Gamma and sine functions for Lie groups and period integrals

We study gamma and sine functions attached to a finite dimensional linear representation of a Lie group including their multiple versions via zeta regularizations. These definitions allow us to understand several multiple gamma and sine functions known so far, such as Barnes' gamma functions, Shintani's sine functions, and their q-analogues, in a unified way. Further, we discuss period integrals for such gamma functions.