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

Lanini, Martina; Ram, Arun

doi:10.1090/ert/524

The Steinberg-Lusztig tensor product theorem, Casselman-Shalika, and LLT polynomials
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by Martina Lanini and Arun Ram

Represent. Theory 23 (2019), 188-204

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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