, we set [t] - [m][m - 1]-.. [m - t + 1]/[t]!. , and [~]i for the corresponding expressions with v replaced by vi. 1 The quantum group attached to (aij) is the Q(r)-algebra U with generators Ei, Fi, Ki, K/--I, i = 1 , . . , n, and relations i) K i K ( -1 = 1 = K ~ I K i and KiKj = K j K i ii) K i E j K ( -1 - v ia i j Ej, iii) EiFj - FjEi iv/ z r+s=l--aij v) E K i F j K i -1 gi-gi - ~ij -- v-iaiJFj -1 vi - -1 v i 8 i (-1)s [1-aiJ 1 F[FjF~-O, r+s--1--aij 8 i#j i It turns out that U is a Hopf algebra with comultiplication A given by A ( K i ) - Ki®Ki, A ( E ~ ) - Ei®I+K~®Ei, A(Fi) - Fi®K~ - l + l ® F i , H.

4) Zr is an exact functor. The reason is that G~/B~ is affine, namely G~/B~ ~_ U¢, so that all higher sheaf cohomology of bundles on G~/Br vavishes. 5) dim Z~ (E) = pNr dim E, because dim k[U¢] = pN~ (cf. 6(iii)). 5 Let A E X(T). i) Z~()~) contains a unique irreducible Gr-submodule, which we denote Lr()~). ii) Any finite dimensional simple G~-module is isomorphic to some such L~()~). iii) If also # E X ( T ) , then L~()~) ~_ Lr(#) iff A - # (mod p X ( T ) ) . 6 Set Xp~(T) = {A E X(T) I 0 _< (A, av> < p" for all simple roots a}.