I am reading the page 38 in this "Post Quantum Cryptography" book (Equations 8 and 9). My question is, why to compute the verification key $Y$, $f$ is applied $2^w-1$ times? Are there any security notions involved?
$w$ is a parameter that can be freely chosen, to maximize performance. Each element of the signature encodes $w$ bits of the message to be signed, so the larger $w$ is, the fewer elements you need to include in the signature.
If you make $w$ large, then signatures can be shorter; however, the tradeoff is that key generation, signing, and verification run slower. If you make $w$ small, then signatures are longer; however, key generation, signing, and verification run faster. So this is a tradeoff between the size of the signatures vs the running time of the scheme. You can choose $w$ to provide the best tradeoff for your application. The scheme will be secure no matter what value of $w$ you choose.