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

Abstract

We consider the tensor power $V=(C^N{)}^{\otimes n}$ of the vector representation of $g{l}_N$ and its weight decomposition $V={\oplus }_{\lambda =({\lambda }_1,...,{\lambda }_N)}V[\lambda ]$. For $\lambda =({\lambda }_1\ge ...\ge {\lambda }_N)$, the trivial bundle $V[λ]\times \C^n\to\C^n$ has a subbundle of q-conformal blocks at level l, where $l={\lambda }_1-{\lambda }_N$ if ${\lambda }_1-{\lambda }_N>0$ and l=1 if ${\lambda }_1-{\lambda }_N=0$. We construct a polynomial section $I_{\lambda }(z_1,...,z_n,h)$ of the subbundle. The section is the main object of the paper. We identify the section with the generating function $J_{\lambda }(z_1,...,z_n,h)$ 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 $I_{\lambda }(z_1,...,z_n,h)$ 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.

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• A Selberg integral for the Lie algebra A_n
• The sl_3 Selberg integral
• A Selberg Integral Type Formula for an sl_2 One-Dimensional Space of Conformal Blocks
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Cited by in corpus (7)

• Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra
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• The ${\mathbb{F}}_p$-Selberg integral of type $A_n$
• $q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type
• Quiver varieties and quantum Knizhnik--Zamolodchikov equation
• Grassmann-Grassmann conormal varieties, integrability, and plane partitions
• Loop Models and $K$-Theory