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TitleThe residual Eisenstein cohomology of $mathrmSp_4$ over a totally real number field.
Author(s) Neven Grbac, Harald Grobner
TypeArticle in Journal
AbstractLet
G
=
Sp
4
/k
be the
k
-split symplectic group of
k
-rank 2, where
k
is a totally real number field. In this paper we compute the Eisenstein co-
homology of
G
with respect to any finite–dimensional, irreducible,
k
-rational
representation
E
of
G
∞
=
R
k/
Q
G
(
R
), where
R
k/
Q
denotes the restriction of
scalars from
k
to
Q
. This approach is based on the work of Schwermer regard-
ing the Eisenstein cohomology for
Sp
4
/
Q
, Kim’s description of the residual
spectrum of
Sp
4
, and the Franke filtration of the space of automorphic forms.
In fact, taking the representation theo
retic point of view, we write, for the
group
G
, the Franke filtration with respect to the cuspidal support, and give
a precise description of the filtration quotients in terms of induced representa-
tions. This is then used as a prerequis
ite for the explicit computation of the
Eisenstein cohomology. The special focus is on the residual Eisenstein coho-
mology. Under a certain compatibility c
ondition for the coefficient system
E
and the cuspidal support, we prove the existence of non–trivial residual Eisen-
stein cohomology classes, which are not square–integrable, that is, represented
by a non–square–integrable residue of an Eisenstein serie
ISSN0002-9947; 1088-6850/e
File
URL http://www.ams.org/journals/tran/2013-365-10/S0002-9947-2013-05796-0/home.html
LanguageEnglish
JournalTrans. Am. Math. Soc.
Volume365
Number10
Pages5199--5235
PublisherAmerican Mathematical Society (AMS), Providence, RI
Year2013
Edition0
Translation No
Refereed No
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