Translation Functors and the Shapovalov Determinant

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 10 February 2015

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

Abstract

Let $M\left(\lambda \right)$ be the Verma module of highest weight $\lambda$ and $L\left(\mu \right)$ the irreducible module of highest weight $\mu \text{.}$ The module $M\left(\lambda \right)\otimes L\left(\mu \right)$ decomposes as a direct sum of submodules ${⨁}_{\left[\nu \right]}{(M\left(\lambda \right)\otimes L\left(\mu \right))}^{\left[\nu \right]},$ where each summand ${(M\left(\lambda \right)\otimes L\left(\mu \right))}^{\left[\nu \right]}$ corresponds to a block $\left[\nu \right]\text{.}$ For a given choice of $\mu$ and $\left[\nu \right],$ the translation functor corresponding to $\mu$ and $\left[\nu \right]$ is the map taking $M\left(\lambda \right)$ to ${(M\left(\lambda \right)\otimes L\left(\mu \right))}^{\left[\nu \right]}\text{.}$

Jantzen [Jan1973] studied translation functors for finite-dimensional semisimple Lie algebras using the Shapovalov determinant, and he computed statistics $a_{\left[\nu \right]}^{\mu }\left(\lambda \right)$ associated to each translation functor. In this dissertation, I compute the analogous statistics for the Virasoro algebra and for quantum groups. I also show that these statistics contain useful representation-theoretic information.

Notes and References

This is an excerpt from the PhD thesis Translation Functors and the Shapovalov Determinant by Emilie Wiesner, University of Wisconsin-Madison, 2005.

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