Abstract. Let L be a finite extension of Qp , and ρL be an n-dimensional semi-stable non-crystalline representation of the absolute Galois group GalL of L. Suppose ρL apb L ) be a p-adic Banach representation of pears on the patched eigenvariety, and let Π(ρ GLn (L) associated to ρL (via global method). In this paper, we prove that some of the Fontaine-Mazur L-invariants of ρL (which we call simple L-invariants) can be found, in terms of Breuil’s L-invariants (as parameters of extensions of certain locally analytic b L ). generalized Steinberg representations), in the locally analytic subrepresentation of Π(ρ

Contents 1. Introduction

2

2. Breuil’s L-invariants

4

2.1. Preliminaries and notations

4

2.2. Some extensions of locally analytic representations

7

3. Fontaine-Mazur L-invariants

17

3.1. Fontaine-Mazur L-invariants

17

3.2. A sub candidate in p-adic local Langlands programme

18

3.3. Colmez-Greenberg-Stevens formula

20

3.4. Trianguline deformation of special (ϕ, Γ)-modules

21

4. Local-global compatibility

25

4.1. Patched eigenvariety

25

4.2. Local-global compatibility

29

References

38

The author is supported by EPSRC grant EP/L025485/1. 1

1. Introduction This paper presents some results in p-adic Langlands programme (cf. [3]), where a fundamental problem is to find the lost information, when passing from an n-dimensional (de Rham) p-adic Galois representation ρL : GalL → GLn (E) to its associated Weil-Deligne representation WD(ρL ), on the automorphic side, e.g. in the Banach representations or locally Qp -analytic representations of GLn (L). Here L is a finite extension of Qp , E is our coefficient field, which is also finite over Qp but sufficiently large. Fontaine-Mazur L-invariants. In this paper, we consider the case where ρL is semistable non-crystalline and non-critical with the monodromy operator N on Dst (ρL ) satisfying N n−1 6= 0. In this case, the lost information can be explicitly described via the so-called Fontaine-Mazur L-invariants. We study some of these Fontaine-Mazur L-invariants in this paper, which are, roughly speaking, those that can be seen in the trianguline deformations of ρL (via the Colmez-Greenberg-Stevens formula), and we call such L-invariants simple L-invariants. In fact, for each i ∈ ∆ = {1, · · · , n − 1} (the set of simple roots of GLn ), one can associate to ρL a dL := [L : Qp ]-dimensional E-vector subspace (cf. §3.1, where L(ρL )i is denoted by L(Di ), and D = Drig (ρL )) (1)

L(ρL )i ⊂ Hom(L× , E),

where the latter denotes the E-vector space of E-valued additive characters on L× and thus is (dL + 1)-dimensional. Note that using certain specific basis of Hom(L× , E), each L(ρL )i will correspond to a dL -tuple (Li,σ )σ∈ΣL ∈ E dL (where ΣL denotes the set of embeddings of L in Qp ), which then give (some of) the usual Fontaine-Mazur L-invariants of ρL (see §3.1 for details). Breuil’s L-invariants. First introduce some notation. Let α ∈ E × such that {α, qL α, · · · , qLn−1 α} are the eigenvalues of ϕfL on Dst (ρL ), where fL denotes the unramified degree of L over Qp , and qL := pfL . Let λ be the dominant weight of T (L) associated to the HodgeTate weights HT(ρL ) of ρL . Let St(λ) (resp. Σ(λ)) denote the locally algebraic (resp. locally Qp -analytic) Steinberg representation of weight λ, and for a parabolic subgroup P ⊃ B (the Borel subgroup of lower triangular matrices), denote by vP∞ (λ) the locally algebraic generalized Steinberg representation of GLn (L) with respect to P (see §2.1.2 for details). Let St(λ, α) := St(λ) ⊗E unr(α) ◦ det, Σ(λ, α) := Σ(λ) ⊗E unr(α) ◦ det, and vP∞ (λ, α) := vP∞ (λ) ⊗E unr(α) ◦ det, where unr(α) denotes the unramified character of L× sending uniformizers to α. For i ∈ ∆, denote by P {i} ) B the minimal parabolic subgroup with respect to i. A key fact is the following isomorphism (cf. Cor. 2.12) (2) Ext1GLn (L) vP∞{i} (λ, α), Σ(λ, α) ∼ = Hom(L× , E), 2

where the Ext1 is taken in the category of locally Qp -analytic representations of GLn (L). Again using certain specific basis of Hom(L× , E), the extensions of vP∞ (λ, α) by Σ(λ, α) {i}

can thus be parametrized by elements in PdL +1 (E), which we call Breuil’s (simple) Linvariants. For ψ ∈ Hom(L× , E), denote by Σi (λ, α, E ·ψ) the associated extension of vP∞ (λ, α) by {i}

Σ(λ, α) via (2). More generally, for an E-vector subspace V of Hom(L× , E) of dimension d, let {ψ1 , · · · , ψd } be a basis of V , and we put Σi (λ, α, V ) := Σi (λ, α, E · ψ1 ) ⊕Σ(λ,α) Σi (λ, α, E · ψ1 ) ⊕Σ(λ,α) · · · ⊕Σ(λ,α) Σi (λ, α, E · ψd ) which is thus an extension of vP∞ (λ, α)⊕d by Σ(λ, α), and is independent of the choice {i}

of the basis of V (only depends on V ). In fact, in this way, we obtain a one-to-one correspondence between d-dimensional E-vector subspaces V of Hom(L× , E) and extensions Σi (λ, α, V ) of vP∞ (λ, α)⊕d by Σ(λ, α) satisfying that any sub-extension in Σi (λ, α, V ) of {i}

vP∞ (λ, α) by Σ(λ, α) is non-split, i.e. socGLn (L) Σi (λ, α, V ) = socGLn (L) Σ(λ, α). {i}

Local-global compatibility. We put Σ(λ, α, L(ρL )) := Σ1 (λ, α, L(ρL )1 ) ⊕Σ(λ,α) Σ2 (λ, α, L(ρL )2 ) ⊕Σ(λ,α) · · · ⊕Σ(λ,α) Σn−1 (λ, α, L(ρL )n−1 ), which is an extension of ⊕i∈∆ vP∞ (λ, α)⊕dL by Σ(λ, α), and contains exactly the informa{i} tion of WD(ρL ), HT(ρL ), {L(ρL )i }i∈∆ . The precedent discussion is purely local. Suppose now ρL appears on the patched eigenvariety [9], in particular, to ρL is associated an admissible unitary Banach representation b L ) in [10] (where the method is global ), which is believed to be the right represenΠ(ρ tation (up to multiplicities) corresponding to ρL . Using the local-global compatibility in the classical local Langlands correspondence, we have (3)

b L ). St(λ, α) ,−→ Π(ρ

The following theorem is the main result of the paper Theorem 1.1 (Local-global compatibility, cf. Thm. 4.11). The injection (3) extends uniquely to a morphism b L ). Σ(λ, α, L(ρL )) −→ Π(ρ Moreover, for i ∈ ∆, ψ ∈ Hom(L× , E), (3) can extend to a morphism b L) Σi (λ, α, E · ψ) −→ Π(ρ if and only if ψ ∈ L(ρL )i . b L )an . Such a result in modular curve In particular, {L(ρL )i }i∈∆ can be found in Π(ρ case was proved by Breuil ([4]) using modular symbols. In [12], we proved similar results 3

for quaternion Shimura curves, using global triangulation theory, and we adapt this strategy in the current paper. Note also that when n ≥ 3, ρL can not be recovered from {WD(ρL ), HT(ρL ), {L(ρL )i }i∈∆ }, and we refer to [6] for a conjectural approach on finding other lost information in the locally Qp -analytic representations of GLn (L). We refer to the body of the text for more detailed and more precise statements (with slightly different notation). Organisation. In §2, we study some extensions of locally analytic generalized Steinberg representations, and in particular, we prove (2) (cf. Cor. 2.12). In §3, we recall the Fontaine-Mazur L-invariants, and the Colmez-Greenberg-Stevens formula. This section also includes a study of trianguline deformations of semi-stable non-crystalline representations (which were often excluded in the previous literature). In §4, we prove Thm. 1.1. Acknowledgement. I would like to thank Benjamin Schraen for answering my questions.

2. Breuil’s L-invariants 2.1. Preliminaries and notations. 2.1.1. Locally analytic representations. Recall some locally analytic representation theory. Let G be a strictly paracompact locally Qp -analytic group (where strictly paracompact means that any open covering of G can be refined into a covering by pairwise disjoint open subsets), and denote by D(G) the locally convex E-algebra of E-valued distributions on G (cf. [23, Prop. 2.3]). Let M(G) denote the abelian category of (abstract) D(G)-modules. Note that one can embed G into D(G) by sending g ∈ G to δg := [f 7→ f (g)] with f ∈ C la (G, E), where C la (G, E) denotes the E-vector spaces of E-values locally analytic functions on G. Let V be a locally (Qp -)analytic representation of G over E (cf. [23, §3]), the continuous dual V ∨ is naturally equipped with a D(G)-module structure: (4)

(µ · w)(v) = µ([g 7→ w(gv)])

for µ ∈ D(G), w ∈ V ∨ , v ∈ V , and note that this action is well defined since the function [g 7→ w(gv)] on G is locally analytic. As in [27, D´ef. 3.1], for locally Qp -analytic representations V , W of G over E, we put (5)

ExtrG (V, W ) := ExtrM(G) (W ∨ , V ∨ ),

where the latter denotes the r-th extension group in the abelian category M(G). Let Z 0 be a locally Qp -analytic closed subgroup of the center Z of G, χ be a locally Qp -analytic character of Z 0 , we denote by M(G)Z 0 ,χ the subcategory of M(G) of D(G)-modules on 4

which Z 0 acts by χ where the Z 0 -action is induced by Z 0 ,→ G ,→ D(G) . For locally Qp -analytic representations V , W of G over E on which Z 0 acts via χ, we put ExtrG,Z 0 =χ (V, W ) := ExtrM(G)Z 0 ,χ (W ∨ , V ∨ ) where the latter denotes the r-th extension group in M(G)Z 0 ,χ . In the case where Z 0 = Z, we denote M(G)χ := M(G)Z,χ , and ExtrG,χ (V, W ) := ExtrG,Z=χ (V, W ). Suppose that G/Z 0 is also a strictly paracompact locally Qp -analytic group, and G ∼ = G/Z 0 × Z 0 as Qp -analytic manifolds, then we have equivalence of categories between M(G)Z 0 ,1 and M(G/Z 0 ), and hence in this case we have Extr 0 (V, W ) ∼ = Extr 0 (V, W ). G,Z =1

G/Z

One easily sees that similar results hold in the case where χ (as a character of Z 0 ) can lift to a character of G. Remark 2.1. (i) Let V be a finite dimensional locally Qp -analytic representation of G over E, then we have (cf. [27, Cor. 3.3]) ExtrG (1, V ) ∼ = Hran (G, V ) where Hran (G, V ) denotes the locally analytic group cohomology defined by Casselman and Wigner. In particular, H1an (G, E) ∼ = Hom(G, E), the E-vector space of continuous Evalued characters on G. (ii) Let V and W be locally analytic representations of G such that W ∨ and V ∨ are finitely presented D(H)-modules (thus they are both coadmissible D(H)-modules, cf. [24, Cor. 3.4 (iv)]) for a certain uniform pro-p open compact subgroup H of G, then any extension of V ∨ by W ∨ is also a finitely presented D(H)-module (thus a coadmissible D(H)-module). Thus in this case, by the equivalence of categories ([24, Thm. 6.3]), Ext1G (V, W ) consists exactly of (admissible) locally analytic representations which are extensions of W by V . Similarly, in this case, Ext1G,Z 0 =χ (V, W ) consists of extensions of W by V on which Z 0 acts via the character χ. Let D∞ (G) denote the strong continuous dual of the E-vector space (equipped with the finest locally convex topology) of locally constant E-valued functions on G, which is in fact the quotient algebra of D(G) by the ideal generated by the Lie algebra of G. The map g 7→ δg induces in fact an injection G ,→ D∞ (G). Let M(G)∞ denote the category of D∞ (G)-modules, which is thus a subcategory of M(G). For a locally analytic representation V of G, it’s easy to see V ∨ ∈ M(G)∞ if and only if the G-action on V is smooth. For a locally Qp -analytic closed subgroup Z 0 ⊆ Z, let χ be a smooth character ∞ 0 of Z 0 , denote by M(G)∞ Z 0 ,χ the category of D (G)-modules on which Z acts via χ, and ∞ we denote M(G)∞ χ := M(G)Z,χ . Let Rep∞ (G) resp. Rep∞ Z 0 =χ (G) denote the category of smooth representations of G over E (resp. smooth representations of G over E on which Z 0 , a locally analytic closed subgroup of Z, acts via the character χ), we have the following functor Rep∞ (G) → M(G)∞ , V 7→ V ∨ 5

where V ∨ denotes the abstract dual of V , which coincides with the continuous dual of V (as D∞ (G)-modules) if V is equipped with the finest locally convex topology, and where the D∞ (G)-action on V ∨ is given by the same way as in (4). This functor is fully faithful. As in [25, §1], we have also the following functor M(G)∞ → Rep∞ (G), M 7→ M sm := lim M U −→

(6)

U

where U runs over compact open subgroups of G (note that M is naturally equipped with a G-action induced by the embedding G ,→ D∞ (G)). This functor is obviously left exact. On the other hand, by another description of this functor as in [25, Lem. 1.3], we see that this functor is also right exact (and hence exact). Recall Rep∞ (G) has enough projective and injective objects. Lemma 2.2. Let V , W ∈ Rep∞ (G), then we have natural isomorphisms Extr ∞ (V, W ) ∼ = Extr ∞ (W ∨ , V ∨ ), ∀r ∈ Z≥0 . Rep (G)

M (G)

0

Similarly, let Z be a locally analytic closed subgroup of Z and χ be a smooth character of Z 0 , suppose V , W ∈ Rep∞ Z 0 =χ (G), then ExtrRep∞0

Z =χ

(G) (V, W )

∼ = ExtrM∞ (G)Z 0 ,χ (W ∨ , V ∨ ), ∀r ∈ Z≥0 .

Proof. It’s sufficient to prove that any injective object V ∈ Rep∞ (G), V ∨ is projective in M∞ (G). Let f : M N be an epimorphism in M∞ (G), and g : V ∨ → N , we prove that g can lift to a morphism from V ∨ to M . Since V ∨ = (V ∨ )sm , g factors through g sm : V ∨ → N sm . Since the functor (6) is exact, f induces a surjective map f sm : M sm → N sm , and by taking the dual, we get an injective morphism (in Rep∞ (G)): (f sm )∨ : (N sm )∨ ,→ (M sm )∨ . Since V is an injective object, (g sm )∨ : (N sm )∨ → V factors through a morphism h : (M sm )∨ → V . Taking again the dual, we get the desired morphism h∨ : V ∨ → M sm → M . The first part of the lemma follows. The second part follows by the same argument. For a Lie algebra g overQ L, σ ∈ ΣL , let gσ := g ⊗L,σ E (which is thus a Lie algebra over E); for J ⊆ ΣL , let gJ := σ∈J gσ . In particular, we have gΣL ∼ = g ⊗Qp E. 2.1.2. Notations. In the following, let G = GLn (L), and Z be the center of GLn (L). For a closed subgroup H of G containing Z, we put H := H/Z. Let ∆ be the set of simple roots of GLn (with respect to the Borel subgroup B of upper triangular matrices), and we identify the set ∆ with {1, · · · , n − 1} such that i ∈ {1, · · · , n − 1} corresponds to the simple root (x1 , · · · , xn ) ∈ t 7→ xi − xi+1 , where t denotes the Lie algebra of the torus T of diagonal matrices. To I ⊂ ∆, we can associate a parabolic subgroup PI (L) of G containing B(L) note P∆ (L) = G, P∅ (L) = B(L) , with NI (L) the nilpotent radical of PI (L), and LI (L) the (unique) Levi subgroup of PI (L) containing T (L). Let P I (L) be the parabolic subgroup opposite to PI (L), and N I (L) be the nilpotent radical of P I (L). Note LI (L) is also the unique Levi subgroup of P I (L) 6

containing T (L). Let pI , nI , lI , pI , nI be the Lie algebra (over L) of PI (L), NI (L), LI (L), P I (L), N I (L) respectively, and let b := p∅ , b := p∅ . Let λ := (λ1,σ , · · · , λn,σ )σ∈ΣL be a weight of tΣL . For I ⊆ ∆ = {1, · · · , n − 1}, we call that λ is PI -dominant (with respect to B) if λi,σ ≥ λi+1,σ for all i ∈ I and σ ∈ ΣL , and we denote by XI+ the set of PI -dominant weights of tΣL . If λ is PI -dominant, there exists a unique irreducible algebraic representation, denoted by Fλ,I , of ResLQp LI with highest weight λ. Denote χλ := Fλ,∅ , and Fλ := Fλ,∆ . We use λ to denote the central character of Fλ . Let λ be a dominant integer weight of tΣL . Put IG (λ) := PI iPGI (λ) :=

Qp −an IndG F , P I (L) λ,I ∞ IndG 1 ⊗E F λ . P I (L)

Note that iG (λ) is the locally algebraic subrepresentation of IG (λ). And if J ⊇ I, we PI PI G G G G have natural injections IP (λ) ,→ IP (λ), iP (λ) ,→ iP (λ). Put J I J I X (λ)/ vPanI (λ) := IG IPGJ (λ) , PI J)I

(λ)/ vP∞I (λ) := iG PI

X

iPGJ (λ) .

J)I

For simplicity, we put IPG := IG (0), iG := iPG (0), vPan := vPan (0), and vP∞ := vP∞ (0). PI PI I I I I I I × × (λ) ⊗ unr(α) ◦ det where unr(α) : L → E is Let α ∈ E × , put IPG (λ, α) := IG E PI I G an unramified character sending uniformizers to α. We define representations iP (λ, α), I vPan (λ, α), vP∞ (λ, α) in a similar way, and we let St(λ, α) := vB∞ (λ, α). I

I

Let I ⊆ ∆. Recall that the Orlik-Strauch functor (cf. [22]) associates, to an object M in pI,Σ

the BGG category Oalg L together with a finite length smooth admissible representation π of LI (L), a locally Qp -analytic representation FPG (M, π). It was proved in [22] that I FPG (M, π) is strongly admissible. The following proposition is due to Schraen. I

Proposition 2.3. Keep the above notation, the continuous dual FPG (M, π)∨ is a finitely I presented D(H)-module for some (or any) uniform prop-p compact open subgroup H of G. Consequently, the continuous dual of all the locally analytic representations that we consider in the following section is finitely presented over D(H), and hence by Rem. 2.1 (ii), the Ext1G defined in (5) will give exactly the extensions that we want. 2.2. Some extensions of locally analytic representations. We study some extensions of locally analytic generalized Steinberg representations in this section. In particular, 7

we show that for any i ∈ ∆, and any dominant weight λ of tΣL , the Ext1 of vPan{i} (λ) by Σ(λ) can be parametrized via Breuil’s L-invariants (cf. Cor. 2.12). Proposition 2.4. We have (7)

ExtiG

G ∼ (λ), (λ) iG I = PI PJ

( Hian LJ (L), E 0

if J ⊆ I , otherwise

and (8)

ExtiG,λ

iG (λ), IPGJ (λ) PI

( i H L (L)/Z, E J an ∼ = 0

if J ⊆ I . otherwise

Proof. We only prove (8) (for the case with central character λ), since (7) follows by the same argument. By [27, Cor. 4.9], we have a spectral sequence: G G (λ)), Fλ,J ⇒ Extr+s E2r,s = ExtrLJ (L),Z=λ Hs (N J , iG G,λ iP I (λ), IP J (λ) . PI By [27, (4.41)], Hs (N J , iG (λ)) = Hs (nJ,ΣL , Fλ ) ⊗E JP J (iPG ) where JP J (iG ) denotes the PI PI I Jacquet module of iG with respect to the parabolic subgroup P J . By [27, Thm. 4.10], PI one has an isomorphism of algebraic representations of lJ,ΣL (thus of ResLQp LJ ): M (9) Hs (nJ,ΣL , Fλ ) = Fw·λ,J . w∈Sn−1 ,lg(w)=s, w·λ∈XJ+

Thus by [27, Prop. 4.7 (1)], for all s > 1, r ≥ 0, ExtrLJ (L),Z=λ Hs (N J , iPGI (λ)), Fλ,J = 0. Consequently, for all r ∈ Z≥0 , we have r G G ∼ (i ExtrG,λ iG (λ), I (λ) Ext ) ⊗ F , F J . = E λ,J λ,J L (L),Z=λ P PI PJ PI J J We first reduce to the case λ = 0: By [27, (4.26)] (with a slight variation), we have r G ∨ ∼ ExtrLJ (L),Z=λ JP J (iG ) ⊗ F , F Ext (i ), F ⊗ F J = E λ,J λ,J λ,J E λJ . LJ (L),Z=1 PJ PI PI ∨ of LJ (L) is semi-simple, in which However, the algebraic representation Fλ,J ⊗E Fλ,J the trivial representation has multiplicity one. Thus by [27, Prop. 4.7 (i)], we get ∨ ∼ ExtrLJ (L),Z=1 JP J (iG ), Fλ,J ⊗E Fλ,J = ExtrLJ (L),Z=1 JP J (iPGI ), 1 . PI As in the proof of [20, Prop. 15], we define a filtration on iPG by P J (L)-invariant subspaces: I n o , Supp(f ) ⊆ ∪ F k iPGI := f ∈ iG P (L)\ P (L)wP (L) , I J w∈WI \W/WJ I PI lg(w)≤k

for k ∈ Z≥0 ,where lg(w) is the length of its Kostant-representative (which is the one of minimal length with its double coset). In the following, we identify the double cosets 8

WI \W/WJ with its Kostant-representatives. As in the proof of [20, Prop. 15], we have M ∞ L (L) ∼ JP J (grk iG ) IndLJ (L)∩(w−1 P (L)w) γw , = PI I J w∈WI \W/WJ , lg(w)=k

where grk iG := F k iG /F k−1 iG and γw is the modulus character of P J (L) ∩ w−1 P I (L)w PI PI PI acting on N J (L)/ N J (L) ∩ w−1 P I (L)w . As in the proof of [20, Prop. 15] (by comparing the central characters), we have LJ (L) ∞ γ ) , 1 =0 (10) ExtrRep∞ (Ind w −1 (L (L)) J L (L)∩(w P w) Z=1 I

J

for all r ∈ Z≥0 , if J * I or w 6= 1. On the other hand, we have a spectral sequence (e.g. see [27, (4.27)]) s ∨ ∨ ExtrM(LJ (L))∞ V , H (l ) , E ⊗ W ⇒ Extr+s J Σ E L LJ (L),Z=1 (W, V ). Z,1 for smooth representations W , V of LJ (L) with trivial character on Z, where lJ denotes the Lie algebra of LJ (L)/Z. By Lem. 2.2, we have ∨ r s s ∨ ∼ ∨ ∞ (L (L)) W ⊗E H ,V . ExtrM(LJ (L))∞ ) , E ⊗ W Ext ) , E V , H (l (l = J Σ E J Σ Rep L L J Z=1 Z,1 These combined with (10) imply L (L)

ExtrLJ (L),Z=1 (IndLJ (L)∩(w−1 P J

I

∞ γ ) , 1 =0 w w)

for all r ∈ Z≥0 , if J * I or w 6= 1. From which, we deduce that if J * I, ExtrLJ (L),Z=1 JP J (iPGI ), 1 = 0, and if J ⊆ I, (see Rem. 2.1 (i) for the second isomorphism) ), 1 ∼ ExtrLJ (L),Z=1 JP J (iG = ExtrLJ (L),Z=1 (1, 1) ∼ = Hran LJ (L), E , PI the proposition follows.

Remark 2.5. (i) Denote by DJ (L) the derived subgroup of LJ (L) which is naturally isomorphic to the derived subgroup of LJ (L) = LJ (L)/Z , ZJ (L) the center of LJ (L), dJ the Lie algebra of DJ (L). Then by [27, Cor. 3.11, (3.28)], we have M (11) Hian (LJ (L), E) ∼ ∧r Hom(ZJ (L), E) ⊗E Hi−r (dJ )ΣL , E , = 0≤r≤i

(12)

Hian (LJ (L), E) ∼ =

M

∧r Hom ZJ (L), E

⊗E Hi−r (dJ )ΣL , E

0≤r≤i

Since dJ is semi-simple, we have Hr (dJ )ΣL , E = 0 for r = 1, 2. (ii) Let J ⊆ I, we describe explicitly the isomorphism in (7) for Ext1 : Let ψ ∈ H1an LJ (L), E ∼ = Hom ZJ (L), E , 9

.

which induces an extension 1ψ of the trivial characters of LJ (L): 1 ψ(a) 1ψ (a) = , 0 1 F ⊗E 1ψ IndG P J (L) λ,J

for all a ∈ LJ (L). The parabolic induction exact sequence

(λ) → IndPGJ (L) Fλ,J ⊗E 1ψ 0 → IG PJ

Qp −an

Qp −an

lies thus in an

pr

(λ) → 0. − → IG PJ

Since J ⊆ I, we have natural injections iG (λ) ,−→ IG (λ) ,−→ IPGJ (λ). PI PI Let IIJ (λ, ψ) := pr−1 iG (λ) , which is thus an extension of iG (λ) by IG (λ). We claim PI PI PJ J that II (λ, ψ) is mapped to ψ via the isomorphism (13) Ext1 iG (λ), IG (λ) ∼ = H1 LJ (L), E . G

PI

an

PJ

In fact, by the discussion in [27, §4.4], we see that (13) factors though the following composition ∼ (14) Ext1G iPGI (λ), IG (λ) −→ Ext1P J (L) iG (λ)|P J (L) , Fλ,J PJ PI ∼ ∼ −→ Ext1LJ (L) JP J (iPGI ) ⊗E Fλ,J , Fλ,J −→ Ext1LJ (L) Fλ,J , Fλ,J . where the first map sends V to the push-forward of V |P J (L) via the natural evaluation map IPG (λ) Fλ,J , f 7→ f (1), the second map is inverse of the map induced by the J natural projection iG (λ)|P J (L) JP J (iG ) ⊗E Fλ,J (note the latter one is no other than PI PI the N J (L)-covariant quotient of the first one), and the last map is induced by the injection Fλ,J ,→ JP J (iG ) ⊗E Fλ,J . It’s sufficient to prove that (14) sends IIJ (λ, ψ) to 1ψ ⊗E Fλ,J . PI The composition map P J (L)-invariant (15)

iG (λ) −→ IG (λ) − Fλ,J PI PJ

factors though (16)

JP J (iPGI ) ⊗E Fλ,J − Fλ,J . (16)

Moreover, it’s straightforward to see the composition Fλ,J ,→ JP J (iG ) ⊗E Fλ,J −−→ Fλ,J PI is the identity map (up to non-zero scalars). Thus the inverse of the last isomorphism in (14) can be induced by the projection (16), and hence the inverse of the composition of the last two maps in (14) is induced by (15), in other words, can factor as 1 G (λ)| , F −→ Ext i (λ)| , F . Ext1LJ (L) Fλ,J , Fλ,J −→ Ext1P J (L) IG λ,J λ,J P (L) P (L) P J (L) PJ PI J J However, by construction, IIJ (λ, ψ) is just the image of 1ψ ⊗E Fλ,J via the above composition, thus we see that (14) maps ψ to 1ψ ⊗E Fλ,J , and the claim follows. One sees moreover that [IIJ (λ, ψ)] ∈ Ext1G,λ iG (λ), IG (λ) is split if and only if the character ψ P P I

J

factors through LJ (L). 10

(iii) Recall that by [20, Prop. 15], we have ( ∧i Hom∞ LJ (L), E if J ⊆ I i G G ExtRep∞ (G) (iP I , iP J ) = , 0 otherwise where Hom∞ (LJ (L), E) denotes the E-vector space of smooth E-valued characters on LJ (L). One can similarly describe as in (ii) the isomorphism for Ext1 . On the other hand, the injection iPG ,→ IG induces an injection PJ J 1 G G G , i ,−→ Ext i , I . Ext1G iG G PI PJ PI PJ Using the explicit description for Ext1 in (ii), we see that the following diagram is commutative (where the horizontal maps are natural injections) Ext1Rep∞ (G)Z,1 iPG , iPG −−−→ Ext1G,1 iPG , IG I J I PJ ∼y ∼y . Hom∞ LJ (L), E −−−→ Hom LJ (L), E We also have similar results for iPG (λ), iPG (λ), IPG (λ). And consequently, using the I J J notation in (ii), we see that IIJ (λ, ψ) is induced from a locally algebraic extension of (λ) by iG (λ) if and only if the character ψ is smooth. iG P P I

J

Corollary 2.6. We have ExtrG

(17)

∼ (λ) vP∞I (λ), IG = PJ

( Hr−|∆\I| LJ (L), E an 0

if I ∪ J = ∆ , otherwise

and ExtrG,λ

(18)

(λ) ∼ vP∞I (λ), IG = PJ

( Hr−|∆\I| LJ (L), E an 0

if I ∪ J = ∆ . otherwise

Proof. We only prove (18), and the proof for (17) is parallel. Recall that we have a long exact sequence of locally algebraic representations 0 → iPG∆ (λ) → ⊕

G I⊆K, iP (λ) K |K|=n−2

G I⊆K, iP (λ) K |K\I|=1

→ ··· → ⊕

→ iG (λ) → vP∞I (λ) → 0. PI

The cohomology groups in the corollary can thus be identified with the hypercohomology groups of the complex (e.g. see the arguments in [27, §5.2]) h i Cn−1−|I| → Cn−2−|I| → · · · → C0 , with Cj = ⊕

G I⊆K, iP (λ). |K/I|=j K

We get thus a spectral sequence

E1r,s = ExtrG,λ ⊕

G G I⊆K, iP (λ), IP (λ) K J |K/I|=s

11

∞ G ⇒ Extr+s G,λ vP I (λ), IP J (λ) .

By the above proposition, for any r ∈ Z≥0 , the r-th row of the E1 -page of the spectral sequence is given by G (λ) (λ), IG ExtrG,λ iP P ∆

Hran

G (λ), IP (λ) ⊕ I∪J⊆K, ExtrG,λ iG P

→

J

K

|∆\K|=1

k LJ (L), E

→

···

J

→ ExtrG,λ iG P

I∪J

k ⊕ I∪J⊆K, Hran LJ (L), E

→

→

···

→

|∆\K|=1

Hran

(λ), IG (λ) P J

k LJ (L), E

If I ∪ J 6= ∆, the above sequence is exact, and hence ExtrG,λ vP∞ (λ), IG (λ) = 0 for all PJ I r ∈ Z≥0 ; if I ∪ J = ∆, only the elements on the |∆ \ I|-th column of the E1 -page can be non-zero, and hence r+|∆\I| Hran LJ (L), E ∼ vP∞I (λ), IPGJ (λ) . (λ), IPGJ (λ) ∼ = ExtrG,λ iG = ExtG,λ P∆ The corollary follows.

Corollary 2.7. We have natural isomorphisms r+|J c ∩I c | ExtrG vP∞I (λ), vPanJ (λ) ∼ Fλ , vPanJ∪I c (λ) , = ExtG r+|J c ∩I c | ExtrG,λ vP∞I (λ), vPanJ (λ) ∼ Fλ , vPanJ∪I c (λ) . = ExtG,λ Proof. We only prove the second isomorphism, and the first one follows by the same argument. By [21, Thm. 4.2], we have a long exact sequence (19) 0 → IG (λ) → ⊕ P∆

G J⊆K IP (λ) K |K|=n−2

G J⊆K IP (λ) K |K\J|=1

→ ··· → ⊕

→ IG (λ) → vPanJ (λ) → 0. PJ

Similarly as in the proof of the above corollary, one can identify the cohomology of vPan (λ) J with the hypercohomology of the complex h i C−(n−1−|J|) → C−(n−2−|J|) → · · · → C0 G J⊆K IP (λ). |K|=i+|J| K

with C−i = ⊕ (20)

E1r,−s = ⊕

J⊆K |K|=s+|J|

We have the following spectral sequence

r−s ∞ an (λ) ⇒ Ext v (λ), v (λ) . ExtrG,λ vP∞I (λ), IG G,λ PK PI PJ

By the above corollary, for r ∈ Z≥0 , the r-th row of the first page of the spectral sequence is thus given by (21) Hr−|∆\I| L∆ (L), E an

→

r−|∆\I| ⊕ J∪I c ⊆K Han LK (L), E

→

···

r−|∆\I| → Han LJ∪I c (L), E

|K|=n−2

k

k

k

r,−(n−1−|J|) E1

r,−(n−2−|J|) E1

r,−|I c ∩J c | E1

→

→

···

→

Consider the long exact sequence 0 → IG (λ) → ⊕ J∪I c ⊆K IG (λ) → · · · → IG (λ) → vPanJ∪I c (λ) → 0, P∆ PK P J∪I c |K|=n−2

which gives a spectral sequence E1r,−s = ⊕

(J∪I c )⊆K |K|=|J∪I c |+s

r−s an F ExtrG,λ Fλ , IG (λ) ⇒ Ext , v (λ) . λ G,λ PK P J∪I c 12

.

For r ∈ Z≥0 , the r-th row of the E1 -page of the spectral sequence is thus L∆ (L), E Hr−|∆\I| an

→

⊕ J∪I c ⊆K Hr−|∆\I| LK (L), E an

→

···

→ Hr−|∆\I| LJ∪I c (L), E an

|K|=n−2

k

k

r,−(n−1−|J∪I c |) E1

r,−(n−2−|J∪I c |) E1

→

→

···

→

k E1r,0

which is equal to the complex in (21). From which, one deduces r−|J c ∩I c | ExtrG,λ (Fλ , vPanJ∪I c (λ)) ∼ vP∞I (λ), vPanJ (λ) = ExtG,λ The corollary follows.

In the following, we calculate some extension groups in some specific cases. Proposition 2.8. Suppose |I| = 1, we have the following commutative diagram consisting of natural isomorphisms: ∼ Hom Z∆\I (L)/Z, E −−−→ Hom Z∆\I (L), E / Hom Z, E ∼y ∼y . ∼ Ext1G,λ vP∞ (λ), Σ(λ) −−−→ Ext1G vP∞ (λ), Σ(λ) I I In particular, dimE Ext1G vP∞ (λ), Σ(λ) = dL + 1. I

Proof. The bottom isomorphism follows from the fact that HomG vP∞ (λ), Σ(λ) = 0. The I top isomorphism is clear. We use the spectral sequence (20) to prove the left isomorphism. Indeed, we see by (18) that only the objects in the (1 − n)-th and (2 − n)-th columns of the E1 -page can be non-zero, where the (n − 1)-th row is given by H1an (L∆ (L)/Z, E) k

−→

E1n−1,1−n

1 −− −−−→

dn−1,1−n

H1an (L∆\I (L)/Z, E) k E1n−1,2−n

and the n-th row is given by H2an (L∆ (L)/Z, E) k

−→

E1n,1−n

−−1−−→

dn,1−n

H2an (L∆\I (L)/Z, E) k . E1n,2−n

From which (together with Rem. 2.5 (i)), the left isomorphism follows. One also has the following spectral sequence (obtained in the same way as (20)): E1r,−s = ⊕ J⊆K ExtrG vP∞I (λ), IG (λ) ⇒ Extr−s vP∞I (λ), vPanJ (λ) . G PK |K|=s+|J|

13

As above, by (17), only the objects in the (1 − n)-th and (2 − n)-th columns of the E1 page can be non-zero, and one has morphisms: dn−1,1−n : H1an (L∆ (L), E) −→ H1an (L∆\I (L), E), 1 dn,1−n : H2an (L∆ (L), E) −→ H2an (L∆\I (L), E). 1 From which (together with Rem. 2.5 (i)), we see that where d1n−1,2−n = 0 )/ Im(d1n−1,1−n ) ∼ Ker(dn−1,2−n = Hom(L∆\I (L), E)/ Hom(Z, E) 1 and d1n,1−n is given by the natural injection ∧2 Hom(Z, E) ,→ ∧2 Hom(Z∆\I , E). The right isomorphism follows. At last, the diagram commutes, since the two spectral sequences are compatible with respect to the natural transformations ExtrG,λ (−, −) → ExtrG (−, −). Remark 2.9. From the proof, one sees that the isomorphism (22) Ext1G,λ vP∞I (λ), Σ(λ) ∼ = Hom(L∆\I (L)/Z, E) is induced by the isomorphisms (see the proof of Cor. 2.6 for the second isomorphism) 1 ∞ G G G ∼ Ext1G,λ vP∞I (λ), Σ(λ) ∼ (λ), (λ) Ext (λ), (λ) v I i I . = Extn−1 = G,λ G,λ PI P ∆\I P∆ P ∆\I But it’s not clear to the author how to describe (22) explicitly. Proposition 2.10. Suppose |I| = 1, we have the following commutative diagram consisting of natural isomorphisms: ∼

Hom(T (L)/Z, E)/Hom(ZI (L)/Z, E) −−−→ Hom(T (L), E)/Hom(ZI (L), E) ∼ ∼ . (23) y y ∼ Ext1G,λ iPG (λ), Σ(λ) −−−→ Ext1G iG (λ), Σ(λ) PI I In particular, dimE Ext1G (iG λ), Σ(λ) = dL + 1. P I

Proof. The bottom isomorphism follows from the fact that HomG iG (λ), Σ(λ) = 0. The PI top isomorphism is clear. Using the long exact sequence (19) as in the proof of Cor. 2.7, we get a spectral sequence (λ), IG (λ) ⇒ Extr−s iG (λ), Σ(λ) , (24) E1r,−s = ⊕|K|=s ExtrG iG G PI PK PI

G resp. E1r,−s = ⊕|K|=s ExtrG,λ iPGI (λ), IPGK (λ) ⇒ Extr−s i (λ), Σ(λ) . G,λ PI

We only prove the right isomorphism of (23), since the left isomorphism follows by the same argument and the commutativity of the diagram follows from the compatibility of the above two spectral sequences via the natural transformations ExtrG,λ (−, −) → ExtrG (−, −). 14

By (7), only the 0-th and (−1)-th columns of the E1 -page of (24) can have non-zero terms. We see the 1-th row is given by Ext1G iPG (λ), IG (λ) −→ Ext1G iG (λ), IPG (λ) P P I I I ∅ k k d1,−1 1

E11,−1

E11,0

−−−→

thus by (7), E11,0 / Im(d1,−1 )∼ = Hom(T (L), E)/ Hom(ZI (L), E). 1 The 2-th row is given by G −→ Ext2G iG (λ), I (λ) Ext2G iPG (λ), IPG (λ) PI P∅ I I k k , d2,−1

E12,−1

E12,0

−−1−→

and we have Ext2G iPGI (λ), IPGI (λ) Ext2G iPGI (λ), IG (λ) P∅

∼ = ∧2 Hom(ZI (L), E), ∼ = ∧2 Hom(T (L), E).

So d12,−1 is injective. The proposition follows.

Remark 2.11. We describe explicitly the right isomorphism in (23). From the proof, this isomorphism is induced via the natural morphism Hom(T (L), E) ∼ = Ext1 iG (λ), IG (λ) −→ Ext1 iG (λ), Σ(λ) . G

PI

G

P∅

PI

Let ψ ∈ Hom(T (L), E), II∅ (λ, ψ) be the extension of IG (λ) by iPG (λ) associated to ψ as in P ∅

I

II∅ (λ, ψ)

via the the quotient map IG (λ) Rem. 2.5 (ii), then the push-forward S(λ, ψ) of P∅ G Σ(λ) gives the extension of Σ(λ) by iP (λ) associated to ψ via the right isomorphism in I (23). This extension is split if and only if ψ is induced from a character on ZI (L). Note also that by Rem. 2.5 (iii), S(λ, ψ) is induced by a locally algebraic extension of iG (λ) PI ∞ by vB (λ) if and only if the character ψ is smooth modulo Hom(ZI (L), E). Corollary 2.12. Suppose |I| = 1, the natural map Ext1G (vP∞I (λ), Σ(λ)) −→ Ext1G (iG (λ), Σ(λ)) PI is an isomorphism. In particular, we have an isomorphism (25)

∼

Hom(T (L), E)/ Hom(ZI (L), E) −→ Ext1G (vP∞I (λ), Σ(λ)).

Proof. Consider the exact sequence pr

0 → Ker(pr) → iG (λ) − → vP∞I (λ) → 0, PI which induces a long exact sequence 0 → HomG (vP∞I (λ), Σ(λ)) → HomG (iPGI (λ), Σ(λ)) → HomG (Ker(pr), Σ(λ)) → Ext1G (vP∞I (λ), Σ(λ)) → Ext1G (iPGI (λ), Σ(λ)) → Ext1G (Ker(pr), Σ(λ)). 15

Since HomG (Ker(pr), Σ(λ)) = 0, we get an injection Ext1G (vP∞I (λ), Σ(λ)) ,−→ Ext1G (iG (λ), Σ(λ)) PI which is moreover an isomorphism since both of these two E-vectors spaces are (dL + 1)dimensional by Prop. 2.8, 2.10. The second part follows thus from Prop. 2.10. Remark 2.13. Let ψ ∈ Hom(T (L), E), I = {i} ⊆ ∆, S(λ, ψ) ∈ Ext1G iG (λ), Σ(λ) PI be the extension associated to ψ as in Rem. 2.11. By Cor. 2.12, the pull-back of S(λ, ψ) via the natural injection Ker(pr) ,→ iG (λ) is split as an extension of Ker(pr) PI by Σ(λ) . Quotient by Ker(pr), one gets thus the extension, denoted by Σiψ (λ), of vP∞ (λ) I by Σ(λ) associated to ψ. Note that this extension only depends on the image of ψ in Hom(T (L), E)/ Hom(ZI (L), E), and Σiψ (λ) is induced from a locally algebraic extension of vP∞ (λ) by vB∞ (λ) if and only if ψ is smooth modulo Hom(ZI (L), E). I

For each i ∈ ∆, we have an isomorphism (26)

∼

ιi : Hom(L× , E) −→ Hom(T (L), E)/ Hom(Z{i} (L), E), ψ 7→ [(a1 , · · · , an ) 7→ ψ(ai /ai+1 )],

and let ςi denote the following projection which naturally factors though the quotient Hom(T (L), E)/ Hom(Z{i} (L), E) (27)

ςi : Hom(T (L), E) −→ Hom(L× , E), (ψ1 , · · · , ψn ) 7→ ψi − ψi+1 .

It’s straightforward to see that ςi ◦ ιi = id. Let (28)

ς := (ςi ) : Hom(T (L), E) −

Y

Hom(L× , E).

i∈∆

which factors through Hom(T (L), E)/ Hom(Z, E). Remark 2.14. Let 1ur ∈ Hom(L× , E) such that 1ur |O× = 0, and 1ur (p) = 1 thus L Hom∞ (L× , E) = E · 1ur ; and for σ ∈ ΣL , let logσ,p ∈ Hom(L× , E) such that logσ,p (p) = 0 log

σ

and logσ,p |O× equals the composition OL× −→ OL → − E. Thus 1ur and {logσ,p }σ∈ΣL form a L × × basis of Hom(L , E). For any non-smooth additive character ψ of L P over E, there exist × hence Lσ ∈ E for all σ ∈ ΣL and a ∈ E such that ψ = a 1 + σ∈ΣL Lσ logσ,p . By Rem. 2.13 and (26), these L-invariants, which we call Breuil’s simple L-invariants, just parametrize the locally Qp -analytic extensions of vP∞ (λ) by Σ(λ) which are not locally {i} algebraic. Finally, Let α ∈ E × , i ∈ ∆, one easily deduces from (25) an isomorphism ∼ Hom(T (L), E)/ Hom(Z{i} (L), E) −→ Ext1G vP∞{i} (λ, α), Σ(λ, α) . For ψ ∈ Hom(T (L), E)/ Hom(Z{i} (L), E), we denote by Σiψ (λ, α) := Σiψ (λ)⊗E unr(α)◦det, which is the induced extension via the above isomorphism. 16

3. Fontaine-Mazur L-invariants 3.1. Fontaine-Mazur L-invariants. First, let D be a rank 2 (ϕ, Γ)-module over RE := † RL ⊗Qp E (where RL := Brig,L denotes the robba ring of L), and suppose D is triangulable, i.e. there exist two continuous characters δ1 , δ2 : L× → E × such that D lies in an exact sequence 0 → RE (δ1 ) → D → RE (δ2 ) → 0. Suppose there exist kσ ∈ Z≥1 for all σ ∈ ΣL (i.e. we only consider the non-critical case in this note) such that Y δ := δ1 δ2−1 = unr(qL−1 ) σ kσ . σ∈ΣL

In this case, we have the following natural perfect pairing (via the cup product) ∪

h·, ·i : H1(ϕ,Γ) (RE (δ)) × H1(ϕ,Γ) (RE ) −→ H2(ϕ,Γ) (RE (δ)) ∼ = E, which induces an isomorphism H1(ϕ,Γ),e (RE (δ)) ∼ = H1(ϕ,Γ),g (RE )⊥ . Note also that we have natural isomorphisms H1 (RE ) ∼ = H1 (GalL , E) ∼ = Hom(L× , E), (ϕ,Γ)

which induces an isomorphism (29)

H1(ϕ,Γ),g (RE ) ∼ = Hom∞ (L× , E) ∼ = E · 1ur .

By our assumption, D0 := D ⊗ RE (δ2−1 ) is always semi-stable, and D0 is crystalline if and only if [D0 ] ∈ H1(ϕ,Γ),e (RE (δ)) ∼ = Hom∞ (L× , E)⊥ . Let L(D) be the E-vector subspace of Hom(L× , E) killed by [D0 ], thus dimE L(D) = dL . Remark 3.1. (i) Note that D is determined by δ1 , δ2 , L(D) . (ii) Recall (cf. Rem. 2.14) that {logp,σ }σ∈ΣL and 1ur form a basis of Hom(L× , E). By (29), if D0 is non-crystalline, then 1ur ∈ / L(D)., and we put for σ ∈ ΣL : Lσ := h[D0 ], logp,σ i/h[D0 ], 1ur i; if D0 is crystalline, then 1ur ∈ L(D), and we can associate to D the following data: (Lσ )σ∈ΣL ∈ PdL (E), with Lσ = h[D0 ], logp,σ i. The dL -tuple L := {Lσ }σ∈ΣL (which is only well defined up to ratio when D0 is crystalline) is called the Fontaine-Mazur L-invariants of D, and we see that D is determined by {δ1 , δ2 , L}. Indeed, when D is moreover associated to a Galois representation ρL of GalL , one can actually prove that our L coincide with the usual Fontaine-Mazur L-invariants of ρL obtained from the associated filtered (ϕ, N )-module Dst (ρL ) (e.g. by comparing the formulas in [29, Thm. 1.1] and [11, Thm. 0.1]). Note also that when D0 is non-crystalline, logσ,p,−Lσ := logp,σ −Lσ 1ur σ∈Σ form a basis of L(D). L

17

(iii) One can also define Fontaine-Mazur L-invariants in critical case and we refer to [11] for details. For technical reasons (e.g. on the geometry of the patched eigenvariety etc.), we only consider the non-critical case in this paper. Now let D be a rank n triangular (ϕ, Γ)-module, with {δi }i=1,··· ,n a trianguline parameter of D. We make the following assumption on (D, {δi }): Hypothesis 3.2. For all i = 1, · · · , n − 1, there exist ki,σ ∈ Z≥1 for all σ ∈ ΣL , such that Y −1 δi δi+1 = unr(qL−1 ) σ ki,σ . σ∈ΣL

For i ∈ ∆ = {1, · · · , n − 1}, let Di be the rank 2 subquotient of D appearing in the triangulation as an extension of RE (δi+1 ) by RE (δi ). Put Y Y L(D) := L(Di ) ⊂ Hom(L× , E). i∈∆

i∈∆

Let Li := (Li,σ )σ∈ΣL be the Fontaine-Mazur L-invariants of Di (cf. Rem. 3.1 (ii)), and we call L := {Li,σ } i∈∆ the Fontaine-Mazur simple L-invariants of D. σ∈ΣL

Remark 3.3. (i) Suppose D is associated to a Galois representation ρL of GalL , these L-invariants of D (thus of ρL ) are lost when passing to the associated Weil-Deligne representation WD(ρL ), and hence are invisible in classical local Langlands correspondence. Indeed, these L-invariants can also be defined in terms of filtered (ϕ, N )-modules (e.g. see [28]). (ii) Note that when n ≥ 3, {δi }i=1,··· ,n and L(D) can not determine D but only the Di ’s. 3.2. A sub candidate in p-adic local Langlands programme. Let ρL : GalL → GLn (E) be a semi-stable representation, D := Drig (ρL ) be the associated (ϕ, Γ)-module (of rank n) over RE . Suppose D is trianguline with a triangulation (δ1 , · · · , δn ) satisfying Hypo. 3.2. In this section, we associate to ρL a locally Qp -analytic representation, by matching Breuil’s simple L-invariants with Fontaine-Mazur simple L-invariants. By assumption, there exist hσ,i for all σ ∈ ΣL , i = 1, · · · , n, and α ∈ E × such that for all σ ∈ ΣL , hσ,1 > hσ,2 > · · · > hσ,n and Y δi = unr(qLi−1 α) σ hσ,i . σ∈ΣL

Note that (−hσ,n , · · · , −hσ,1 ) are the σ-Hodge-Tate weights of ρL where we use the convention that the Hodge-Tate weight of the cyclotomic character is −1 . Let λσ,i := hσ,i + i − 1 for σ ∈ ΣL , i = 1, · · · , n, we see that λ := (λ1,σ , · · · , λn,σ )σ∈ΣL is a dominant weight of tΣL . Note that the smooth representation St(α) := vB∞ (0, α) is associated to the Weil-Deligne representation of ρL in classical local Langlands correspondence in the case when all the Di ’s are semi-stable non-crystalline, and that St(λ, α) ∼ = vB∞ (λ, α) = vB∞ (0, α)⊗E Fλ is the locally algebraic subrepresentation of Σ(λ, α) = vBan (λ, α). 18

Let i ∈ ∆, and let ψi,1 , · · · , ψi,dL be a basis of ιi L(Di ) which is a dL -dimensional E-vector subspace of Hom(T (L), E)/ Hom(Zi (L), E) (cf. (26)). Thus for each i ∈ ∆, j = 1, · · · , dL , we get an extension Σiψi,j (λ, α) of vP∞{i} (λ, α) by Σ(λ, α). Put for i ∈ ∆: (30)

Σi (λ, α, L(Di )) := Σiψi,1 (λ, α) ⊕Σ(λ,α) Σiψi,2 (λ, α) ⊕Σ(λ,α) · · · ⊕Σ(λ,α) Σiψi,d (λ, α). L

i

vP∞{i} (λ, α)⊕dL

i

Remark 3.4. (i) By definition, Σ (λ, α, L(D )) is an extension of which only depends on L(Di ) and is independent of the choice of the basis.

by Σ(λ, α),

(ii) By Rem. 2.11 and (29), the locally algebraic subrepresentation of Σi (λ, α, L(Di )) is St(λ, α) if and only if Di is non-crystalline. If so, {logσ,p,−Li,σ }σ∈ΣL form a basis of L(Di ) cf. Rem. 3.1 (ii) , and thus the extension Σi (λ, α, L(Di )) can be parametrized by the L-invariants {Li,σ }σ∈ΣL . (iii) We know socGLn (L) Σi (λ, α, L(Di )) = socGLn (L) Σ(λ, α), however, for this moment the author does not know how to prove socGLn (L) Σ(λ, α) = St(λ, α) (which is however known for GL2 (L) and GL3 (Qp ) by Schraen’s work [26] [27]). Finally, we put L(ρL ) := L(D), and (31) Σ(λ, α, L(ρL )) := Σ(λ, α, L(D)) := Σ1 (λ, α, L(D1 )) ⊕Σ(λ,α) Σ2 (λ, α, L(D2 )) ⊕Σ(λ,α) · · · ⊕Σ(λ,α) Σn−1 (λ, α, L(Dn−1 )). In the case where all the Di ’s are non-crystalline, we illustrate the structure of Σ(λ, α, L(ρL )) in the following picture: vP∞{1} (λ, α) . ....... ....... ....... ....... . . . . . . ... ....... ....... ....... ....... ....... . . . . . . . 1................ ....... ....... ....... . . . . . . ... ....... ....... ......... ....... .................... ....... ....... dL .............................................. . . . . . . ....... ..... . . . . . . . . . . . . . . . . . . . . . . . . .......... .... .................... ....... .................... .................... .................... .................... ....... .................... ....... .................... 1 ....... .................... ....... .................... ....... .................... ....... .................... ....... ... ....... ....... ....... ....... ....... dL ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... .....

L1,σ

vP∞{1} (λ, α)

L1,σ

Σ(λ, α)

.. .

Ln−1,σ

.. .

Ln−1,σ

vP∞{n−1} (λ, α) .. .

vP∞{n−1} (λ, α),

where the line denotes an extension with the left object the sub and the right object the quotient, and “Li,σ ” signifies that the i corresponding extension is associated to the additive character ιi logσ,p,−Li,σ ∈ ιi (L(D )). Remark 3.5. When n ≥ 3, the locally analytic representation Σ(λ, α, L(ρL )) can not determine D (and thus can not determine ρL ). We refer to [6] (see in particular [6, 19

Rem. 4.4.2 (ii)]) for a conjectural approach in finding all the information of ρL on the automorphic side. 3.3. Colmez-Greenberg-Stevens formula. Recall the Colmez-Greenberg-Stevens formula. Let A be an affinoid E-algebra, DA be a rank n triangulable (ϕ, Γ)-module over b Qp A of trianguline parameter {δA,i }i=1,··· ,n , i.e. DA is a successive extension RA := RL ⊗ of RA (δA,i ), which is the rank 1 (ϕ, Γ)-module over RA associated to δA,i : L× → A× . Let z ∈ Spm A be an E-point, D := DA |z , δi := δA,i |z , and suppose that D satisfies the hypothesis 3.2 and we use the notation in §3.1. Recall (cf. [11, Thm. 0.1]) Theorem 3.6. Let i ∈ ∆ = {1, · · · , n − 1}, if Di is semi-stable non-crystalline (resp. crystalline) then the differential form (see the remark below for the weights of characters) X −1 −1 d log δi δi+1 (p) + Li,σ d wt(δi δi+1 )σ ∈ Ω1A/E σ∈ΣL

resp.

X

Li,σ d

−1 wt(δi δi+1 )σ

∈

Ω1A/E

,

σ∈ΣL

vanishes at the point z. Remark 3.7. Recall that a locally analytic character δ : L× → A× induces a Qp -linear map (where L is viewed as the Lie algebra of L× ) d L −→ E, x 7→ δ(exp(tx))|t=0 , dt and hence induces an E-linear map Y wt(δ) = (wt(δ)σ )σ : L ⊗Q E ∼ E −→ E, = p

L

σ∈ΣL

which is called the weight of δ (and wt(δ)σ is called the σ-weight of δ). Corollary 3.8. Let t : Spec E[]/2 → Spm A be an element in the tangent space of A at z, δei := t∗ δA,i : L× → (E[]/2 )× . For i ∈ ∆, let ψ : L× → E be the additive character −1 −1 such that δei δei+1 = δi δi+1 (1 + ψ), then ψ ∈ L(Di ). We also need a “converse” version of this formula. Proposition 3.9. For i = 1, · · · , n − 1, let ψi ∈ L(Di ), δe1 : L× → (E[]/2 )× with δe1 ≡ δ1 −1 −1 (mod ), and let δei+1 : L× → (E[]/2 )× be such that δei δei+1 = δi δi+1 (1 + ψi ). Then there exists a trianguline (ϕ, Γ)-module DE[]/2 over RE[]2 of parameter δe1 , · · · , δen such that DE[]/2 ≡ D (mod ). Proof. We use induction on the rank of D. The rank 1 case is trivial. Suppose the statement holds for rank (n − 1) (ϕ, Γ)-modules (satisfying Hypo. 3.2), and let Dn−1 be the (ϕ, Γ)-submodule of D of rank (n−1), which is trianguline of parameter (δ1 , · · · , δn−1 ), and (Dn−1 )E[]/2 a lifting of Dn−1 , which is a trianguline (ϕ, Γ)-module over RE[]/2 of 0 parameter δe1 , · · · , δen−1 . Let δen be as in the proposition, put Dn−1 := Dn−1 ⊗ RE (δn−1 ), 20

0 0 (Dn−1 )E[]/2 := Dn−1 ⊗ RE[]/2 (δen−1 ), and D0 := D ⊗ RE (δn−1 ) ∈ H1(ϕ,Γ) Dn−1 . Put 1 n−1 −1 n−1 0 n−1 −1 is the subquotient of D (D ) := D ⊗ RE (δn ) ∈ H(ϕ,Γ) RE (δn δn−1 ) recall D as an extension of RE (δn ) by RE (δn−1 ) . We have the following commutative diagram of (ϕ, Γ)-modules 0 −−−→

0 Dn−1 y

0 −−−→ (Dn−1 )E[]/2 −−−→ y

0 Dn−1 y

−−−→ 0

0 −−−→ RE (δn−1 δn−1 ) −−−→ RE (δen−1 δen−1 ) −−−→ RE (δn−1 δn−1 ) −−−→ 0 where the vertical maps are natural projections. Taking cohomology, we get p0 δ 0 0 0 ) ) −−−→ H2(ϕ,Γ) (Dn−1 )E[]/2 −−−→ H1(ϕ,Γ) (Dn−1 H1(ϕ,Γ) (Dn−1 p1 y p2 y p3 y . p00 δ0 H1(ϕ,Γ) δen−1 δen−1 −−− → H1(ϕ,Γ) (δn−1 δn−1 ) −−−→ H2(ϕ,Γ) (δn−1 δn−1 ) It’s sufficient to prove [D0 ] ∈ Im(p0 ). However, we know p3 is an isomorphism, p1 , p2 are 0 surjective since H2(ϕ,Γ) (M ) = 0 if M is the kernel of Dn−1 RE (δn−1 δn−1 ) or the kernel 0 )E[]/2 RE (δen−1 δen−1 ) . We are reduced to prove [(Dn−1 )0 ] ∈ Im(p00 ), however, of (Dn−1 this follows from our assumption on ψn−1 and [11, (2.3)]. 3.4. Trianguline deformation of special (ϕ, Γ)-modules. Let D be a rank n trianguline (ϕ, Γ)-module of parameter (δ1 , · · · , δn ). Recall the trianguline deformation functor n o local E-algebras FD : Artinian −→ Sets with residue field E sends A to the set FD (A) = {(DA , δA,1 , · · · , δA,n )} where (1) DA is a rank n (ϕ, Γ)-module over RA such that DA (mod mA ) ∼ = D, × × (2) δA,i are continuous A -valued characters of L such that δA,i ≡ δi (mod mA ) and that DA is trianguline of parameter {δA,i }i=1,··· ,n . Suppose D satisfies Hypo. 3.2, and we study the functor FD in this section (note that this case was often excluded in the previous literature). We assume moreover End(ϕ,Γ) (D) = E, 0 ∨ ∼ where End(ϕ,Γ) (D) = H(ϕ,Γ) D ⊗RE D denotes the E-vector space of strict endormorphisms of D. Proposition 3.10. Keep the notation and assumption, the functor FD is pro-representable, and formally smooth. Proof. The first part follows from [19, Cor. 2.38]. We prove that the functor is formally smooth. It’s sufficient to prove that for any small extension A A/I (i.e. I = () and mA = 0), the natural map FD (A) → FD (A/I) is surjective. We make induction on the rank of D: 21

When n = 1, the hypothesis 3.2 is empty, and one needs to prove that any continuous character L× → (A/I)× can lift to a character L× → A× , but this follows from the fact that the character space of L× is smooth. Suppose the theorem holds if D is of rank less than n − 1, and suppose now D is of rank n, let Dn−1 be the rank n − 1 saturated (ϕ, Γ)-submodule of D which is a successive extension of RE (δi ) for i = 1, · · · , n − 1. Let DA/I ∈ FD (A/I) be of trianguline parameter (δA/I,1 , · · · , δA/I,n ), and (Dn−1 )A/I be the saturated (ϕ, Γ)-submodule of rank n − 1 over RA/I of DA/I which a successive extension of RA/I -modules RA/I (δA/I,i ) for i = 1, · · · , n− 1, thus (Dn−1 )A/I ∈ FDn−1 (A/I). By the induction hypothesis, there exists (Dn−1 )A ∈ FDn−1 (A) such that (Dn−1 )A ≡ (Dn−1 )A/I (mod I). 0 0 Let δA,n : L× → A× be a lifting of δA/I,n , and put Dn−1 := Dn−1 ⊗RE (δn−1 ), (Dn−1 )A/I := −1 −1 0 0 0 0 (Dn−1 )A/I ⊗ RA/I (δA/I,n ) ∈ FDn−1 (A/I), (Dn−1 )A := (Dn−1 )A ⊗ RA (δA,n ) ∈ FDn−1 (A) 0 × × which is a lifting of (Dn−1 )A/I . For any character χ : L → A , χ ≡ 1 (mod I), we 0 0 0 )A/I by the natural projection (by )A ⊗ RA (χ) ∈ FDn−1 (A) is mapped to (Dn−1 see (Dn−1 0 0 (A/I). So in this way, we obtain an injection (A) FDn−1 assumption) pr : FDn−1 0 V := χ ∈ Hom(L× , A× ) | χ ≡ 1 (mod I) ,−→ pr−1 E · (Dn−1 )A/I ,

and hence an injection (of E-vector spaces) 0 0 (32) Hom(L× , E) ,−→ pr−1 E · (Dn−1 )A/I , ψ 7→ (Dn−1 )A ⊗ RA (1 + ψ)χ . Since A → A/I is a small extension, one has an exact sequence a7→a

0 → E −−−→ A → A/I → 0. 0 0 0 )A/I , we have an exact sequence of (A) of (Dn−1 Thus for any lifting (Dn−1 )0A ∈ FDn−1 (ϕ, Γ)-modules 0 0 0 0 → Dn−1 → (Dn−1 )0A → (Dn−1 )A/I → 0.

Taking cohomology, we get (33)

δ

0 0 0 H1(ϕ,Γ) ((Dn−1 )0A ) −→ H1(ϕ,Γ) ((Dn−1 )A/I ) −→ H2(ϕ,Γ) (Dn−1 ).

Consider the cup product h·,·i 2 0 0 0 0 H1(ϕ,Γ) ((Dn−1 )A/I ) × Ext1(ϕ,Γ) (Dn−1 )A/I , Dn−1 − −→ H(ϕ,Γ) (Dn−1 )∼ = E, 0 the connecting map δ in (33) equals thus h·, [(Dn−1 )A ]i up to non-zero scalars. −1 0 0 ) ∈ H1(ϕ,Γ) ((Dn−1 )A/I ), we claim that there exists χ ∈ For DA/I := DA/I ⊗ RA/I (δA,n 0 0 V such that DA/I lies in the image of the natural map H1(ϕ,Γ) (Dn−1 )A ⊗ RA (χ) → 0 0 0 H1(ϕ,Γ) ((Dn−1 )A/I ). Note that by this claim, one can choose a preimage DA of DA/I in 0 0 H1(ϕ,Γ) (Dn−1 )A ⊗ RA (χ) ∼ )A ⊗ RA (χ) , = Ext1RA RA , (Dn−1 0 then DA := DA ⊗ RA (δA,n χ−1 ) ∈ FD (A) is thus a lifting of DA/I , from which the proposition follows.

22

We prove the claim. By the precedent discussion, it’s sufficient to prove that there exists χ : L× → A× , χ ≡ 1 (mod I) such that

0 0 (34) [DA/I ], [(Dn−1 )A ⊗ RA (χ)] = 0. 0 0 However, the following E-vector subspace of Ext1(ϕ,Γ) (Dn−1 )A/I , Dn−1 0 0 0 | [DA/I ], v = 0 )A/I , Dn−1 v ∈ Ext1(ϕ,Γ) (Dn−1 is of codimension at most 1, while dimE Hom(L× , E) = dL + 1 > 1. By the injection (32), we see that there exists ψ ∈ Hom(L× , E), such that (34) is satisfied for χ = 1 + ψ. This concludes the proof. Remark 3.11. We discuss a little on the difference between the trianguline deformation in our case and that in [1, Prop. 2.3.10] and [19, Prop. 2.39]. We use the notation in the proof. Given a lifting (Dn−1 )A of (Dn−1 )A/I , and an arbitrary lifting RA (δA,n ) of RA/I (δA/I,n ), under the assumption in [1, Prop. 2.3.10] (and [19, Prop. 2.39]), one can always find an extension of RA (δA,n ) by (Dn−1 )A which is a lifting of DA/I ; however, in our case, one needs in general to modify the lifting RA (δA,n ) to get a desired DA . In if an extension (Dn−1 )A RA (δA,n ) is a lifting of DA/I = other words, in our case, (Dn−1 )A/I RA/I (δA/I,n ) , then there should be some extra condition on the lifting δA,n (of δA/I,n ) contrary to [1, Prop. 2.3.10 (ii)] . In fact, when A/I ∼ = E, A ∼ = E[]/2 , this extra condition gives exactly the Colmez-Greenberg-Stevens formula. Keep the notation and assumption, and consider the following composition Y ς Hom(L× , E) κ : FD (E[]/2 ) −→ Hom(T (L), E) −→ i∈∆

where the first map sends (DE[]/2 , δE[]/2 ,1 , · · · , δE[]/2 ,n ) to and see (28) for ς.

δE[]/2 ,i δi−1 − 1 / i=1,··· ,n ,

Proposition 3.12. We have dimE FD (E[]/2 ) = 1 + n(n+1) dL , and κ factors through a 2 surjective map Y κ : FD (E[]/2 ) − L(Di ) = L(D). i∈∆

Proof. We use induction on the rank of D. If rk D = 1, we have FD (E[]/2 ) ∼ = Hom(L× , E). The statements thus hold (in this case κ = 0). Suppose the statements hold if rk D ≤ n − 1, and suppose now rk D = n. Let Dn−1 be the (ϕ, Γ)-submodule of D of trianguline parameter (δ1 , · · · , δn−1 ), we have a natural map pr1 : FD (E[]/2 ) −→ FDn−1 (E[]/2 ), 23

which is in fact surjective by Prop. 3.9 (and the proof). Moreover, one can describe Ker(pr1 ) as the following E-vector space DE[]/2 ∈ FD (E[]/2 ) DE[]/2 is an extension of Dn−1 ⊗RE RE[]/2 by RE[]/2 (δen ) with δen ≡ δn (mod ) . By Colmez-Greenberg-Stevens formula and Prop. 3.9, the map Ker(pr1 ) −→ Hom(L× , E), DE[]/2 7→ (δen δn−1 − 1)/

(35)

factors though L(Dn−1 ) and induces a projection pr2 : Ker pr1 − L(Dn−1 ). Thus the second part of the proposition follows. The kernel of pr2 is the following E-vector space DE[]/2 ∈ FD (E[]/2 ) DE[]/2 is an extension of Dn−1 ⊗R RE[]/2 by RE (δn ) . E

0 0 0 e n−1 ⊗RE RE[]/2 ∼ := Dn−1 := Dn−1 ⊗ RE (δn−1 ), D0 := D ⊗ RE (δn−1 ), and D Let Dn−1 = 0 0 0 0 e Dn−1 ⊕Dn−1 as (ϕ, Γ)-modules over RE . The natural projection of Dn−1 → Dn−1 (induced by RE[]/2 RE ) gives a spit exact sequence 0 e 0 → D0 → 0 0 → Dn−1 →D n−1 n−1

which induces pr3 1 0 0 e0 0 → H1(ϕ,Γ) (Dn−1 ) → H1(ϕ,Γ) D −→ H(ϕ,Γ) (Dn−1 ) → 0. n−1 − On the other hand, one has a natural map 0 e0 7 D e 0 ⊗ RE[]/2 (δn ), pr−1 3 (E · [D ]) −→ Ker(pr2 ), D →

which is surjective, and the kernel of this map is the one-dimensional E-vector space generated by [D0 ⊗RE RE[]/2 ]. So 1 0 0 dimE Ker(pr2 ) = dimE pr−1 3 (E · [D ]) − 1 = dimE H(ϕ,Γ) (Dn−1 ) − 1

= ((n − 1)dL + 1) − 1 = (n − 1)dL , where the third equation follows from the assumption on D. Hence dimE Ker(pr1 ) = ndL , and finally by induction hypothesis, we have dimE FD (E[]/2 ) = dimE FDn−1 (E[]/2 ) + ndL = 1 + This concludes our proof.

n(n + 1) dL . 2

Remark 3.13. For the generic case as in [1, Prop. 2.3.10] and [19, Prop. 2.39], the map (35) is surjective (thus the image will be one dimensional more than that in our case), 0 however pr−1 3 (E · [D ]) will be one dimensional less. Consequently, the tangent space has the same dimension as in our case. 24

4. Local-global compatibility In this section, we prove some local-global compatibility result in semi-stable noncrystalline case. We choose to work with the patched Banach representations ([10]) rather than those obtained via classical p-adic automorphic representations (e.g. see (36)), roughly because the geometry of the resulted so-called patched eigenvariety ([9]) seems easier to study than the genuine eigenvariety (for our concern), and similar results in the classical setting could be in general deduced from those in the patched setting (e.g. see the discussion in §4.1.1). 4.1. Patched eigenvariety. 4.1.1. Patched Banach representation of GLn (L). Recall the patched Banach representation in [10]. We follow the notation of loc. cit. except for the CM field F (denoted by Fe in loc. cit.). Suppose p - 2n, and let r : GalL → GLn (kE ) be a continuous representation, such that r admits a potentially crystalline lift of regular weight which is potentially diagonalisable. Let rpot.diag : GalL → GLn (OE ) be such a lift, and suppose that rpot.diag has weight ξ and inertial type τ (see [10]). In this case, we can associate to r a triple (F, F + , ρ) where (F, F + ) was denoted by (Fe, Fe+ ) in loc. cit. where F is an imaginary CM field with maximal totally real subfield F + such that the extension F/F + is unramified at all finite places, and ρ is a suitable globalisation of r. e is a definite unitary We use the notation and assumption in [10, §2.3], in particular, G + + group over F , v1 is a finite place of F prime to p (satisfying certain properties as in [10, §2.3]), p is a place of F + above p, Sp is the set of places of F + above p, and e ∞+ ) (which only varies {Um = U p Up,m }m∈Z≥0 is a tower of compact open subgroups of G(A F at p-level), where we refer to [10, §2.3] for details on {Um }, but note that for v ∈ Sp \ {p}, Uvp ∼ = GLn (OL ). For the weight ξ and inertial type τ , we have a finite free OE -module Lξ,τ with a locally algebraic action of GL we let Wξ,τ := ⊗v∈Sp \{p} Lξ,τ be the finite Q n (OL ), and p free OE -module with an action of v∈Sp \{p} Uv where Uvp acts on the factor corresponding to v via Uvp ∼ = GLn (OL ). Put e + )\G(A e ∞+ ) → Wξ,τ ⊗O OE /$k | Sξ,τ (Um , OE /$Ek ) := f : G(F E F E −1 e ∞+ ), u ∈ Um f (gu) = u f (g) for g ∈ G(A F Q where Um acts on Wξ,τ ⊗OE via the projection Um v∈Sp \{p} Uvp . And put Sbξ,τ (U p , OE ) := lim lim Sξ,τ (Um , OE /$Ek ), ←− −→ m

k

p

Sbξ,τ (U , E) := Sbξ,τ (U p , OE ) ⊗OE E, where the latter is an admissible unitary Banach representation of GLn (L) and equipped with certain Hecke operators (which form the polynomial Hecke OE -algebra TSp ,univ ). 25

Moreover, one can associate a maximal ideal m of TSp ,univ to the global Galois representation ρ. The action of TSp ,univ on the localisation Sbξ,τ (U p , OE )m induces an action of S Tξ,τp (U ℘ , OE )m (see [10, §2.8] for the definition). For v ∈ Sp ∪ {v1 } (which thus splits in F ), denote by ve a place of F such that if v ∈ Sp , then ρ|GalFve ∼ = r (the existence of ve follows from that ρ is a suitable globalisation of r). We denote by Rve the maximal reduced and p-torsion free quotient of the universal OE -lifting ring of ρ|GalFve , and for v ∈ Sp \ {p}, denote by Rve,ξ,τ for the reduced and p-torsion free quotient of Rve corresponding to potentially crystalline lifts of weight ξ and inertial type τ. Let S denote the global deformation problem as in [10], RSuniv the universal deformation ring, and ρuniv the universal deformation. Note that one has a natural morphism RSuniv → S S Tξ,τp (U ℘ , OE )m , in particular, Sbξ,τ (U p , OE )m is naturally equipped with an action of RSuniv . Let b ⊗ b Sp \{p} Rve,ξ,τ ⊗R b ve1 , Rloc := Rep ⊗ R∞ := Rloc Jx1 , · · · , xg K, S∞ := OE Jz1 , · · · , zn2 (|Sp |+1) , y1 , · · · , yq K, where q ≥ [F : Q] n(n−1) is the integer as in [10], g = q − [F : Q] n(n−1) , and xi , yi , zi are 2 2 formal variables. By [10], we have (1) a continuous R∞ -admissible unitary representation Π∞ of G = GLn (L) over E together with a G-stable and R∞ -stable unit ball Πo∞ ⊂ Π∞ ; (2) a morphism of local OE -algebras S∞ → R∞ such that M∞ := HomOL (Πo∞ , OE ) is finite projective as S∞ JGLn (OL )K-module; (3) an isomorphism R∞ /aR∞ ∼ = RSuniv and a G × R∞ /aR∞ -invariant isomorphism Π∞ [a] ∼ = Sbξ,τ (U p , E)m , where R∞ acts on Sbξ,τ (U p , E)m via R∞ /aR∞ ∼ = RSuniv .

4.1.2. Patched eigenvariety. Recall the patched eigenvariety of Breuil-Hellmann-Schraen, and we refer to [9] for details. Indeed, although our input is slightly different from that in [9], one sees easily that all the arguments in loc. cit. apply in our case. ∞ −an Let ΠR denote the locally R∞ -analytic vector of Π∞ , which are locally Qp -analytic ∞ vectors for the action G × Zsp with respect to one (or any) presentation OE JZsp K R∞ .

Applying the Jacquet-Emerton functor (with respect to the upper triangular Borel ∞ −an subgroup B), we get a locally Qp -analytic representation JB (ΠR ) of T (L) equipped ∞ with a continuous action of R∞ , which is moreover an essentially admissible locally Qp analytic representation of T (L) × Zsp (with respect to a chosen presentation). 26

rig := O X∞ , and let Tb denote the character space of T (L) Let X∞ := (Spf R∞ )rig , R∞ i.e. the rigid space parametrizing continuous characters of T (L) . The strong dual R∞ −an ∨ rig b JB Π∞ is a coadmissible R∞ ⊗E O(Tb)-module, which corresponds to a coherent sheaf M∞ over X∞ × Tb such that ∨ ∞ −an . Γ X∞ × Tb, M∞ ∼ = JB ΠR ∞ Let Xp (ρ) be the support of M∞ on X∞ × Tb, called the patched eigenvariety. We see that, for x = (y, χ) ∈ X∞ × Tb, x ∈ Xp (ρ) if and only if the corresponding eigenspace ∞ −an JB ΠR [my , T (L) = χ] 6= 0, ∞ where my denotes the maximal ideal of R∞ [ p1 ] corresponding to y. Recall Theorem 4.1 (cf. [9, Cor. 3.12, Thm. 3.19, Cor. 3.20], [8, Lem. 3.8]). (1) The rigid space Xp (ρ) is reduced and equidimensional of dimension q + n2 (|Sp | + 1) + ndL = g +

n(n − 1) + [F : Q] + n2 (|Sp | + 1) + ndL . 2

(2) The coherent sheaf M∞ over Xp (ρ) is Cohen-Macauley. (3) The set of very classical points (cf. [9, Def. 3.17]) is Zariski-dense in Xp (ρ) and is an accumulation set. Let Xtri (r) be the trianguline variety associated to r (cf. [9, D´ef. 2.4]), which is b in particular a closed reduced rigid subspace of X r × T , equidimensional of dimension rig n(n+1) n2 + 2 dL , where X . r := Spf Rr p rig p b v∈Sp \{p} Rve,ξ,τ ⊗R b ve , R∞ Let Rp := ⊗ := Rp Jx1 , · · · , xg K, X ρp := (Spf R ) , and U be 1 g ∼ p rig the open unit ball in A1 , we have thus (Spf R∞ )rig ∼ × X = X r. ρp × U × Xr = (Spf R∞ ) Put Y Y ζ := unr(qL1−n ) ⊗ · · · ⊗ unr(qLi−n ) σ i−1 ⊗ · · · ⊗ σ n−1 σ∈ΣL

σ∈ΣL

as a character of T (L), and denote by ιL the following isomorphism ∼ ιL : Tb −→ Tb, δ 7→ δζ. b b Let ι−1 L Xtri (r) := Xtri (r)×Tb,ιL T , which is also a closed reduced rigid subspace of Xr × T . Recall Theorem 4.2 ([9, Thm.3.21]). The natural embedding g b (36) Xp (ρ) ,−→ (Spf R∞ )rig × Tb ∼ = X ρp × U × Xr × T factors though (37)

Xp (ρ) ,−→ Xρp × Ug × ι−1 L Xtri (r) ,

and induces an isomorphism between Xp (ρ) and a union of irreducible components (e−1 g quipped with the reduced closed rigid subspace structure) of Xρp × U × ιL Xtri (r) . 27

4.1.3. Trianguline variety at special points. Let ρL : GalL → GLn (E) be a semi-stable representation, and D := Drig (ρL ). Suppose (1) D is trianguline of parameter (δ1 , · · · , δn ) which satisfies Hypo. 3.2 (we put δ = δ1 ⊗ · · · δn ) (2) End(ϕ,Γ) (D) = E, (r). (3) x := (ρL , δ) ∈ Xtri Let X be a union of irreducible components of an open subset of Xtri (r), and suppose X satisfies the accumulation property (cf. [8, Def. 2.12]) at x. Consider the composition Y ς (38) TX,x −→ Hom(T (L), E) −→ Hom(L× , E), i∈∆

where the first map is the tangent map of the natural morphism X → Tb Proposition 4.3. Keep the above notation and assumption, then X is smooth at the point x, and (38) factors though a surjective map TX,x − L(D). Proof. As in [8, (4.1)], one has an exact sequence: (39)

f

0 → K(ρL ) ∩ TX,x → TX,x → − Ext1GalL (ρL , ρL )

where K(ρL ) is an E-vector space of dimension n2 − dimE EndGalL (ρL ) = n2 − 1 (by the assumption (2)). On the other hand, since X satisfies the accumulation property, by global triangulation theory ([18, Prop. 4.3.5]), we know the image of f is contained in dL , and hence the E-vector space FD (E[]/2 ), which is of dimension 1 + n(n+1) 2 n(n + 1) dL = dim X. 2 Thus X is smooth at the point x, and Im(f ) = FD (E[]/2 ). The second part of the proposition follows then from Prop. 3.12. dimE TX,x ≤ n2 +

p rig Keep the assumption, and assume moreover there exists xp ∈ (Spf R∞ ) such that p rig b x = (xp , ρL , χ) ∈ Xp (r) ,−→ (Spf R∞ ) × X r × T,

where χ := δζ −1 . We have the following composition Y ς (40) TXp (ρ),x −→ Hom(T (L), E) −→ Hom(L× , E), i∈∆

where the first map is the tangent map at x of the natural morphism Xp (ρ) −→ Tb. Corollary 4.4. Keep the notation, Xp (ρ) is smooth at the point x, and (40) factors through a surjective map TXp (ρ),x − L(D). 28

Proof. Let X be an irreducible component of Xp (ρ) containing x, which has thus the form X p × ι−1 L (Xp ) where Xp is an irreducible component of Xtri (r) containing xp := (ρL , δ). By (the proof of) [9, Thm. 3.19], Xp satisfies the accumulation property at xp , and hence by Prop. 4.3, Xp is smooth at xp , and the second part holds. However, by [17, Thm. 3.3.8] and [10, Lem. 2.5] (seealso [10, Cor. A.2]), X p is also smooth at xp , the smoothness of X and hence of Xp (ρ) at x follows. 4.2. Local-global compatibility. Let ρL : GalL → GLn (E) be a continuous representation and suppose ρL appears in the patched eigenvariety Xp (ρ), i.e. there exist xp ∈ (Spf Rp )rig , χ ∈ Tb such that ∞

(xp , ρL , χ) ∈ Xp (r). Let my be the maximal ideal of R∞ [1/p] corresponding to the point y := (xp , ρL ) of (Spf R∞ )rig . We see that (where the object on the right hand side denotes the subspace of vectors annihilated by my ) b L ) := Π∞ [my ] Π(ρ is an admissible unitary Banach representation of GLn (L), which is believed to be the right one (up to multiplicities) corresponding to ρL in p-adic local Langlands correspondence (we refer to [10] for more general cases). We make moreover the following assumptions on ρL : (1) D := Drig (ρL ) is of trianguline parameter (δ1 , · · · , δn ) satisfying Hypo. 3.2, (2) the monodromy operator N on Dst (ρL ) satisfies N n−1 6= 0. In the following, we use the notation of §3.2, in particular, Y Y (41) δζ = unr(qL1−n α) σ hσ,1 ⊗ · · · ⊗ unr(qL2i−n−1 α) σ hσ,i +i−1 ⊗ σ∈ΣL

σ∈ΣL

· · · ⊗ unr(qLn−1 α)

Y

σ hσ,n +n−1 = δB χλ unr(α) ◦ det

σ∈ΣL

where δB = unr(qL1−n )⊗· · ·⊗unr(qL2i−n−1 )⊗· · ·⊗unr(qLn−1 ) denotes the modulus character of B(L). Remark 4.5. (i) The assumption (2) is equivalent to that all the Di ’s (cf. §3.1) are noncrystalline. Indeed, if there exists i ∈ ∆ such that Di is crystalline, let Di−1 denote the (ϕ, Γ)-submodule of D of trianguline parameter (δ1 , · · · , δi−1 ), and consider the natural exact sequence 0 → Di−1 → D → D/Di−1 → 0. i n−i Since D is crystalline, N = 0 on Dst (D/Di−1 ); on the other hand, we have N i−1 = n−1 0 on Dst (Di−1 ), and thus N = 0. Conversely, suppose that all the Di ’s are noncrystalline, we use induction on rk D to show N n−1 6= 0: If rk D = 2, it’s trivial, and suppose that it holds if rk D = n − 1. Assume rk D = n, consider the natural exact sequence 0 → Dn−1 → D → D/Dn−1 → 0, 29

and apply the function Dst (·). Let en ∈ Dst (D) be a ϕfL -eigenvector of eigenvalue qLn−1 α (recall fL is the unramified degree of L over Qp ), which is thus a lifting of a non-zero element in Dst (D/Dn−1 ). By assumption, N (en ) ∈ Dst (Dn−1 ) is non-zero since Dn−1 is non-crystalline, and ϕfL (N (en )) = qLn−2 α. By induction hypothesis, N n−2 (N (en )) 6= 0 and hence N n−1 (en ) 6= 0. (ii) The assumption (2) also implies End(ϕ,Γ) (D) = EndGalL (ρL ) = E. Indeed, under this assumption, using the (unique) triangulation of D, one easily sees any non-zero element f in End(ϕ,Γ) (D) is an isomorphism; now consider the induced isomorphism f on Dst (ρL ), and let en ∈ Dst (D) be as above, then f is determined by its restriction in EndE (E · en ) = E. Since all the Di ’s are non-crystalline, the triangulation δ of D is unique. By assumption p rig (in the beginning of this section), there exist χ : T (L) → E × and xp ∈ (Spf R∞ ) such ∞ −an [m ])[T (L) = that (xp , ρL , χ) ∈ Xp (ρ), or equivalently, such that the eigenspace JB (ΠR y ∞ p χ] is non-zero, with y = (x , ρL ). Since ρL is non-critical with a unique triangulation, the following lemma follows directly from the global triangulation theory [16][18]. Lemma 4.6. Keep the notation and assumption, for a continuous character χ : T (L) → E × , the eigenspace R∞ −an JB (Π∞ [my ])[T (L) = χ] 6= 0 if and only if χ = δζ. Equivalently, (y, χ) ∈ Xp (ρ) if and only if χ = δζ. In the following, let χ := δζ, we fix the point y of (Spf R∞ )rig , and let x := (y, χ). Proposition 4.7. We have a bijection note χδB−1 = χλ unr(α) ◦ det ∼ G R∞ −an ∞ −an − → Hom (42) HomT (L) χ, JB ΠR [m ] I α), Π [m ] . (λ, y y GL (L) n ∞ ∞ B ∞ −an Proof. By [9, Prop. 3.8] (and the proof), ΠR is a direct summand of C Qp −an Zps × ∞ GLn (OL ), E as GLn (OL )-representations where s = n2 (|Sp | + 1) + q. Let m ⊂ S∞ be ∞ −an the preimage of my via the morphism S∞ → R∞ . Then V := ΠR [m] is an admissible ∞ Banach representation of GLn (L) equipped with a continuous action of R∞ , satisfying that V ∨ |H ∼ = C Qp −an (H, E)⊕r for certain r ∈ Z≥1 , where H is a pro-p uniform compact open subgroup of GLn (OL ). By the same argument as in the proof of [6, Prop. 6.3.3], we have (43) Hi bΣL , V = 0, ∀i ∈ Z≥1 .

We modify the proof of [2, Thm. 4.8] (see also [7, §5.6]) to obtain the bijection in (42). Indeed, our V satisfies the first hypothesis in [2, Thm. 4.8], and we let U := χλ , π := unr(α) ◦ det. Since ρL is non-critical, in the terminology of loc. cit., (U, π) is non-critical with respect to V [my ] (rather than with respect to V ) by Lem. 4.6. We have a similar commutative diagram but with V replaced by V [my ] as in the proof of [2, Thm. 4.8] (before [2, Prop. 4.9]). And one reduces to show the following natural 30

maps are bijective (44)

Hom(gΣL ,B(L)) M (λ)∨ ⊗E Ccsm N (L), πδB−1 , V [my ] η1 −→ Hom(gΣL ,B(L)) L(λ) ⊗E Ccsm N (L), πδB−1 , V [my ] η2 −→ Hom(gΣL ,B(L)) M (λ) ⊗E Ccsm N (L), πδB−1 , V [my ]

where M (λ) := U(gΣL ) ⊗U(bΣL ) λ, M (λ)∨ is the dual of M (λ) in the BGG category ObΣL , L(λ) is the unique semi-simple quotient of M (λ) and thus the unique semi-simple sub of M (λ)∨ , Ccsm N (L), πδB−1 denotes the space of locally constant (πδB−1 )-valued functions on N (L) with compact support, and the sequence is induced by the natural composition M (λ) L(λ) ,→ M (λ)∨ . We discuss a little the topology. For M ∈ ObΣL , we equip M with the finest locally convex topology, which realizes M as an E-vector space of compact type since M is of countable dimension. And Ccsm N (L), πδB−1 , with the topology defined in [13, §3.5], is also an E-vector space of compact type. One can check in our case that this topology coincides with the finest locally convex topology on Ccsm N (L), πδB−1 . Finally, we equip M ⊗E Ccsm N (L), πδB−1 with the inductive (or equivalently, productive) tensor product topology, which is of compact type and coincides in fact with the finest locally convex topology. Note that all the maps in the sets in (44) are continuous. Since (U, π) is non-critical with respect to V [my ], one can show as in [2, Prop. 4.9] that η2 is bijective and η1 is injective. Since all the irreducible constituents of M (λ) are of form L(s · λ) with s an element in the Weyl group (Sn−1 )⊕dL of ResLQp GLn , and L(λ) has multiplicity one, to prove that η1 is surjective, it’s sufficient to prove the following claim (which is similar as [2, Thm. 4.11] but with V replaced by V [my ]): Claim: Let M, M 0 ∈ ObΣL such that M ⊂ M 0 and M 0 /M ∼ = L(s · λ) with s 6= 1, then the restriction map −1 0 sm Hom(gΣL ,B(L)) M ⊗E Cc N (L), πδB , V [my ] −→ Hom(gΣL ,B(L)) M ⊗E Ccsm N (L), πδB−1 , V [my ] is surjective. We use the arguments in [7, §5.6] and in the proof of [2, Thm. 4.11] to prove the claim. Let λ0 := s · λ. We equip M 0 ⊗E Ccsm N (L), πδB−1 with an R∞ [1/p]-action via R∞ [1/p] R∞ [1/p]/my ∼ = E. Given a (gΣL , B(L))-equivariant morphism which is also equivariant under the R∞ [1/p]-action, where the left object is equipped with the R∞ [1/p] action via R∞ [1/p] R∞ [1/p]/my ∼ =E f : M ⊗E Ccsm N (L), πδB−1 −→ V [my ] ,−→ V, 31

let Ve denote the push-forward of M 0 ⊗E Ccsm N (L), πδB−1 via f . which sits thus in an exact sequence of (gΣL , B(L))-modules κ (45) 0 → V → Ve → − L(λ0 ) ⊗E Ccsm N (L), πδB−1 → 0, and is equipped with a natural locally convex topology and a natural continuous R∞ [1/p]0 action (satisfying that (45)) is invariant under the R∞ [1/p]-action). Since L(λ ) ⊗E −1 sm Cc N (L), πδB is equipped with the finest locally convex topology, the extension (45) is split as extension of topological E-vector spaces, and hence in particular, Ve is also an E-vector space of compact type. By (43), taking the vectors fixed by N (OL ), one obtains an exact sequence (e.g. see the proof of [6, Prop. 6.3.4]) N (OL ) 0 → V N (OL ) → Ve N (OL ) → χλ0 ⊗E Ccsm N (L), πδB−1 → 0. Note that this exact sequence (of E-vector spaces of compact type) is equivariant under the T (L)+ × R∞ [1/p]-action, where T (L)+ := {diag(a1 , · · · , an ) ∈ T (L) | |a1 | ≤ |a2 | ≤ · · · ≤ |an |}, and we refer to [13, §(3.4)] for the Hecke action of T (L)+ . Let W be the preimage in Ve N (OL ) of N (OL ) χλ0 π ∼ = χλ0 ⊗E JB C sm N (L), πδ −1 ,−→ χλ0 ⊗E C sm N (L), πδ −1 c

c

B

B

(cf. [13, Lem. 3.5.2] for the first isomorphism, and [13, (3.4.8)] for the second lifting map) which is thus a closed subspace of Ve N (OL ) stable by T (L)+ × R∞ [1/p], and sits in an exact sequence (46)

κ

1 0 → V N (OL ) → W −→ χλ0 π → 0.

Let $L be a uniformizer of OL , and U$L := diag($Ln−1 , $Ln−1 , · · · , 1) ∈ T (L)+ , which acts as a compact operator on V N (OL ) by the proof of [13, Prop. 4.2.33]. And hence U$L acts as a compact operator on W since χλ0 π is one dimensional. By the theory of compact operators (cf. [13, §2.3], see in particular [13, Prop. 2.3.6]), the following subspace W [tΣL = λ0 ]{T (L)+ = χλ0 π, my } ⊆ W where “[−]” signifies the corresponding eigenspace, and “{·}” signifies the corresponding generalized eigenspace is thus finite dimensional over E, and projects onto χλ0 π. However, since ρL is non-critical, by Lem. 4.6, this space has zero intersection with V N (OL ) . We thus obtain a (T (L)+ , R∞ [1/p])-invariant section of κ1 : j1 : χλ0 π ,−→ W [my ] = (Ve [my ])N (OL ) , which, by [13, Thm. 3.5.6], induces a B(L)-invariant map χλ0 ⊗E Ccsm (N (L), πδB−1 ) −→ Ve [my ], and hence a (gΣL , B(L))-invariant map M (λ0 ) ⊗E Ccsm (N (L), πδB−1 ) −→ Ve [my ]. 32

However, this map has to factor through (47)

L(λ0 ) ⊗E Ccsm (N (L), πδB−1 ) −→ Ve [my ],

since otherwise, one will get a non-zero (gΣL , B(L))-invariant map L(λ00 ) ⊗E Ccsm (N (L), πδB−1 ) −→ V [my ] with L(λ00 ) a subquotient of M (λ0 ), which, however, will contradict to the fact (U, π) is non-critical with respect to V [my ] (e.g. see the proof of [2, Prop. 4.9]). One can check (e.g. by [13, (0.4)]) that (47) gives a section of κ in (45). Now consider the corresponding section Ve [my ] V [my ], the composition M 0 ⊗E Ccsm (N (L), πδB−1 ) −→ Ve [my ] − V [my ] gives thus the desired lifting of f . This concludes the proof.

Corollary 4.8. The following restriction map is bijective ∼ R∞ −an R∞ −an [my ] −→ HomGLn (L) iG [my ] . (λ, α), Π∞ (λ, α), Π∞ (48) HomGLn (L) IG B B Proof. Since ρL is non-critical, by Lem. 4.6 and [5, Cor. 3.4], any irreducible constituent of R∞ −an IG (λ, α) which does not appear in iG (λ, α) can not be a subrepresentation of Π∞ [my ], B B from which the injectivity follows. On the other hand, applying the Jacquet-Emerton functor, we obtain a natural map from the set on the right hand side of (48) to the set on the left hand side of (42), which, composed with (42), gives an inverse of (48). Corollary 4.9. Let W be a subrepresentation of Σ(λ, α) containing υB∞ (λ, α), the following commutative diagram of natural maps consists of bijections: ∼ R∞ −an R∞ −an −−−→ HomGLn (L) vB∞ (λ, α), Π∞ [my ] HomGLn (L) W, Π∞ [my ] x

∼

∼ R∞ −an ∞ −an HomGLn (L) Σ(λ, α), ΠR [my ] −−−→ HomGLn (L) vB∞ (λ, α), Π∞ [my ] . ∞ ∼y ∼y ∼ G R∞ −an R∞ −an (λ, (λ, α), Π [m ] − − − → Hom i α), Π [m ] HomGLn (L) IG y y GL (L) n ∞ ∞ B B Proof. Since N n−2 6= 0, by local-global compatibility in classical local Langlands corre∞ −an spondence for any P ) B, vP∞ (λ, α) can not be a subrepresentation of ΠR [my ] . The ∞ right bottom vertical bijection follows. The bijectivity of the left bottom map follows by similar arguments. Together with Cor. 4.8, we see that the middle horizontal map is bijective. The injectivity of the top horizontal map and the left top map follows by the same argument in the (first part of the) proof of Cor. 4.8. However, since the middle horizontal map is bijective, we deduce that these two maps are actually bijective. 33

Let Iy ⊆ my be a closed ideal of R∞ [1/p] such that dimE (R∞ [1/p]/Iy ) < +∞ and that my is the unique closed maximal ideal containing Iy . ∞ −an Corollary 4.10. Let W be as in Cor. 4.9, given a morphism f : W → ΠR [Iy ], if its ∞ R∞ −an R∞ −an ∞ [my ], then f has image in Π∞ [my ]. restriction on vB (λ, α) has image in Π∞

∞ −an [my ] such that f 0 |v∞ (λ,α) = f |v∞ (λ,α) . Proof. By Cor. 4.9, there exists f 0 : W → ΠR ∞ B B Since ρL is non-critical, the irreducible constituents of W other than vB∞ (λ, α) can not be R∞ −an R∞ −an [Iy ]. [my ], and hence can not be subrepresentations of Π∞ subrepresentations of Π∞ 0 From which, we deduce f = f , and the corollary follows.

Theorem 4.11. The following restriction map R∞ −an ∞ −an (49) HomGLn (L) Σ(λ, α, L(ρL )), Π∞ [my ] −→ HomGLn (L) vB∞ (λ, α), ΠR [my ] ∞ is bijective. Moreover, let ψ ∈ Hom(L× , E), and i ∈ ∆, an injection ∞ −an [my ] f : vB∞ (λ, α) ,−→ ΠR ∞ ∞ −an can extend to a non-zero map Σiιi (ψ) (λ, α) → ΠR [my ] if and only if ψ ∈ L(Di ). ∞

Remark 4.12. The author does not know how to prove socGLn (L) Σ(λ, α, K(ρL )) ∼ = vB∞ (λ, α) for the moment (which however is known for GL2 (L) and GL3 (Qp ) by [26] [27]), and hence does not know whether any non-zero map in the left hand side set is injective. The rest of the paper is to prove Thm. 4.11, and we use the strategy of [12]. The injectivity of (49) follows from the fact ρL is non-critical and the local-global compatibility in classical local Langlands correspondence using the same argument as in the proof of Cor. 4.8 and 4.9. We show it’s surjective. By definition (31) and (30), it’s sufficient to show that for any i ∈ ∆, ψ ∈ L(Di ), the restriction map R∞ −an ∞ ∞ −an (50) HomGLn (L) Σiιi (ψ) (λ, α), ΠR [m ] −→ Hom (λ, α), Π [m ] v y y GL (L) n ∞ ∞ B is surjective. Let t : Spec E[]/2 → Xp (ρ) be an element in TXp (ρ),x , such that the i-th factor of image of t under (40) equals ψ, and the j-th factors for all j 6= i are zero (where the existence of t follows from Prop. 4.4). Let Ψ ∈ Hom(T (L), E) be the image of t of first map in (40) thus Ψ|Z{j} (L) = 1 for j 6= i . Since Xp (ρ) is smooth at x, M∞ is locally free in a certain neighborhood of x. Let r = dimE z ∗ M∞ . We have the following facts: ⊕r (1) (t∗ M∞ )∨ ∼ =: χ e⊕r resp. (z ∗ M∞ )∨ ∼ = χ(1+Ψ) = χ⊕r as T (L)-representations (recall χ = δζ, see (41)), ∞ −an (2) (t∗ M∞ )∨ ⊆ JB ΠR [It ] where It denotes the kernel of the morphism R∞ [1/p] → ∞ 2 E[]/ induced by t, (3) we have natural injections (51)

R∞ −an (z ∗ M∞ )∨ ,−→ (t∗ M∞ )∨ ,−→ JB (Π∞ [It ]).

34

∞ −an Lemma 4.13. The morphisms of T (L)-representations: (z ∗ M∞ )∨ ,→ JB (ΠR [my ]) ∞ ∗ ∨ R∞ −an and (t M∞ ) ,→ JB (Π∞ [It ]) are balanced (in the sense of [14, Def. 0.8], see also [15, Def. 5.17] ).

Proof. It’s sufficient to prove the kernel of the (U(gΣL ), B(L))-equivariant map R∞ −an U(gΣL ) ⊗U(bΣL ) Ccsm N (L), (z ∗ M∞ )∨ −→ Π∞ [my ] R∞ −an resp. U(gΣL ) ⊗U(bΣL ) Ccsm N (L), (t∗ M∞ )∨ −→ Π∞ [It ] , contains the kernel of the (U(gΣL ), B(L))-equivariant map see [14, Def. 2.5.21] for Cclp (N (L), −) U(gΣL ) ⊗U(bΣL ) Ccsm N (L), (z ∗ M∞ )∨ −→ Cclp N (L), (z ∗ M∞ )∨ resp. U(gΣL ) ⊗U(bΣL ) Ccsm N (L), (t∗ M∞ )∨ −→ Cclp N (L), (t∗ M∞ )∨ , Let π := unr(α) ◦ det, thus χ = χλ π. The natural exact sequence 0 → χ → χ e→χ→0 gives the following commutative diagram 0 y

0 y

i1 −−− → Cclp N (L), χ y i2 U(gΣL ) ⊗U(bΣL ) Ccsm N (L), χ e −−− → Cclp N (L), χ e . y y i1 M (λ) ⊗E Ccsm N (L), πδB−1 −−− → Cclp N (L), χ y y M (λ) ⊗E Ccsm N (L), πδB−1 y

0

0

And i1 induces an injection L(λ) ⊗E Ccsm N (L), πδB ,→ Cclp (N (L),χ) ∼ = M (λ)∨ ⊗E Ccsm N (L), πδB−1 . So Ker(i1 ) consists of L(γ · λ) ⊗E Ccsm N (L), πδB−1 for γ 6= 1. We claim so does Ker(i2 ). Indeed, the commutative diagram can also be obtained from the following commutative diagram by tensoring with Ccsm (N (L), E): −1

0 −−−→ U(gΣL ) ⊗U(bΣL ) χ −−−→ U(gΣL ) ⊗U(bΣL ) χ e −−−→ U(gΣL ) ⊗U(bΣL ) χ −−−→ 0 j1 y j2 y , y 0 −−−→ C Qp −pol (N (L), χ) −−−→ C Qp −pol N (L), χ e −−−→ C Qp −pol (N (L), χ) −−−→ 0 and thus Ker(i2 ) ∼ = Ccsm (N (L), E) ⊗E Ker(j2 ). The claim follows since Ker(j2 ) consists of L(γ · λ) ⊗E π with γ 6= 1. 35

Since ρL is non-critical, as in the proof of [2, Prop. 4.9], we have by Lem. 4.6 (which also holds if my is replaced by It ) R∞ −an Hom(gΣL ,B(L)) L(γ · λ) ⊗E Ccsm N (L), πδB−1 , Π∞ [Iy ] = 0 for any γ 6= 1, where Iy = my or Iy = It . And thus R∞ −an [my ] = 0 Hom(gΣL ,B(L)) Ker(i1 ), Π∞

R∞ −an resp. Hom(gΣL ,B(L)) Ker(i2 ), Π∞ [It ] = 0 .

The lemma follows.

By [14, Thm. 0.13], the injections in (51) induce thus ⊕r ⊕r (52) IBG χδB−1 ,−→ IBG χ eδB−1 −→ Π∞ [It ]. where as in loc. cit., for a locally Qp -analytic representation V of T (L), IBG (V ) denotes the closed G-subrepresentation of (IndG V )Qp −an generated by V via the composition B(L) G G Qp −an −→ (IndB(L) V )Qp −an , V −→ JB (IndB(L) V ) where the second map denotes the canonical lifting with respect to a chosen compact open subgroup No of N (L) and by [14, Lem. 2.8.3], IBG (V ) is independent of the choice of No . We have the following easy lemma. (λ, α). Lemma 4.14. (1) IBG (χδB−1 ) ∼ = iG B (2) The natural exact sequence 0 → χ → χ e → χ → 0 induces a sequence (which is not exact in general) IBG (χδB−1 ) ,−→ IBG (e χδB−1 ) − IBG (χδB−1 ). Proof. (1) follows easily from [14, Prop. 2.8.10]. (2) follows by definition.

By the local-global compatibility in classical local Langlands correspondence, (52) induces ⊕r X e −1 )/ (53) v ∞ (λ, α)⊕r ,−→ I G (δδ v ∞ (λ, α) −→ ΠR∞ −an [It ]. B

B

B

P

∞

P )B ∞ −an [my ], moreover, any map Note that the composition is injective and has image in ΠR ∞ in the right hand set of (49) is in fact induced by this composition. Let X −1 G ∞ W := IB (e χδB )/ vP (λ, α) ∩ Σ(λ, α),

P )B

36

where the intersection is taken inside (IndG χ eδB−1 )Qp −an . By Cor. 4.10, the composition B(L) ⊕r X ∞ −an W ⊕r ,−→ IBG (e χδB−1 )/ vP∞ (λ, α) [It ] −→ ΠR ∞ P )B R∞ −an [my ], and by Cor. 4.9, can extend uniquely to a morphism also has image in Π∞ R∞ −an ∞ −an Σ(λ, α)⊕r −→ ΠR [my ] ,−→ Π∞ [It ]. ∞

Putting these together, we get the following morphisms where the composition has image R∞ −an in Π∞ [my ] ⊕r X −1 ⊕r G ∞ ∞ −an Σ(λ, α) ,−→ IB (e χδB )/ vP (λ, α) ⊕W Σ(λ, α) −→ ΠR [It ]. ∞ P )B

Consider the commutative diagram χδB−1 ) −−−→ iG (λ, α) IBG (e B

y

I∅∅ (λ, α, Ψ) −−−→ iG (λ, α) B where the horizontal maps are natural projections (induced by χ e χ), I∅∅ (λ, α, Ψ) := I∅∅ (λ, Ψ) ⊗E unr(α) ◦ det (see Rem. 2.5 (ii)), and the left vertical map is injective and induced from the natural injection IBG (e χδB−1 ) ,→ (IndG χ eδB−1 )Qp −an (e.g. by Lem. 4.14 B(L) (2)). From which, we deduce X χδB−1 )/ vP∞ (λ, α) ⊕W Σ(λ, α) ∼ IBG (e = I∅∅ (λ, α, Ψ). P )B

By definition, Cor. 2.12, and Rem. 2.13, Σiιi (ψ) (λ, α) is a subrepresentation of I∅∅ (λ, α, Ψ) containing Σ(λ, α). Finally, we deduce from (53) the following morphisms (54)

∞ −an Σ(λ, α)⊕r ,−→ Σiιi (ψ) (λ, α)⊕r −→ ΠR [It ], ∞

∞ −an [my ]. We claim the image of the second morphism is also sending Σ(λ, α)⊕r to ΠR ∞ R∞ −an [my ]. Indeed, for any a ∈ my , the composition contained in Π∞

(55)

v7→av

R∞ −an ∞ −an Σiιi (ψ) (λ, α)⊕r −→ Π∞ [It ] −−−→ ΠR [It ], ∞

factors through Σiιi (ψ) (λ, α)⊕r /Σ(λ, α)⊕r ∼ = vP∞{i} (λ, α)⊕r ∞ −an since (54) has image in ΠR [my ]. However, vP∞{i} (λ, α) can not be a subrepresentation of ∞ ∞ −an ΠR [It ] by the local-global compatibility in classical local Langlands correspondence, ∞ thus (55) is zero. The claim follows.

So (54) induces morphisms (56)

R∞ −an vB∞ (λ, α)⊕r ,−→ Σiιi (ψ) (λ, α)⊕r −→ Π∞ [my ].

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From which the surjectivity of (50) follows note that all the maps in the right hand set of (50) are induced from the composition (56) . We prove the second part of Thm. 4.11. “If” is implied by the bijection (49). Now suppose ψ ∈ / L(Di ), then E · ψ + L(Di ) = Hom(L× , E) and consequently Hom∞ (L× , E) ⊂ E · ψ + L(Di ). By (49) and the middle horizontal isomorphism in Cor. 4.9, f induces a morphism (57)

∞ −an Σ(λ, α, L(Di )) ⊕Σ(λ,α) Σiψ (λ, α) −→ ΠR [my ]. ∞

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