Extended Joseph polynomials, quantized conformal blocks, and a [formula omitted]-Selberg type integral

Abstract

We consider the tensor product 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]×Cn→Cn 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 Di Francesco and Zinn-Justin (2005) [11] and Knutson and Zinn-Justin (2009) [12].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.