On the existence of twisted Shalika periods: the Archimedean case

Let $\text{K}$ be an archimedean local field. We investigate the existence of the twisted Shalika functionals on irreducible admissible smooth representations of ${\text{GL}}_{2n}(\text{K})$ in terms of their L-parameters. As part of our proof, we establish a Hochschild-Serre spectral sequence for nilpotent normal subgroups and a Kunneth formula in the framework of Schwartz homology. We also prove the analogous result for twisted linear periods using theta correspondence. The existence of twisted Shalika functionals on representations of ${\text{GL}}_{2n}^{+}(\mathbb{R})$ is also studied, which is of independent interest.