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Lie Groups, Differential Equations, and Geometry pp 41–70Cite as

Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras

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Abstract

This essay is meant as an introduction to a very successful approach towards understanding the structure of certain highest weight categories appearing in algebraic Lie theory. For simplicity, the focus lies on the case of the category \(\mathcal{O}\) of representations of a simple complex Lie algebra. We show how the approach yields a proof of the classical Kazhdan–Lusztig conjectures that avoids the theory of D-modules on flag varieties.

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Notes

  1. 1.

    To be precise, one should replace \(\mathfrak{g}\) here by its Langlands dual \(\mathfrak{g}^{\vee }\).

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Acknowledgements

The author was partially supported by the DFG grant SP1388.

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Authors and Affiliations

  1. Department Mathematik, FAU Erlangen–Nürnberg, Cauerstraße 11, 91058, Erlangen, Germany

    Peter Fiebig

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  1. Peter Fiebig
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Correspondence to Peter Fiebig .

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Editors and Affiliations

  1. Dipartimento di Matematica e Informatica, University of Palermo, Palermo, Italy

    Giovanni Falcone

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Fiebig, P. (2017). Character, Multiplicity, and Decomposition Problems in the Representation Theory of Complex Lie Algebras. In: Falcone, G. (eds) Lie Groups, Differential Equations, and Geometry. UNIPA Springer Series. Springer, Cham. https://doi.org/10.1007/978-3-319-62181-4_3

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