Journal of the Ramanujan Mathematical Society
Volume 32, Issue 3, September 2017 pp. 259–297.
Shintani Functions on SL(2,C) and Heun's
Differential Equations
Authors:
Kohta Gejima
Author institution:Department of Mathematics, Graduate School
of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Summary: 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.
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