Partition functions of $p$-forms from Harish-Chandra characters

We show that the determinant of the co-exact $p$-form on spheres and anti-deSitter spaces can be written as an integral transform of bulk and edge Harish-Chandra characters. The edge character of a co-exact $p$-form contains characters of anti-symmetric tensors of rank lower to $p$ all the way to the zero-form. Using this result we evaluate the partition function of $p$-forms and demonstrate that they obey known properties under Hodge duality. We show that partition function of conformal forms in even $d+1$ dimensions, on hyperbolic cylinders can be written as integral transforms involving only the bulk characters. This supports earlier observations that entanglement entropy evaluated using partition functions on hyperbolic cylinders do not contain contributions from the edge modes. For conformal coupled scalars we demonstrate that the character integral representation of the free energy on hyperbolic cylinders and branched spheres coincide. Finally we propose a character integral representation for the partition function of $p$-forms on branched spheres.