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- Title
Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials.
- Authors
Naito, Satoshi; Sagaki, Daisuke
- Abstract
In this paper, we give a characterization of the crystal bases $$\mathcal {B}_{x}^{+}(\lambda )$$ , $$x \in W_{\mathrm {af}}$$ , of Demazure submodules $$V_{x}^{+}(\lambda )$$ , $$x \in W_{\mathrm {af}}$$ , of a level-zero extremal weight module $$V(\lambda )$$ over a quantum affine algebra $$U_{q}$$ , where $$\lambda $$ is an arbitrary level-zero dominant integral weight, and $$W_{\mathrm {af}}$$ 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 $$\mathcal {B}(\lambda )$$ of the level-zero extremal weight module $$V(\lambda )$$ and the crystal $${\mathbb {B}}^{\frac{\infty }{2}}(\lambda )$$ of semi-infinite Lakshmibai-Seshadri paths of shape $$\lambda $$ , which is obtained in our previous work. As an application, we obtain a formula expressing the graded character of the Demazure submodule $$V_{w_0}^{+}(\lambda )$$ in terms of the specialization at $$t=0$$ of the symmetric Macdonald polynomial $$P_{\lambda }(x\,;\,q,\,t)$$ .
- Publication
Mathematische Zeitschrift, 2016, Vol 283, Issue 3/4, p937
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-016-1628-7