Character of induced representation

Question

Let $H$ be a subgroup of a group $G$ and let $\rho^G$ be the representation of $G$ induced by a representation $\rho$ of $H$. My book Algebra Vol. 2 by Cohn states that if the character of $\rho$ is $\alpha$, then the character of $\rho^G$ is $$ \alpha^G(x)=\sum_\lambda\operatorname{tr}\rho(t_\lambda xt_\lambda^{-1})=\sum_\lambda\dot{\alpha}(t_\lambda xt_\lambda^{-1}),\qquad \dot{\alpha}(x):=\begin{cases}\alpha(x)&\mathrm{~if~}x\in H,\\0&\text{ otherwise,}&\end{cases} $$ where the first sum is over all $\lambda$ such that $t_\lambda xt_\lambda^{-1}\in H$. This all seems to make sense, but what I do not understand is the following. Since $\alpha$ is a class function on $H$, we have $$\tag{1} \alpha(ht_\lambda xt_\lambda^{-1}h^{-1})=\alpha(t_\lambda xt_\lambda^{-1}) $$ for any $h\in H$ and it follows that $$\tag{2} |H|\alpha^G(x)=\sum_{h\in H}\sum_\lambda \dot{\alpha}(ht_\lambda xt_\lambda^{-1}h^{-1})=\sum_{g\in G}\dot{\alpha}(gxg^{-1}). $$ Why does equation $(2)$ follow from equation $(1)$?

Answer 1

Assuming that $\{t_{\lambda}\}$ is a set of coset representatives for $G/H$.

First of all, we need to verify that $\dot{\alpha}$ satisfies something like equation $(1)$, namely we want to check that $\dot{\alpha}(x) = \dot{\alpha}(hxh^{-1})$ for all $x \in G$, $h \in H$.

Indeed, if $x \in H$ then $hxh^{-1} \in H$ and we have $\dot{\alpha}(x) = \alpha(x) =^{(1)} \alpha(hxh^{-1}) = \dot{\alpha}(hxh^{-1})$

OTOH, if $x \not\in H$, then for any $h \in H$ we have $hxh^{-1} \not\in H$ (since $hxh^{-1} \in H \iff x \in h^{-1}Hh = H$) and so $\dot{\alpha}(x) = \dot{\alpha}(hxh^{-1}) = 0$

Call this $\dot{(1)}$. Then $\dot{(1)}$ gives us the first equals sign in the line marked $(2)$, since $$\sum_{h\in H}\sum_\lambda \dot{\alpha}(ht_\lambda xt_\lambda^{-1}h^{-1}) =^{\dot{(1)}} \sum_{h\in H}\sum_\lambda \dot{\alpha}(t_\lambda xt_\lambda^{-1}) = |H|\sum_\lambda \dot{\alpha}(t_\lambda xt_\lambda^{-1}) = |H|\alpha^G(x)$$

The second equals sign in line $(2)$ is a result of the fact that every element in $g \in G$ can be written uniquely in the form $h \cdot t_{\lambda}$ for some $h \in H$ and coset representative $t_{\lambda}.$