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A very useful tool for the calculations to come is the group ring Zƒ of the weight lattice ƒ considered as an additive group with coefficients in Z. Elements of the group ring Zƒ are appropriately thought of as finite formal sums of terms ce with c 2 Z and 2 ƒ, because the naive multiplication of such formal sums agrees with the standard convolution product in the group ring Zƒ. dim E /e ch V WD 2ƒ where E is the generalized complex eigenspace of the action of the maximal torus on V : E WD fv 2 V ˝R C j H ?

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