## 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 ${\u2a01}_{\left[\nu \right]}{(M\left(\lambda \right)\otimes L\left(\mu \right))}^{\left[\nu \right]},$
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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