## The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials

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- by Martina Lanini and Arun Ram PDF
- Represent. Theory
**23**(2019), 188-204 Request permission

## Abstract:

In this paper we establish a Steinberg-Lusztig tensor product theorem for abstract Fock space. This is a generalization of the type A result of Leclerc-Thibon and a Grothendieck group version of the Steinberg-Lusztig tensor product theorem for representations of quantum groups at roots of unity. Although the statement can be phrased in terms of parabolic affine Kazhdan-Lusztig polynomials and thus has geometric content, our proof is combinatorial, using the theory of crystals (Littelmann paths). We derive the Casselman-Shalika formula as a consequence of the Steinberg-Lusztig tensor product theorem for abstract Fock space.## References

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## Additional Information

**Martina Lanini**- Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3FD United Kingdom
- MR Author ID: 990628
- Email: lanini@mat.uniroma2.it
**Arun Ram**- Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
- MR Author ID: 316170
- Email: aram@unimelb.edu.au
- Received by editor(s): April 15, 2018
- Received by editor(s) in revised form: January 31, 2019
- Published electronically: April 2, 2019
- Additional Notes: This research was partially supported by grants DP1201001942 and DP130100674.

The first author was partially supported by Australian Research Council grant DP150103525. The first author also acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. - © Copyright 2019 American Mathematical Society
- Journal: Represent. Theory
**23**(2019), 188-204 - MSC (2010): Primary 17B37; Secondary 20C20
- DOI: https://doi.org/10.1090/ert/524
- MathSciNet review: 3933899

Dedicated: Dedicated to Friedrich Knop and Peter Littelmann on the occasion of their 60th birthdays