Specializations of nonsymmetric Macdonald–Koornwinder polynomials

Abstract

This work records the details of the Ram-Yip formula for nonsymmetric Macdonald-Koornwinder polynomials for the double affine Hecke algebras of not-necessarily-reduced affine root systems. It is shown that the t → 0 equal-parameter specialization of nonsymmetric Macdonald polynomials admits an explicit combinatorial formula in terms of quantum alcove paths, generalizing the formula of Lenart in the untwisted case. In particular our formula yields a definition of quantum Bruhat graph for all affine root systems. For mixed type the proof requires the Ram-Yip formula for the nonsymmetric Koornwinder polynomials. A quantum alcove path formula is also given at t → ∞. As a consequence we establish the positivity of the coefficients of nonsymmetric Macdonald polynomials under this limit, as conjectured by Cherednik and the first author. Finally, an explicit formula is given at q → ∞, which yields the p-adic Iwahori-Whittaker functions of Brubaker, Bump, and Licata.2.2. Root data. Given a Cartan datum (I, A), a root datum is a triple (X,where X is a lattice (free Z-module) containing elements α i (simple roots) for i ∈ I, and α ∨ i (simple coroots) are elements in X * = Hom Z (X, Z) such that α ∨ i , α j = a ij for i, j ∈ I (2.1) 1 In [1], the authors use E λ (X; q −1 ; t −1 ) and correspondingly send q → 0; [16] also uses this convention for q and t.

Generalized Weyl modules for twisted current algebras

“…The properties of generalized Weyl modules are closely related to the theory of nonsymmetric Macdonald polynomials E λ (x, q, t) [Ch1,Ch2,OS]. More precisely, we prove the following Theorem (see [FM3] for the untwisted case).…”

“…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 .…”

Nonsymmetric Macdonald polynomials and PBW filtration: Towards the proof of the Cherednik–Orr conjecture

“…More precisely, the t → 0 limit E λ (x, q, 0) coincides with the character of the corresponding level one Demazure module. In the recent papers [5][6][7]26] the t → ∞ limit of the nonsymmetric Macdonald polynomials was studied. In particular, it was shown that E λ (x, q, ∞) are polynomials in x and q −1 with non-negative integer coefficients.…”

“…We are working on such a construction and we hope to present the result elsewhere. We note also that in [26] the authors derived a nice formula for the polynomials E λ (x, q −1 , ∞), which uses the alcove paths and quantum Bruhat graphs. The great advantage of the Orr-Shimozono formula is that it works in any type and it is very tempting to use it in order to establish the desired property of the Macdonald polynomials.…”

“…The great advantage of the Orr-Shimozono formula is that it works in any type and it is very tempting to use it in order to establish the desired property of the Macdonald polynomials. However, at the moment we do not know how to use the results of [26] in order to study the Cherednik-Orr conjecture.…”