On extensions of principal series representations

We compute ${\mathrm{E}\mathrm{x}\mathrm{t}}_B^1({\chi }_1,{\chi }_2)$ between two characters ${\chi }_1,{\chi }_2$ of a Borel subgroup $B$ of a split reductive group $G$ over a finite field ${\mathbb{F}}_q,$ and make an application to the calculation of ${\mathrm{E}\mathrm{x}\mathrm{t}}_G^1({\pi }_1,{\pi }_2)$ between principal series representations ${\pi }_1,{\pi }_2$ of $G({\mathbb{F}}_q)\mathrm{.}$