Compatibility of t_D^+ and γ_D^+ and determination of the Hodge filtration

Establish that, for every u in the set of minimal length representatives _n^{S_0}, the identity t_D^+ ∘ γ_D^+|_{\overline{Ext}^1_{S_0(u)}(D,D)} = f_D ∘ ι_{S_0(u)} holds under the splitting f_D of Proposition 4.?. Here t_D^+ is the surjection t_D^+: Ext^1_G(π_{\natural}^{}(\underline{φ},), π_1^+(\underline{φ},)) → Ext^{1,∘}_{\varphi^f}(D[1/t], D[1/t]) ⊕ End_{Hodge-filtration}(D) defined in the Further comments subsection, γ_D^+ is the map from the direct sum of parabolic deformation spaces \bigoplus_{u∈_n^{S_0}} \overline{Ext}^1_{S_0(u)}(D,D) to Ext^1_G(π_{\natural}^{}(\underline{φ},), π_1^+(\underline{φ},)), ι_{S_0(u)} is the natural inclusion \overline{Ext}^1_{S_0(u)}(D,D) ↪ \overline{Ext}^1(D,D), and g_D^+ is the natural surjection \bigoplus_{u∈_n^{S_0}} \overline{Ext}^1_{S_0(u)}(D,D) → \overline{Ext}^1(D,D). In particular, show that g_D^+ factors through γ_D^+ with g_D^+ = t_D^+ ∘ γ_D^+. Furthermore, ascertain that the kernel of t_D^+ determines the Hodge filtration ^H_\bullet(D) of the semistable (φ,Γ)-module D.

Background

The paper develops methods to capture p-adic Hodge parameters of a non-critical semistable (φ,Γ)-module D via explicit locally analytic representations and extension groups. A key construction is a conjectural representation π1^+(\underline{φ},) together with a surjective map t_D^+ from Ext^1_G(π{\natural}^{}(\underline{φ},), π_1^+(\underline{φ},)) to a direct sum of de Rham deformation data, which is expected to control Hodge-theoretic information.

The maps γD^+ and g_D^+ relate parabolic deformation spaces of D to the global extension space Ext^1_G(π{\natural}^{}(\underline{φ},), π_1^+(\underline{φ},)) and to \overline{Ext}^1(D,D), respectively, while f_D is a splitting of the natural exact sequence identifying de Rham and monodromy contributions to \overline{Ext}^1(D,D). The conjecture asks for precise compatibilities among these maps and for the reconstruction of the Hodge filtration from ker(t_D^+), which would strengthen the connection between locally analytic representation theory and p-adic Hodge theory in the semistable setting.