reductive groups – Restriction of a line bundle on $G/B$ to a fibre which is isomorphic to $mathbb{P}_1$

Let $G$ be a reductive group over a field $k$ of characteristic zero with maximal split torus $T$, Borel $B supset T$ and Weyl group $W$. Set $X:=G/B$ and $C_w:=BwB/B subset X$ for $w in W$ the associated schubert cell. Furthermore fix a character $lambda in X^*(T)$ to which we associate the line bundle $mathcal{L}_lambda=G times_B k_lambda$ (for example as in Jantzens “Representations of algebraic groups”).

Fix $w, w’ in W$ with $w=s_alpha w’$ for a simple root $alpha$ with respect to $B$ and $l(w)=l(w’)+1$. In “Differenatial operators on flag varieties” Brylinski explains on page 52 how to get a proper smooth morphism
$$p:C_w cup C_{w’} rightarrow C_{w’}$$ with each fibre isomorphic to $mathbb{P}^1$.
Namely as the geometric quotient by the $SL_2$-action coming from the identification $C_w cup C_{w’}=P_{alpha}w’B/B$ where $P_alpha$ is the parabolic subgroup generated by $B$ and a subgroup $L_alpha$ which is isomorphic to $SL_2$.

He then considers for a point $y in C_{w’}$ the restriction $mathcal{L}_{lambdamid p^{-1}(y)}$. As $p^{-1}(y)$ is isomorphic to $mathbb{P}^1$ the line bundle $mathcal{L}_{lambdamid p^{-1}(y)}$ has to be isomorphic to some $mathcal{O}(n)$.

On page 53 he only considers the structure sheaf, but on page 55 he says that the arguments used on page 53 go through for dominant $lambda$. As he deals with the vanishing of $H^1(p^{-1}(y),mathcal{L}_{lambdamid p^{-1}(y)})$ he seems to know how $mathcal{L}_{lambdamid p^{-1}(y)}$ looks like, but he says nothing about it.

For me it’s not clear. How can we determine $n$?