Generalized Weyl modules, alcove paths and Macdonald polynomials

Abstract

Classical local Weyl modules for a simple Lie algebra are labeled by dominant weights. We generalize the definition to the case of arbitrary weights and study the properties of the generalized modules. We prove that the representation theory of the generalized Weyl modules can be described in terms of the alcove paths and the quantum Bruhat graph. We make use of the Orr-Shimozono formula in order to prove that the t = ∞ specializations of the nonsymmetric Macdonald polynomials are equal to the characters of certain generalized Weyl modules.1 Remark 1.5. The coroots β k (w) comprise the set of all positive affine coroots which are mapped to the negative roots by w. We note also that {β k (w)} is the sequence of labels of walls crossed by a shortest walk from the alcove w −1 to the initial alcove of the current sheet (see example on page 6 in [RY]). Letb = (b 1 , . . . , b l ) ∈ 0, 1 l be a binary word and let J = {i|b i = 0}, J = {j 1 < · · · < j r }. We call J the set of foldings. For an element u ∈ W a we set z 0 = uw, z k+1 = z k s β j k+1 , k = 0, . . . , r − 1.We denote this data by an alcove path p J , so p J can be written as z 0 β j 1

Generalized Weyl modules for twisted current algebras

“…The proof is analogous to the proof for untwisted algebras by interchanging roots and coroots (see [FM3], Lemma 2.12).…”

“…It is easy to see that the generalized Weyl modules are well defined, i.e. W µ does not depend on the choice of σ and λ such that σ(λ) = µ (see Lemma 2.2 in [FM3]). We note that the algebra n af does not contain the finite Cartan subalgebra h. However, sometimes it is convenient to add extra operators from h acting on W µ .…”

“…Using this action we define the generalized Weyl modules for dual untwisted types analogously to the untwisted definition from [FM3].…”

“…So we have n af = n af ′ and hence there is nothing to prove. The third case is ∆ ′ being the root system of type A 2 ; then n af ′ is the positive affine nilpotent algebra of type A ) 3 is equivalent to the untwisted C 2 situation from [FM3]. The only difference is that we apply Corollary 1.5, iii), to the roots instead of coroots.…”

“…Let W a and W e be the affine and the extended affine Weyl groups for g ∨ 0 . In particular, W e = W a ⋊ Π, Π = W e /W a and we have the natural embedding X ⊂ W e (see [OS,FM3]). For λ ∈ X we denote by t λ the corresponding element in W e .…”