  The character integral representation of one loop partition functions is
useful to establish the relation between partition functions of conformal
fields on Weyl equivalent spaces. The Euclidean space $S^a\times AdS_b$ can be
mapped to $S^{a+b}$ provided $S^a$ and $AdS_b$ are of the same radius. As an
example, to begin with, we show that the partition function in the character
integral representation of conformally coupled free scalars and fermions are
identical on $S^a\times AdS_b$ and $S^{a+b}$. We then demonstrate that the
partition function of higher derivative conformal scalars and fermions are also
the same on hyperbolic cylinders and branched spheres. The partition function
of the four-derivative conformal vector gauge field on the branched sphere in
$d=6$ dimension can be expressed as an integral over `naive' bulk and `naive'
edge characters. However, the partition function of the conformal vector gauge
field on $S^1_q\times AdS_5$ contains only the `naive' bulk part of the
partition function. This follows the same pattern which was observed for the
partition of conformal $p$-form fields on hyperbolic cylinders. We use the
partition function of higher derivative conformal fields on hyperbolic
cylinders to obtain a linear relationship between the Hofman-Maldacena
variables which enables us to show that these theories are non-unitary.
