space of sections of homogenious spaces

Question

Let $G/H$ be a homogeneous space and then for homogeneous line bundle $L$ of $G/H$ the space of sections can be written as functions related to character of $H$. what about $\Gamma (G/H, L^2)$. then can we write each section as function?ces

Answer 1

$L$ corresponds to a one-dimensional representation, i.e. an abelian character, of $H$, and $L^2$ just corresponds to the square of this character.

Answer 2

This is a special case of a more general situation (sometimes it is easier to understand the more general case). Let $P\to M$ be a principal $H$-bundle and $\rho:H\to GL(V)$ a linear representation. The associated vector bundle is $E=P\times V/H,$ with $H$ acting diagonaly. Then there is a bijection between sections of $E\to M$ and $H$-equivariant functions $f:P\to V$, ie $f(ph)=\rho(h^{-1})f(p)$ (please verify this claim). If you take $V\otimes V$ then the vector bundle is $E\otimes E$ and the sections are equivariant maps $P\to V\otimes V$. If $M$ is homogeneous, ie $M=G/H$, then $P=G$, and $G$ acts on everything in sight ($P$, $E$, sections of $E\to M$).