标题： Ruijsenaars波函数作为模群矩阵系数 显示英文标题

标题： Ruijsenaars wavefunctions as modular group matrix coefficients

摘要： 我们给出了Hallnäs--Ruijsenaars在2粒子双曲Ruijsenaars系统中的本征函数作为矩阵系数，这些矩阵系数对应于作用在穿孔环面的$GL(2)$量子Teichmüller理论的希尔伯特空间上的4阶元素$S\in SL(2,\mathbb{Z})$。 然后，$GL(2)$麦克唐纳多项式作为这些矩阵系数的解析延拓的特殊值获得。 证明中使用的主要工具是带有框架的$GL(2)$局部系统在穿孔环面上的模空间上的簇结构，以及$GL(2)$球形DAHA到相应簇泊松流形的量化坐标环的$SL(2,\mathbb{Z})$-等变嵌入。 显示英文摘要

摘要： We give a description of the Halln\"as--Ruijsenaars eigenfunctions of the 2-particle hyperbolic Ruijsenaars system as matrix coefficients for the order 4 element $S\in SL(2,\mathbb{Z})$ acting on the Hilbert space of $GL(2)$ quantum Teichm\"uller theory on the punctured torus. The $GL(2)$ Macdonald polynomials are then obtained as special values of the analytic continuation of these matrix coefficients. The main tool used in the proof is the cluster structure on the moduli space of framed $GL(2)$-local systems on the punctured torus, and an $SL(2,\mathbb{Z})$-equivariant embedding of the $GL(2)$ spherical DAHA into the quantized coordinate ring of the corresponding cluster Poisson variety.