令 G 是一個定義在 q 個元素的有限體上的線性的代數群，G是其有理點所構成的有限群。s是一個在 G* 裡面的半單元素，T∗是一個在G∗裡面的最大環面，其中 G∗ 是 G 的對偶群。我們定義關連到 s 的 Lusztig 序列 E(G)s 為一個 G 的那些不可約的特徵標，滿足每一個特徵標滿足它和RT∗,s的內積不等於零，RT∗,s是Deligne-Lusztig的虛擬特徵標。根據Lusztig對應，我們有一個從E(G)s到E(CG∗(s))1的雙射，而且包含所有G的不可約的特徵標的集合可以被寫成所有Lusztig序列的不相交聯集。
在這篇論文中，我們考慮G=Sp4(q)，從Srinivasan的工作中，我們知道了所有G的不可約特徵標，我們想要討論如何將這些特徵標寫成Lusztig序列的聯集。

Let $G$ be the group of rational points of a linear algebraic group $\mathbf{G}$ over a finite field with $q$ elements, $s \in G^{*}$ be a semisimple element, $T^{*}$ be a maximal torus of $G^{*}$ such that $s \in T^{*}$, where $G^{*}$ is the dual group of $G$. Let the Lusztig series ${\cal E}(G)_{s}$ of $s$ be the set of all irreducible character $\chi$ of $G$ such that the inner product of $\chi$ and $R_{T^{*}, s}$ is not zero, where $R_{T^{*}, s}$ is the Deligne-Lusztig virtual character. According to Lusztig correspondence, we know that there is a bijection from ${\cal E}(G)_{s}$ to ${\cal E}(C_{G^{*}}(s))_{1}$ and the set of all irreducible characters of $G$ can be written as the disjoint union of all Luisztig series on $G$. In this paper, we consider $G = {\rm Sp}_{4}(q)$. From Srinivasan's work, we obtain all irreducible characters of $G$. Then we want to discuss how to classify these characters into a union of ${\cal E}(G)_{s}$.

1 Introduction: 1
1.1 Representations and characters of a nite group . . . . . . . 1
1.2 Deligne-Lusztig virtual characters and unipotent characters . 4
1.3 Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Lusztig series of Sp(4, q): 8
2.1 Centralizer of a semisimple element . . . . . . . . . . . . . 8
2.2 Unipotent characters . . . . . . . . . . . . . . . . . . . . . 10
2.3 Irreducible characters of Sp4(q) . . . . . . . . . . . . . . . 15
2.4 Classification of characters of Sp4(q) . . . . . . . . . . . . 27
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