$\newcommand{\Ind}{{\text{Ind}}}$ $\newcommand{\Res}{{\text{Res}}}$ $\newcommand{\ds}{{\displaystyle}}$ $\newcommand{\inv}{{^{-1}}}$
I am doing Exercise 3.16 from Fulton Harris. http://bit.ly/JeTz1J
I have already read http://bit.ly/HYm9Fx, but it's much too advanced an answer.
Goals: a) $U \otimes \Ind(W) =\Ind (\Res (U) \otimes W) $ and b)$\Ind^G_H(W) =\Ind^G_K (\Ind^K_H(W))$ where H
My attempts:
a) $U \otimes \Ind W = U \otimes \ds\bigoplus_{\sigma \in G/H} \sigma W = \ds\bigoplus_{\sigma \in G/H} U \otimes \sigma W = \ds\bigoplus_{\sigma \in G/H} \sigma( \sigma\inv U \otimes W)$. Now $\Ind (\Res (U) \otimes W) = \ds\bigoplus_{\sigma \in G/H} \sigma (\Res U \otimes W) = \ds\bigoplus_{\sigma \in G/H} \sigma \Res U \otimes \sigma W$.
In particular if you let $W$ be the trivial representation then $\Res U$ can be expressed as $\Res( U) \otimes W$ and thus $\Ind(\Res (U))= \Ind(\Res (U) \otimes W) = U \otimes \Ind W= U \otimes P$ Using example 3.13.
b) By defintion we can expand $\Ind^G_H(W) = \bigoplus _{\sigma \in G/H} \sigma W$ and
$\Ind^G_K (\Ind^K_H(W)) = \Ind^G_K (\bigoplus _{\tau \in K/H} \tau W) =\bigoplus _{\gamma\in G/K} \gamma (\bigoplus _{\tau \in K/H} \tau W) = \bigoplus _{\gamma\in G/K} \bigoplus _{\tau\in K/H} \gamma \tau W$.