Given a complex Lie group $G$ and a maximal torus $T$ with associated Borel subgroup $B$ and a character $\lambda : T \to U(1)$ what is the canonical extension to the Borel subgroup?
The only thing I can think of is to lift the character to the Cartan subgroup $\lambda:\mathfrak{h}\to i\mathbb{R}$ then we can extend this trivially to the Borel subalgebra and exponentiate to get a character for the Borel subgroup. However this is trivial on the off-diagonal elements which strikes me as incorrect.