我们将主要考虑下面两类问题。第一类问题是给出量子仿射代数的Borel子代数的q特征之间的关系式。对于型A,B,C,D的无扭量子仿射代数，利用ODE/IM对应所给出的联络系数之间的关系式，我们给出了其对应的Borel子代数的某一类表示的q特征之间的猜想多项式关系。我们想把这些结果推广到无扭的其它型的量子仿射代数和有扭的量子仿射代数这两种情形。第二类问题是对于一般的A型单李代数,给出其在任意点处有任意Poincaré阶的合流型KZ方程。

We will mainly consider the following two problems. First we will propose conjectural polynomial relations among the q-characters of the Borel subalgebra. For the non-twisted quantum affine algebra of type A,B,C,D, guided by the ODE/IM correspondece in quantum integrable models, we proposed conjectural polynomial relations among the q-characters of certain representations of the Borel subalgebra. We would like to extend the results mentioned above to the rest cases of non-twisted quantum affine algebras and the cases of twisted quantum affine algebras. Second for the simple Lie algebra of type A, we would like to construct confluent KZ equations with arbitary Poincaré rank at arbitray singularities.