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研究生： 劉玉蓮
Liu, Yu-Lien
論文名稱： O 2(F), Sp2(F)在有限體上的Theta對應
The Theta Correspondence of O2 (F), Sp2(F)
指導教授： 潘戍衍
Pan, S. Y.
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： 2005
畢業學年度： 93
語文別： 英文
論文頁數： 55
中文關鍵詞： 在有限體上的Theta對應
外文關鍵詞： Theta correspondence
相關次數： 點閱：87  下載：3 
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    • 在這篇論文中,考慮一個特徵數不是2 的一個有限體。我們個別地
    分解正交群和辛群的表現，然後我們得到在正交群和辛群的THETA 對應。

    In this thesis, F is a finite field whose characteristic is not two. We
    decompose the Weil representation of O2(F) and Sp2(F) respectively.
    And then we get the theta correspondence between the reductive dual
    pairs (O1(F),Sp2(F))in Sp2(F) and (O2(F),Sp2 (F)), (O1,1(F),Sp2(F)) in Sp4 (F).

    1 Introduction 3 1.1 Representation of a finite group . . . . . . . . . . . . . . . . . 3 1.2 Finite classical groups . . . . . . . . . . . . . . . . . . 7 2 Weil Representations and Reductive Dual Pairs 10 2.1 The Heisenberg group . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The Weil representation of Sp2n(W) . . . . . . . . 1 2.3 Reductive dual pair . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Theta correspondence . . . . . . . . . . . . . . . . . . . . . . . 13 3 Theta correspondence of (O1(F); Sp2(F)) 14 3.1 The irreducible representation of GL2(F) . . . . . . . . . . . . 13 3.2 The irreducible representation of SL2(F) . . . . . . . . . . . . 21 3.3 The irreducible representation of O1(F) . . . . . . . . . . . . 25 3.4 The Weil representation of Sp2(F) . . . . . . . . . . . . . . . . 26 3.5 The theta correspondence of (O1(F); Sp2(F)) . . . . . . . . . . 28 4 Theta correspondence of (O1;1(F); Sp2(F)) and (O2(F); Sp2(F)) 31 4.1 The irreducible representations of O1;1(F) and O2(F) . . . . . 31 4.2 The Weil representation of Sp4(F) . . . . . . . . . . . . . . . . 33 4.3 Theta correspondence of (O1;1(F); Sp2(F)) . . . . . . . . . . . 36 4.4 Theta correspondence of (O2(F); Sp2(F)) . . . . . . . . . . . . 45

    [1] W. Fulton and J. Harris, Representation Theory Springer-Verlag, New
    York, 1991.
    [2] Paul G´erardin, Weil representations associated to finite fields, J. Algebra
    46 (1977), 54-101.
    [3] J. E. Humphreys, Representations of SL(2; p), Amer. Math. Monthly 82
    (1975), 21-39.
    [4] Chi-Wei Hong, The Theta Correspondence of (U1;U2) over a Finite
    Field, Master Thesis, National Cheng Kung University, 2003.
    [5] Feng-Chu Kau, The Theta Correspondence of (GL1;GL2) over a Finite
    Field, Master Thesis, National Cheng Kung University, 2003.
    [6] 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.
    [7] Brooks Roberts, lecture note, University of Maryland, College Park,
    1994.
    [8] Jean-Pierre Serre, Linear representations of finite groups (translated by
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    54
    [9] B. Srinivasan, The characters of the finite symplectic group Sp(4,q),
    Trans. Amer. Math. Soc. 131 (1968), 488-525.
    [10] Steven H. Weintraub, Representation Theory of Finite Group: Algebra
    and Arithmetic, Graduate Studies in Mathematics, 2003.

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