  We construct a certain class of Arakawa--Kaneko zeta-functions associated
with $GL_2(\mathbb{C})$, which includes the ordinary Arakawa--Kaneko
zeta-function. We also define poly-Bernoulli polynomials associated with
$GL_2(\mathbb{C})$ which appear in their special values of these
zeta-functions. We prove some functional relations for these zeta-functions,
which are regarded as interpolation formulas of various relations among
poly-Bernoulli numbers. Considering their special values, we prove difference
relations and duality relations for poly-Bernoulli polynomials associated with
$GL_2(\mathbb{C})$.
