Generalized Weyl modules and nonsymmetric q-Whittaker functions

Abstract

We introduce generalized global Weyl modules and relate their graded characters to nonsymmetric Macdonald polynomials and nonsymmetric q-Whittaker functions. In particular, we show that the series part of the nonsymmetric q-Whittaker function is a generating function for the graded characters of generalized global Weyl modules.

Representation theoretic realization of non-symmetric Macdonald polynomials at infinity

“…Note that the Cartan subalgebra h is included in I. In [FeMa1,FMO] the authors used the algebra n af = n + ⊕ g ⊗ zC[z] ⊂ g ⊗ C[z] instead of I. The only difference is that the Cartan part h is missing in n af , i.e.…”

“…The final results are presented in subsection 3.5. The main ideas are to find a surjection W λ → U λ , and refine the decomposition procedure from [FeMa1,FMO] to identify U λ with a module with prescribed characters and defining equations (Theorem 3.20, Theorem 3.22, and Corollary 3.21).…”

“…Remark 3.2. Let W σ(λ − ) be the generalized Weyl module ([FeMa1,FMO,No]). Then W σ(λ − ) is the cyclic I-module with the same set of relations as for U σ(λ − ) with the last two lines replaced with…”

“…In [Kat], the weight (λ + ) w is defined for each λ + ∈ P + and w ∈ W as λ + − j:wα j <0 ω j . Our notation (λ − ) σ (that follows [FeMa1,FMO]) is different from [Kat], and the relation between two notations are: w → σw 0 , and λ + → w 0 λ − . Note that we have σλ − = wλ + , and one has…”

“…To prove part (ii) we use the same inductive argument as in the proof [FMO,Theorem 3.16] (that ultimately relies on the counting paths in the quantum Bruhat graph in [FeMa1, Theorem 2.21] through [FMO,Lemma 3.12]). The only modification needed is [FMO,Lemma 3.13]. Namely, the crucial point in this Lemma is to figure out if a coroot −β • is simple.…”