By Bent Ørsted and Henrik Schlichtkrull (Eds.)

ISBN-10: 0126254400

ISBN-13: 9780126254402

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Then A[V] has even the structure of a unitary highest weight module. Moreover, except for easily described special cases, D[V] coincides with U(g)/Anna[V]. Conversely, every unitary highest weight module L is associated to a parabolic subalgebra with a commutative nilradical m +, and remarkably Annu(m-) L is a prime ideal and hence the ideal of definition of the closure of an orbital variety contained in m+. The proof of these results will be indicated in Lectures 3-6. Some further perspectives including the use of q u a n t u m groups and of differential operators are given in Lectures 7 and 8.

Set 8w(C) - {(u, B 1 , B 2 ) E 8 ( C ) ] (B1, B2) E Z~}. These form a finite disjoint union of 8(C). Set U - B N b/ and U~, - U n ~U. The fibers of the projection 8 ~ ( d ) isomorphic to d n U~. ~ 13 x 13 are all Since Zw identifies with G / ( B n ~B), it follows that dim 8~(C) - d i m ( G / B N ~ B ) + dim(C N U~) _< dim G - rk G, with equality if and only if C N U~ is dense in U~, equivalently if (GU~) - C. We denote the subset of w E W satisfying this last condition by St -1 (C). In this case, C n U~ is irreducible.

3 (1990). 257-296. [Lu2] ~ , Introduction to quantum groups, Progress in Mathematics, BirkhSuser, 1993. [Ma] O. Mathieu, Filtrations of G-modules, Ann. Sci. Ecole Norm. Sup. 23 (1990), 625-644. [Pa] J. Paradowski, Filtrations of modules over quantum algebras, Proc. Symp. Pure Math. 56 (1994), Part 2, 93-108. [Hi] C. M. Ringel, The category of modules with good filtrations over a quasi-hereditary algebra has almost split sequences, Math. Z. 208 (1991), 209-223. [RT] N. Reshetikhin and V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent.

### Algebraic and Analytic Methods in Representation Theory by Bent Ørsted and Henrik Schlichtkrull (Eds.)

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