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Let $F$ be a number field, and $G$ be a semisimple linear algebraic group over $F$. We let $P_0 \subseteq G$ be a minimal $F$-rational parabolic subgroup. Let $P$ be a standard (i.e. containing $P_0$) parabolic subgroup. Let $X^*$ denote the group of characters defined over $F$. Then in the paper I am reading it states: any element $\chi \in X^*(P)$ defines a line bundle $L_{\chi}$ on $P \backslash G$. But I am not getting this part and I would appreciate any explanation of this. Thank you.

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    $\begingroup$ Denote the quotient of $G$ by $P$ as $Y$. By fppf descent relative to $G\to Y$, an invertible sheaf on $Y$ is determined by a pair $(L,\phi)$ of an invertible sheaf $L$ on $G$ and an isomorphism $\phi:\text{pr}_1^*L\to \text{pr}_2^*L$ on the fiber product $G\times_Y G$ satisfying the cocycle condition. Choose $L$ to be the structure sheaf. Also observe that $G\times_Y G$ is canonically isomorphic to $P\times G$ by the definition of the group quotient. Thus, $\phi$ is equivalent to an invertible regular function on $P\times G$. The cocycle condition is that $\phi$ is a character of $P$. $\endgroup$ Jan 18, 2019 at 14:42
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    $\begingroup$ A general fact is that line bundle on $G/P$ pulls back to a unique $P$ equivariant line bundle on $G$, and every $P$ equivariant line bundle arises in this way. The character $\chi$ gives a $P$ action on the trivial line bundle over $G$. $\endgroup$ Jan 18, 2019 at 14:43
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    $\begingroup$ The same in very-very simple language: Consider $L'=G\times_F \mathbb{G}_a$. Consider the left action of $P$ on $L'$ by $$p*(g,a)=(pg,\chi(p)a).$$ Set $L=P\backslash L'$ and consider the map $$\pi\colon L\to P\backslash G,\quad [g,a]\mapsto [g].$$ Then $(L,\pi)$ is the desired line bundle $L_\chi$ on $P\backslash G$. $\endgroup$ Jan 18, 2019 at 16:59
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    $\begingroup$ You might like to read Bott, Homogeneous vector bundles, Ann. of Math., 1957, where this idea is followed up (over the complex numbers) to give a nice understand of these line bundles, and their cohomology. $\endgroup$
    – Ben McKay
    Jan 21, 2019 at 16:57

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