How can I typeset a group action?

You can use the [t] option of the aligned environment.

\documentclass{article}
\usepackage{amsmath,amssymb}
\begin{document}
\begin{align*}
    \sigma \colon T^k(V) & \longrightarrow T^k(V) \\
                  T & \longmapsto
    \begin{aligned}[t]
        \sigma T \colon V \times \dots \times V & \longrightarrow \mathbb R \\
        (v_1, \dots, v_k) & \longmapsto T(v_{\sigma(1)}, \dots, v_{\sigma(k)})
    \end{aligned}
\end{align*}
\end{document}

I would simply use an alignedat environment, and the \mathrlap command from mathtools. I think the layout will look better if the last line is closer to the secng than the latter from the first:

\documentclass{article}
\usepackage{mathtools, amssymb}

\begin{document}

\begin{alignat*}{2}
    \sigma \colon T^k(V) & \longrightarrow \mathrlap{T^k(V)} \\
                  T & \longmapsto& \sigma T \colon V \times \dots \times V & \longrightarrow \mathbb R \\[-0.8ex]
         && (v_1, \dots, v_k) & \longmapsto T(v_{\sigma(1)}, \dots, v_{\sigma(k)})
\end{alignat*}

\end{document}