Suppose $G$ is an algebraic group, $P=LU$ is a parabolic subgroup with Levi factor and unipotent subgroup $L$ and $U$, respectively. Let $R$ is a commutative, unital ring and $e_U=\frac{1}{|U|}\sum_{u\in U}u$, so I'm assuming $R$ has characteristic such that $|U|^{-1}$ makes sense.
Why are $RG\otimes_{RP} RL$ and $RGe_U$ isomorphic as $(RG,RL)$-bimodules?
The map $RG\times RL\to RGe_U$ sending $(x,y)\mapsto xye_U$ is $RP$-bilinear, and clearly surjective, so it induces a surjection $RG\otimes_{RP}RL\to RGe_U$ via $x\otimes y\mapsto xye_U$, but I'm not sure if it's injective. That it's a map of left $RG$-modules seems okay, but since $RG\otimes_{RP}RL$ is a right $RL$-module by multiplication, this would require $xye_U\ell=xy\ell e_U$, for $\ell\in RL$, which doesn't seem quite right.