Extended Joseph polynomials, quantized conformal blocks, and a q-Selberg type integral

TL;DR

This paper constructs polynomial sections of q-conformal blocks related to tensor powers of vector representations of gl_N, linking them to extended Joseph polynomials and deriving a q-Selberg integral identity.

Contribution

It introduces a new polynomial section of q-conformal blocks, identifies it with extended Joseph polynomials, and establishes a q-Selberg integral representation for specific cases.

Findings

The polynomial section is flat with respect to the quantum KZ connection.

For N=2, l=1, the polynomial admits a multidimensional q-hypergeometric integral representation.

The integral evaluates to an explicit polynomial, establishing a q-Selberg type identity.

Abstract

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{\lambda=(\lambda_1,...,\lambda_N)}V[\lambda]$. For $\lambda = (\lambda_1 \geq ... \geq \lambda_N)$, the trivial bundle $V[\lambda]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l = \lambda_1-\lambda_N$ if $\lambda_1-\lambda_N> 0$ and l=1 if $\lambda_1-\lambda_N=0$. We construct a polynomial section $I_\lambda(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_\lambda(z_1,...,z_n,h)$ of the extended Joseph polynomials of orbital varieties, defined in [DFZJ05,KZJ09]. For l=1, we show that the subbundle of q-conformal blocks has rank 1 and $I_\lambda(z_1,...,z_n,h)$ is flat with respect to the quantum Knizhnik-Zamolodchikov discrete connection. For N=2 and l=1, we represent our polynomial as a multidimensional q-hypergeometric integral and obtain a q-Selberg type identity, which says that the integral is an explicit polynomial.