19 research outputs found
On the SL(2) period integral
Let E/F be a quadratic extension of number fields. For a cuspidal
representation of SL(2,A_E), we study the non-vanishing of the period
integral on SL(2,F)\SL(2,A_F). We characterise the non-vanishing of the period
integral of in terms of being generic with respect to characters of
E\A_E which are trivial on A_F. We show that the period integral in general is
not a product of local invariant functionals, and find a necessary and
sufficient condition when it is. We exhibit cuspidal representations of
SL(2,A_E) whose period integral vanishes identically while each local
constituent admits an SL(2)-invariant linear functional. Finally, we construct
an automorphic representation on SL(2,A_E) which is abstractly SL(2,A_F)
distinguished but none of the elements in the global L-packet determined by
is distinguished by SL(2,A_F)
Distinguished representations, base change, and reducibility for unitary groups
We show the equality of the local Asai L-functions defined via the
Rankin-Selberg method and the Langlands-Shahidi method for a square integrable
representation of GL(n,E). As a consequence we characterise reducibility of
certain induced representations of U(n,n), and the image of the base change map
from U(n) to GL(n,E) in terms of GL(n,F)-distinguishedness.Comment: 13 page
Toric periods for a -adic quaternion algebra
Let be a compact group with two given subgroups and . Let be
an irreducible representation of such that its space of -invariant
vectors as well as the space of -invariant vectors are both one dimensional.
Let (resp. ) denote an -invariant (resp. -invariant) vector of
unit norm in a given -invariant inner product on
. We are interested in calculating the correlation coefficient
In this paper, we compute the
correlation coefficient of an irreducible representation of the multiplicative
group of the -adic quaternion algebra with respect to any two tori. In
particular, if is such an irreducible representation of odd minimal
conductor with non-trivial invariant vectors for two tori and , then its
root number is and is non-vanishing
precisely when
Distinction inside L-packets of SL(n)
If is a quadratic extension -adic fields, we first prove that the
-distinguished representations inside a distinguished unitary
L-packet of are precisely those admitting a degenerate
Whittaker model with respect to a degenerate character of . Then we
establish a global analogue of this result. For this, let be a quadratic
extension of number fields and let be an
-distinguished square integrable automorphic
representation of . Let be the unique
pair associated to , where is a cuspidal representation of
with . Using an unfolding argument, we
prove that an element of the L-packet of is distinguished with respect to
if and only if it has a degenerate Whittaker
model for a degenerate character of type of
which is trivial on , where is
the group of unipotent upper triangular matrices of . As a first
application, under the assumptions that splits at infinity and is
odd, we establish a local-global principle for
-distinction inside the L-packet of . As a
second application we construct examples of distinguished cuspidal automorphic
representations of such that the period
integral vanishes on some canonical copy of , and of everywhere locally
distinguished representations of such that their
L-packets do not contain any distinguished representation.Comment: Merged with withdrawn arXiv:1906.11560. We simplified some arguments
and removed an unnecessary Grunwald-Wang assumptio
Distinguished representations for SL(2)
Let E/F be a quadratic extension of p-adic fields. We compute the multiplicity of the space of SL2(F)-invariant linear forms on a representation of SL2(E). This multiplicity varies inside an L-packet similar in spirit to the multiplicity formula for automorphic representations due to Labesse and Langlands
A local global question in automorphic forms
In this paper, we consider the \SL(2) analogue of two well-known theorems
about period integrals of automorphic forms on \GL(2): one due to
Harder-Langlands-Rapoport, and the other due to Waldspurger.Comment: 28 page