Help with a proof of a lemma from Humphreys about reflections

Question

I want to understand this proof, I don't get why he says that the minimal polinomial of $\tau$ divides $(T-1)^l$. And at the end how can I explain why gcd$(T^k-1,(T-1)^l)=T-1$?

Answer 1

If all eigenvalues of $\tau$ are equal to $1$, then the matrix of $\tau$ with respect to some basis of $\mathrm E$ is upper triangular having only $1$'s in the main diagonal; therefore $(\tau-\operatorname{Id})^{\dim E}=0$, and so the minimal polynomial of $\tau$ has to divide $(T-1)^l$ (note that $l=\dim\mathrm E$).

On the other hand,$$T^k-1=(T-1)\left(T^{k-1}+T^{k-2}+\cdots+T+1\right)$$and $1$ is not a root of $T^{k-1}+T^{k-2}+\cdots+T+1$. Therefore, the only common factor of $T^k-1$ and $(T-1)^l$ is indeed $T-1$.