Let i be a level-zero fundamental weight for an affine Lie algebra g over Q, and let B( i ) be the crystal of all Lakshmibai-Seshadri paths of shape i . First, we prove that the crystal graph of B( i ) is connected. By combining this fact with the main result of our previous work, we see that B( i ) is, as a crystal, isomorphic to the crystal base B( i ) of the extremal weight module V ( i ) over a quantum affine algebra U q (g) over Q(q) of extremal weight i . Next, we obtain an explicit description of the decomposition of the crystal B(m i ) of all Lakshmibai-Seshadri paths of shape m i into connected components. Furthermore, we prove that B(m i ) is, as a crystal, isomorphic to the crystal base B(m i ) of the extremal weight module V (m i ) over U q (g) of extremal weight m i .

Abstract. We lift the parabolic quantum Bruhat graph into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analogue of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the quantum Bruhat graph on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the quantum Bruhat graph to the parabolic one; this lemma compares paths between two vertices in the former graph.The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules.

We introduce semi-infinite Lakshmibai-Seshadri paths by using the semi-infinite Bruhat order (or equivalently, Lusztig's generic Bruhat order) on affine Weyl groups in place of the usual Bruhat order. These paths enable us to give an explicit realization of the crystal basis of an extremal weight module of an arbitrary level-zero dominant integral extremal weight over a quantum affine algebra. This result can be thought of as a full generalization of our previous result (which uses Littelmann's Lakshmibai-Seshadri paths), in which the levelzero dominant integral weight is assumed to be a positive-integer multiple of a level-zero fundamental weight. * 2010 Mathematics Subject Classification. Primary: 17B37, Secondary: 17B67, 81R50, 81R10. and ̟ i is the i-th level-zero fundamental weight for i ∈ I, over the quantum affine algebra U q (g af ) in terms of Lakshmibai-Seshadri (LS for short) paths of shape m i ̟ i in the sense of [Li2]; however, in [NS4], we showed that it is impossible to give a realization of the crystal basis B(λ) of the extremal weight module V (λ) of a general level-zero dominant integral extremal weight λ in terms of Littelmann's LS paths of shape λ. The purpose of this paper is to overcome this difficulty, and to give an explicit realization of the crystal basis of the extremal weight module V (λ) for a level-zero dominant integral weight λ in full generality; however, we assume that an affine Lie algebra g af is of untwisted type throughout this paper.Extremal weight modules over the quantized universal enveloping algebras of symmetrizable Kac-Moody algebras were introduced by Kashiwara [Kas1] in his study of the level-zero part of the modified quantized universal enveloping algebra (see [Lu2]) of an affine Lie algebra.Let λ be an integral weight for an affine Lie algebra g af . If λ is of positive (resp., negative) level, then the extremal weight module V (λ) is just the integrable highest (resp., lowest) weight module over U q (g af ). However, in the case when λ is of level-zero, the structure of V (λ) is much more complicated than in the case of positive or negative level. In fact, it is known ([N2, Remark 2.15]; see also [CP, Proposition 4.5]) that V (λ) is isomorphic to the quantum Weyl module W q (λ) introduced by Chari and Pressley ([CP]). Also, in the case when g af is an untwisted affine Lie algebra of type A, D, or E, an extremal weight module can be thought of as a universal standard module. Here standard modules M P , parametrized by Drinfeld polynomials P , were constructed by Nakajima ([N1]) by use of quiver varieties, as a new basis of the Grothendieck ring Rep U q (Lg) of finite-dimensional modules (of type 1) over the quantum loop algebra U q (Lg), where g ⊂ g af is the canonical finite-dimensional subalgebra (of type A, D, or E), and Lg = C[t, t −1 ] ⊗ C g; unique irreducible quotients L P of the standard modules M P form another basis of Rep U q (Lg).More precisely, for a level-zero dominant integral weight λ = i∈I m i ̟ i with m i ∈ Z ≥0 , i ∈ I, the universal standard mod...

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).

Abstract. Let g be a complex finite-dimensional simple Lie algebra. Given a positive integer k and a dominant weight λ, we define a preorder on the set P + (λ, k) of k-tuples of dominant weights which add up to λ. Let ∼ be the equivalence relation defined by the preorder and P + (λ, k)/ ∼ be the corresponding poset of equivalence classes. We show that if λ is a multiple of a fundamental weight (and k is general) or if k = 2 (and λ is general), then P + (λ, k)/ ∼ coincides with the set of S k -orbits in P + (λ, k), where S k acts on P + (λ, k) as the permutations of components. If g is of type An and k = 2, we show that the S2-orbit of the row shuffle defined by Fomin et al in [FFLP05] is the unique maximal element in the poset.Given an element of P + (λ, k), consider the tensor product of the corresponding simple finite-dimensional g-modules. We show that (for general g, λ, and k) the dimension of this tensor product increases along . We also show that in the case when λ is a multiple of a fundamental minuscule weight (g and k are general) or if g is of type A2 and k = 2 (λ is general), there exists an inclusion of tensor products along with the partial order on P + (λ, k)/ ∼. In particular, if g is of type An, this means that the difference of the characters is Schur positive.