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Extra info for Algebraic and Analytic Methods in Representation Theory
Whenever this quantization is not straightforward, we have added a note or a remark. We shall assume that G is a connected reductive group over an algebraically closed field k. The corresponding quantum group is Uq, where q E C is a n / t h root of unity, 1 odd. 9). 2 Since H°(0) - k, the trivial module is a tilting mod- ule. The same is true more generally whenever H°(A) is irreducible (because then H°(A) * = L(A)* ~_ L(-w0A) = H°(-w0A)). Elementary weight considerations show that H°(£) is irreducible for all p whenever )~ is minuscule.
7 Let )~ E Xl. Then Qq(£) = Dq(2(1- 1)p + w0A). (Contrary to our usual convention we have stated this result in its quantized version. 10 says that Qq()~) has a good filtration. Since Stl is self-dual, we see that Qq(/~)* is also injective (cf. 8), and hence Qq()~)* has also a good filtration. Thus, Qq(A) E ~t. By construction, we furthermore see that Qq()~) has highest weight 2 ( / - 1)p + w0£ and that this weight occurs with multiplicity 1.  We conclude by some results whose quantized version play a role in the construction of invariants of 3-manifolds by Reshetikhin and Turaev [RT].
Phys. 149 (1992), 149-159. [AJSI H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Asterisque 220 (1994). [AP] H. H. Andersen and J. Paradowski, Fusion categories arising from semisimple Lie algebras, Comm. Math. Phys. 169 (1995), 563-588. [APW] H. H. Andersen, P. Polo, and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), 1-59. [Bo] R. Bott, Homogeneous vector bundles, Ann.