Write $\mathbb T$ for the circle group, $C_n$ for the cyclic group of order $n$, and fix embeddings $C_n \hookrightarrow \mathbb T$. Let $\lambda^i$ be the $\mathbb T$-representation with underlying vector space $\mathbb C$, such that
$$\mathbb T\times \mathbb C \ni (z,w) \mapsto z^i w\in\mathbb C$$
In particular $\lambda^0$ is the trivial representation, and $\lambda=\lambda^1$ is the standard representation. I am interested in the representation
$$[n]_\lambda = \lambda^0 \oplus \lambda^1 \oplus \dotsb \oplus \lambda^{n-1} $$
This has the property that its restriction to $C_n$ is the complex regular representation:
\begin{equation}\tag{$\ast$} \mathrm{Res}^{\mathbb T}_{C_n} [n]_\lambda = \mathbb C[C_n] = \mathrm{Ind}^{C_n}_e \mathbb C \end{equation}
Is there a description of $[n]_\lambda$ as a $\mathbb T$-representation—ideally as an induced or coinduced representation—which formally implies $(*)$ for all $n$?
