Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians

We study the difference analog of the quotient differential operator from [Tarasov V., Uvarov F., Lett. Math. Phys. 110 (2020), 3375-3400, arXiv:1907.02117]. Starting with a space of quasi-exponentials $W=\langle \alpha_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $\alpha_{i}\in{\mathbb C}^{*}$ and $p_{ij}(x)$ are polynomials, we consider the formal conjugate $\check{S}^{\dagger}_{W}$ of the quotient difference operator $\check{S}_{W}$ satisfying $\widehat{S} =\check{S}_{W}S_{W}$. Here, $S_{W}$ is a linear difference operator of order $\dim W$ annihilating $W$, and $\widehat{S}$ is a linear difference operator with constant coefficients depending on $\alpha_{i}$ and $\deg p_{ij}(x)$ only. We construct a space of quasi-exponentials of dimension $\operatorname{ord} \check{S}^{\dagger}_{W}$, which is annihilated by $\check{S}^{\dagger}_{W}$ and describe its basis and discrete exponents. We also consider a similar construction for differential operators associated with spaces of quasi-polynomials, which are linear combinations of functions of the form $x^{z}q(x)$, where $z\in\mathbb C$ and $q(x)$ is a polynomial. Combining our results with the results on the bispectral duality obtained in [Mukhin E., Tarasov V., Varchenko A., Adv. Math. 218 (2008), 216-265, arXiv:math.QA/0605172], we relate the construction of the quotient difference operator to the $(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$-duality of the trigonometric Gaudin Hamiltonians and trigonometric dynamical Hamiltonians acting on the space of polynomials in $kn$ anticommuting variables.