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

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We consider the tensor power V=(Cⁿ)⊗ ⁿ of the vector representation of glₙ and its weight decomposition V=⊕λ₌₍λ₁,...,λₙ₎V[λ]. For λ= (λ₁ ≥... ≥ λₙ), the trivial bundle V[λ]× ⁿ→ⁿ has a subbundle of q-conformal blocks at level l, where l = λ₁-λₙ if λ₁-λₙ> 0 and l=1 if λ₁-λₙ=0. We construct a polynomial section Iλ(z₁,...,zₙ,h) of the subbundle. The section is the main object of the paper. We identify the section with the generating function Jλ(z₁,...,zₙ,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 Iλ(z₁,...,zₙ,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.