Shintani Functions on Sl(2,C) and Heuna Differential Equations

Abstract

We study Shintani functions attached to non-unitary principalseries representations of SL(2, C). Let 7r[v.inJ be a non-unitary principalseries representation with (1ml + })-dimensional minimal SU(2)-type. Byusing Zuckerman tensoring, we give an inductive formula which givesa relation between the Shintani functions attached to 7r[v,ml and thoseattached to 7r[v+l,m+IJ · From this, we obtain explicit formulas of Shintanifunctions attached to 7r[v.mJ(lml = 1, 2). For Jml = 2, we observe that thesystem of differential equations satisfied by Shintai functions gives us apinteresting example of Heun's differential equations.