Parabolic eigenvarieties via overconvergent cohomology

Let $G'$ be a connected reductive group over $\mathbb{Q}$ such that $G = G'/\mathbb{Q}_p$ is quasi-split, and let $Q \subset G$ be a parabolic subgroup. We introduce parahoric overconvergent cohomology groups with respect to $Q$, and prove a classicality theorem showing that the small slope parts of these groups coincide with those of classical cohomology. This allows the use of overconvergent cohomology at parahoric, rather than Iwahoric, level, and provides flexible lifting theorems that appear to be particularly well-adapted to arithmetic applications. When $Q$ is the Borel, we recover the usual theory of overconvergent cohomology, and our classicality theorem gives a stronger slope bound than in the existing literature. We use our theory to construct $Q$-parabolic eigenvarieties, which parametrise $p$-adic families of systems of Hecke eigenvalues that are finite slope at $Q$, but that allow infinite slope away from $Q$.


Context.
Hida and Coleman families describe the variation of automorphic representations as their weight varies p-adic analytically. They have become ubiquitous in many areas of number theory, and are vital tools in the study of the Langlands program and the Bloch-Kato conjectures. Their behaviour is captured geometrically in the theory of eigenvarieties. To construct and study an eigenvariety, one requires: • a rigid analytic weight space W, encoding p-adic analytic variation of weights; • for each λ ∈ W, a space M λ that varies analytically in λ, and which carries an action of a suitable Hecke algebra; • and a notion of 'classical structure/classicality', relating finite-slope systems of Hecke eigenvalues appearing in M λ to those arising from p-refinements of automorphic representations of weight λ.
The eigenvariety is then a rigid analytic space E, with a weight map w : E → W, whose points lying above a weight λ parametrise finite-slope systems of Hecke eigenvalues that appear in M λ . Via the classical structure these relate to eigensystems attached to automorphic representations.
Let G be a connected reductive group over Q, and suppose G . . = G /Qp is quasi-split. In this case Hansen [Han17] has constructed eigenvarieties for G by taking M λ to be overconvergent cohomology groups; his work generalises earlier constructions of Ash-Stevens and Urban [AS08,Urb11]. Cohomological automorphic representations of G(A) of weight λ arise in the cohomology of locally symmetric spaces S K for G, of level K, with coefficients in an algebraic representation V ∨ λ of weight λ. Overconvergent cohomology is defined by replacing V ∨ λ with an (infinite-dimensional) module D G λ of p-adic distributions. The classical structure is then furnished by a classicality theorem, which says that the 'non-critical/small slope' parts of the overconvergent and classical cohomology coincide, so that non-critical slope systems of Hecke eigenvalues in M λ are classical. Here the slope of an eigensystem is the p-adic valuation of the U eigenvalue (for an appropriate 'controlling operator' U ). A slope 0 eigensystem is ordinary.
This classicality theorem was first introduced in [Ste94] for modular forms, and is a cohomological analogue of Coleman's classicality criterion [Col96]. It has, in its own right, had far-reaching arithmetic consequences: to give a brief flavour, it has been used to construct p-adic L-functions [PS11], to study L-invariants [GS93], to construct Stark-Heegner points [Dar01], and to give conjectural analogues of class field theory over real quadratic fields [DV21].

Parabolic families and classicality.
In the usual theory, p-adic families for G encode variation with respect to a Borel subgroup B ⊂ G. In particular, U is a B-controlling operator in the sense of §2.5, the natural generalisation of the U p operator for modular forms. Then the eigenvariety encodes U -finite-slope eigensystems, and the non-critical slope bound depends on U .
All of the above is defined using the Iwahori subgroup at p. When applying this to the study of an automorphic representation π, this forces one to work at Iwahoric level, studying 'full' p-refinements of π. In practice, however, it is frequently more natural to work only at parahoric level for a parabolic subgroup Q ⊂ G, corresponding to a weaker p-refinement. In this setting, passing further to full Iwahoric level often requires stronger hypotheses and a loss of information.
In this paper, we present a refined version of overconvergent cohomology which applies to Qparahoric level, and prove a classicality theorem for this refined theory. We vary this in p-adic families and use it to construct 'parabolic eigenvarieties', parametrising parabolic families of automorphic representations. This approach brings two further benefits: • the criterion for non-critical slope is weaker, giving more control in the classicality theorem; • the resulting parabolic families parametrise Q-finite-slope eigensystems, without requiring finite slope away from Q. This is offset by the fact that these spaces vary over smaller-dimensional weight spaces.
A very special case of this is as follows. Suppose F is a real quadratic field in which p splits as pp, and let G = Res F/Q GL 2 . Then G = GL 2 × GL 2 , and U p = U p U p is a B-controlling operator. Let E/F be a modular elliptic curve with good ordinary reduction at p and bad (additive) reduction at p. The attached system of Hecke eigenvalues has infinite slope for U p and hence U p , and does not appear in the (2-dimensional) Hilbert eigenvariety. However, we may take a parabolic Q = B 2 × GL 2 ⊂ G, where B 2 is the Borel in GL 2 ; then U p is a Q-controlling operator, and the ordinary p-refinement of E satisfies the Q-classicality theorem, giving a 1-dimensional 'p-adic family' through E. Moreover, this classicality yields a class in the p-adic overconvergent cohomology attached to E, which has been used to construct p-adic points on E [GMŞ15].

Methods and results.
Our parahoric overconvergent cohomology groups are defined using parahoric 1 distribution modules. Any weight λ is naturally a character on the torus T (Z p ); we are most interested in those that are algebraic dominant, and call these classical. The typical coefficient modules used in overconvergent cohomology are: • overconvergent coefficients D G λ , dual to the locally analytic induction of λ to the Iwahori subgroup of G(Z p ), • and classical coefficients V ∨ λ , dual to the algebraic induction of λ to G(Z p ).
We consider a hybrid construction, defining spaces D Q λ by taking the algebraic induction of λ to the Levi subgroup L Q of Q, then (locally) analytically inducing to the parahoric subgroup for Q, then taking the dual. These groups are naturally quotients of D G λ . Moreover if we take Q = B to be the Borel, we recover D G λ ; and if we take Q = G the 'trivial' parabolic we recover V ∨ λ . All of this is described in §3, and summarised in Table 3.1.
In §4, we construct a parahoric version of Jones-Urban's locally analytic BGG resolution. This is an analytic version of the main result of [Lep77], and provides a tool for our main result, which is a Q-classicality theorem giving an isomorphism between the small-slope parts of cohomology with D Q λ and V ∨ λ coefficients. In particular, in Theorem 4.4 we prove: Theorem A. Let Q = P 0 ⊂ P 1 ⊂ · · · ⊂ P m = G be a maximal chain of parabolics containing Q, and let U Q be a Q-controlling operator which factorises as U Q = U 1 · · · U m , where each U i · · · U m is a P i−1 -controlling operator. Let φ be a system of Hecke eigenvalues and λ a classical weight.
There exist precise bounds h i ∈ Q >0 , depending on λ, such that if v p (φ(U i )) < h i for each i, then the φ-parts of the weight λ classical and Q-overconvergent cohomology are isomorphic.
If v p (φ(U i )) < h i for each i, we say φ has Q-non-critical slope. The notion of being a controlling operator, and the precise values of h i , are described in terms of root data and Weyl groups, which we recap in §2. We describe this theorem in a number of explicit cases in Examples 4.5.
Remark. If Q is the Borel, the most general classicality theorems for (Iwahoric) overconvergent cohomology that currently appear in the literature -for example, [Urb11,Prop. 4.3.10] -require v p (φ(U Q )) < min i (h i ), so even in this case we give a significant improvement on the known range of non-critical slopes. Such improved ranges were known to exist in other settings (for example, see [Eme06,§4.4]), and we believe an analogue for overconvergent cohomology was expected by experts. However, it does not appear in the literature, which we aim to rectify here.
The parahoric overconvergent cohomology groups can be naturally varied analytically in the weight, from which the construction of p-adic families and eigenvarieties -and their basic properties -is fairly standard. In particular, we construct rigid analytic spaces whose points parametrise Qfinite slope systems of eigenvalues, and coherent sheaves on these spaces that interpolate Q-finite slope eigenspaces in classical cohomology. We describe this in §5. In §5.4, we give sufficient conditions for the existence of parabolic families of cuspidal automorphic representations.

A note on assumptions.
We will use results from [Urb11] and [Han17], which work in slightly different settings to us. In [Urb11], Urban's main applications are in the case where G is quasi-split at p and satisfies the Harish-Chandra condition at infinity (i.e. G der (R) admits discrete series). The Harish-Chandra condition is assumed only to control the geometry of the eigenvarieties he constructs. In particular it is not used anywhere in §2,3 of [Urb11], which covers the results we use; in these sections Urban sets up the theory of (Iwahoric) overconvergent cohomology assuming only G /Qp is quasi-split. (We indicate briefly where the Harish-Chandra condition is useful in our setting. In the notation of Definition 5.11 below, it implies that at any Q-non-critical slope cuspidal point x we have ℓ Q (x) = 0; and thus by Proposition 5.12, any irreducible component of the parabolic eigenvariety through x has the same dimension as the weight space. Without the Harish-Chandra condition this might not be true).
In [Han17], Hansen works under the assumption that G /Qp is split instead of quasi-split. This appears to have been done only for convenience, since (as explained in [Urb11, §3.1.1, §3.2]) the formalism of locally analytic distributions goes through equally well when G is quasi-split, up to keeping track of a finite field extension (the field L for us). Moreover Hansen's main tools -the spectral sequences -require only formal properties of distributions that hold in the quasi-split case.
In fact, as remarked on p.1712, footnote 16 of [Urb11], it should be possible to drop the quasi-split requirement altogether if one uses Bruhat-Tits buildings. One then replaces the parahoric subgroup with any open compact subgroup with a Bruhat-Iwahori decomposition. This approach is taken in [Loe11] and [HL11], where there are no assumptions at all on G at p. We have opted to stick to the notationally much simpler, but still very general, quasi-split setting.
Finally, we choose to use compactly supported cohomology throughout this paper as it best suits our future applications, but all of the results go through identically replacing this with singular cohomology (and, in §5, Borel-Moore homology with singular homology).

Comparison to the literature.
Constructions of parabolic families/eigenvarieties have been previously given using methods different to this paper. The theory was introduced for Hida families in [Hid98], and other papers on this subject include [Loe11] (for unitary groups), [Pil12] (Hida theory for Siegel modular forms), and in particular [HL11], which treats a very general setting using Emerton's completed cohomology. They are also related to the µ-ordinary setting of [EM21]. Parabolic families have important applications in arithmetic: for example, in the case of G = GSp 4 , Siegel-parabolic families are used in [LZ,§17], where new cases of the Bloch-Kato conjecture are proved; when G is a definite unitary group, parabolic eigenvarieties were used in [Che20] to attach Galois representations to certain regular, polarised automorphic representations of GL n ; and parabolic Hida families are used in upcoming work of Caraiani-Newton to answer deep questions about local-global compatibility for Galois representations.
In this spirit, the main motivation for giving a new version of this theory comes through arithmetic applications, for which parahoric overconvergent cohomology appears particularly well-suited; it adapts a very useful arithmetic tool (overconvergent cohomology) to a setting of increasing arithmetic interest (parahoric level/families).
This utility is illustrated in the example of GL 2 over a number field F , where special cases of the above theory have appeared repeatedly: -In the case where F is totally real, partial p-adic families were used in [BSDJ] and [JN19], with applications to the trivial zero and parity conjectures respectively.
-For more general F , versions of Theorem A have been proved and used to construct Stark-Heegner points on elliptic curves [Tri06, GM14,GMŞ15], and when F is imaginary quadratic, to construct conjectural Stark-Heegner cycles attached to Bianchi modular forms [VW]. It was also used in [BSW19a] to construct p-adic L-invariants and prove an exceptional zero conjecture for Bianchi modular forms.
-Moreover, versions of the refined slope conditions given by Theorem A were used in [Wil17] and [BSW19b] to construct p-adic L-functions attached to automorphic forms for GL 2 .
In forthcoming work with Dimitrov, we use Theorem A in the setting of GL 2n over totally real fields, using the parabolic Q with Levi GL n × GL n , to construct p-adic L-functions attached to Q-noncritical conjugate-symplectic automorphic representations of GL 2n . We use the results of the present paper to give stronger non-critical-slope and growth conditions than could be achieved with Iwahoric overconvergent cohomology. We also vary this construction in Q-families.
These methods also appear well-adapted to the study of the general automorphic L-invariants defined in [Geh], in which parabolic subgroups arise very naturally. In addition to the examples for GL 2 above, a combination of parahoric overconvergent cohomology with recent work of Gehrmann and Rosso [GR] should, in nice examples (such as the setting of conjugate-symplectic GL 2n ) yield arithmetic interpretations of automorphic L-invariants. For GL 2 , such interpretations are already crucial in the construction of the Stark-Heegner points/cycles mentioned above.
Finally, we note the recent related work of Loeffler [Loe] on universal deformation spaces, which can be described as 'big' parabolic eigenvarieties. The eigenvarieties we construct are the 'small' automorphic eigenvarieties of §6.2 op. cit.; as yet there is no 'big' automorphic analogue.

Acknowledgements.
We are very grateful to Mladen Dimitrov, who helped us work out these definitions explicitly for GL 2n , and to David Loeffler, who gave valuable comments and suggestions on an earlier draft. We are also indebted to the referee for their careful reading of the paper, and for their valuable comments and corrections. D.B.S. was supported by the FONDECYT PAI 77180007. C.W. was funded by an EPSRC Postdoctoral Fellowship EP/T001615/1.

Global notation.
Let F be a number field, and for each non-archimedean place v let F v denote its completion at v, with ring of integers O v and uniformiser ̟ v . Let G ′ be a connected reductive group over F , and G . . = Res F/Q G ′ be the Weil restriction of scalars. We will be fundamentally interested in the cohomology of locally symmetric spaces attached to G. Let K ⊂ G(A f ) be an open compact subgroup, where A f denotes the finite adeles of Q, let C ∞ (resp. Z ∞ ) be the maximal compact subgroup (resp. centre) of G(R), and let K ∞ = C ∞ Z ∞ . Then let If M is a right K-module such that the centre Z(K ∩ G(Q)) acts trivially, then we get an associated local system on S K given by the fibres of the projection with action γ(g, m)uk = (γguk, m|u).

Local notation and root data at p.
Let G = G /Qp . We assume that G is quasi-split, and splits over a (fixed) finite Galois extension E/Q p . As far as possible we will suppress E from notation. We take G ′ /F v and G to have (henceforth fixed) models over O v and Z p respectively. Let T be a maximal torus in G, and B a Borel subgroup containing T . Let B − denote the opposite Borel, and N, N − the unipotent radicals of B, B − . Attached to all of these groups we have corresponding Lie algebras g, t, b, b − , n, n − over Q p . Let be the lattices of algebraic characters and cocharacters of the torus, and , the canonical pairing on X • (T ) ⊗ X • (T ). Let R ⊂ X • (T ) denote the set of roots for (G, T ). For each root α, let H α ∈ t and α ∨ ∈ X • (T ) be the corresponding coroots, defined so that α, α ∨ = α(H α ) = 2. We fix a basis X α of g α . . = {X ∈ g : ad(t) · X = α(t)X for all t ∈ T } normalised so that [X α , X −α ] = H α in g. Our choice of Borel fixes a set of positive roots R + ⊂ R and a set ∆ ⊂ R + of simple roots. We say a character λ ∈ X Let W G denote the Weyl group of (G, T ), generated by reflections w α for α ∈ ∆, acting on X • (T ) by λ wα = λ − λ(H α )α. Also define the * -action of W G on X • (T ) by (2.2) Example. To anchor this general framework, we keep in mind the familiar example of GL n /Q.

Parabolic subgroups.
There is a well-known correspondence between the standard parabolic subgroups B ⊂ Q ⊂ G and subsets of the simple roots: if q . .= Lie(Q), we let (2.4) The correspondence Q ↔ ∆ Q is inclusion-preserving: in particular, ∆ B = ∅ and the maximal standard parabolics correspond to excluding a single simple root. It is convenient (if non-standard) to allow G to be the 'trivial' parabolic subgroup, equal to its Levi subgroup and with ∆ G = ∆.
Let L Q denote the Levi group attached to Q, and N Q the unipotent radical of Q, so that Q = L Q N Q . Note ∆ Q can be identified with ∆ LQ . Also let Q − and N − Q be the opposite groups. Define the parahoric subgroup at Q to be For non-trivial Q we have a parahoric decomposition and for g ∈ J Q , we write this as g = n − g · t g · n g . If the context is clear, we sometimes drop the subscript g. Note that when Q = B is the Borel, J B is the usual Iwahori subgroup and we recover the Iwahori decomposition [Mat77, Prop. 5.3.3].

The Hecke algebra.
Fix a parabolic subgroup Q, and let To define the (Q-parahoric) Hecke algebra at p, we define (2.6) which has a basis indexed by the positive roots R + . We obtain co-ordinates {x β (n) ∈ Q p : β ∈ R + } for any n ∈ N (Q p ), with the property that for any β, β ′ ∈ R + , we have Let B t be the matrix of conjugation by t in this basis; it is diagonal with value β −1 (t) at (β, β). By the valuation condition, we have v p (β −1 (t)) 0 for all t. Now, the subgroup N (Z p ) is exactly the subspace of n such that x β (n) ∈ Z p for all β, and this is clearly preserved by B t .
But this space is preserved by the action of t ∈ T + by the arguments above. Finally (iii) is immediate since B(Z p ) = T (Z p )N (Z p ) and T + commutes with T (Z p ).

Definition 2.2.
• We define H p (K p ) to be the commutative Q p -algebra generated by • For the (all but finitely many) places v of F at which K v is hyperspecial maximal compact and If S is a Q p -algebra, then a system of Hecke eigenvalues over S is a non-trivial algebra homo- Remark. We could take other choices of ramified Hecke algebra, altering the local geometry of the eigenvariety to suit particular arithmetic applications. The construction and results we present here go through for any reasonable choice of ramified Hecke algebra.

Controlling operators.
In the general theory, the role of U p operator for modular forms is played by controlling operators. Let Q be a parabolic subgroup. For s . If t ∈ T + , then by Proposition 2.1 we know conjugation by t preserves N Q (Z p ). We define Proof. Suppose v p (α(t)) < 0 for all α ∈ ∆\∆ Q , and let n ∈ N Q (Z p ). In the notation of the proof of Proposition 2.1, the set R + \R + Q is precisely the set of β ∈ R + whose decomposition β = α i into simple roots (in G) has at least one of the α i ∈ ∆\∆ Q . Then v p (β(t)) < 0 for all β ∈ R + \R + Q , and every entry of Consider the case of G = GL n . For Q the parabolic with Levi GL n−r × GL r , the element t = diag(1, ..., 1, p, ..., p), with p's in the last r entries, defines a Q-controlling operator, but not a B-controlling operator. The element t = diag(1, p, . . . , p n−2 , p n−1 ) defines a B-controlling operator and hence a Q-controlling operator for any standard parabolic Q.

Parahoric overconvergent cohomology
We now introduce the coefficient modules for overconvergent cohomology, using a more flexible notion of 'parahoric distributions' defined relative to a parabolic Q. When Q = B is the Borel, this specialises to the usual definition of locally analytic distributions; and when Q = G, we recover classical coefficient modules. Cohomology with coefficients in Q-parahoric distributions is more easily controlled (but varies over smaller weight spaces) as Q gets larger.

Weight spaces.
Let Definition 3.1 (Weights for T ). Define the weight space of level K for G to be the Q p -rigid analytic space whose L-points, for L ⊂ C p any sufficiently large extension of Q p , are given by This space has a natural group structure, and has dimension dim T (Z p ) − dim Z(K). It is usually more convenient to identify a weight λ ∈ W K (L) with the corresponding character on T (Z p ) that is trivial under Z(K), and we do this freely throughout. The condition that characters be trivial on Z(K) ensures the local systems we define later are well-defined, as discussed before (2.1). Since K will typically be fixed, we will henceforth mostly drop it from the notation.
Definition 3.2. Each λ ∈ X • (T ) induces a character on T (Z p ); let X • (T ) K be the subspace of such λ trivial on Z(K). There is a natural inclusion X • (T ) K ⊂ W(L), and we call this the subspace of algebraic weights. Via §2.2, the algebraic weights carry the * -action of the Weyl group and can be paired naturally, via −, − , with X • (T ). A classical weight is a dominant algebraic weight.
When using the standard notion of distributions with respect to the Borel subgroup, it is possible to define distributions over arbitrary affinoids in W (see, for example, [Han17, §2.2]). The additional flexibility we obtain with parahoric distributions, i.e. weaker notions of finite-slope families and noncriticality, come at the cost of less flexibility when defining distributions in families. In particular, they vary only over the following smaller weight spaces.

Definition 3.3 (Weights for Q).
Let Q be a standard parabolic subgroup.
(i) For K and L as above, let W Q (L) be the Q p -rigid analytic space with L-points (ii) For λ 0 ∈ W(Q p ) a fixed classical weight, define W Q λ0 to be the coset λ 0 W Q inside W, which hence obtains the structure of a Q p -rigid space. We have Again, we identify these weights with characters on L Q (Z p ) that are trivial under Z(K). This space has dimension dim(L ab Q (Z p )) − dim(Z(K)), which is at most dim(W K ). Whilst we encode λ 0 in the notation, the space W Q λ0 evidently only depends on λ 0 up to translation by W Q . Example. Let G = GL 2n , and Q the standard parabolic with Levi L Q = GL n × GL n embedded diagonally. Then W(L) comprises 2n-tuples λ = (λ 1 , ..., λ 2n ) of characters Z × p → L × (that are trivial on Z(K)), and W Q (L) is the subspace where λ 1 = · · · = λ n and λ n+1 = · · · = λ 2n .

Parahoric distributions.
Locally analytic induction modules for a group G, as for example seen in [AS08,Jon11,Urb11], are usually defined through p-adic analytic functions on the Iwahori subgroup, and are uniquely defined by their restriction to N (Z p ). For G = GL n , for example, this translates into functions that are locally analytic in n(n − 1)/2 variables, corresponding to the off-diagonal entries in N (Z p ).
We now define 'partially overconvergent' distribution modules, defined with respect to the parabolic Q, where we only allow analytic variation in some subset of the variables in N (Z p ) and dictate algebraic variation in the others. For this, we first algebraically induce up to the Levi L Q , and then analytically induce to the parahoric J Q . This is explained in explicit detail for GL 3 /Q in [Wil18, §4.3]; the concrete setting op. cit. simplifies the concepts whilst retaining the key ideas.
We recap standard results on locally analytic induction. As G splits over E, all our coefficient modules come from representations of g /E . For the rest of the paper, fix L/Q p finite containing E, and an L-Banach algebra R.

Algebraic induction and highest weight representations.
Let λ ∈ X • (T ) ⊂ W(L) be a classical weight for the group G. We have a finite-dimensional irreducible representation V G λ of highest weight λ, whose L-points can be realised as the algebraic induction module by right translation, and we denote this action by · λ . Any f ∈ V G λ (L) is determined by its restriction to the (open, dense) Iwahori subgroup J B , and thus (by the transformation property and (2.5)) by its restriction to N (Z p ). Moreover, it is standard 2 that any algebraic f : has a unique algebraic extension to G(Z p ).

Analytic function spaces.
Let X ⊂ Z r p be open compact and L and R be as above. A function f : X → R is analytic if it can be written as a convergent power series for some (a 1 , ..., a r ) ∈ X. We write the space of such functions as A 0 (X, R); note that as the a n converge to zero, A 0 (X, R) ∼ = A 0 (X, L) ⊗ L R is the completed tensor product. We say f is algebraic if a n = 0 for all but finitely many n, and denote the subspace of such f as V (X, R) ⊂ A 0 (X, R). For any integer s, we say f : X → R is s-analytic (resp. s-algebraic) if it is analytic (resp. algebraic) on each open disc of radius p −s in X (inside Z r p ), and write A s (X, R) for the space of s-analytic functions. Note 0-analytic is the same as analytic, so the notation is consistent.

Analytic induction modules.
Let Q = L Q N Q be a parabolic. We may identify J Q with an open compact subset of Z r p for some r, and thus apply the above formalism of analytic functions on J Q . Let M be a finite Banach R-module with a left-action of L Q (Z p ). We extend this action to We write LAInd Q M for the space of such functions f such that f ∈ A(J Q , M ).
Note that any such function f is uniquely determined by its restriction to N Q (Z p ) by (3.1) and the parahoric decomposition (2.5). Recall from (2.7) we have an explicit realisation of N Q (Z p ) as an open compact subset of Z t p via the product decomposition N Q (Z p ) ∼ = β∈R + \R + Q Z p X β . Note then that a function on N Q (Z p ) is s-analytic if and only if it is analytic on each N s Q (Z p )-coset.

Locally analytic induction at single weights.
We recap the usual locally analytic modules. Here we take Q to be the Borel B, with Levi T . Let λ ∈ W(L) be a classical weight.
• Denote the s-analytic induction of λ by realised as functions f : Definition 3.6. •

Integral structures.
All of the above Banach spaces have natural integral structures, where we replace L with O L ; in particular, as in [Urb11, 3.2.13] we define

Analytic functions in families.
We now vary these spaces in families. Fix a classical weight λ 0 ∈ W(L), and let U ⊂ W Q λ0 be an affinoid (which we always take to be admissible in the sense of [Con08, Def. 2.2.6], so that it is open in the Tate topology on W Q λ0 ). If λ ∈ U(L), then by definition Proof. The character λλ −1 0 can itself, as an irreducible representation of L Q , be viewed as the highest weight representation V is a subrepresentation of the tensor product; but the tensor product of an irreducible representation with a character is irreducible.
Crucial for variation is the fact that the underlying spaces of V LQ λ (L) and V LQ λ0 (L) are the same: only the L Q (Z p )-action is different. We now vary the action analytically.
As W is a rigid analytic group, translation by λ 0 defines a rigid analytic automorphism of W. Let U 0 . .= λ −1 0 U ⊂ W Q ; this translation identifies U 0 isomorphically with U, so it is an affinoid defined over L. Attached to such an affinoid, there exists a tautological/universal character χ U0 : Remark 3.11. The choice of λ 0 fixes an identification of U and U 0 , and hence of O(U) and O(U 0 ), which is compatible with our normalisation of specialisation maps. Henceforth we work only with U, and implicitly the transfer of structure is with respect to this choice of identification.

Distributions in families.
Since O(U)-duals are not as well-behaved as L-duals, we have to work harder to study the distributions in this setting. See e.g. [

Since O(U) is a contractive Banach L-algebra, this space is an ONable Banach
To see this is well-defined, note {e i ⊗ 1} is an ON basis of A Q U ,s by Lemma 3.13. As the inclusion A Q U ,s ⊂ A Q U ,s+1 is compact, the sequence e i ⊗ 1 tends to zero in A Q U ,s+1 , and µ(e i ⊗ 1) → 0; hence the sum in (3.4) converges in the completed tensor product. Commutativity of (3.3) follows easily from the definitions. Proof. The first isomorphism is standard, and the second isomorphism (between inverse limits) follows from Lemma 3.14. We conclude since lim ← −s D Q U ,s is compact Fréchet by definition.
Remark 3.17. If λ ∈ W Q λ0 is any (possibly non-classical) weight, then we may still define an Hence we can define D Q λ,s (L) and D Q λ (L) identically to Definition 3.6. Note V LQ λ (L) is independent of the choice of base weight λ 0 , since if λ ′ 0 is another choice, by Lemma 3.8 (in the first isomorphism) we have Hence D Q λ,s (L) and D Q λ (L) are also independent of the choice of λ 0 .

Summary of notation.
The notation in the above is heavy. To ease notation, henceforth we will fix a coefficient field L/Q p , containing the fixed splitting field E of G, and drop it from the notation, writing A Q λ,s = A Q λ,s (L), V G λ = V G λ (L), etc. In Table 3.1 we give a brief key of our notation in the language of §3.2.2. Note that all of the analytic function spaces can be characteristed uniquely by their restrictions to a unipotent subgroup, valued in some Banach module, and then extended uniquely to J B or J Q using the weight action. For a classical weight λ and any s 0, we get the chain of modules (3.5) The notation we maintain is that A Q means Q-parabolic induction and A G means full induction. Modules with subscripts s are Banach modules, and s denotes the degree of analyticity; those without a subscript s are Fréchet modules. Despite the equality A B λ,s = A G λ,s , we choose to maintain the separate notation A and A both for clarity and because the modules A LQ λ,s play a crucial role in the sequel.

Module
On unipotent Extension Dual Nomenclature loc. an. on N Q , loc. alg. on L Q

The action of Σ Q and local systems.
Definition 3.19. Let Σ Q denote the monoid in G(Q p ) generated by J Q and T + .
Let ⋄ denote either a single classical weight λ or an affinoid U in W Q λ0 for a fixed classical λ 0 . The parahoric J Q acts on itself by right multiplication, which then give rise to left actions of J Q on A Q ⋄,s and A Q ⋄ and dual right actions on D Q ⋄,s and D Q ⋄ . The action of T + is more subtle; we note that any function f ∈ A Q ⋄,s is uniquely determined by its restriction to B(Z p ), upon which t ∈ T + acts by b → t −1 bt (by Proposition 2.1(iii)). In itself, this is not compatible with the action of J Q above due to the left multiplication by t −1 . To rectify this, note that our choice of uniformisers defines a splitting Now if t ∈ T (Z p ) = T (Q p ) ∩ J Q , then σ(t)t −1 = 1 and (3.7) coincides with right translation by t. *  t), and extend to J Q via (3.1). A simple check shows t * f ∈ A Q ⋄,s is well-defined, giving a left action of T + on A Q ⋄,s and a right action on D Q ⋄,s .
Notation 3.20. If g ∈ Σ Q , denote the action of g on f ∈ A Q ⋄,s by g * f , and on µ ∈ D Q ⋄,s by µ * g.
Lemma 3.21. The image of the map r s : Proof. We can argue exactly as in [Bel12,Rem. 3.1]. Alternatively, we can directly write down an action on D Q U ,s : let j ∈ J Q , t ∈ T + and µ ⊗ α ∈ D Q λ0,s (L) ⊗ L O(U 0 ), which we identify with D Q U ,s via restriction to N Q (Z p ). Write j = j − ℓ j n j under (2.5). On pure tensors, define extended by continuity. One may check explicitly that (3.2) is equivariant for the * -actions.
Via projection to K p , these spaces of locally analytic distributions are K-modules which then, via (2.1), give local systems over the locally symmetric space, which in a slight abuse of notation we denote by the same symbols.
The action of t ∈ T + then allows us to define Hecke operators U t on the parahoric overconvergent cohomology groups, exactly as in [Han17, §2.1]. We extend this to an action of H(K) by letting Remarks 3.23. (i) Note that more or less by definition, the * -action of Σ Q defined here preserves the integral subspaces D Q λ,s (O L ) of §3.2.5. (ii) The * -action also preserves algebraic subspaces. In particular, we get a * -action of Σ Q on . But any f ∈ V G λ (L) extends uniquely from G(Z p ) to G(Q p ), from which we get a natural 'algebraic' action of G(Q p ) defined by (t · f )(g) . . = f (gt). From the definition, we find that for f ∈ V G λ and t ∈ T + , we have . The ·-action does not preserve V G λ (O L ), and the * -action can be viewed as an 'optimal' integral normalisation of it.
(iii) For GL 2 , it is easy to write down the ·-action on V G,∨ λ explicitly, and one easily sees that this explicit action extends to distributions; this is done, for example, in [PS11,Bel12,BSW19b]. We warn the reader, however, that this does not give the * -action of T + on distributions defined here: in particular, it does not preserve integrality (see [BSW19b, §9.1]).
For the remainder of this paper, unless explicitly stated, all actions will be the * -actions.

Compact operators and slope decompositions.
We now recap the (standard) arguments that show the parahoric overconvergent cohomology groups admit slope decompositions with respect to Q-controlling operators. Proof. At a single weight λ, we follow [Urb11, Lemma 3.2.8]. Firstly, since by definition of T ++ If M is a module admitting a slope h decomposition with respect to an operator U (see, for example, [Han17, Definition 2.3.1]), we write it as (3.9) Let H • c denote compactly supported (Betti) cohomology, dual to the Borel-Moore homology. The following adaptation of [AS08, §4] is the main reason we introduced the (ONable) spaces D Q U ,s .
be an open affinoid, let h 0, and let t ∈ T ++ Q . Then, possibly up to replacing U with a smaller affinoid neighbourhood of λ:

and for any s
Proof. These results are all standard, so we only give analogous references. The modules we have defined give rise to compactly supported chain complexes C • c (K, D Q λ,s ) and C • c (K, D Q U ,s ), as at the end of [Han17,§3], and the compactness of t on distributions lifts to compactness of t on the complex. The cohomology of this complex gives rise to the compactly supported cohomology groups in which we are primarily interested. Since the D Q λ,s and D Q U ,s are ONable, Propositions 2.3.3-2.3.5 of [Han17] then show part (i). Part (ii) is the parahoric analogue of Proposition 3.1.5 op. cit., arguing identically using instead the parahoric chain complexes. Part (iii) follows in the inverse limit (using Lemma 3.16 for distributions over U).
Note that, directly from the definitions, if M is a Q p -module that admits a slope decomposition with respect to an operator U , and β ∈ Q p , then (3.10)

Parahoric classicality theorems
We now prove our central result, a relative classicality theorem for parahoric overconvergent cohomology. This encompasses the analogous theorem for lifting from fully algebraic to fully analytic coefficients, and indeed we expect that it gives a numerically optimal slope bound for such a result. Our main tool is a parahoric version of Jones and Urban's locally analytic Bernstein-Gelfand-Gelfand (BGG) resolution for classical weights λ (Corollary 4.17), which we develop in §4.2-4.4. This can also be considered as a locally analytic version of the main result of [Lep77].
As in §3.3, we fix a coefficient field L/Q p , containing E splitting G, and drop it from notation.

The parahoric classicality theorem.
Fix throughout this section a parabolic Q ⊂ G, an open compact K ⊂ G(A f ) with K p ⊂ J Q and a classical weight λ. Dualising the natural inclusion , and a corresponding map on cohomology: (4.1) Definition 4.1. Let φ be a system of Hecke eigenvalues (for H(K)) occurring in H • c (S K , V G,∨ λ ). We say φ is Q-non-critical if the map ρ restricts to an isomorphism of φ-generalised eigenspaces Such systems φ naturally arise from 'p-refined' automorphic representationsπ; see §4.6. We say such aπ is Q-non-critical if the associated φ is. We observe that for finite slope systems, this definition has no dependence on the radius of analyticity s, so is well-defined; and in fact we may pass to distributions that are fully locally analytic in Q: Lemma 4.2. Let φ be a Q-non-critical system of Hecke eigenvalues, and assume φ has Q-finite slope (i.e. φ(U t ) = 0 for some t ∈ T ++ Q ). Then for any s 0, we have Proof. This follows from Proposition 3.25 applied with some h v p (φ(U t )).
This provides a numerical criterion for Q-non-criticality. Define a maximal chain of parabolics The rest of §4 will be dedicated to proving:

Theorem 4.4. Let φ be as in Definition 4.1. Suppose φ is Q-non-critical slope in the sense that
for all i = 1, ..., m. Then φ is Q-non-critical.
Remark 4.6. Our definition of Q-non-critical uses cohomology with compact support H • c ; to be more precise, we could call this Q-non-critical for H • c . It is also common to use Betti cohomology (without support) H • , as in for example [Urb11,Han17], giving a (directly analogous) notion of Qnon-critical for H • . It seems natural to expect that the two notions are equivalent, but it does not a priori appear obvious that this is the case. However, Theorem 4.4 applies equally well to both cases: so Q-non-critical slope implies both flavours of Q-non-criticality. Henceforth, unless specified otherwise, our notion of non-critical should be clear from the underlying setting.

Analytic BGG for the Borel.
We recap the usual locally analytic BGG resolution (Theorem 4.7). Recall A(J B , L) is the space of locally L-analytic functions on the Iwahori J B , and . This action is L-analytic, and thus induces an analytic action of g. Explicitly, X ∈ g acts by This extends in a natural way to an action of the universal enveloping algebra U(g).
α for a classical weight λ, and we have a map This is J B -equivariant and (recalling ζ from (3.6)) transforms under t ∈ T + as The following describes the first few terms of the locally analytic BGG resolution. Let V G λ,loc ⊂ A G λ be the subspace of functions that are locally L-algebraic on J B , that is, the union of the subspaces of s-algebraic functions over all s 0.
The action of g on A(J B , L) preserves A 0 (J B , L) (as we can define it on this space directly). Hence we have maps X α : Corollary 4.8. Let λ be a classical weight. There is an exact sequence

Theta operators on parahoric distributions.
We now describe A Q λ,0 as a canonical subspace of A G λ,0 . If f ∈ A Q λ,0 and n ∈ N Q (Z p ), then by definition [f (n) : Proposition 4.9. There is an injective Σ B -equivariant map ι Q : A Q λ,0 ֒→ A G λ,0 defined by Using (3.1) for f and the L Q -action on V LQ λ , we have so ι Q (f ) has the right transformation property and ι Q is well-defined. Since f is uniquely determined by its restriction to From now on, we freely identify A Q λ,0 with its image ι Q (A Q λ,0 ) in A G λ,0 . We can give an intrinsic criterion for an element of A G λ,0 to be in this subset.
Definition 4.10. Let n ∈ N Q (Z p ). Define a map By definition of ∆ Q , if α ∈ ∆ Q then X α ∈ l Q = Lie(L Q ), so α is a simple root of L Q and we get a well-defined map Θ α : Proof. It suffices to prove that R n commutes with the action of X α on A 0 (J B , L). But if f ∈ A 0 (J B , L), then for all ℓ ∈ L Q (Z p ) ∩ J B , we have Lemma 4.14. Suppose α ∈ ∆ Q . Then A Q λ,0 ⊂ ker(Θ α ).
Proof. If f ∈ A Q λ,0 , then R n (f ) ∈ V LQ λ for all n ∈ N Q (Z p ) by Proposition 4.11; thus is also algebraic, the equality being Lemma 4.13. Then 4.11 again says Θ α (f ) ∈ A Q wα * λ,0 . As α is a root for L Q , the weight w α * λ is not dominant for L Q . It follows that V LQ wα * λ = 0, which forces A Q wα * λ,0 = 0 by definition. It follows that A Q λ,0 ⊂ ker(Θ α ).
We saw if α ∈ ∆ Q , then Θ α (A Q λ,0 ) ⊂ A Q wα * λ,0 . We want to prove this for α / ∈ ∆ Q . Such an α is not a root of L Q , so we cannot follow the same strategy. Instead, we argue directly: Proof. Choose a set of co-ordinates y i on L Q (Z p ) ∩ J B that identify it as a subset of Z r p . We also have a set of co-ordinates z j on N Q (Z p ), indexed by j ∈ R + \R + Q as in (2.7). Let f ∈ A Q λ,0 . If g ∈ Q(Z p ) ∩ J B , write it as We may write f (g) = f (y i (ℓ g ), z j (n g )) in the co-ordinates above; then by definition, f is algebraic in the y i and analytic in the z j .
To show the proposition, by Proposition 4.11 we must show that R n (Θ α (f )) is algebraic on the last equality since Θ α respects the * action of J B . Replacing f with n * f , it then suffices to prove that the restriction of Θ α (f ) to L Q (Z p ) ∩ J B lies in V LQ λ . By definition, this is the function Since α / ∈ ∆ Q , a sufficiently small neighbourhood U of 0 in Q p X α ⊂ n is contained in n Q (Z p ). For t in such a U , we have exp(−tX α ) ∈ N Q (Z p ). This is a normal subgroup in G(Z p ), so in particular, for any ℓ ∈ L Q (Z p ) ∩ J B we have exp(−tX α )ℓ = ℓe(ℓ, t) with e(ℓ, t) = ℓ −1 · exp(−tX α ) · ℓ ∈ N Q (Z p ). Then we have The co-ordinates z j (e(ℓ, t)), which are linear functions in t, are algebraic in the y i (ℓ) (since inverse and multiplication operations are algebraic on a reductive group). We know f is algebraic in the y i (ℓ), and analytic in the z j (e(ℓ, t)); and by above the coefficient of the linear term in t is algebraic in the y i (ℓ). We deduce that Θ α (f )(ℓ) = d dt f (ℓe(ℓ, t))| t=0 is algebraic in the y i (ℓ), as required.

The parahoric analytic BGG resolution.
Proposition 4.16. For a classical weight λ, there is an exact sequence Proof. That A Q λ,0 ⊂ ker(Θ α ) is an immediate consequence of Lemma 4.14. To see the converse, suppose f ∈ ker . .= ker(Θ α ). Then for all n ∈ N Q (Z p ), by Lemma 4.13 we have Θ α (R n (f )) = R n (Θ α (f )) = 0 for any α ∈ ∆ Q . Thus we have a diagram where exactness of the bottom row is Corollary 4.8 for the group L Q , noting that ∆ Q is precisely the set of simple roots for L Q corresponding to the Borel B ∩ L Q . But then R n (f ) ∈ V LQ λ for any n; thus by Proposition 4.11 we have f ∈ A Q λ,0 , as required.
Corollary 4.17. Let P ⊂ Q be two standard parabolics, with ∆ P ∪ {β} = ∆ Q (that is, there is no parabolic P ′ with P P ′ Q). There is an exact sequence Proof. We restrict the map ⊕Θ α of 4.16 from A G λ,0 to A P λ,0 . It is clear that the kernel of this restriction is A Q λ,0 ∩ A P λ,0 = A Q λ,0 , the equality following by Remark 4.12. If α ∈ ∆ Q is not equal to β, then α ∈ ∆ P , so A P λ,0 ⊂ ker(Θ α ) by Lemma 4.14. In particular, the direct sum ⊕ α∈∆Q Θ α collapses, with Θ β the only non-zero term. The image lands in A P w β * λ,0 by Proposition 4.15, giving the claimed exact sequence.

Proof of Theorem 4.4.
We can finally prove our main result. Recall from Theorem 4.4 that Q = P 0 ⊂ · · · ⊂ P m = G is a maximal chain of parabolics, Proof. (Theorem 4.4). First we make sense of taking U i -slope decompositions on D Pi -cohomology. Note that t Q = t 1 · · · t m is in T ++ Q ⊂ T ++ Pi by Proposition 2.3, hence it acts compactly on each D Pi λ,0 by Lemma 3.24; we get a Q-controlling operator U aux = U tQ on H • c S K , D Pi λ,0 for each i, and we can take slope decompositions. Let h aux ≫ v p (φ(U aux )), so that for each i, we have By the theory of slope decompositions, the right-hand side is Hecke-stable and finite-dimensional over L; thus we may take further slope decompositions for U i , as they always exist on finitedimensional spaces. Define the local piece of the eigenvariety at (U, h) to be the rigid analytic space The natural structure map O(U) → T Q, * U ,h gives rise to a map w : E Q, * U ,h → U, which we call the weight map. We get the following key property essentially by definition.

Proposition 5.2. There is a bijection between:
• L-points x = x(φ) of the rigid space E Q, * U ,h with w(x) = λ, and • systems of Hecke eigenvalues φ : H(K) → L that occur in the localisation where m λ ⊂ O(U) is the maximal ideal corresponding to λ.
Proof. Such a point x corresponds to a maximal ideal in m x ⊂ T Q, * U ,h with residue field L, and we obtain a surjective algebra homomorphism which by definition occurs in H * c (S K , D Q U ) h . To say that w(x) = λ means that the contraction m x ∩ O(U) = m λ , and thus φ x occurs in the stated localisation.

The global eigenvariety.
These local pieces glue into a 'global' eigenvariety over the weight space W Q λ0 . This is straightforward using the 'eigenvariety machine' of [Han17, §4.2]; although non-minimal parabolics do not feature in Hansen's paper, the formalism of this machine carries over to this case with little (and often no) modification. As such, our treatment of the material will be terse. The key will be to identify an eigenvariety datum, from which we may apply Theorem 4.2.2 op. cit. to obtain our global eigenvariety.

Fredholm power series and hypersurfaces.
The modules of analytic functions from previous sections give rise to Borel-Moore chain complexes C BM * (K, A Q U ,s ) (dual to the compactly supported complex with distributions defined previously). The proofs of Propositions 3.1.2-3.1.5 of [Han17] hold in our setting with no modification, showing that the (small slope) homology and cohomology of these complexes is compatible with changing the affinoid U.  we see that the rigid localisation of this sheaf at x -which is a faithfully flat extension of the algebraic localisation -must be zero. Thus, perhaps after shrinking the neighbourhoods U and V, this vanishing lifts to V. Thus for any y ∈ V, we have H • ∂ (S K , D Q U ) φy = 0; and localising the boundary spectral sequence at y, we deduce that H • ∂ (S K , D Q λy (L)) φy = 0 and φ y is strongly interior. Now suppose y ∈ V cl . Since φ y is Q-non-critical slope, by Lemma 5.14 it is interior. But for regular weights, a class is interior if and only if it is cuspidal [LS04, Prop. 5.2, §5.3], so φ y appears in the cuspidal cohomology, as required.
As a special case where the dimension hypothesis on V will always be satisfied, we have: Corollary 5.16. Suppose G der (R) admits discrete series, and let x ∈ E Q, * λ0 satisfy (C1-3). Every irreducible component of E Q, * λ0 through x contains a Zariski-dense set of classical cuspidal points.
Proof. The conditions on G and x = x(φ) ensure that φ appears in only one degree of classical cohomology (e.g. [LS04, §4-5]); and then Proposition 5.12 ensures that any such irreducible component has dimension dim W Q λ0 . We conclude by Proposition 5.15.
Remark 5.17. The assumptions on regular weights ensure control over the classical cohomology, and in situations where we have a more complete understanding of the classical cohomology -for example, the case of GL 2 -we may relax these conditions.
For B-families, every affinoid neighbourhood of a classical weight λ 0 contains a Zariski-dense set of regular classical weights. If λ 0 is not regular, this is not necessarily true in the parahoric case. For example, consider G = GL 4 , and λ 0 = (0, 0, 0, 0), and Q with Levi GL 2 × GL 2 . Then every weight λ = (λ 1 , ..., λ 4 ) ∈ W Q λ0 has λ 1 = λ 2 and λ 3 = λ 4 , so this space contains no regular weights. Remark 5.18. Suppose G der (R) does not admit discrete series. Then if a point x is cuspidal Qnon-critical, then ℓ Q (x) 1. When Q = B, [Han17,Thm. 4.5.1] says that irreducible components through such x never have maximal dimension (that is, dimension equal to dim W), and conjecturally the inequality of Proposition 5.12 is an equality. This conjecture is false in the general parahoric setting. Indeed, in [BSDW] examples are given of Q-parabolic families of dimension dimW Q λ0 in the setting of G = Res F/Q GL 2n , even though ℓ Q (x) = n − 1. Conceptually these families arise through transfer from GSpin 2n+1 (where we do have discrete series).