Generalized affine Springer fibres (Q2894444)

The paper under review finds two generalizations of affine Springer theory adapted to the root valuation strata of Goresky-Kottwitz-MacPherson, both with pros and cons. Let \(G\) be a connected reductive group over \(\mathbb{C}\), \(A\) be a maximal torus in \(G\), \(\mathfrak{a}\) be the Lie algebra of \(A\), \(R\) be the set of roots. Let \(\mathcal{O}\) be the formal power series ring \(\mathbb{C}[[\epsilon]]\) and \(F\) be its fraction field \(\mathbb{C}((\epsilon))\). Fix an Iwahori subgroup \(I\) containing \(A(\mathcal{O})\). Let \(r\) be a given root valuation function.NEWLINENEWLINEIn the first generalization, the authors consider the subset \(C_r\) of \(v\in \mathfrak{g}(F)\) such that the relative position of Lie \(I\) and ad\((v)\)Lie \(I\) is less than or equal to the data of values of \(r\), in a certain suitable sense. The the generalized affine Springer fibre for a regular \(u\in\mathfrak{a}(F)\) is of the form NEWLINE\[NEWLINE Y_r(u)=\{g\in G(F)/I: g^{-1}u g\in C_r\}. NEWLINE\]NEWLINE The second generalization is easier to understand, which is of the form NEWLINE\[NEWLINE Z_{r,\lambda}(u)=\{g\in G(F)/K_{r,\lambda}: g^{-1}ug\in\Lambda_{r,\lambda}\}, NEWLINE\]NEWLINE where \(\Lambda_{r,\lambda}\) is what the authors call a root valuation lattice for \(r\), and \(K_{r,\lambda}\) is the connected normalizer of \(\Lambda_{r,\lambda}\) in \(G(F)\).NEWLINENEWLINEBesides, the paper also develops various linear versions of Katz's Hodge-Newton decomposition.