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Explicit formulas for local factors of supercuspidal representations of $GL_n$ and their applications

Ye, Rongqing
http://orcid.org/0000-0003-0382-194X
http://rave.ohiolink.edu/etdc/view?acc_num=osu1558359289350258

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
We compute the local Rankin-Selberg factors and the local Jacquet-Shalika exterior square gamma factors of special classes of supercuspidal representations of $GL_n(F)$ for a $p$-adic local field $F$. For the local Rankin-Selberg gamma factor of a pair of level zero supercuspidal representations, we prove that it is equal to the Rankin-Selberg gamma factor of the pair of corresponding cuspidal representations over the residue field $\mathfrak{f}$ of $F$. Using this equality, we further show that such a gamma factor over $\mathfrak{f}$ can be written as a product of Gauss sums, which solves a conjecture by Nien and Zhang. For the Jacquet-Shalika exterior square gamma factors, we show the existence of exterior square gamma factors for irreducible cuspidal representations of $GL_n(\mathfrak{f})$ which have no non-zero Shalika vectors, and prove the equality between exterior square gamma factor of a level zero supercuspidal representation and that of its corresponding cuspidal representation. We show that the Jacquet-Shalika exterior square gamma factors match those from the Langlands-Shahidi method in the $GL_3(F)$ case. We also compute the local (twisted) Jacquet-Shalika exterior square gamma factors for simple supercuspidal representations, which we use to prove a local converse theorem for simple supercuspidal representations.
James Cogdell (Advisor)
Roman Holowinsky (Committee Member)
Wenzhi Luo (Committee Member)
155 p.
Mathematics
local gamma factors; level zero supercuspidal representation; simple supercuspidal representation; local converse theorem

Recommended Citations

Citations

  • Ye, R. (2019). Explicit formulas for local factors of supercuspidal representations of $GL_n$ and their applications [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1558359289350258

    APA Style (7th edition)

  • Ye, Rongqing. Explicit formulas for local factors of supercuspidal representations of $GL_n$ and their applications. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1558359289350258.

    MLA Style (8th edition)

  • Ye, Rongqing. "Explicit formulas for local factors of supercuspidal representations of $GL_n$ and their applications." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1558359289350258

    Chicago Manual of Style (17th edition)

osu1558359289350258
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© 2019, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.