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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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The Hom-spaces between projective functors
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by Erik Backelin PDF
Represent. Theory 5 (2001), 267-283 Request permission


The category of projective functors on a block of the category $\mathcal O(\mathfrak g)$ of Bernstein, Gelfand and Gelfand, over a complex semisimple Lie algebra $\mathfrak g$, embeds to a corresponding block of the category $\mathcal O(\mathfrak g \times \mathfrak g)$. In this paper we give a nice description of the object $V$ in $\mathcal O(\mathfrak g \times \mathfrak g)$ corresponding to the identity functor; we show that $V$ is isomorphic to the module of invariants, under the diagonal action of the center $\mathcal Z$ of the universal enveloping algebra of $\mathfrak g$, in the so-called anti-dominant projective. As an application we use Soergel’s theory about modules over the coinvariant algebra $C$, of the Weyl group, to describe the space of homomorphisms of two projective functors $T$ and $T’$. We show that there exists a natural $C$-bimodule structure on $\operatorname {Hom}_{\{\operatorname {Functors}\}}(T, T’)$ such that this space becomes free as a left (and right) $C$-module and that evaluation induces a canonical isomorphism $k \otimes _C \operatorname {Hom}_{\{\operatorname {Functors}\}} (T, T’) \cong \operatorname {Hom}_{\mathcal O(\mathfrak g)}(T(M_e), T’(M_e))$, where $M_e$ denotes the dominant Verma module in the block and $k$ is the complex numbers.
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Additional Information
  • Erik Backelin
  • Affiliation: Sorselevägen 17, 16267 Vällingby, Stockholm, Sweden
  • Email:
  • Received by editor(s): May 16, 2000
  • Received by editor(s) in revised form: May 2, 2001
  • Published electronically: September 10, 2001
  • © Copyright 2001 American Mathematical Society
  • Journal: Represent. Theory 5 (2001), 267-283
  • MSC (2000): Primary 17B10, 18G15, 17B20
  • DOI:
  • MathSciNet review: 1857082