Comparison results for certain periods of cusp forms on GL(2n) over a totally real number field

This article grew out of my talk in "The Legacy of Srinivasa Ramanujan" conference where I spoke about some techniques to prove algebraicity results for the special values of symmetric cube L-functions attached to the Ramanujan \Delta-function. If one wishes to compare these different techniques, then one needs to compare various automorphic periods attached to the symmetric cube transfer of \Delta. Motivated by this problem, in this article we provide comparison results for Whittaker-Betti periods, Shalika-Betti periods and certain relative periods attached to a given cohomological cuspidal automorphic representation of GL(2n) over a totally real number field.