Title: On some problems related to several integrals involving the exceptional group G_2

Abstrct: Integral representation is an important method to study L-functions and automorphic periods. Several integrals involving the smallest exceptional group G_2 were discovered to represent some interesting L-functions, which includes the standard L-function of G_2, adjoint L-function of GL_3, and the L-function of \widetile{SL}_2\times GL_2. There are many problems arising from those integral representations. In this talk, I will briefly report our recent work on some of these problems, including existence of local gamma factors for G_2, a converse theorem for finite G_2, and non-vanishing of certain periods on G_2. Certain parts of the results are joint work with Baiying Liu.