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

TL;DR

This paper generalizes the Steinberg-Lusztig tensor product theorem to abstract Fock space, providing a combinatorial proof using crystal theory and deriving the Casselman-Shalika formula as a consequence.

Contribution

It extends the Steinberg-Lusztig tensor product theorem to a broader setting of abstract Fock space with a combinatorial proof approach.

Findings

Established a tensor product theorem for abstract Fock space.

Derived the Casselman-Shalika formula from the theorem.

Connected geometric and combinatorial aspects via crystal theory.

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.