Dimension of zero weight space: An algebro-geometric approach

TL;DR

This paper investigates how the dimension of zero weight spaces in irreducible representations of a simple algebraic group varies, showing it is a piecewise polynomial function over a cone of weights using an algebro-geometric approach.

Contribution

It establishes that zero weight space dimensions are piecewise polynomial functions on a polyhedral cone of dominant weights, providing a new geometric perspective.

Findings

Zero weight space dimensions are piecewise polynomial functions.

The variation of these dimensions is characterized over a polyhedral cone.

The approach uses algebro-geometric methods to analyze representation weights.

Abstract

Let G be a connected, adjoint, simple algebraic group over the complex numbers with a maximal torus T and a Borel subgroup B containing T. The study of zero weight spaces in irreducible representations of G has been a topic of considerable interest; there are many works which study the zero weight space as a representation space for the Weyl group. In this paper, we study the variation on the dimension of the zero weight space as the irreducible representation varies over the set of dominant integral weights for T which are lattice points in a certain polyhedral cone. The theorem proved here asserts that the zero weight spaces have dimensions which are piecewise polynomial functions on the polyhedral cone of dominant integral weights.