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On the SL(2) Period Integral

U. K. Anandavardhanan and Dipendra Prasad
American Journal of Mathematics
Vol. 128, No. 6 (Dec., 2006), pp. 1429-1453
Stable URL: http://www.jstor.org/stable/40068040
Page Count: 25
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On the SL(2) Period Integral
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Abstract

Let E/F be a quadratic extension of number fields. For a cuspidal representation π of ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ , we study in this paper the integral of functions in π on ${\rm{SL}}_{\rm{2}} ({\rm{F}})\backslash {\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ . We characterize the nonvanishing of these integrals, called period integrals, in terms of π having a Whittaker model with respect to characters of ${\rm{E}}\backslash {\Bbb A}_{\rm{E}}$ which are trivial on ${\Bbb A}_{\rm{F}}$ . We show that the period integral in general is not a product of local invariant functionals, and find a necessary and sufficient condition when it is. We exhibit cuspidal representations of ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ whose period integral vanishes identically while each local constituent admits an ${\rm{SL}}_{\rm{2}}$ -invariant linear functional. Finally, we construct an automorphic representation π on ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{E}} )$ which is abstractly ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ distinguished but for which none of the elements in the global L-packet determined by it is distinguished by ${\rm{SL}}_{\rm{2}} ({\Bbb A}_{\rm{F}} )$ .

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