Abstract
We discuss some of the analytic properties of lens space indices for 4d $ \mathcal{N}=2 $ theories of class $ \mathcal{S} $ . The S-duality properties of these theories highly constrain the lens space indices, and imply in particular that they are naturally acted upon by a set of commuting difference operators corresponding to surface defects. We explicitly identify the difference operators to be a matrix-valued generalization of the elliptic RuijsenaarsSchneider model. In a special limit these difference operators can be expressed naturally in terms of Cherednik operators appearing in the double affine Hecke algebras, with the eigenfunctions given by non-symmetric Macdonald polynomials.
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