## Contravariant forms on Whittaker modules

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- by Adam Brown and Anna Romanov PDF
- Proc. Amer. Math. Soc.
**149**(2021), 37-52 Request permission

## Abstract:

Let $\mathfrak {g}$ be a complex semisimple Lie algebra. We give a classification of contravariant forms on the nondegenerate Whittaker $\mathfrak {g}$-modules $Y(\chi , \eta )$ introduced by Kostant. We prove that the set of all contravariant forms on $Y(\chi , \eta )$ forms a vector space whose dimension is given by the cardinality of the Weyl group of $\mathfrak {g}$. We also describe a procedure for parabolically inducing contravariant forms. As a corollary, we deduce the existence of the Shapovalov form on a Verma module, and provide a formula for the dimension of the space of contravariant forms on the degenerate Whittaker modules $M(\chi , \eta )$ introduced by McDowell.## References

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## Additional Information

**Adam Brown**- Affiliation: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
- ORCID: 0000-0002-0955-0119
- Email: abrown@ist.ac.at
**Anna Romanov**- Affiliation: School of Mathematics and Statistics, Camperdown NSW 2006, Australia
- MR Author ID: 1397543
- ORCID: 0000-0002-9146-7217
- Email: anna.romanov@sydney.edu.au
- Received by editor(s): October 22, 2019
- Received by editor(s) in revised form: February 17, 2020, and April 21, 2020
- Published electronically: October 9, 2020
- Additional Notes: The first author acknowledges the support of the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 754411.

The second author was supported by the National Science Foundation Award No. 1803059. - Communicated by: Sarah Witherspoon
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**149**(2021), 37-52 - MSC (2010): Primary 17B10, 20G05, 22E47
- DOI: https://doi.org/10.1090/proc/15205
- MathSciNet review: 4172584