Generalized exponents of small representations. II

TL;DR

This paper provides explicit non-negative formulas for generalized exponents of small representations across all types, extending classical formulas and utilizing Fourier coefficients of the degenerate Cherednik kernel.

Contribution

It introduces a new formula for generalized exponents of small weights, based on minimal root expressions and Fourier coefficients, extending previous classical results.

Findings

Derived a formula for small weights' generalized exponents

Connected Fourier coefficients of Cherednik kernels to root combinatorics

Extended classical exponents formulas to all types

Abstract

This is the second paper in a sequence devoted to giving manifestly non-negative formulas for generalized exponents of small representations in all types. It contains a first formula for generalized exponents of small weights which extends the Shapiro-Steinberg formula for classical exponents. The formula is made possible by a computation of Fourier coefficients of the degenerate Cherednik kernel. Unlike the usual partition function coefficients, the answer reflects only the combinatorics of minimal expressions as a sum of roots.