In a Hilbert space $\mathcal{H}$, let $T\in \mathcal{L}(\mathcal{H})$ be such that there exists an integer $k$ such that $T^k=\mathbb{I}$; let $\zeta := $exp$(2\pi i/k)$ and let $$P_\mu=\frac{1}{k}\sum_{j=0}^{k-1}\zeta^{-\mu j}T^j .$$ I would like to prove that $P_\mu \circ P_\eta = P_\mu$ when $\mu = \eta$ (mod $2k$) and $0$ otherwise.
I was only able to say this:
if $\mu = \eta$ (mod $2k$), then lets say that $h\in \mathbb{Z}$ is such that $\eta = \mu + 2kh$, then i came up with $$P_\mu \circ P_\eta = \frac{1}{k^2}\sum_{j=0}^{k-1}\sum_{l=0}^{k-1}\frac{T^{j+l}}{\zeta^{\mu (j+l)+2khl}} = \frac{1}{k^2}\sum_{j,l=0}^{k-1}\frac{1}{\zeta^{2khl}}\frac{T^{j+l}}{\zeta^{\mu (j+l)}} = \frac{1}{k^2}\sum_{j,l=0}^{k-1}\frac{T^{j+l}}{\zeta^{\mu (j+l)}}$$ and considering the fact that $T$ has finite order $$= \frac{1}{k^2}\sum_{j=0}^{k-1}\Big( \sum_{l=0}^j + \sum_{l=j+1}^{k-1} \Big)\Bigg(\frac{T}{\zeta^\mu}\Bigg)^j = \frac{1}{k^2}\sum_{j=0}^{k-1}\Big(k \Big)\Bigg(\frac{T}{\zeta^\mu}\Bigg)^j = P_\mu$$
Any hints on how to continue? Thanks in advance.