(Sp2,O2)與(Sp2,O3)的Weil表現｜國立清華大學博碩士論文庫

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研究生：	辜銘壕 Gu, Ming Hao
論文名稱：	(Sp2,O2)與(Sp2,O3)的Weil表現 The Weil representations of (Sp2,O2) and (Sp2,O3)
指導教授：	潘戍衍 Pan, Shu Yen
口試委員:	鄭志豪 Teh, Jyh Haur 楊一帆 Yang,Yi Fan
學位類別：	碩士 Master
系所名稱：	理學院 - 數學系 Department of Mathematics
論文出版年：	2016
畢業學年度：	104
語文別：	英文
論文頁數：	38
中文關鍵詞：	韋爾表現
外文關鍵詞：	Weil representation
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在這篇文章的前半段，我們會介紹群表現基本的定義與定理，接著會介紹最後一章所需要的群，例如：辛群、正交群。在第三章會用Deligne-Lusztig虛擬特徵的表示，將特徵表列出。最後會介紹辛群與正交群上的Weil表現，並利用已知公式求其分解的結果。

In the previous parts of this thesis, we introduce the basic definitions and theorems of the finite group representation theory. After that we state the classical groups what we need in the last section, e.g. symplectic groups Sp2, and orthogonal groups O2, O3. In the third part, we would list the character tables by the Deligne-Lusztig virtual characters. For the rest, we introduce the Weil representations and find the decompositions of them.

Contents
Chapter 1-----------------------------------------------1
1.1.  Representations ----------------------------------1
1.2.  Characters----------------------------------------2
1.3.  Induced representations---------------------------3
Chapter 2-----------------------------------------------5
2.1.  Representations of cyclic groups and dihedral groups --------------------------------------------------------5
2.2.  Bilinear forms and quadratic forms ---------------6
2.3.  Symplectic groups  -------------------------------7
2.4.  Orthogonal groups   ------------------------------8
2.5.  Hermitian bilinear forms and unitary groups -----11
Chapter 3----------------------------------------------13
3.1.  Deligne-Lusztig virtual characters---------------13
3.2.  The character table of GL(2;Fq)  ----------------15
3.3.  The character table of SL(2;Fq)  ----------------17
3.4.  The character table of PGL(2;Fq) ----------------21
3.5.  The character tables of SO(3;Fq) and O(3;Fq)-----24
Chapter 4 ---------------------------------------------28
4.1.  Weil representations  ---------------------------28
4.2.  The Weil representations of Sp(2;Fq) and O(2;Fq)-28
4.3.  The Weil representations of Sp(2;Fq) and O(3;Fq)-32
References --------------------------------------------37
Index -------------------------------------------------38
                                

[1] Jerey Adams. Character tables for gl (2), sl (2), pgl (2) and psl (2) over anite eld.Lecture Notes, University of Maryland, 2002.
[2] Peter J Cameron. Notes on classical groups.Lecture Notes for the Universityof London, 2000.
[3] Roger William Carter.Finite groups of Lie type: Conjugacy classes and com-plex characters, volume 5. John Wiley & Sons, 1985.
[4] Roger William Carter.Simple groups of Lie type, volume 22. John Wiley &Sons, 1989.
[5] Francois Digne and Jean Michel.Representations of nite groups of Lie type,volume 21. Cambridge University Press, 1991.
[6] Larry C Grove.Classical groups and geometric algebra, volume 39. AmericanMathematical Soc., 2002.
[7] Shu-Yen Pan. Weil representations of nite symplectic groups and nite odd-dimensional orthogonal groups.Journal of Algebra, 453:291{324, 2016.
[8] Jean-Pierre Serre.Linear representations of nite groups, volume 42. SpringerScience & Business Media, 2012.
[9] Bhama Srinivasan. Weil representations of nite classical groups.Inventionesmathematicae, 51(2):143{153, 1979

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