Extended Joseph polynomials, quantized conformal blocks, and a q-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=(λ 1≥⋯≥λ N), the trivial bundle V[λ]×Cn→Cn 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 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 I λ(z 1,.....