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Contravariant forms on Whittaker modules


Authors: Adam Brown and Anna Romanov
Journal: Proc. Amer. Math. Soc. 149 (2021), 37-52
MSC (2010): Primary 17B10, 20G05, 22E47
DOI: https://doi.org/10.1090/proc/15205
Published electronically: October 9, 2020
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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.


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Additional Information

Adam Brown
Affiliation: Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria
Email: abrown@ist.ac.at

Anna Romanov
Affiliation: School of Mathematics and Statistics, Camperdown NSW 2006, Australia
Email: anna.romanov@sydney.edu.au

DOI: https://doi.org/10.1090/proc/15205
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
Article copyright: © Copyright 2020 American Mathematical Society