Faithful representations of \(\text{SL}_2\) over truncated Witt vectors. (Q1400183)

Let \(k\) be an algebraically closed field of characteristic \(p>0\) and let \({\mathcal W}_2={\mathcal W}(k)/p^2{\mathcal W}(k)\) be the ring of Witt vectors truncated at length two, viewed as a two dimensional affine ring variety over \(k\). We are interested in faithful representations \((\rho,V)\) of the affine algebraic group \(\Gamma_2=\text{SL}_2({\mathcal W}_2)\). Here faithful means that \(\rho\) defines a closed embedding \(\Gamma_2\to\text{GL}(V)\). So \(\rho\) should not just be faithful on the group \(\Gamma_2(k)\) of \(k\)-rational points of \(\Gamma_2\). Theorem 1 tells that if \(\dim V\leq p+2\) then \(\rho(u^p)\) is trivial for each unipotent element \(u\) of \(\Gamma_2(k)\). For instance, the six dimensional adjoint representation is far from faithful. Using the theorem one sees that the analogue of the Jacobson-Morozov theorem fails badly for \(\Gamma_2\). Theorem 2 provides a faithful representation of dimension \(p+3\) provided \(p>2\). A similar theorem is proved for \(\Gamma_2(\mathbb{F}_p)=\text{SL}_2(\mathbb{Z}/p^2\mathbb{Z})\) and more generally for \(\Gamma_2(\mathbb{F}_q)\).