Questions about Lusztig's $\mathbf a$-function

Question

In chapter 13 of Lusztig's Hecke Algebra with Unequal Parameters, the function $\mathbf a$ is defined to be $$\mathbf a(z) = \max_{x,y} \deg h_{x,y,z},$$ for any $z$ in the Coxter group, where the definition for $h_{x,y,z}$ is $$c_xc_y = \sum_u h_{x,y,u}c_u.$$

Similary, $$T_xT_y = \sum_u f_{x,y,u}T_u.$$

I can prove that if the group is bounded, then $$\max_{x,y,z} \deg h_{x,y,z} = \max_{x,y,z} \deg f_{x,y,z}. $$

Then there are my questions:

1. For fixed $z$, does $$\max_{x,y} \deg h_{x,y,z} = \max_{x,y} \deg f_{x,y,z}?$$
2. Let $(W,S,L)$ be a Coxeter group ($S$ being the set of simple reflections, $L$ being the weight function), and $(W',S',L')$ be a parabolic subgroup generated by $S' \subset S$, where $L'$ is the restriction of $L$ to $W'$. Is it true that $$\mathbf a(z) \text{ (in } W \text{)} = \mathbf a(z) \text{ (in } W' \text{)},$$ for any $z \in W'$?

In fact, it can be seen by definition that $$\max_{x,y \in W'} \deg f_{x,y,z} = \max_{x,y \in W} \deg f_{x,y,z}$$ for any parabolic subgroup $W'$ containing $z$ with the restricted weight function. So if the answer to the first question is affirmative, the second is answered.

Thanks to everyone.