Showing that $ \text{Ind}_H^G W \cong \text{Ind}_K^G(\text{Ind}_H^K W)$

Question

Suppose we have the tower $H \subseteq K \subseteq G$. I'm trying to show that $ \text{Ind}_H^G W \cong \text{Ind}_K^G(\text{Ind}_H^K W)$. I first wrote it out more explicitly, i.e. $$ \text{Ind}_H^G W = \bigoplus_{\sigma \in G/H} W^\sigma$$ $$ \text{Ind}_K^G(\text{Ind}_H^K W) = \bigoplus_{\lambda \in G/K} \left( \bigoplus_{\sigma \in K/H} W^\sigma \right)^\lambda $$ Note that the $\sigma$ and $\lambda$ below the direct sums mean coset representatives rather than actual cosets. Also, $W^\sigma = W$, etc. I thought that perhaps we need to use a $G$-intertwiner; mine is as follows: $$ T : \bigoplus_{\lambda \in G/K} \left( \bigoplus_{\sigma \in K/H} W^\sigma \right)^\lambda \rightarrow \bigoplus_{\sigma \in G/H} W^\sigma $$ $$ T \left( \sum_{\lambda \in G/K}\left( \sum_{\sigma \in K/H} v_\sigma \right)_\lambda \right)(x) = \sum_{x \in G/H} v_x $$ where the subscripts mean for example $v_\sigma \in W^\sigma$. The action of $G$ is given by $gw_\sigma = hw_\tau$, where $h \in H$ and the $\tau$ subscript means that $w$ now lies in a different copy of $W$. For $T$ to be an intertwiner, we want "$gT(v)=T(gv)$". Checking this, we have: $$ gT \left( \sum_{\lambda \in G/K}\left( \sum_{\sigma \in K/H} v_\sigma \right)_\lambda \right)(x) = g \sum_{x \in G/H} v_x = \sum_{\tau \in G/H} hv_\tau $$ and $$ T \left( g \sum_{\lambda \in G/K}\left( \sum_{\sigma \in K/H} v_\sigma \right)_\lambda \right)(x) = T \left( \sum_{\lambda \in G/K}\left( \sum_{\tau \in K/H} hv_\tau \right)_\lambda \right)(x) = \sum_{x \in G/H} hv_x $$ So $T$ is an intertwiner. Before I attempt an inverse, would anybody be able to let me know if any of the above is correct? Also, I feel that if the above is correct, then bijectivity seems relatively clear (?) so we don't even need to show it has an inverse?

Many thanks for any help.

Edit: I've just realised that this is a slight duplicate, but the answer in that question uses category theory and my working is a little shorter, so I'd like to know if any of my method is correct.

Answer 1

You are doing too much work, and so is that other answer. Here is a general fact about adjoints:

If $R_1 : C_1 \to C_2$ and $R_2 : C_2 \to C_3$ are functors with left adjoints $L_1 : C_2 \to C_1$ and $L_2 : C_3 \to C_2$, then the composite $R_2 \circ R_1 : C_1 \to C_3$ has left adjoint $L_1 \circ L_2 : C_3 \to C_1$.

The proof is very straightforward:

$$\text{Hom}(L_1(L_2(-)), (-)) \cong \text{Hom}(L_2(-), R_1(-)) \cong \text{Hom}((-), R_2(R_1(-)).$$

Syntactically it's identical to the proof that the adjoint of a composition of linear operators is the composition of the adjoints in the opposite order.

Now, as Andreas observes in the comments, the restriction functor from $G$ to $H$ is the composite of the restriction from $G$ to $K$ and the restriction from $K$ to $H$, and induction is the left adjoint of restriction. So we're done.

This is a nice example of the value of using universal properties instead of constructions to prove things.