On extensions of principal series representations

We compute $\mathrm{Ext}^{1}_B(χ_1,χ_2)$ between two characters $χ_1,χ_2$ 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 $\mathrm{Ext}^1_G(π_1,π_2)$ between principal series representations $π_1,π_2$ of $G(\mathbb{F}_q).$

💡 Research Summary

The paper investigates first‑order extensions between characters of a Borel subgroup B of a split reductive group G defined over a finite field 𝔽_q, and then applies these results to principal series representations of the finite group G(𝔽_q).

After fixing a maximal split torus T⊂B⊂G, the authors recall that any B‑module which is simple is one‑dimensional, so a character of B is just a character of T (N acts trivially). The unipotent radical N of B satisfies