= Solution
Let $T$ be a <standard Young tableau> of shape $\lambda\vdash n$, and let $\mu$ be the shape formed by its entries $1,\ldots,k$. Since <Young–Jucys–Murphy elements> act by contents, the <eigenvalue> of $\xi_k$ on its <Gelfand–Tsetlin basis> vector is
$$
\prod_{r=2}^k c_T(r).
$$
If $\mu$ is not a <hook partition>, it contains $(2,2)$, whose content is zero, so this product is zero. If $\mu=(a+1,1^b)$ with $a+b+1=k$, its nonzero contents are $1,\ldots,a$ and $-1,\ldots,-b$. Their product is independent of the order of the labels. Therefore \b[the <eigenvalue> is]
$$
\boxed{
\xi_kw_T=
\begin{cases}
(-1)^b a!b!\,w_T,&\mu=(a+1,1^b),\\
0,&\mu\text{ is not a hook}.
\end{cases}}
$$
The empty product at $k=1$ gives $1$. When $k=n$, $\mu=\lambda$, giving exactly the hook-versus-nonhook distinction suggested by the hint. For $k<n$, it is the intermediate shape $\mu$, rather than the final shape $\lambda$, that controls the answer.
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