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研究生：	蔡誌軒 Tsai, Chih-Hsuan
論文名稱：	有限域上的線性群 (GL(4)) 中的 Shalika 模型 Shalika models of GL(4) over a finite field
指導教授：	粘珠鳳 Nien, Chufeng
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系應用數學碩博士班 Department of Mathematics
論文出版年：	2018
畢業學年度：	106
語文別：	英文
論文頁數：	27
中文關鍵詞：	Shalika model
外文關鍵詞：	Shalika model
相關次數：	點閱：65 下載：18
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這篇論文旨在研究如何去辨別維度4 的線性群的不可約群表示是否具有Shalika 模型，我們將不可分群表示分為兩部分：cuspidal 群表示及noncuspidal 群表示。
對於cuspidal 群表示的部分，我們使用了印度數學家Dipendra Prasad 在2000 年發表的一篇文章，在這篇文章的結果下，我們可以很容易去看出一個cusipidal 群表示是否具有Shalika models。
對於noncuspidal 群表示的部分，由cuspidal 群表示的定義，我們可以得知此群表示為一個parabolic induction 的子表示，所以我們考慮所有在線性群中的parabolic induction。分析一個parabolic induction 的過程使用到了有限群著名的定理Mackey’s 定理，在此定理的作用下我們可以得到在那些條件下，一個parabolic induction 會具有Shalika models。並依這些條件我們整理出了一些結論。

The goal of this paper is to determine whether a irreducible representation of GL(4) over a finite field admits a shalika model. We analysis it for two parts:cuspidal representations and noncuspidal representations.
For cuspidal representations, we use a result published from mathematician Dipendra Prasad at 2000. From the result, it is easy to determine whether a cuspidal representation has a Shalika model.
For noncuspidal representations, by the definition of cuspidal, we see a noncuspidal representation as a subrepresentation of an parabolic induction from some parabolic induction subgroup of GL(4). We consider all parabolic inductions of GL(4) and use a well-known theorem Mackey’s theorem to determined whether a parabolic induction has a Shalika model. From Mackey’s theorem, we get some condition for a parabolic induction has a Shalika model and use it to get some result.

Introduction..........................................1
Known Fact............................................1
1 Parabolic Induction.................................1
2 Shalika model.......................................2
3 Mackey’s Theorem....................................2 
4 cuspidal representation.............................3
5 Intertwining operators for induced representations..3
6 Green’s Construction................................4
7 Analysis for cuspidal representation................5
The Shalika model in GL(4)............................5
1 Subrepresentations of parabolic induction from  partition (3,1).....................6
1.1 (id)..............................................7
1.2 (14)..............................................7
1.3 (24).............................................8
1.4 (34).............................................8
1.5 Summary..........................................9
2 Subrepresentations of parabolic induction from partition (2,2).....................9
2.1 (id).............................................9
2.2 (13).............................................10
2.3 (24).............................................10
2.4 (14).............................................10
2.5 (23).............................................11
2.6 (13)(24).........................................11
2.7 Summary..........................................12
3 Subrepresentations of parabolic induction from partition (2,1,1).................12
3.1 (id).............................................12
3.2 (34).............................................13
3.3 (13).............................................13
3.4 (243)............................................13
3.5 (14).............................................13
3.6 (234)............................................14
3.7 (143)............................................14
3.8 (23).............................................15
3.9 (134)............................................15
3.10 (24)............................................15
3.11 (13)(24)........................................15
3.12 (14)(23)........................................16
3.13 Summary.........................................16
4 Subrepresentations of parabolic induction from partition (1,1,1,1).............16
4.1 (12)(34)........................................16
4.2 (id)..............................................17
4.3 (12)..............................................17
4.4 (34)..............................................17
4.5 (13)..............................................18
4.6 (1432)...........................................18
4.7 (14)..............................................19
4.8 (1342)...........................................19
4.9 (1243)...........................................20
4.10 (23).............................................20
4.11 (1234).........................................20
4.12 (24).............................................20
4.13 (123)...........................................20
4.14 (243)...........................................21
4.15 (124)...........................................21
4.16 (234)...........................................21
4.17 (134)...........................................21
4.18 (142)...........................................22
4.19 (143)...........................................22
4.20 (132)...........................................22
4.21 (1423).........................................22
4.22 (1324).........................................23
4.23 (13)(24).......................................23
4.24 (14)(23).......................................23
4.25 Summary.........................................24
5 cuspidal case......................................24
Conclusion...........................................24
References...........................................26

                                    

[1] Chufeng Nien: Uniqueness of Shalika Models. Canadian Journal of Mathematics, V61, No6, 2009, 1325-1340
[2] Chufeng Nien: n 1 local gamma factors and Gaussian sums. Finite Fields and Their Applications, Volume 46, July 2017, Pages 255–270
[3] Green, J. A.: The characters of the finite general linear groups. Trans. Amer. Math. Soc. 80 (1955), 402-447.
[4] Dipendra Prasad: The Space of Degenerate Whittaker Models for General Linear Groups over a Finite FieldIMRN, vol. 11, 579-595 (2000)
[5] Daniel Bump: Notes on representations of GLr over a finite field.
[6] Roditty, Edva-Aida: On gamma factors and Bessel functions for representations of general linear groups over finite field. M.s.c. Thesis (2010), Tel-Aviv University.

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