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Extremal part of the PBW-filtration and nonsymmetric Macdonald polynomials. (English) Zbl 1378.17040
Summary: Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials in the limit \(t\to\infty\) and for antidominant weights, which is an important ingredient of the new theory of nonsymmetric \(q\)-Whittaker function. These coefficients are pure \(q\)-powers and their degrees are expected to coincide in the untwisted setting with the extremal degrees of the so-called PBW-filtration in the corresponding finite-dimensional irreducible representations of the simple Lie algebras for any root systems. This is a particular case of a general conjecture in terms of the level-one Demazure modules. We prove this coincidence for all Lie algebras of classical type and for \(G_2\), and also establish the relations of our extremal degrees to minimal \(q\)-degrees of the extremal terms of the Kostant \(q\)-partition function; they coincide with the latter only for some root systems.

MSC:
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20C08 Hecke algebras and their representations
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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