Extremal part of the PBW-filtration and E-polynomials

Abstract

Given a reduced irreducible root system, the corresponding nil-DAHA is used to calculate the extremal coefficients of nonsymmetric Macdonald polynomials, also called E-polynomails, in the limit t=infinity 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 algebra 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.

References in corpus (1)

• Favourable modules: Filtrations, polytopes, Newton-Okounkov bodies and flat degenerations

Cited by in corpus (5)

• PBW filtration: Feigin-Fourier-Littelmann modules via Hasse diagrams
• Specializations of nonsymmetric Macdonald-Koornwinder polynomials
• Nonsymmetric Macdonald polynomials, Demazure modules and PBW filtration
• PBW-degenerated Demazure modules and Schubert varieties for triangular elements
• PBW-type filtration on quantum groups of type $A_n$