By C. Faith, S. Wiegand
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For all ƒ. 24 (Kashiwara–Saito). Let B be a g-crystal, b0 2 B an element of weight 0. b0 / D 0, for all i 2 I , As is well-known, Lusztig constructed the basis by geometrizing Ringel’s work when the generalized Cartan matrix is symmetric. b/ 19 is finite, for all i 2 I and b 2 B, (iv) there exists a strict embedding ‰i W B ! a/ j b 2 B; a 2 Z<0 g. 1/. Assume there exists also a seminormal crystal D, a dominant integral weight ƒ and an element dƒ 2 D of weight ƒ such that (v) dƒ is the unique element of D of weight ƒ, (vi) there is a strict epimorphism ˆ W B ˝ Tƒ !
Finite dimensional Hecke algebras 27 As V D H ˚ CvH is a decomposition into a direct sum of WH -modules, and WH acts trivially on H , CvH affords a faithful representation of WH . 2 1=eH /. In other words, we define wH D ti if H D Hi and wH D sij I˛ if H D Hij I˛ . 3. H;k D eH 1ak=eH /wH , for 0 Ä k < eH . 2 p The WH -module C H;k affords the representation wH 7! exp. 2 1k=eH /. 4. Let R be a commutative C-algebra. We suppose that parameters Ä1 ; : : : ; Äd 1 ; h 2 R are given. hi /i2Z=2Z by extending the kappa parameters by Ä0 D Äd D 0 and by h1 D h, h0 D h2 D 0.
For M 2 O, define M prim D fm 2 M j V m D 0g. P Note that M prim ¤ 0 whenever M ¤ 0, and M prim Â E 2Irr W M . 16. Suppose that R is a Noetherian local ring such that the residue field F contains C. Then any M 2 O is a direct sum of finitely many indecomposable objects of O. In fact, if we had a strictly increasing infinite sequence of submodules M1 Â M1 ˚ M2 Â Â M then we have a strictly increasing sequence prim M1 prim Â M1 prim ˚ M2 Â Â M prim ; which is a contradiction since M prim is a Noetherian R-module.
Module Theory by C. Faith, S. Wiegand