Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

Abstract

<jats:title>Abstract</jats:title><jats:p>We study the non-symmetric Macdonald polynomials specialized at infinity from various points of view. First, we define a family of modules of the Iwahori algebra whose characters are equal to the non-symmetric Macdonald polynomials specialized at infinity. Second, we show that these modules are isomorphic to the dual spaces of sections of certain sheaves on the semi-infinite Schubert varieties. Third, we prove that the global versions of these modules are homologically dual to the level one affine Demazure modules for simply-laced Dynkin types except for type <jats:inline-formula id="j_crelle-2019-0011_ineq_9999_w2aab3b7e5289b1b6b1aab1c14b1b1Aa"><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_crelle-2019-0011_eq_0769.png" /><jats:tex-math>{\mathrm{E}_{8}}</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>

Generalized Weyl modules and nonsymmetric q-Whittaker functions

“…More precisely, for a composition λ we consider certain left module D λ and a right module U o λ of the Iwahori algebra of type gl n . Both modules are (gl n -versions of) the generalized global Weyl modules with characteristics (see [FMO,FKM,NS,NNS1,NNS2]). The modules D λ and U o λ are endowed with the action of the graded (commutative) algebra A D λ (the so-called highest weight algebra) commuting with the action of the current algebra gl n [z] [CFK]).…”