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University of Oklahoma

1.
VerNooy, Colin.
K-types and Invariants for the Representations of *GSp*(*4*,R).

Degree: PhD, 2019, University of Oklahoma

URL: http://hdl.handle.net/11244/321122

Automorphic representations of the adelic group GSp (4 ,A Q ) are of importance
in their relation to Siegel modular forms of degree 2. Given an automorphic
representation π of GSp (4 ,A Q ), it decomposes into a product of admissible
representations at each place. In the non-archimedean case, many useful results
have been produced by Roberts and Schmidt. Here we find some invariants for
the case of GSp (4 ,R ), including the K -type structure, the L- and epsilon -factors, and
the Gelfand-Kirillov dimension for all irreducible admissible representations.
*Advisors/Committee Members: Schmidt, Ralf (advisor), Heyck, Hunter (committee member), Przebinda, Tomasz (committee member), Kujawa, Jonathan (committee member), Roche, Alan (committee member).*

Subjects/Keywords: representation theory; GSp(4); sympleptic group

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

VerNooy, C. (2019). K-types and Invariants for the Representations of GSp(4,R). (Doctoral Dissertation). University of Oklahoma. Retrieved from http://hdl.handle.net/11244/321122

Chicago Manual of Style (16^{th} Edition):

VerNooy, Colin. “K-types and Invariants for the Representations of GSp(4,R).” 2019. Doctoral Dissertation, University of Oklahoma. Accessed January 28, 2021. http://hdl.handle.net/11244/321122.

MLA Handbook (7^{th} Edition):

VerNooy, Colin. “K-types and Invariants for the Representations of GSp(4,R).” 2019. Web. 28 Jan 2021.

Vancouver:

VerNooy C. K-types and Invariants for the Representations of GSp(4,R). [Internet] [Doctoral dissertation]. University of Oklahoma; 2019. [cited 2021 Jan 28]. Available from: http://hdl.handle.net/11244/321122.

Council of Science Editors:

VerNooy C. K-types and Invariants for the Representations of GSp(4,R). [Doctoral Dissertation]. University of Oklahoma; 2019. Available from: http://hdl.handle.net/11244/321122

The Ohio State University

2.
Chan, Ping Shun.
Invariant representations of *GSp*(2).

Degree: PhD, Mathematics, 2005, The Ohio State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1132765381

Let F be a number field or a p-adic field. We
introduce in Chapter 2 of this work two reductive rank one
F-groups, H1, H2, which are twisted endoscopic groups of GSp(2)
with respect to a fixed quadratic character varepsilon of the idele
class group of F if F is global, F* if F is local. If F is global,
Langlands functoriality predicts that there exists a canonical
lifting of the automorphic representations of H1, H2 to those of
GSp(2). In Chapter 4, we establish this lifting in terms of the
Satake parameters which parametrize the automorphic
representations. By means of this lifting we provide a
classification of the discrete spectrum automorphic representations
of GSp(2) which are invariant under tensor product with varepsilon.
The techniques through which we arrive at our results are inspired
by those of Kazhdan’s in [K]. In particular, they involve comparing
the spectral sides of the trace formulas for the groups under
consideration. We make use of the twisted extension of Arthur’s
trace formula, and Kottwitz-Shelstad’s stabilization of the
elliptic component of the geometric side of the twisted trace
formula. If F is local, in Chapter 5 we provide a classification of
the irreducible admissible representations of GSp(2, F) which are
invariant under tensor product with the quadratic character
varepsilon of F*. Here, our techniques are also directly inspired
by [K]. More precisely, we use the global results from Chapter 4 to
express the twisted characters of these invariant representations
in terms of the characters of the admissible representations of
Hi(F) (i = 1, 2). These (twisted) character identities provide
candidates for the liftings predicted by the local component of the
conjectural Langlands functoriality. The proofs rely on
Sally-Tadic’s classification of the irreducible admissible
representations of GSp(2, F), and Flicker’s results on the lifting
from PGSp(2) to PGL(4).
*Advisors/Committee Members: Flicker, Yuval (Advisor).*

Subjects/Keywords: Mathematics; Automorphic representations; Langlands Functoriality; Lifting; Harmonic analysis on p-adic groups; Symplectic group of similitudes; GSp(2); { m GSp}(2); GSp(4); { m GSp}(4)

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Chan, P. S. (2005). Invariant representations of GSp(2). (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1132765381

Chicago Manual of Style (16^{th} Edition):

Chan, Ping Shun. “Invariant representations of GSp(2).” 2005. Doctoral Dissertation, The Ohio State University. Accessed January 28, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1132765381.

MLA Handbook (7^{th} Edition):

Chan, Ping Shun. “Invariant representations of GSp(2).” 2005. Web. 28 Jan 2021.

Vancouver:

Chan PS. Invariant representations of GSp(2). [Internet] [Doctoral dissertation]. The Ohio State University; 2005. [cited 2021 Jan 28]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1132765381.

Council of Science Editors:

Chan PS. Invariant representations of GSp(2). [Doctoral Dissertation]. The Ohio State University; 2005. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1132765381

The Ohio State University

3.
Young, Justin N.
The twisted tensor L-function of *GSp*(*4*).

Degree: PhD, Mathematics, 2009, The Ohio State University

URL: http://rave.ohiolink.edu/etdc/view?acc_num=osu1244049123

We construct an integral representation for the
twisted tensor L-function of a globally generic cuspidal
automorphic representation of GSp(4) over a number field. We prove
that the integral is Eulerian, i.e., has an infinite product
expansion. We compute the unramified integrals and show by way of a
branching result (from GL(4) to Sp(4)) that these integrals
calculate the correct local L-factor. This gives a new proof of the
analogous identity in D. Jiang's thesis. Finally, we show all the
local integrals are absolutely convergent in a right half-plane and
that they are non-vanishing for appropriate choice of data. We
close with some remarks about poles of our global integral and
possible future applications to period integrals and quadratic base
change for GSp(4).
*Advisors/Committee Members: Rallis, Stephen (Advisor).*

Subjects/Keywords: Mathematics; automorphic forms; L-functions; GSp(4); integral representations; Rankin-Selberg

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Young, J. N. (2009). The twisted tensor L-function of GSp(4). (Doctoral Dissertation). The Ohio State University. Retrieved from http://rave.ohiolink.edu/etdc/view?acc_num=osu1244049123

Chicago Manual of Style (16^{th} Edition):

Young, Justin N. “The twisted tensor L-function of GSp(4).” 2009. Doctoral Dissertation, The Ohio State University. Accessed January 28, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1244049123.

MLA Handbook (7^{th} Edition):

Young, Justin N. “The twisted tensor L-function of GSp(4).” 2009. Web. 28 Jan 2021.

Vancouver:

Young JN. The twisted tensor L-function of GSp(4). [Internet] [Doctoral dissertation]. The Ohio State University; 2009. [cited 2021 Jan 28]. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1244049123.

Council of Science Editors:

Young JN. The twisted tensor L-function of GSp(4). [Doctoral Dissertation]. The Ohio State University; 2009. Available from: http://rave.ohiolink.edu/etdc/view?acc_num=osu1244049123