, Volume 249, Issue 2, pp 357-386
Date: 15 Oct 2004

On Arkhipov’s and Enright’s functors

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We give a description of Arkhipov’s and (Joseph’s and Deodhar-Mathieu’s versions of) Enright’s endofunctors on the category associated with a fixed triangular decomposition of a complex finite-dimensional semi-simple Lie algebra, in terms of (co)approximation functors with respect to suitably chosen injective (resp. projective) modules. We establish some new connections between these functors, for example we show that Arkhipov’s and Joseph’s functors are adjoint to each other. We also give several proofs of braid relations for Arkhipov’s and Enright’s functors.

Mathematical Subject Classification (2000): 16G10, 17B10, 16G20, 17B35
Acknowledgement An essential part of the research which led to the results written in this paper was done during the visit of the authors to The Fields Institute for Research in Mathematical Sciences, whose financial support is gratefully acknowledged. The first author thanks the German Research Council (DFG) for financial support within the scope of the project “Projektive Funktoren und Hecke algebra”. The second author was partially supported by The Royal Swedish Academy of Science and The Swedish Research Council. Some parts of the paper were written during the visit of the first author to Uppsala and of the second author to Leicester. We thank the host Universities for their hospitality and The Royal Swedish Academy of Sciences and EPSRC for financial support of these visits. We are especially indebted to Catharina Stroppel for many helpful and stimulating discussions and for various remarks, which led to the improvements in this paper. We also thank Wolfgang Soergel for helpful discussions and remarks.