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On Shalika periods and a theorem of Jacquet-Martin
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 132, Number 2, April 2010
- pp. 475-528
- 10.1353/ajm.0.0109
- Article
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Let $\pi$ be a cuspidal automorphic representation of $GL_4(\Bbb A)$ with
central character $\mu^2$. It is known that $\pi$ has Shalika period with
respect to $\mu$ if and only if the $L$-function $L^S(s, \pi,
\bigwedge^2\otimes\mu^{-1})$ has a pole at $s=1$. Jacquet and Martin
considered the analogous question for cuspidal representations $\pi_D$ of
the inner form $GL_2(D)(\Bbb A)$, and obtained a partial result via the
relative trace formula. In this paper, we provide a complete solution to
this problem via the method of theta correspondence, and give necessary
and sufficient conditions for the existence of Shalika period for $\pi_D$.
We also resolve the analogous question in the local setting.