Graded multiplicities in the exterior algebra of the little adjoint module

TL;DR

This paper uses double affine Hecke algebras to compute graded multiplicities of modules in the exterior algebra of the little adjoint module for certain Lie algebras, revealing new formulas involving root system exponents.

Contribution

It introduces a novel application of double affine Hecke algebras to determine graded multiplicities in Lie algebra modules, especially for non-simply laced types.

Findings

Multiplicities expressed via special exponents of positive long roots.

Explicit formulas for types B, C, and F.

Connection between algebraic multiplicities and root system exponents.

Abstract

As a first application of the double affine Hecke algebra with unequal parameters on Weyl orbits to representation theory of semisimple Lie algebras, we find the graded multiplicities of the trivial module and of the little adjoint module in the exterior algebra of the little adjoint module of a simple Lie algebra $\,\mathfrak{g}\,$ with a non-simply laced Dynkin diagram. We prove that in type $\,B, C\,$ or $\,F\,$ these multiplicities can be expressed in terms of special exponents of positive long roots in the dual root system of $\,\mathfrak{g}.\,$