On the SL(2) period integral

Anandavardhanan, U. K. ; Prasad, Dipendra (2006) On the SL(2) period integral American Journal of Mathematics, 128 (6). pp. 1429-1453. ISSN 0002-9327

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Official URL: http://muse.jhu.edu/journals/ajm/summary/v128/128....

Related URL: http://dx.doi.org/10.1353/ajm.2006.0000


Let E/F be a quadratic extension of number fields. For a cuspidal representation π of SL2(AE), we study in this paper the integral of functions in π on SL2(F)\SL2(AF). We characterize the nonvanishing of these integrals, called period integrals, in terms of π having a Whittaker model with respect to characters of E\AE which are trivial on AF. 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 SL2(AE) whose period integral vanishes identically while each local constituent admits an SL2-invariant linear functional. Finally, we construct an automorphic representation π on SL2(AE) which is abstractly SL2(AF) distinguished but for which none of the elements in the global L-packet determined by it is distinguished by SL2(AF).

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