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| Title: | Automorphic forms on certain affine symmetric spaces. |
| Authors: | Zhang, Lei |
| Keywords: | Automorphic forms
Distinguished tame supercuspidal representation
Gelfand pairs
Number theory
special value of L-function
Mathematics |
| Issue Date: | May-2011 |
| Abstract: | In this thesis, we consider automorphic periods associated to certain affine
symmetric spaces such as the symmetric pairs
In this thesis, we consider automorphic periods associated to certain affine
symmetric spaces such as the symmetric pairs
(Sp4n; ResK=kSp2n) and (GSp4n; ResK=kGSp2n); where k is a number field and K is an Etale algebra over k of dimension 2. We
consider the period integral of a cusp forms of Sp4n(Ak) against with an Eisenstein
series of the symmetric subgroup ResK=kSp2n. We expect to establish an identity
between this period integrals and the special value of the spin L-function of the
symplectic group.
In the local theory, using Aizenbud and Gourevitch's generalized Harish-Chandra
method and traditional methods, i.e. the Gelfand-Kahzdan theorem, we can
prove that these symmetric pairs are Gelfand pairs when Kv is a quadratic extension
field over kv for any n, or Kv is isomorphic to kv x kv for n <_ 2. Since
(U(J2n; kv(
p
)); Sp2n(kv)) is a descendant of (Sp4n(kv); Sp2n(kv) Sp2n(kv)), we
prove that it is a Gelfand pair for both archimedean and non-archimedean fields.
According to the Yu' construction in [76] of irreducible tame supercuspidal representations,
we give a parametrization of the distinguished tame supercuspidal
representation of symplectic groups in this thesis. Applying the dimension formula
of the space HomH(; 1) given by Hakim and Murnaghan [28], we prove that
if (G;H) is the symmetric pair (U(J2n;Kv); Sp2n(kv)) there is no H-distinguished
tame supercuspidal representation, where Kv is a quadratic extension over kv. In
addition, for the symmetric pair (Sp4n(kv); Sp2n(Kv)), we give the sufficient and
necessary conditions of generic cuspidal data such that the corresponding tame
supercuspidal representations are H-distinguished. Note that our case is the first
case worked out with none of G and H being the general linear groups. Furthermore,
motived by a sub-question, we also give an example for the distinguished representations of finite groups of Lie Type in a low rank case. In particular,
we show that 10 is the unique SL2(Fq2)-distinguished cuspidal representation of
Sp4(Fq).
Note: See PDF abstract for the correct interpretation of the mathematical symbols |
| Description: | University of Minnesota Ph.D. dissertation. June 2011. Major: Mathematics. Advisor: Prof. Dr. Dihua Jiang. 1 computer file (PDF); ii, 131 pages. |
| Permanent URL: | http://purl.umn.edu/109867 |
| Appears in Collections: | Dissertations
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Files in This Item:
| File |
Description |
Size | Format |
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| Zhang_umn_0130E_12024.pdf | | 917Kb | PDF | View/Open |
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