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MR2592088 (Review) 11F70 (11F85 22E50)
Matringe, Nadir (4EANG)
Distinguished representations and exceptional poles of the AsaiLfunction. (English
summary)
Manuscripta Math. 131 (2010), no. 34, 415426.
Let K/F be a quadratic extension of padic fields. An irreducible smooth representation of
GLn(K) is said to be distinguished with respect to GLn(F) if it admits a GLn(F)invariant
nontrivial linear form. If the representation is square integrable, it is known that the representation
is distinguished if and only if its Asai Lfunction has a pole at zero. It is easy to see that the
Asai Lfunction of a nonsquare integrable representation can have a pole at zero without the
representation being distinguished. In the paper under review, the author characterizes distinction
for any irreducible generic representation in terms of poles of the Asai Lfunction by making use
of the notion of an exceptional pole introduced by Cogdell and PiatetskiShapiro.
If the Lfunction has a pole of order d at zero, then the zeta integral which is used to define the
Lfunction has a Laurent expansion
B(W, )
