p. via the construction of some interesting locally analytic representations. Let E be a sufficiently large finite extension of Q p and ρ p be a p-adic semi-stable representation Gal(Q p /Q p ) → GL 3 (E) such that the associated Weil-Deligne representation WD(ρ p ) has rank two monodromy and the associated Hodge filtration is non-critical. A computation of extensions of rank one (ϕ, Γ)-modules shows that the Hodge filtration of ρ p depends on three invariants in E. We construct a family of locally analytic representations Σ min (λ, L 1 , L 2 , L 3 ) of GL 3 (Q p ) depending on three invariants L 1 , L 2 , L 3 ∈ E, such that each representation in the family contains the locally algebraic representation Alg ⊗ Steinberg determined by WD(ρ p ) (via classical local Langlands correspondence for GL 3 (Q p )) and the Hodge-Tate weights of ρ p . When ρ p comes from an automorphic representation π of a unitary group over 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 π) 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 Π in Breuil's family embeds into the Hecke eigenspace above, the embedding of Π extends uniquely to an embedding of a Σ min (λ, L 1 , L 2 , L 3 ) into the Hecke eigenspace, for certain L 1 , L 2 , L 3 ∈ E uniquely determined by Π. This gives a purely representation theoretical necessary condition for Π to embed into completed cohomology. Moreover, certain natural subquotients of Σ min (λ, L 1 , L 2 , L 3 ) give an explicit complex of locally analytic representations that realizes the derived object Σ(λ, L ) in (1.14) of [Ann. Sci.Éc. Norm. Supér. 44 (2011), pp. 43-145]. Consequently, the locally analytic representation Σ min (λ, L 1 , L 2 , L 3 ) gives a relation between the higher 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 Σ(λ, L ) in [Ann. Sci.Éc. Norm. Supér. 44 (2011), pp. 43-145].

Let p be prime number and K be a p-adic field. We systematically compute the higher Ext-groups between locally analytic generalized Steinberg representations (LAGS for short) of GLn(K) via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each Ext-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their Ext-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an Ext-group, and then describe the cup product maps between such Ext-groups using these bases. As an application, we generalize Breuil's L -invariants for GL2(Qp) and Schraen's higher L -invariants for GL3(Qp) to GLn(K). Along the way, we also establish a generalization of Bernstein-Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik-Strauch, generalizing a result in Schraen's thesis for GL3(Qp).

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