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MR2592088 (Review) 11F70 (11F85 22E50)
Matringe, Nadir (4-EANG)
Distinguished representations and exceptional poles of the Asai-L-function. (English
summary)
Manuscripta Math. 131 (2010), no. 3-4, 415426.
Let K/F be a quadratic extension of p-adic 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 L-function has a pole at zero. It is easy to see that the
Asai L-function of a non-square 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 L-function by making use
of the notion of an exceptional pole introduced by Cogdell and Piatetski-Shapiro.
If the L-function has a pole of order d at zero, then the zeta integral which is used to define the
L-function has a Laurent expansion
B(W, )

  

Source: Anandavardhanan, U. K. - Department of Mathematics, Indian Institute of Technology Bombay

 

Collections: Mathematics