Dilogarithm and higher $\mathscr {L}$-invariants for $\mathrm {GL}_3(\mathbf {Q}_p)$

The primary purpose of this paper is to clarify the relation between previous results in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145], [Amer. J. Math. 141 (2019), pp. 661–703], and [Camb. J. Math. 8 (2020), p. 775–951] via the construction of some interesting locally analytic representations. Let $E$ be a sufficiently large finite extension of $\mathbf {Q}_p$ and $\rho _p$ be a $p$-adic semi-stable representation $\mathrm {Gal}(\overline {\mathbf {Q}_p}/\mathbf {Q}_p)\rightarrow \mathrm {GL}_3(E)$ such that the associated Weil–Deligne representation $\mathrm {WD}(\rho _p)$ has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one $(\varphi , \Gamma )$-modules shows that the Hodge filtration of $\rho _p$ depends on three invariants in $E$. We construct a family of locally analytic representations $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ of $\mathrm {GL}_3(\mathbf {Q}_p)$ depending on three invariants $\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3 \in E$, such that each representation in the family contains the locally algebraic representation $\mathrm {Alg}\otimes \mathrm {Steinberg}$ determined by $\mathrm {WD}(\rho _p)$ (via classical local Langlands correspondence for $\mathrm {GL}_3(\mathbf {Q}_p)$) and the Hodge–Tate weights of $\rho _p$. When $\rho _p$ comes from an automorphic representation $\pi$ of a unitary group over $\mathbf {Q}$ which is compact at infinity, we show (under some technical assumption) that there is a unique locally analytic representation in the above family that occurs as a subrepresentation of the Hecke eigenspace (associated with $\pi$) in the completed cohomology. We note that [Amer. J. Math. 141 (2019), pp. 611–703] constructs a family of locally analytic representations depending on four invariants ( cf. (4) in that publication ) and proves that there is a unique representation in this family that embeds into the Hecke eigenspace above. We prove that if a representation $\Pi$ in Breuil’s family embeds into the Hecke eigenspace above, the embedding of $\Pi$ extends uniquely to an embedding of a $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ into the Hecke eigenspace, for certain $\mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3\in E$ uniquely determined by $\Pi$. This gives a purely representation theoretical necessary condition for $\Pi$ to embed into completed cohomology. Moreover, certain natural subquotients of $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ give an explicit complex of locally analytic representations that realizes the derived object $\Sigma (\lambda , \underline {\mathscr {L}})$ in (1.14) of [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145]. Consequently, the locally analytic representation $\Sigma ^{\mathrm {min}}(\lambda , \mathscr {L}_1, \mathscr {L}_2, \mathscr {L}_3)$ gives a relation between the higher $\mathscr {L}$-invariants studied in [Amer. J. Math. 141 (2019), pp. 611–703] as well as the work of Breuil and Ding and the $p$-adic dilogarithm function which appears in the construction of $\Sigma (\lambda , \underline {\mathscr {L}})$ in [Ann. Sci. Éc. Norm. Supér. 44 (2011), pp. 43–145].