Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials

Abstract

In this paper, we give a characterization of the crystal bases B + x (λ), x ∈ W af , of Demazure submodules V + x (λ), x ∈ W af , of a level-zero extremal weight module V (λ) over a quantum affine algebra U q , where λ is an arbitrary level-zero dominant integral weight, and W af denotes the affine Weyl group. This characterization is given in terms of the initial direction of a semi-infinite Lakshmibai-Seshadri path, and is established under a suitably normalized isomorphism between the crystal basis B(λ) of the level-zero extremal weight module V (λ) and the crystal B ∞ 2 (λ) of semi-infinite Lakshmibai-Seshadri paths of shape λ, which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule V + w 0 (λ) in terms of the specialization at t = 0 of the symmetric Macdonald polynomial P λ (x ; q, t).

A UNIFORM MODEL FOR KIRILLOV–RESHETIKHIN CRYSTALS III: NONSYMMETRICMACDONALD POLYNOMIALS AT t = 0 AND DEMAZURE CHARACTERS

“…Theorem 3.3. We know from[NS2, Theorem 7.2.2 (1)] that there exists a subsetB(Z + w• (λ)) of B(λ) such that G(b) | b ∈ B(Z + w• (λ)) is the global basis of Z + w• (λ). Also, recall that G(b) | b ∈ B + w (λ) is the global basis of V + w (λ).…”

“…Therefore, G(b) mod Z + w• (λ) | b ∈ B(U + w (λ)) := B + w (λ) \ B(Z + w• (λ)) is the global basis of U + w (λ), which is the image of V + w (λ) under the canonical projection V + w• (λ) ։ V + w• (λ)/Z + w• (λ). We know from[NS2, Theorem 7.2.2 (2)] that there exists an isomorphism Ψ ∨ λ : B(λ) ∼ → B ∞ 2 (λ) of crystals, which maps B(U + w (λ)) ⊂ B(λ) to π η | η ∈ QLS(λ) such that w ≥ ι(π η ) ⊂ B ∞ 2 (λ).…”

A Uniform Model for Kirillov–Reshetikhin Crystals II. Alcove Model, Path Model, and $P=X$

“…There have been several developments related to the work in this paper. Based on our results, an interpretation of the specialization P λ (x; q, 0) above is given in [INS,NS7], in terms of the crystal bases of level-zero extremal weight modules over quantum affine algebras. On another hand, our work was used in [CSSW] to provide the character of a stable level-one Demazure module associated to type B…”

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

“…It has been demonstrated recently that the t = ∞ specialization of the nonsymmetric Macdonald polynomials is very meaningful as well (see [CO1,CO2,OS,Kat,FeMa1,FeMa2,NS,NNS]). The study of the "opposite" t = ∞ specialization has lead to various discoveries of representation theoretic, combinatorial and geometric nature.…”