Shintani Functions on SL(2,C) and Heun’s Differential Equations

We study Shintani functions attached to non-unitary principal series representations of SL(2,C). Let π_[ν,m] be a non-unitary principal series representation with (|m| + 1)-dimensional minimal SU(2)-type. By using Zuckerman tensoring, we give an inductive formula which gives a relation between the Shintani functions attached to π_[ν,m] and those attached to π_[ν+1,m+1]. From this, we obtain explicit formulas of Shintani functions attached to π_[ν,m](|m| = 1, 2). For |m| = 2, we observe that the system of differential equations satisfied by Shintai functions gives us an interesting example of Heun’s differential equations.