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

We consider the tensor product V=(CN)⊗nV=(CN)⊗n of the vector representation of glNglN and its weight decomposition V=⊕λ=(λ1,…,λN)V[λ]V=⊕λ=(λ1,…,λN)V[λ]. For λ=(λ1⩾⋯⩾λN)λ=(λ1⩾⋯⩾λN), the trivial bundle V[λ]×Cn→CnV[λ]×Cn→Cn has a subbundle of qq-conformal blocks at level ℓℓ, where ℓ=λ1−λNℓ=λ1−λN if λ1−λN>0λ1−λN>0 and ℓ=1ℓ=1 if λ1−λN=0λ1−λN=0. We construct a polynomial section Iλ(z1,…,zn,h)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)Jλ(z1,…,zn,h) of the extended Joseph polynomials of orbital varieties, defined in Di Francesco and Zinn-Justin (2005) [11] and Knutson and Zinn-Justin (2009) [12].For ℓ=1ℓ=1, we show that the subbundle of qq-conformal blocks has rank 1 and Iλ(z1,…,zn,h)Iλ(z1,…,zn,h) is flat with respect to the quantum Knizhnik–Zamolodchikov discrete connection.For N=2N=2 and ℓ=1ℓ=1, we represent our polynomial as a multidimensional qq-hypergeometric integral and obtain a qq-Selberg type identity, which says that the integral is an explicit polynomial.