By Leslie C. Glaser
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Example text
In O, respectively. For 2 hdom set P ´ 2W P . /. P /op . 4 BGG reciprocity and quasi-hereditary structure A module N 2 O is said to have a standard filtration or Verma flag if there is a filtration of N whose subquotients are Verma modules. /. 5 (BGG reciprocity). (a) Every projective module in O has a standard filtration. (b) If some N 2 O has a standard filtration, then for any 2 h the multiplicity ŒN W M. / of M. / as a subquotient of a standard filtration of N does not depend on the choice of such filtration.
0 (a) For every W -antidominant 2 W there is a unique indecomposable projective functor  ; such that  ; . / D P . /. (b) Every indecomposable projective functor from O to O 0 is isomorphic to  ; for some W -antidominant 2 W 0 . 2 implies that an indecomposable projective functor  is completely determined by its value  . / on the corresponding dominant Verma module . /. Moreover, as  . / is projective and projective modules form a basis of ŒO 0 (as O 0 , being quasi-hereditary, has finite global dimension), the functor  is already uniquely determined by ŒÂ .
8. For every 2 h there is a unique (up to isomorphism) indecomposable module T . r/ such that . / T . / and the cokernel of this inclusion admits a standard filtration. L For 2 hdom the module T ´ 2W T . / is called the characteristic tilting module. 9. The module T is ext-selforthogonal, has finite projective dimension and there is an exact sequence 0 ! P ! Q0 ! Q1 ! Qk ! T / for all i. The (opposite of the) endomorphism algebra of T is called the Ringel dual of B . The Ringel dual is defined for any quasi-hereditary algebra and is again a quasihereditary algebra.