58 research outputs found
Distinguished principal series representations for GLn over a p-adic field
In the following article, we give a description of the distingushed
irreducible principal series representations of the general linear group over a
p-adic field in terms of inducing datum. This provides a counter-example to a
conjecture of Jacquet about distinction (Conjecture 1 in U.K Anandavardhanan,
"Distinguished non-Archimedean representations ", Proc. Hyderabad Conference on
Algebra and Number Theory, 2005, 183-192)
On the local Bump-Friedberg L-function
Let be a -adic field. If be an irreducible representation of
, Bump and Friedberg associated to an Euler fator
in \cite{BF}, that should be equal to
, where is the
Langlands' parameter of . The main result of this paper is to show that
this equality is true when , for in \C. To prove
this, we classify in terms of distinguished discrete series, generic
representations of which are -distinguished by the Levi
subgroup , for
, where is a character
of of real part between -1/2 and 1/2. We then adapt the technique of
\cite{CP} to reduce the proof of the equality to the case of discrete series.
The equality for discrete series is a consequence of the relation between
linear periods and Shalika periods for discrete series, and the main result of
\cite{KR}.Comment: We fixed a problem in the proof of Theorem 3.1, at the cost of making
the assumption that belongs to in the statement. This
does not affect any other resul
Conjectures about distinction and Asai -functions of generic representations of general linear groups over local fields
Let be a quadratic extension of p-adic fields. The
Bernstein-Zelevinsky's classification asserts that generic representations are
parabolically induced from quasi-square-integrable representations. We show,
following a method developed by Cogdell and Piatetski-Shapiro, that the
equality of the Rankin-Selberg type Asai -function of generic
representations of and of the Asai -function of the Langlands
parameter, is equivalent to the truth of a conjecture about classification of
distinguished generic representations in terms of the inducing
quasi-square-integrable representations. As the conjecture is true for
principal series representations, this gives the expression of the Asai
L-function of such representations
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