On the theta correspondence for \((\mathrm{GSp}(4), \mathrm{GSO}(4,2))\) and Shalika periods (Q2922910)

The paper studies theta correspondence between the similitude groups \(\mathrm{GSp}(4)\) and \(\mathrm{GSO}(4,2)\). It covers both the global theory and local theory. Globally, using the accidental isomorphism between \(\mathrm{GSO}(4,2)\) and \(\mathrm{GU}(2,2)\), the author shows that the non-vanishing of the Shalika period for a cuspidal representation of \(\mathrm{GU}(2,2)\) is equivalent to be in the image of the theta correspondence of a generic cuspidal representation of \(\mathrm{GSp}(4)\). Locally the paper proves a Howe duality in this setting, and also shows the analogue of global result: a relation between local representations of \(\mathrm{GSO}(4,2)\) with a Shalika functional and the genericity of its theta lift.