Varagnolo-Vasserot and Rouquier proved that, in a symmetric generalized Cartan matrix case, the simple modules over the quiver Hecke algebra with a special parameter correspond to the upper global basis. In this note we show that the simple modules over the quiver Hecke algebras with a generic parameter also correspond to the upper global basis in a symmetric generalized Cartan matrix case.

In this paper we give an integral representation for a Whittaker function of an non holomorphic Eisenstein series which is a non holomorphic Sigel modular form of degree 2. Our integral representation is very useful to the theory of the theta lifting of automorphic forms.