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Some homological properties of the category $\mathcal {O}$, II

Author: Volodymyr Mazorchuk
Journal: Represent. Theory 14 (2010), 249-263
MSC (2000): Primary 16E10, 16E30, 16G99, 17B10
Published electronically: March 1, 2010
MathSciNet review: 2602033
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Abstract: We show, in full generality, that Lusztig’s $\mathbf {a}$-function describes the projective dimension of both indecomposable tilting modules and indecomposable injective modules in the regular block of the BGG category $\mathcal {O}$, proving a conjecture from the first paper. On the way we show that the images of simple modules under projective functors can be represented in the derived category by linear complexes of tilting modules. These complexes, in turn, can be interpreted as the images of simple modules under projective functors in the Koszul dual of the category $\mathcal {O}$. Finally, we describe the dominant projective modules and also the projective-injective modules in some subcategories of $\mathcal {O}$ and show how one can use categorification to decompose the regular representation of the Weyl group into a direct sum of cell modules, extending the results known for the symmetric group (type $A$).

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Additional Information

Volodymyr Mazorchuk
Affiliation: Department of Mathematics, Uppsala University, SE-751 06, Uppsala, Sweden
MR Author ID: 353912

Keywords: Category $\mathcal {O}$; tilting module; Lusztig’s $\mathbf {a}$-function; complex; Koszul dual; categorification; projective dimension
Received by editor(s): September 15, 2009
Received by editor(s) in revised form: October 3, 2009
Published electronically: March 1, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.