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

Abstract

We consider the tensor power $V=(C^N{)}^{\otimes n}$ of the vector representation of $g{l}_N$ and its weight decomposition $V={\oplus }_{\hat{I}»=(\hat{I}{»}_1,...,\hat{I}{»}_N)}V[\hat{I}»]$. For $\hat{I}»=(\hat{I}{»}_1\ge ...\ge \hat{I}{»}_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l=\hat{I}{»}_1-\hat{I}{»}_N$ if $\hat{I}{»}_1-\hat{I}{»}_N>0$ and l=1 if $\hat{I}{»}_1-\hat{I}{»}_N=0$. We construct a polynomial section $I_{\hat{I}}»(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_{\hat{I}}»(z_1,...,z_n,h)$ 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 $I_{\hat{I}}»(z_1,...,z_n,h)$ 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.