On the theta correspondence for (𝐺𝑆𝑝(4),𝐺𝑆𝑂(4,2)) and Shalika periods

Morimoto, Kazuki

doi:10.1090/S1088-4165-2014-00451-5

On the theta correspondence for $(\mathrm {GSp}(4), \mathrm {GSO}(4,2))$ and Shalika periods
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by Kazuki Morimoto

Represent. Theory 18 (2014), 28-87

DOI: https://doi.org/10.1090/S1088-4165-2014-00451-5

Published electronically: April 16, 2014

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Abstract:

We consider both local and global theta correspondences for $\mathrm {GSp}_4$ and $\mathrm {GSO}_{4,2}$. Because of the accidental isomorphism $\mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2}$, these correspondences give rise to those between $\mathrm {GSp}_4$ and $\mathrm {GU}_{2,2}$ for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on $\mathrm {GU}(2,2)$ by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.

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