差分算子与三角Gaudin算子和动力学Hamilton算子的对偶性
Difference Operators and Duality for Trigonometric Gaudin and Dynamical Hamiltonians
摘要：
本文研究了[Tarasov V.，Uvarov F.，Lett. 110（2020），3375-3400，arXiv：1907.02117]。从拟指数空间$W=\langle α_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$出发，考虑商差算子$\check{S}_{W}$的形式共轭$\check{S}^{\dagger}_{W}$满足$\widehat{S} =\check{S}_{W}S_{W}$ .这里，$S_{W}$是使$W$零化的$\dim W$阶的线性差分算子，而$\widehat{S}$是具有仅取决于$α_{i}$和$°p_{ij}(x)$的常系数的线性差分算子。构造了一个由$\check{S}^{\dagger}_{W}$零化的$\operatorname{ord} \check{S}^{\dagger}_{W}$维拟指数空间，并刻画了它的基和离散指数.我们还考虑了一个类似的建设与准多项式，这是线性组合的形式$x^{z}q(x)$，其中$z\in\mathbb C$和$q(x)$是一个多项式的函数的空间微分算子。将我们的结果与[Mukhin E.，塔拉索夫五世，瓦琴科·A Adv.Math.218（2008），216-265，arXiv：math.QA/0605172]，我们将商差算子的构造与作用在$kn$反对易变量中的多项式空间上的三角Gaudin哈密顿算子和三角动力学哈密顿算子的$(\mathfrak{gl}_{k},\mathfrak{gl}_{n})$ -对偶相关联。
英文摘要：
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 $W=\langle α_{i}^{x}p_{ij}(x),\, i=1,\dots, n,\, j=1,\dots, n_{i}\rangle$, where $α_{i}\in{\mathbb C}^{*}$ 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 $α_{i}$ and $°p_{ij}(x)$ 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.
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