臺灣博碩士論文加值系統

這篇論文裡，我們有興趣的是有限Unitary 群的不可約尖點群表現的
Theta 對應。首先要知道有限酉群哪些是unipotent 不可約尖點群表現，必須使用Lusztig 方法來判斷，然後我們使用酉群共軛類的centralizer 來決定有限酉群的不可約群表現哪些是尖點的，之後介紹Induction 法則和Preservation 法則，最後藉由使用Induction 原則和Preservation 法則討論有限酉群的不可約尖點群表現之間的Theta 對應，特別是對於這些有限酉群U0(q)，U1(q)，U2(q)，和U3(q)相互之間Theta對應關係的結果。

In this paper we mainly discuss the theta corrspondence of irreducible cuspidal characters of finite unitary groups. Firstly, we distinguish irreducible unipotent characters of finite unitary groups by Lusztig's discussion. We use
the centralizers of semisimple elements of finite unitary groups to know which irreducible characters of finite unitary groups are cuspidal. Then we introduce the induction principle and preservation principle. Finally, we clarify the theta correspondence between unitary groups by the induction principle and preservation principle, especially for U0(q), U1(q), U2(q), and U3(q).

Contents
1 Introduction . . . . . . . . . . . . . . . . . .2
1.1 Linear Representations of Finite Groups . . . . 2
1.2 Symmetric and Skew-Symmetric Bilinear Forms and Hermitian Forms . . . . . . . . . . . . . . . . . . 4
1.3 The Classical Finite Groups of Lie Type Un . . 5
2 Definition of Theta Correspondence . . . . . . . .7
2.1 Weil Representation of Spn . . . . . . . . . . .7
2.2 Theta Correspondence . .. . . . . . . . . . . . 8
3 Induction Principle and Preservation Principle . 11
3.1 Witt Towers . . . . . . . . . . . . . . . . . 11
3.2 Induction Principle . . . . . . . . . . . . .. 12
3.3 Preservation Principle . . . . . . . . . . . . 12
4 The Chain of Irreducible Cuspidal representations of Finite Unitary Groups . . . . . . . . . . . . . . 15
4.1 Introduction . . . . . . . . . . . . . . . . . 15
4.2 The Unitary Group U0(q) . . . . . . . . . . . 16
4.3 The Unitary Group U1(q) . . . . . . . . . . . .16
4.4 The Unitary Group U2(q) . . . . . . . . . . . 17
4.5 The Unitary Group U3(q) . . . . . . . . . . . 20
reference . . . . . . . . . . . . . . . . . . . . .27

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