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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 $\mathcal{L}_{\lambda\mid p^{-1}(y)}$. As $p^{-1}(y)$ is isomorphic to $\mathbb{P}^1$ the line bundle $\mathcal{L}_{\lambda\mid p^{-1}(y)}$ 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 $H^1(p^{-1}(y),\mathcal{L}_{\lambda\mid p^{-1}(y)})$ he seems to know how $\mathcal{L}_{\lambda\mid p^{-1}(y)}$ looks like, but he says nothing about it.

For me it's not clear. How can we determine $n$ and what is it?

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  • $\begingroup$ The rational curve is a closure of an orbit of a one-parameter subgroup (a torus). You can compute $n$ by equivariant localization with respect to the subgroup: the integer $n$ is the difference of the torus weights for the two fixed points of the torus action on the rational curve. $\endgroup$
    – Jason Starr
    Commented Sep 15, 2021 at 14:32
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    $\begingroup$ Thanks for the quick answer! That approach is really new for me. Therefor some questions popping up. At first how do I get the the torus in question? Furthermore your explanation seems independent of the line bundle $\mathcal{L}_{\lambda\mid p^{-1}(y)}$ resp. $\lambda$ or where does it enter? $\endgroup$
    – KKD
    Commented Sep 15, 2021 at 14:56
  • $\begingroup$ The "torus weights" in my previous comment refer to the weights at the two fixed points for the natural linearized action of the torus on the invertible sheaf $\mathcal{L}_{\lambda}$. As $\lambda$ varies in the character group of the maximal torus $X^*(T)$, the integer $n$ defines an (Abelian) group homomorphism from $X^*(T)$ to $\mathbb{Z}$, i.e., an element of the dual group of cocharacters. You can choose the torus to be any torus in $L_\alpha$ that is not contained in the maximal torus of $B\cap L_\alpha$. $\endgroup$
    – Jason Starr
    Commented Sep 15, 2021 at 15:05
  • $\begingroup$ So it seems for me that you refer to Lemma 2.1 from arxiv.org/pdf/math/0112183.pdf . But there appears also the weight from a tangent space. Is there a reason you havn't mentioned it in your comment? Another thing that I don't understand is why I cannot take the maximal torus of $B\cap L_{\alpha}$. At least by Lemma 3.2 of the mentioned paper, we get that $n=h_{\alpha}(\lambda)$ in the fibre of $w'B/B$ for $w'=id$. $\endgroup$
    – KKD
    Commented Sep 16, 2021 at 15:57
  • $\begingroup$ You are correct that you need to "normalize" the torus so that the open orbit in $\mathbb{P}^1$ is isomorphic to the torus as a variety with an action of the torus (i.e., do not act by the square, cube, etc., of the "standard" action). You can normalize that by multiplying the previous weight by $2$ and dividing by the weight of the tangent space at the point. However, you can also normalize by choosing an appropriate one-parameter subgroup so that the stabilizer of a general point in $\mathbb{P}^1$ is trivial. $\endgroup$
    – Jason Starr
    Commented Sep 16, 2021 at 16:57

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