1. Specialization of nonsymmetric Macdonald polynomials at t = infinity and Demazure submodules of level-zero extremal weight modules
- Author
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Fumihiko Nomoto, Satoshi Naito, and Daisuke Sagaki
- Subjects
Pure mathematics ,Macdonald polynomials ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,0103 physical sciences ,Specialization (functional) ,Zero (complex analysis) ,010307 mathematical physics ,0101 mathematics ,01 natural sciences ,Mathematics - Abstract
In this paper, we give a representation-theoretic interpretation of the specialization E w ∘ λ ( q , ∞ ) E_{w_\circ \lambda } (q,\infty ) of the nonsymmetric Macdonald polynomial E w ∘ λ ( q , t ) E_{w_\circ \lambda }(q,t) at t = ∞ t=\infty in terms of the Demazure submodule V w ∘ − ( λ ) V_{w_\circ }^- (\lambda ) of the level-zero extremal weight module V ( λ ) V(\lambda ) over a quantum affine algebra of an arbitrary untwisted type. Here, λ \lambda is a dominant integral weight, and w ∘ w_\circ denotes the longest element in the finite Weyl group W W . Also, for each x ∈ W x \in W , we obtain a combinatorial formula for the specialization E x λ ( q , ∞ ) E_{x \lambda } (q, \infty ) at t = ∞ t=\infty of the nonsymmetric Macdonald polynomial E x λ ( q , t ) E_{x \lambda } (q,t) and also a combinatorical formula for the graded character gch V x − ( λ ) \operatorname {gch} V_{x}^- (\lambda ) of the Demazure submodule V x − ( λ ) V_{x}^- (\lambda ) of V ( λ ) V(\lambda ) . Both of these formulas are described in terms of quantum Lakshmibai-Seshadri paths of shape λ \lambda .
- Published
- 2018