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

摘要

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 ?, where ?=λ _1-λ _N if λ _1-λ _N>0 and ?=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 ?=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 ? = 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.