# 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 $$n$$?