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研究生:高鳳珠
研究生(外文):Fong-Chu Kao
論文名稱:(GL1,GL2)在有限體上的Theta對應
論文名稱(外文):The Theta Correspondence of (GL1, GL2) over a Finite Field
指導教授:潘戍衍
指導教授(外文):Shu-Yen Pan
學位類別:碩士
校院名稱:國立成功大學
系所名稱:數學系應用數學碩博士班
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2003
畢業學年度:91
語文別:英文
論文頁數:26
中文關鍵詞:群表現
外文關鍵詞:epresentationtheta correspondence
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Representation of a finite group in a vector space is a homomophism from the finite group into the vector space.
Clearly, it is a square matrice. Weil representation is a kind of special representation. By decomposing Weil representation of Sp2 and Sp4, we can find the theta correspondence of (GL1(F),GL1(F)) and (GL1(F),GL2(F)) over a finite field F. In the process, one can know the correspondence between GL1(F) and GL1(F) (resp. GL1(F) and GL2(F)) in Sp2 (resp. Sp4).
1. Introduction 2
1.1. Representations of a finite group 3
1.2. Notation 4
2. Reductive dual pair and Weil representation 6
2.1. Definition of Sp2n(F) 6
2.2. Reductive dual pair in Sp2mn(F) 7
2.3. Heisenberg group 8
2.4. Definition of the Weil representation 9
2.5. Theta correspondence 10
3. Theta correspondence of (GL1(F),GL1(F)) 11
3.1. Sp2(F) and SL2(F) 11
3.2. The representations of GL2(F) 12
3.3. The representations of SL2(F) 14
3.4. The connection between Weil representation varpi and the representation of Sp2(F) 15
3.5. Theta correspondence of (GL1(F),GL1(F)) 17
4. Theta correspondence of (GL1(F),GL2(F)) 22
4.1. GL1(F) in Sp4(F) 22
4.2. Theta correspondence of (GL1(F),GL2(F)) 24
References 26
W. Fulton and J. Harris, Representation theory,
Springer-Varlag, New York, 1991.
P. Gerardin, Weil representations associated to finite
fields, J. Algebra. 46 (1977), 54--101.
R. Howe, Invariant theory and duality for classical groups
over finite fields with applications to their singular representation
theory, preprint.
H-C. Li, A short course on linear representations of finite
groups, preprint.
S-Y. Pan, Depth preservation in local theta correspondence,
Duke Math. Journal. 113 (2002), 531--592.
S-Y. Pan, Local theta correspondence of depth zero
representations and theta dichotomy, J. Math. Soc. Japan.
54 (2002), 793--845.
D. Prasad, Weil representation, Howe duality, and the theta
correspondence, Centre de Recherches Mathematiques CRM Proceedings and
Lecture Notes. 1 (1993), 105--111.
B. Roberts, The Weil representation and dual pair, Lecture
notes in the 1994 university of maryland conference
on the theta correspondence, dual pairs and automorphic forms.
B. Srinivasan, The characters of the finite sympiectic
group Sp(4,q), Trans. Amer. Math. Soc. 131 (1968), 488--525.
J.-P. Serre, Linear representations of finite groups
(translated by L. Scott), Springer-Varlag, New York, 1977.
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