Let $G$ be a reductive algebraic group and $I$ the set of vertices of the Dynkin diagram of $G$. Let $J \subset I$ and $P_J = BW_JB$ the parabolic subgroup of $G$ containing $B$, where $B$ is a Borel subgroup of $G$ and $W_J$ is the subgroup of the Weyl group $W$ of $G$ generated by $s_j, j \in J$.

Let $L_J$ be the Levi subgroup of $P_J$. Then there is a projection $\pi_J: P_J \to L_J$. Is there a projection from $G$ to $L_J$? Thank you very much.

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    $\begingroup$ I'm not sure what properties you want your projection to have, but I doubt there will be such a projection w/ any reasonable properties. For example, taking $G$ to be $SL_2(\mathbb{C})$ and $P=B$, you don't have any projection $G\rightarrow T$ that sends $T$ to itself (here I'm identifying $L$ and $T$ with diagonal matrices.) To see this, note that the map $\pi_1(T)\rightarrow\pi_1(G)$ is not injective. $\endgroup$ – dhy May 20 '18 at 13:56
  • $\begingroup$ @dhy, thank you very much. The only property I need is the image of the projection $G \to L_J$ is $L_J$. Does such projection exist? $\endgroup$ – Jianrong Li May 20 '18 at 15:25
  • $\begingroup$ What do you mean by a projection? A homomorphism of algebraic groups? A morphism of varieties? Or what? $\endgroup$ – Mikhail Borovoi May 21 '18 at 16:19
  • $\begingroup$ @Mikhail, thank you very much. I mean a homomorphism of algebraic groups. $\endgroup$ – Jianrong Li May 21 '18 at 20:03
  • $\begingroup$ In the case $G={\rm SL}_{n,\mathbb{C}}$, $L\subsetneqq G$, there is no nontrivial homomorphism $G\to L$ (for example, because $G$ is a simple group and the dimension of $L$ is smaller than the dimension of $G$). $\endgroup$ – Mikhail Borovoi May 21 '18 at 20:14

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