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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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