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研究生： 洪志瑋
Hong, Jhih-Wei
論文名稱： (U1, U2)在有限體上的Theta對應
The Theta Correspondence of (U1, U2) over a Finite Field
指導教授： 潘戍衍
Pan, Shu-Yen
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： 2003
畢業學年度： 91
語文別： 英文
論文頁數： 34
外文關鍵詞： Weil representation, Theta correspondence
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    • In this thesis, we study the theta correspondence of the reductive dual pairs (U_1(F),U_1(F)) and (U_1(F),U_2(F)) over a finite field F whose characteristic is not two by decomposing the Weil representations of the symplectic Sp_2(F) and Sp_4(F) respectively.

    1 Introduction 2 1.1 Linear representations of finite groups 2 1.2 Notations 6 2 Weil Representations and Reductive Dual Pairs 7 2.1 Weil representation 7 2.2 The dimension of the Weil representation of Spn(F) 8 2.3 Reductive dual pair 10 2.4 Theta correspondence 11 3 Theta Correspondence of (U1(F),U1(F)) 13 3.1 Irreducible representations of SL2(F) 13 3.2 The Weil representation of Sp2(F) 16 3.3 Theta correspondence of (U1(F),U1(F)) 18 4 Theta Correspondence of (U1(F),U2(F)) 23 4.1 The Weil representation of Sp4(F) 23 4.2 Representations of U2(F) 25 4.3 The restriction of representations of U2(F) to U1(F) 27 4.4 Theta Correspondence of (U1(F),U2(F)) 31

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    [3] W. Fulton and J. Harris, Representation Theory,
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    [4] Paul Gerardin, Weil representations associated to finite fields,
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    [6] Brooks Roberts, Lecture notes, University of Maryland, College Park, 1994.

    [7] Jean-Pierre Serre, Linear representations of finite groups (translated by L. Scoot),
    Springer-Verlag, New York, 1977.

    [8] D. Prasad, Weil representation, Howe duality, and the theta correspondence, in Theta functions, from the classical to the modern, Amer. Math. Soc., Providence, 1993, 105--127.

    [9] B. Srinivasan, The characters of the finite symplectic group Sp(4,q),
    Trans. Amer. Math. Soc., 131 (1968), 488--525.

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