Fixed-point group conjugacy classes of unipotent elements in low-dimensional symmetric spaces of special linear groups over a finite field (Q6601870)

In a series of papers [Commun. Algebra 45, No. 12, 5123--5136 (2017; Zbl 1434.20032); Assoc. Women Math. Ser. 15, 69--77 (2018; Zbl 1544.20074); Note Mat. 37, No. 2, 1--10 (2017; Zbl 1434.20031); Commun. Algebra 48, No. 4, 1744--1757 (2020; Zbl 1480.20117)], the authors investigate finite symmetric spaces. For a number theorist who has considered a very particular case of finite upper half planes, there are many surprises, for example, the number of possible spaces to consider for \(\mathrm{SL}(n,k) \) if \(k\) is a finite field, \(n\geq4\). Thus the reviewer wonders if there are finite analogs of the non-Euclidean distance for the special (and general) linear groups over the real numbers and whether there are associated interesting graphs.\N\NSymmetric spaces come from involutions \(\theta\) of the group \(G=\mathrm{SL}(n,k)\). If \(H=\{ g\in G\mid \theta(g)=g\}\), the symmetric space is \(G/H\). If \(k\) is a finite field of odd characteristic and \(n>2\),\Nthe isomorphism classes of involutions are characterized in [\textit{A. G. Helminck} et al., Acta Appl. Math. 90, No. 1--2, 91--119 (2006; Zbl 1100.14040)]. For \(n=3\), there is only one class derived from the inner automorphism associated to the matrix\N\[\NY_{1}=\left(\begin{array} [c]{ccc}\N1 & 0 & 0\\\N0 & 1 & 0\\\N0 & 0 & -1\N\end{array}\N\right).\N\]\NFor \(n=4\), there are 3 classes of involutions. They come from the inner automorphisms associated with the matrices\N\[\NY_{1}=\left(\N\begin{array}\N[c]{cccc}\N1 & 0 & 0 & 0\\\N0 & 1 & 0 & 0\\\N0 & 0 & 1 & 0\\\N0 & 0 & 0 & -1\N\end{array}\N\right) ,\; Y_{2}=\left(\N\begin{array}\N[c]{cccc}\N1 & 0 & 0 & 0\\\N0 & 1 & 0 & 0\\\N0 & 0 & -1 & 0\\\N0 & 0 & 0 & -1\N\end{array}\N\right) ,\; L=\left(\N\begin{array}\N[c]{cccc}\N0 & 1 & 0 & 0\\\Nn_{s} & 0 & 0 & 0\\\N0 & 0 & 0 & 1\\\N0 & 0 & n_{s} & 0\N\end{array}\N\right) ,\N\]\Nwhere \(n_{s}\) denotes a non-square in \(k.\)\N\NThe object of the paper under review is to characterize the \(H\)-orbits of unipotent elements of \(G\) in the symmetric space. There are thus several cases to consider. In the case that \(n=3,\) the authors find \(3+\gcd(q-1,3)\) orbits,\Nwhere \(q\) namely the number of element of the finite field \(k\) is always assumed to be odd.\N\NWhen \(n=4\) and the inner automorphism is from \(Y_{1}\), there are \(4\) orbits. When \(n=4\) and the inner automorphism is from \(Y_{2}\), there are \(19\) orbits if \(-1\) is a square in \(k\) and \(15\) orbits if \(-1\) is not a square in \(k\). When \(n=4\) and the inner automorphism comes from \(L\), there are \(3\) orbits.