Norms of eigenfunctions to trigonometric KZB operators

TL;DR

This paper constructs eigenfunctions for trigonometric KZB operators using Bethe ansatz, introduces a scalar product making these operators symmetric, and relates eigenfunction norms to the Hessian of a master function.

Contribution

It introduces a scalar product that renders the trigonometric KZB operators symmetric and links eigenfunction norms to the Hessian of the master function at critical points.

Findings

Eigenfunctions constructed via Bethe ansatz.

Scalar product makes operators symmetric.

Norms of eigenfunctions equal Hessian of master function.

Abstract

Let $g$ be a simple Lie algebra and $V[0]=V_1\otimes...\otimes V_n[0]$ the zero weight subspace of a tensor product of $g$-modules. The trigonometric KZB operators are commuting differential operators acting on $V[0]$-valued functions on the Cartan subalgebra of $g$. Meromorphic eigenfunctions to the operators are constructed by the Bethe ansatz. We introduce a scalar product on a suitable space of functions such that the operators become symmetric, and the square of the norm of a Bethe eigenfunction equals the Hessian of the master function at the corresponding critical point.