Codimensions of Newton strata for \(\text{SL}_3(F)\) in the Iwahori case.
Let \(F\) denote the field of Laurent power series over the algebraic closure \(\overline k\) of a finite field. One has the skew polynomial ring over \(F\) generated by the Frobenius automorphism \(\sigma\). An isocrystal is a tuple \((V,\Phi)\) where \(V\) is a finite-dimensional vector space over \(F\) and \(\Phi\) is a \(\sigma\)-linear automorphism of \(V\). As \(V\) is cyclic as a module over the afore-mentioned skew polynomial ring \(F[\sigma]\), there is a polynomial \(f\) over it such that \(V\cong F[\sigma]/F[\sigma]f\). One has the Newton polygon associated to \(f\); this is the convex hull of the points \((0,0)\) and \((i,-v(a_i))\) where \(a_i\)'s are the coefficients of \(f\) (this notation differs slightly from the usual one where one considers \((i,v(a_i))\)). The author considers the group \(\text{SL}_3(F)\) and looks at a stratification of it which is associated to the slope sequence of a Newton polygon. Indeed, if \(A\in\text{SL}_3(F)\), then \(\Phi=A\sigma\) gives an isocrystal on \(F^3\) and the Newton polygon is considered for the associated characteristic polynomial. Thus, \(\text{SL}_3(F)\) is the disjoint union of the so-called Newton strata \(G_\lambda\), parametrized by different slope sequences \(\lambda\) arising as \(A\) varies in \(\text{SL}_3(F)\). The set \(\mathcal N\) of slope sequences is in bijective correspondence with \(\sigma\)-conjugacy classes in the group. The author provides two versions of a formula for the codimensions of Newton strata inside each component of the affine Bruhat decomposition on \(I\widetilde WI\) where \(I\) is the Iwahori subgroup of \(\text{SL}_3(F)\) and \(\widetilde W\) is the affine Weyl group. For a particular double coset \(IxI\), there is a corresponding subset \(\mathcal N(x)\) of the finitely many possible slope sequences \(\lambda\) such that \(G_\lambda\cap IxI\neq\emptyset\). The author calls certain subsets of \(IxI\) to be admissible; one can define their codimension. For slopes \(\mu\leq\lambda\) in \(\mathcal N(x)\), the notion of length from \(\mu\) to \(\lambda\) turns out to be well-defined; it is the length of any maximal chain from \(\mu\) to \(\lambda\). For any element \(x\) of the affine Weyl group, one has a unique maximal element \(\nu_x\) in \(\mathcal N(x)\); one has \(\lambda\leq\nu_x\) for all \(\lambda\in\mathcal N(x)\). With these notations, the author proves: Fix \(x\) in the affine Weyl group. Then, for any slope sequence \(\lambda\in\mathcal N(x)\), the subset \(G_{\leq\lambda}\cap IxI\) of \(IxI\) is admissible, and \[ \text{codim}((G_{\leq\lambda}\cap IxI)\subseteq IxI)=\text{length}[\lambda,\nu_x]. \] Further, the closure of a given Newton stratum \(G_\lambda\cap IxI\) in \(IxI\) is precisely \(G_{\leq\lambda}\cap IxI\). Interpreting the above theorem root-theoretically, the author deduces alternative formulae for the codimensions. For the group \(\text{SL}_3(F)\) studied by the author, the theorem above gives information about the Deligne-Lusztig varieties \(X_x(b):=\{g\in\text{SL}_3(F)/I:g^{-1}b\sigma g\in IxI\}\) for \(b\in\text{SL}_3(F)\). Finally, the author's results give rise to a conjectural relationship between the dimension of the Deligne-Lusztig variety \(X_x(b)\) and the codimension of the associated set of Newton strata.
scientific article; zbMATH DE number 5620112
scientific article