Isocrystals with G-Structure
For a connected reductive group \(G\) over \(F\), an isocrystal with \(G\)-structure (or \(G\)-isocrystal) is a tensor-functorial generalisation of an isocrystal, classified by \(B(G)\). They are the \(p\)-adic analogue of representations of \(G\) and the natural setting for the Newton stratification of Shimura varieties.
Definition
Let \(F\) be a non-archimedean local field with residue field \(\mathbb{F}_q\), \(\breve F\) the completion of \(F^{\mathrm{unr}}\), \(\sigma\) the Frobenius automorphism.
\(\mathrm{GL}_n\)-case (= classical isocrystal)
An isocrystal of dimension \(n\) over \(F\) is a pair \((V, F_V)\) where:
- \(V\) is a finite-dimensional \(\breve F\)-vector space of dim \(n\);
- \(F_V : V \to V\) is a \(\sigma\)-linear bijection.
Isomorphism classes (under \(\sigma\)-linear \(\breve F\)-linear maps) form a category.
Dieudonné-Manin classification. \(\mathrm{GL}_n\)-isocrystals up to isomorphism are classified by Newton polygons in \([0, n]\) (= multisets of \(n\) rational numbers with denominator \(\le n\)).
General \(G\)
A \(G\)-isocrystal is a tensor functor \[ \mathcal{I}_b : \mathrm{Rep}_F(G) \;\longrightarrow\; \mathrm{Isoc}(F), \] from finite-dimensional \(F\)-representations of \(G\) to isocrystals, compatible with tensor products and duals.
Concretely: given \(b \in G(\breve F)\), define \(\mathcal{I}_b(V, \rho) = (V \otimes \breve F, \rho(b) \circ \sigma_V)\). Two such functors are isomorphic iff the corresponding \(b\)'s are \(\sigma\)-conjugate in \(G(\breve F)\):
\[ \{G\text{-isocrystals}\} / \cong \;\stackrel{\sim}{\longrightarrow}\; B(G). \]
Equivalence via Tannakian formalism
The classification reduces to: the groupoid of \(G\)-isocrystals is equivalent to the action groupoid of \(\sigma\)-conjugacy classes \(B(G)\), with morphisms = elements of stabilisers \(\{g : g b \sigma(g)^{-1} = b\}\).
This makes the structure of \(B(G)\) intrinsically Tannakian: it classifies fiber functors on \(\mathrm{Rep}_F(G)\) with values in \(\mathrm{Isoc}(F)\).
Slopes and Newton
Given a \(G\)-isocrystal \(\mathcal{I}_b\):
- Apply to a representation \(\rho : G \to \mathrm{GL}_n\): \(\rho_*(\mathcal{I}_b)\) is a classical isocrystal with Newton polygon a specific multiset.
- The slope decomposition of \(\rho_*(\mathcal{I}_b)\) refines the \(G\)-data.
- The Newton point of \(b\) (as a rational cocharacter modulo \(W\)) is the universal slope datum, recovering all \(\rho_*\)-Newton polygons by composition.
Examples
\(\mathrm{GL}_n\)-isocrystals
Slope decomposition: \(V = \bigoplus_\lambda V_\lambda\) where \(V_\lambda\) is the maximal subspace on which \(F^k\) has all eigenvalues of valuation \(k \cdot \lambda\) (Dieudonné-Manin).
For a single rational slope \(\lambda = p/q\) (\(\gcd(p, q) = 1\)): the unique "simple" isocrystal of slope \(\lambda\) has dimension \(q\).
\(G = T\) a torus
\(T\)-isocrystals = \(\sigma\)-conjugacy classes in \(T(\breve F)\) \(\cong X_*(T)^\Gamma\); the Newton map is the inclusion \(X_*(T)^\Gamma \hookrightarrow X_*(T)_\mathbb{Q}^\Gamma\).
\(\mathrm{Sp}_{2n}\)-isocrystals
Slope decompositions are constrained by the symplectic structure: the multiset of slopes \(\{\lambda_i\}\) satisfies \(\lambda_i + \lambda_{2n+1-i} = 0\) (symmetric around \(0\)).
Hodge structure analogues
A Hodge \(G\)-structure on an \(G\)-isocrystal: a filtration of the underlying vector spaces compatible with the \(G\)-action. For \(\mathrm{GL}_n\), this is the classical Hodge filtration.
Mazur-Kottwitz inequality: the Hodge polygon lies below the Newton polygon (with equal endpoints). This is a refinement of Mazur's classical Hodge-Newton theorem to \(G\)-structures.
Role in Langlands
- Shimura varieties: the reduction modulo \(p\) of a Shimura variety \(\mathrm{Sh}_K\) is stratified by the Newton stratification of \(B(G)\); each stratum corresponds to a class of \(G\)-isocrystals.
- Rapoport-Zink spaces: the deformation spaces of (\(p\)-divisible groups with \(G\)-structure) are parametrised by elements of \(B(G)\); their geometry is studied via the affine Deligne-Lusztig variety.
- Local Langlands and isocrystals: the Fargues-Fontaine curve realises \(G\)-isocrystals geometrically and gives a new approach to local Langlands.
References
- Kottwitz, Isocrystals with additional structure I, II (Compositio 1985, 1997).
- Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996).
- Rapoport–Zink, Period spaces for p-divisible groups (Annals of Math.\ Studies 1996).
- Fargues–Fontaine, Courbes et fibrés vectoriels en théorie de Hodge p-adique (Astérisque 2018).