令$\mathbb{F}_q$為一個特徵為奇數的有限體以及約化對偶對$(G_n, G'_{n'})$ 為 $({\rm Sp}_{2n}(\mathbb{F}_{q}),$${\rm O}^{\epsilon}_{2n'}(\mathbb{F}_{q}))$ 或 $({\rm O}^{\epsilon}_{2n}(\mathbb{F}_{q}), {\rm Sp}_{2n'}(\mathbb{F}_{q}))$ 或 $({\rm Sp}_{2n}(\mathbb{F}_{q}), {\rm O}_{2n'+1}(\mathbb{F}_{q}))$ 或 $({\rm O}_{2n+1}(\mathbb{F}_{q}), {\rm Sp}_{2n'}(\mathbb{F}_{q}))$，其中$\epsilon=+$ 或 $\;-$。令$\Theta_{G_n, G'_{n'}}$為對於$(G_n, G'_{n'})$的theta對應。在這篇論文中，我們利用潘戍衍的結果，這些結果顯示$\Theta_{G_n, G'_{n'}}$對應可以被對於一些Lusztig的符號以及半單共軛類的一些組合的關係所刻劃。我們描述了一些有趣的$n'$與$\left\{\rho'\in{\rm Irr}(G'_{n'})\;\middle|\;(\rho, \rho')\in\Theta_{G_n, G'_{n'}} \right\}$之間的關係(當固定住$G_n$與$\rho\in{\rm Irr}(G_n)$時)，以及利用這些組合的關係來證明它們。另外，當$(G_n, G'_{n'})$在穩定範圍時，我們給出一個限制在群$G_n\times G'_{n'}$上的Weil表現的分解，這個分解是Felipe Montealegre-Mora 與 David Gross的結果的一個推廣。

Let $\mathbb{F}_q$ be a finite field of odd characteristic. Let the
reductive dual pair $(G_n, G'_{n'})$ be $({\rm Sp}_{2n}(\mathbb{F}_{q}), {\rm O}^{\epsilon}_{2n'}(\mathbb{F}_{q}))$ or $({\rm O}^{\epsilon}_{2n}(\mathbb{F}_{q}), {\rm Sp}_{2n'}(\mathbb{F}_{q}))$ or $({\rm Sp}_{2n}(\mathbb{F}_{q}), {\rm O}_{2n'+1}(\mathbb{F}_{q}))$ or $({\rm O}_{2n+1}(\mathbb{F}_{q}),$ ${\rm Sp}_{2n'}(\mathbb{F}_{q}))$, where $\epsilon=+$ or $-$. Let $\Theta_{G_n, G'_{n'}}$ be the theta correspondence for the pair $(G_n, G'_{n'})$. In this paper, we use the results from Shu-Yen Pan which showed that the correspondence $\Theta_{G_n, G'_{n'}}$ can be characterized by some combinatorial conditions on some symbols and some semisimple conjugacy classes corresponding to $G_n$ and $G'_{n'}$. We describe some interesting relations between $n'$ and the set $\left\{\rho'\in{\rm Irr}(G'_{n'})\;
\middle|\;(\rho, \rho')\in\Theta_{G_n, G'_{n'}} \right\}$ (with a fixed $G_n$ and a fixed $\rho\in{\rm Irr}(G_n)$), and use these combinatorial conditions to prove them. In addition, when $(G_n, G'_{n'})$ is in the stable range, we give a decomposition of the Weil representation restricted to the group $G_n\times G'_{n'}$, which is a generalization of a result from Felipe Montealegre-Mora and David Gross.