Existence and Orthogonality of Generalized Jack Polynomials and Its $q$-Deformation
Yusuke Ohkubo
TL;DR
The paper proves the existence and orthogonality of generalized Jack symmetric functions by embedding them in a $q$-deformed framework of generalized Macdonald symmetric functions, whose eigenstructure is governed by the operator $X_0$. By analyzing the limit $q\to1$ with $t=q^{\beta}$, it shows that generalized Macdonald functions converge to generalized Jack functions $J_{\uvec{\lambda}}$, and that the nondegeneracy of eigenvalues in the beta-deformed regime yields orthogonality under the $\beta$-inner product. This yields a rigorous foundation for generalized Jack functions, along with a constructive pathway via $q$-deformations, and connects to Hubbard-Stratanovich duality interpretations of 5D AGT relations. The work also provides concrete level-1 and level-2 examples for $N=2$, illustrating the explicit structure and the $q\to1$ transition to $M_{\beta}^i$ forms. Overall, it strengthens the mathematical underpinning of AGT-related symmetric functions and their role in Nekrasov partition functions.
Abstract
We investigate the existence and the orthogonality of the generalized Jack symmetric functions which play an important role in the AGT relations. We show their orthogonality by deforming them to the generalized Macdonald symmetric functions.