The study of arithmetic invariants associated to Galois representations has often relied on the construction of a special family of elements in their Galois cohomology groups.

For instance, it has been a crucial ingredient in the work of Kato in the proof of special cases of the conjecture of Birch and Swinnerton-Dyer and the Iwasawa main conjecture for modular forms.

In this talk, I will describe how to construct elements in the Iwasawa cohomology of Galois representations associated to a product of two cohomological cuspidal automorphic representations of the similitude symplectic group GSp(2n) and give some directions on the range of applications that this construction has. Emphasis will be put on the case where our automorphic representations come from smaller groups.