The nonarchimedean theta correspondence for 𝐺𝑆𝑝(2) and 𝐺𝑂(4)

Roberts, Brooks

doi:10.1090/S0002-9947-99-02126-1

The nonarchimedean theta correspondence for $\mathrm {GSp}(2)$ and $\mathrm {GO}(4)$
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by Brooks Roberts

Trans. Amer. Math. Soc. 351 (1999), 781-811

DOI: https://doi.org/10.1090/S0002-9947-99-02126-1

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Abstract:

In this paper we consider the theta correspondence between the sets $\operatorname {Irr} (\operatorname {GSp} (2,k))$ and $\operatorname {Irr} (\operatorname {GO} (X))$ when $k$ is a nonarchimedean local field and $\dim _{k} X =4$. Our main theorem determines all the elements of $\operatorname {Irr} (\operatorname {GO} (X))$ that occur in the correspondence. The answer involves distinguished representations. As a corollary, we characterize all the elements of $\operatorname {Irr} (\operatorname {O} (X))$ that occur in the theta correspondence between $\operatorname {Irr} (\operatorname {Sp} (2,k))$ and $\operatorname {Irr} (\operatorname {O} (X))$. We also apply our main result to prove a case of a new conjecture of S.S. Kudla concerning the first occurrence of a representation in the theta correspondence.

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