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

We consider the tensor power $V=(C^N)^{\otimes n}$ of the vector representation of $gl_N$ and its weight decomposition $V=\oplus_{λ=(λ_1,...,λ_N)}V[λ]$. For $λ= (λ_1 \geq ... \geq λ_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l = λ_1-λ_N$ if $λ_1-λ_N> 0$ and l=1 if $λ_1-λ_N=0$. We construct a polynomial section $I_λ(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_λ(z_1,...,z_n,h)$ 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 $I_λ(z_1,...,z_n,h)$ 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.

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