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 ${\mathcal{B}}_x^{+}(\lambda )$, $x\in W_{\mathrm{af}}$, 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 $P_{\lambda }(x;q,t)$.

References in corpus (3)

• A uniform model for Kirillov-Reshetikhin crystals II. Alcove model, path model, and P=X
• Specializations of nonsymmetric Macdonald-Koornwinder polynomials
• Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras

Cited by in corpus (4)

• Chevalley formula for anti-dominant weights in the equivariant $K$-theory of semi-infinite flag manifolds
• Semi-infinite Lakshmibai-Seshadri path model for level-zero extremal weight modules over quantum affine algebras
• Positive level, negative level and level zero
• Level-zero van der Kallen modules and specialization of nonsymmetric Macdonald polynomials at $t=\infty$