## Translation Functors and the Shapovalov Determinant

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 ${⨁}_{\left[\nu \right]}{\left(M\left(\lambda \right)\otimes L\left(\mu \right)\right)}^{\left[\nu \right]},$ where each summand ${\left(M\left(\lambda \right)\otimes L\left(\mu \right)\right)}^{\left[\nu \right]}$ 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 ${\left(M\left(\lambda \right)\otimes L\left(\mu \right)\right)}^{\left[\nu \right]}\text{.}$

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.