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MR2275907 (2008b:22014) 22E55 (11F70)
Anandavardhanan, U. K. (6IIT); Prasad, Dipendra (6TIFR)
On the SL(2) period integral. (English summary)
Amer. J. Math. 128 (2006), no. 6, 14291453.
The study of period integrals of automorphic forms is an important subject in the Langlands
program. More precisely, given a cuspidal representation on a reductive lgroup G containing
a subgroup H, one says that is Hdistinguished if the period integral over H defines a nonzero
linear functional on . When E is a quadratic extension of a number field F and (G, H) =
(GLn(E), GLn(F)), this question has been much studied. In this case, a cuspidal representation
is Hdistinguished iff is conjugate selfdual and its Asai Lfunction has a pole at s = 1. We
should mention that there is an analogous local question whose study often illuminates the global
question above. In the local setting, one would like to know the dimension of the space of H
invariant linear functionals on . If, for example, this dimension is at most 1 for all places, then
the global period integral admits a factorization into the product of the local functionals.
In the paper under review, the authors examine the question of distinguishedness for (G, H) =
(SL2(E), SL2(F)). The situation is quite delicate because of the nontriviality of Lpackets for
