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


Authors: Martina Lanini and Arun Ram
Journal: Represent. Theory 23 (2019), 188-204
MSC (2010): Primary 17B37; Secondary 20C20
DOI: https://doi.org/10.1090/ert/524
Published electronically: April 2, 2019
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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.


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Additional Information

Martina Lanini
Affiliation: School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh EH9 3FD United Kingdom
Email: lanini@mat.uniroma2.it

Arun Ram
Affiliation: Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010 Australia
Email: aram@unimelb.edu.au

DOI: https://doi.org/10.1090/ert/524
Keywords: Quantum groups, affine Lie algebras, Hecke algebras, symmetric functions
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.
Dedicated: Dedicated to Friedrich Knop and Peter Littelmann on the occasion of their 60th birthdays
Article copyright: © Copyright 2019 American Mathematical Society