用户: Lieriheart/mackey theorem

1THEOREM AND PROOF

定理 1.1.

Let G be a finite group, and let $H_1$, $H_2$ be two subgroups of G. Let$$G=\bigcup _{i=1}^r{H}_1{g}_i{H}_2,$$be double coset decomposition, where $g_1,...,g_r\in G$ are representives. Let ($\sigma ,V$) be a representation of $H_1$. put ${\sigma }^{g_i}={\sigma }^{\prime }{|}_{g_i{H}_1{g}_i^{-1}}$, then we have$${Ind}_{H_1}^G\sigma {|}_{H_2}=\underset{i=1}{\overset{r}{⨁}}{Ind}_{H_1\cap g_i^{-1}{H}_2{g}_i}^{H_2}{\sigma }^{g_i}$$where ${\sigma }^{\prime }$ is the representation given by $$H_{1}GL(V)g_i{H}_1{g}_i^{-1}$$

证明. Define $K_i=H_2\cap g_i{H}_1{g}_i^{-1}.$Taking the left representives system $x_{ij}$ of $H_2/K_i$ , we obtain the decomposition of G: $$G=\bigcup _{i=1}^m\bigcup _{j=1}^{m_i}{x}_{ij}{g}_i{H}_2,$$where $m=[G:H_1]$. Then $$\mathbb{C}[G]=\underset{i=1}{\overset{m}{⨁}}\underset{j=1}{\overset{m_i}{⨁}}{x}_{ij}{g}_i\mathbb{C}[H_2]$$Thus, $${\text{Ind}}_{H_1}^G\sigma {|}_{H_2}=\underset{i=1}{\overset{m}{⨁}}\underset{j=1}{\overset{m_i}{⨁}}{x}_{ij}{g}_i\mathbb{C}[H_2]{\otimes }_{\mathbb{C}[H_2]}\sigma .$$and we have $${\text{Ind}}_{H_1\cap g_i^{-1}{H}_2{g}_i}^{H_2}{\sigma }^{g_i}=\underset{j=1}{\overset{m_i}{⨁}}{x}_{ij}\mathbb{C}[K_i]{\otimes }_{\mathbb{C}[K_i]}{\sigma }^{\prime }.$$Then we have $$x_{ij}{g}_i\mathbb{C}[H_2]{\otimes }_{\mathbb{C}[H_2]}\sigma \cong x_{ij}{\otimes }_{\mathbb{C}[H_2]}{g}_i\sigma \cong x_{ij}{\otimes }_{\mathbb{C}[K_i]}{g}_i\sigma \cong x_{ij}\mathbb{C}[K_i]{\otimes }_{\mathbb{C}[K_i]}{g}_i\sigma \cong x_{ij}\mathbb{C}[K_i]{\otimes }_{\mathbb{C}[K_i]}\sigma$$and from the definition of ${\sigma }^{\prime }$, we have $\sigma \cong {\sigma }^{\prime }$
Thus, we have$$x_{ij}\mathbb{C}[K_i]{\otimes }_{\mathbb{C}[K_i]}\sigma \cong x_{ij}\mathbb{C}[K_i]{\otimes }_{\mathbb{C}[K_i]}{\sigma }^{\prime }$$

$\square$