A strong splitting of the Frobenius morphism on the algebra of distributions of $SL_2$

Abstract

Let $p$ be a prime number, and let $Dist(SL_2)$ be the algebra of distributions, supported at $1$, on the algebraic group $SL_2$ over $\mathbb{F}_p$. The Frobenius map $Fr:SL_2\to SL_2$ induces a map $Fr:Dist(SL_2)\to Dist(SL_2)$ which is in particular a surjective algebra homomorphism. In this note, we construct a section of this map, whenever $p\geq 3$. The main ingredient of this construction is a certain congruence modulo $p^3$, reminiscent of the congruence $\binom{np}{p}\equiv n\mod p^3$.