P-adic interpolation of metaplectic forms of cohomological type
PP -ADIC INTERPOLATION OF METAPLECTIC FORMS OFCOHOMOLOGICAL TYPE RICHARD HILL AND DAVID LOEFFLER
Abstract.
Let G be a reductive group over a number field k . It is shown howEmerton’s methods may be applied to the problem of p -adically interpolatingthe metaplectic forms on G , i.e. the automorphic forms on metaplectic coversof G , as long as the metaplectic covers involved split at the infinite places of k . Introduction
Metaplectic groups.
Let k be an algebraic number field, A the ad`ele ring of k , and G a connected reductive group over k . Let µ be a finite abelian group. Bya metaplectic cover of G by µ , we shall mean a topological central extension1 → µ → ˜ G ( A ) → G ( A ) → , such that the subgroup G ( k ) of rational points lifts to a subgroup ˆ G ( k ) of ˜ G ( A ).We shall recall some results on the construction and properties of metaplectic coversin section 2.1 below.This paper is concerned with p -adic interpolation of automorphic representationsof a metaplectic group ˜ G ( A ) (specifically, those representations which show upin the cohomology groups of its arithmetic quotients.) More precisely we provemetaplectic versions of the results of [Eme06b], [Eme06a] and [Hil07], which wereoriginally proved there for automorphic representations of G ( A ).We shall write G ∞ for the Lie group G ( k ⊗ Q R ) and G ◦∞ for the connectedcomponent of the identity in G ∞ . We have a topological central extension of realLie groups(1) 1 → µ → ˜ G ◦∞ → G ◦∞ → , where ˜ G ◦∞ is the pre-image of G ◦∞ in ˜ G ( A ). We shall divide metaplectic covers intotwo kinds: • Type 1 : The extension (1) splits as a direct sum; • Type 2 : The extension (1) does not split.In this article we shall be concerned with metaplectic groups of type 1. As we pointout in Proposition 2.1.1 below, there are many groups G for which all metaplecticcovers of G are of type 1, so this restriction is not too onerous. Our methods willnot apply to type 2 covers, since such covers do not possess genuine metaplecticforms of cohomological type. However we note that various authors have dealt withthe p -adic interpolation of metaplectic forms of type 2 in certain cases, using verydifferent methods (see for example [Kob86, Ram06, Ram08]). Mathematics Subject Classification.
Mathematics Subject Classification 2010: 11F33,11F55, 22E50.
Key words and phrases.
Metaplectic forms, p -adic interpolation, eigenvarieties.The second author’s research has been supported by EPSRC Postdoctoral FellowshipEP/F04304X/2. The notation ˜ G ( A ) is standard but a little misleading, since this group is a topological group,but is not the group of adelic points of any linear group. a r X i v : . [ m a t h . N T ] O c t RICHARD HILL AND DAVID LOEFFLER
Cohomology of arithmetic quotients.
Assume now that we have a type 1metaplectic cover of G . We fix once and for all a lift ˆ G ( k ) of G ( k ) to ˜ G ( A ). Wealso fix a maximal compact subgroup K ◦∞ of G ◦∞ . For a compact open subgroup K f of G ( A f ), we write Y ( K f ) for the corresponding arithmetic quotient of G , i.e. Y ( K f ) = G ( k ) \ G ( A ) /K ◦∞ K f . We shall describe Y ( K f ) in a different way, which is more useful to us. Supposethat K f is chosen small enough so that it lifts to a subgroup ˆ K f of ˜ G ( A ). Theexistence of such a K f follows from topological considerations, and we shall fix sucha lift ˆ K f .As ˜ G is of type 1, the group K ◦∞ lifts uniquely to a maximal compact subgroupˆ K ◦∞ of ˜ G ◦∞ . Since ˆ K ◦∞ is connected, it follows that ˆ K f commutes with ˆ K ◦∞ , and sowe have the following alternative description of Y ( K f ): Y ( K f ) = ˜ G ( k ) \ ˜ G ( A ) / ˆ K ◦∞ ˆ K f , where ˜ G ( k ) is the pre-image of G ( k ) in ˜ G ( A ).This allows us to define a local system on Y ( K f ) for any representation of ˜ G ( k ) ∼ = G ( k ) ⊕ µ . In particular, if W is a representation of G ( k ) (over some coefficientfield E ) and ε : µ → E × is a character, we may consider the representation W ⊗ ε of ˜ G ( k ). The corresponding local system V W ⊗ ε is given by V W ⊗ ε = ˜ G ( k ) \ (cid:16) ( ˜ G ( A ) / ˆ K ◦∞ ˆ K f ) × ( W ⊗ ε ) (cid:17) . The cohomology groups of these local systems have an action of the Hecke algebraof ˜ G ( A f ) with respect to ˆ K f . This article will be concerned with the interpolationof the systems of Hecke eigenvalues appearing in these cohomology groups. Moreprecisely, we shall fix a prime p of k above the rational prime p , and take W to be analgebraic representation of the group G = Rest k p Q p ( G × k k p ), where Rest k p Q p denotesrestriction of scalars. We regard such a representation W as a representation of G ( k ) via the inclusion G ( k ) (cid:44) → G ( k p ) ∼ = G ( Q p ). Our goal is to show that the Heckeeigenvalues appearing in these spaces move in p -adic families as the weight of W varies.We will find it more convenient to work with the corresponding smooth repre-sentations. Let us write G for the group G ( Q p ). For a prime q of k we shall oftenwrite G q for the group G ( k q ). Thus there is a canonical identification G = G p . Fora subgroup U ⊂ G ( A ), we shall write ˜ U for the preimage of U in ˜ G ( A ). If U is asubgroup of K f , then we shall write ˆ U for the lift of U to ˆ K f , so that ˆ U = ˜ U ∩ ˆ K f .We shall assume that K f = K p K p , where K p is a compact open subgroup of G p and K p is a compact open subgroup of G ( A p f ). The group K p will be called the “tamelevel” and will remain fixed throughout the paper. We thus have a correspondingdecomposition ˆ K f = ˆ K p ˆ K p .We define for a fixed tame level: H • cl ,ε ( ˆ K p , W ) := lim −→ U ⊂ K p H • ( Y ( U K p ) , V W ⊗ ε ) . Let H p be the Hecke algebra of ˜ G ( A p f ) with respect to ˆ K p , and with coefficientsin E . The vector spaces H • cl ,ε ( ˆ K p , W ) have commuting actions of ˜ G = ˜ G p and H p . As a ˜ G -module, H • cl ,ε ( ˆ K p , W ) is smooth and admissible. We may recover thefinite level cohomology H • ( Y ( K p K p ) , V W ⊗ ε ) as the subspace of ˆ K p -invariants in H • cl ,ε ( ˆ K p , W ). -ADIC INTERPOLATION OF METAPLECTIC FORMS 3 Classical points and Eigenvarieties.
We may replace µ by µ/ ker ε withoutchanging the classical cohomology groups. We shall therefore assume from now onthat the map ε : µ → E × is injective. We shall also assume that the coefficientfield E is a complete and discretely valued subfield of C p , and that G is split over E . Our results are most complete (and easiest to state) when G is quasi-split over Q p .Let us assume for now that this is the case, and choose a Borel subgroup B = T N of G defined over Q p , with unipotent radical N and Levi factor T . The group N of Q p -valued points of N has a unique lift ˆ N to ˜ G . For a smooth representation V of˜ G , the smooth Jacquet module V ˆ N is defined to be the space of ˆ N -coinvariants. ThisJacquet module is a smooth representation of the group ˜ T , which is the pre-image in˜ G of the group T of Q p -valued points of T . Applying this smooth Jacquet functor to H • cl ,ε ( ˆ K p , W ), we obtain a representation H • cl ,ε ( ˆ K p , W ) ˆ N of ˜ T ×H p , which is smoothand admissible as a representation of ˜ T . If W is irreducible, then the contragredientrepresentation W (cid:48) is also irreducible, and hence by the highest weight theorem(cf. [Eme06a, Theorem 1.1.1]) ( W (cid:48) ) N is a 1-dimensional algebraic representation of T . Tensoring with this representation, we obtain a locally algebraic representationrep s ( W ) = rep s ( ˆ K p , ε, W ) of ˜ T × H p defined as follows:rep s ( W ) = ( W (cid:48) ) N ⊗ E H s cl ,ε ( ˆ K p , W ) ˆ N . By a classical point of cohomological dimension s , we shall mean an absolutely ir-reducible (and hence finite dimensional) subquotient of rep s ( W ) for some W . Undera certain hypothesis (see definition 1.4.1 below) we shall construct a rigid analyticspace, called the eigenvariety, containing all the classical points (independently ofthe choice of irreducible representation W ).Each classical point is of the form π = π p ⊗ π p , where π p is an irreducible locallyalgebraic representation of ˜ T and π p is an irreducible representation of H p . Let Z be the centre of ˜ T . Recall that by the Stone-von Neumann Theorem for smoothrepresentations of metaplectic tori [Wei09, Theorem 3.1], the representation π p isdetermined by its restriction to Z (here we have used the assumption that ε isinjective). We may therefore regard π p as a point of the rigid analytic space (cid:98) Z oflocally analytic characters of Z .There is a decomposition H p = H sph ⊗ H ramified , where H sph is commutativeand H ramified is finitely generated over E . This gives us a further decomposition π p = π sph ⊗ π ramified , where π sph is in Spec( H sph ). Thus a classical point gives riseto a point ( π p , π sph ) of (cid:98) Z × Spec( H sph ). We shall show that there is a rigid analyticsubspace Eig s ⊂ (cid:98) Z × Spec( H sph ) containing all the classical points of cohomologicaldimension s , such that the projection Eig s → (cid:98) Z is finite, with discrete fibres. Inparticular, the dimension of Eig s is at most the dimension of T .We shall see that there is a representation J s = J s ( ˆ K p , ε ) of ˜ T ×H p , together withinjective homomorphisms rep s ( W ) → J s for each W , so that the representation J s interpolates all of the classical points. As a representation of ˜ T , J s is locallyanalytic, and we shall prove that J s is essentially admissible in the sense of [Eme04](see definition 3.8.1 below). For such a representation, there is a correspondingcoherent sheaf E on (cid:98) Z , such that the fibre at χ ∈ (cid:98) Z is isomorphic (as an H p -module)to the contragredient of the χ -eigenspace in J s . The eigenvariety is defined to bethe relative spectrum Eig s = Spec( A ), where A is the image of H sph in the sheaf ofendomorphisms of E . Localizing E over A , we obtain a canonical sheaf M on Eig s .The sheaf M allows us to recover the action of H ramified on J s .Note that each classical point on the eigenvariety has a (cid:98) Z -coordinate which islocally algebraic. It would be useful to know whether the converse is true, i.e. RICHARD HILL AND DAVID LOEFFLER given a point of the eigenvariety whose (cid:98) Z -coordinate is locally algebraic, can weconclude that the point is a classical point? We prove that if χ ∈ (cid:98) Z is locally( W (cid:48) ) N -algebraic and has non-critical slope (see definition 5.4.3 below), then themap of χ -eigenspaces rep s ( W )[ χ ] → J s [ χ ] is bijective. Hence every point on theeigenvariety with (cid:98) Z -coordinate χ is classical.The representation J s together with the maps rep s ( W ) → J s are defined in twosteps: completed cohomology and the locally analytic Jacquet functor. We shallbriefly discuss these constructions in the next two paragraphs of this introduction.1.4. Completed cohomology.
Recall that we have fixed a prime p of k above p .Let O E be the valuation ring of E .We define the completed cohomology spaces ¯ H s = ¯ H s ( ˆ K p , E ) as follows: Thechoice of ˆ K f determines a µ -covering space ˜ Y ( ˆ K f ) of Y ( K f ), defined by˜ Y ( ˆ K f ) = ˆ G ( k ) \ ˜ G ( A ) / ˆ K ◦∞ ˆ K f . We set ¯ H s ( ˆ K p , E ) = (cid:32) lim ←− n lim −→ K p H s ( ˜ Y ( ˆ K p ˆ K p ) , O E /p n ) (cid:33) ⊗ O E E. The completed cohomology spaces are naturally Banach spaces over E , with com-muting actions of ˜ G = ˜ G p and the Hecke algebra H p . We show in section 4.1 that¯ H s is an admissible continuous representation of ˜ G in the sense of [ST02a] (see def-inition 3.3.1 below). We shall write ¯ H sε ( ˆ K p , E ) for the ε (cid:48) -eigenspace for the actionof µ on ¯ H s ( ˆ K p , E ).Let g be the Lie algebra of G over Q p . There is a natural action of g on thesubspace ¯ H sε, la of Q p -locally analytic vectors in ¯ H sε . We show (see Corollary 4.2.3)that there is a spectral sequence relating this action to the classical cohomologyspaces for any algebraic representation W :Ext r g ( W (cid:48) , ¯ H sε, la ) ⇒ H r + s cl ,ε ( ˆ K p , W ) . Here W (cid:48) denotes the contragredient of W . In particular, we have an edge map(2) H s cl ,ε ( ˆ K p , W ) → Hom g ( W (cid:48) , ¯ H sε, la ) . This map is a homomorphism of smooth ˜ G × H p -representations. Written anotherway, this gives a homomorphism of locally analytic representations:(3) W (cid:48) ⊗ H s cl ,ε ( ˆ K p , W ) → ¯ H sε, la . Definition 1.4.1.
We say the tuple ( ˜ G , p , ε, ˆ K p , s ) satisfies Emerton’s edge mapcriterion if for every W the map (2) is an isomorphism, or equivalently if the imageof (3) is the set of all W (cid:48) -locally algebraic vectors in ¯ H sε . Our results concerning eigenvarieties described above depend on the edge mapcriterion. It is therefore rather important to know that this criterion holds in anumber of cases. It is clear that the criterion is satisfied for s = 0. We provethe following result for s = 1 and s = 2, generalizing the results of [Hil07] to themetaplectic case. Theorem 1.4.2.
Suppose G is semi-simple and simply connected and has positivereal rank (i.e. G ∞ is not compact). Then the edge map (2) is an isomorphism indimension s = 1 . If in addition G has finite congruence kernel and ε is non-trivialthen the edge map is an isomorphism in dimension s = 2 . The definition of these spaces generalises the spaces denoted by ˜ H s ( ... ) in [Eme06b]. Wedenote them here with a horizontal line rather than a tilde in order to avoid conflict with our useof the tilde sign to denote preimages in ˜ G ( A ) of subgroups of G ( A ). -ADIC INTERPOLATION OF METAPLECTIC FORMS 5 The locally analytic Jacquet module.
We then turn to finding an analoguein this situation of the locally analytic Jacquet module construction of [Eme06a].Let P be a parabolic subgroup of G defined over Q p , with unipotent radical N andLevi factor M . In line with our earlier notation, we shall write P , N and M forthe groups of Q p -valued points of P , N and M respectively. For each such P , wedefine a left-exact functor J P from locally analytic representations of ˜ P to those of˜ M .Any smooth or locally algebraic representation may be regarded as a locallyanalytic representation, and so we may apply J P to such representations. We shallprove the following, which determines J P on locally algebraic representations: Theorem 1.5.1. If X is a smooth admissible representation of ˜ G and W is analgebraic representation of G , then J P ( X ⊗ W ) = X ˆ N ⊗ W N , where ˆ N is the unique lift to ˜ G of N . In particular, the locally analytic Jacquet functor coincides with the smoothJacquet functor on smooth representations.Suppose again that G is quasi-split over Q p and let B be a Borel subgroup withLevi subgroup T . The representation J s discussed above is defined as follows: J s = J B (cid:16) ¯ H sε ( ˆ K p , E ) la (cid:17) . Applying the Jacquet functor J B to the map (3) we get the required map rep s ( W ) → J s . Assuming the edge map criterion, we know that (3) is injective with closed im-age. Hence by left exactness, we conclude that the map rep s ( W ) → J s is injective.Roughly speaking, the eigenvariety Eig s defined above is the set of characters of (cid:98) Z × H sph appearing in J s , where H sph is the spherical part of the Hecke algebraof ˆ K p . If the edge map criterion holds, then since J s contains rep s ( W ) for every W , this space contains all characters arising from automorphic representations of˜ G ( A ) which are cohomological in degree s , which have a ˆ K p -fixed vector, andwhose local factor at p is principal series. We also give a “classicality criterion”,i.e. a sufficient condition for a point ( χ, λ ) of Eig( J s ) to appear in the classicalcohomology rep s ( W ).The completed cohomology ¯ H sε is a continuous admissible representation of ˜ G .From this, it follows that its subspace ¯ H sε, la of locally analytic vectors is a stronglyadmissible locally analytic representation of ˜ G in the sense of [Eme04]. In order toshow that J s is essentially admissible, we prove the following. Theorem 1.5.2. If V is an essentially admissible locally analytic representationof ˜ G , then J P ( V ) is an essentially admissible locally analytic representation of ˜ M . Representations of metaplectic tori.
A novel aspect of the metaplecticcase, compared to the theory for algebraic groups, is that the representations J s constructed in the above fashion are representations of a non-abelian group (a meta-plectic extension of a torus). Essentially admissible locally analytic representationsof abelian locally analytic groups are well-understood, and may be interpreted ascoherent sheaves on a rigid-analytic space, which is the method used in [Eme06b]in order to construct eigenvarieties.In the method described above, we have restricted our representation J s of ˜ T to the centre Z of ˜ T in order to construct the eigenvariety. This constructionhas certain drawbacks. The first problem is to do with the ramified part of theHecke algebra. Let us suppose that we have an absolutely irreducible ˜ T × H p -subrepresentation π p ⊗ π sph ⊗ π ramified of J s , and let χ : Z → E × be the central RICHARD HILL AND DAVID LOEFFLER character of π p . We have a corresponding point ( χ, π sph ) in the eigenvariety. Onewould ideally like recover the representation π ramified in the dual space of the fibreof the sheaf M at the point ( χ, π sph ). However, with the construction describedabove the contribution to the fibre is ( π ramified ) ∗ d , where d is the dimension of π p .In a sense, this means that the sheaf M is d times as big as we would like.The second drawback is that the field of definition of a point of the eigenvarietyis not exactly the same as the field of definition of the corresponding absolutelyirreducible representation. More precisely, suppose that χ : Z → E × is a locallyanalytic character, and let π p be the corresponding absolutely irreducible locallyanalytic representation of ˜ T . It can quite easily happen that π p is not defined over E , but only over some finite extension. The field of definition of a point on theeigenvariety will only see the field of definition of χ , since we have restricted to Z in our construction. To some extent this problem is unavoidable, since π p will oftenhave no unique minimal field of definition.In section 6, we show that both of these problems may be resolved assuming acertain “tameness” condition on the group ˜ T . Assuming the tameness condition, weshow that there is an equivalence of categories between the essentially admissiblelocally analytic representations of ˜ T extending ε , and the essentially admissiblelocally analytic representations of Z extending ε . In particular, this implies thatthat character χ and the corresponding representation π p have the same field ofdefinition. Applying this equivalence of categories to J s instead of simply restrictionof representations, we obtain a slightly different coherent sheaf E on (cid:98) Z . This givesrise to a slightly different sheaf M on the eigenvariety, and this new sheaf has thecorrect multiplicities of representations of H ramified . Finally, we show that if G issemi-simple, simply connected and split over k p and if p does not divide the orderof µ , then ˜ T satisfies the tameness condition.1.7. Relation to the work of Emerton.
In this paper we rely heavily on thework of Emerton. The results and definitions of this paper are analogues of thoseobtained by Emerton in the case of algebraic groups (as opposed to metaplec-tic groups). The locally analytic Jacquet functor for representations of reductivealgebraic groups was introduced in [Eme06a], and completed cohomology was in-troduced in [Eme06b], where it was used to construct eigenvarieties for algebraicgroups. In many parts of this paper – particularly in section 5 – the proofs ofour results very closely follow those of Emerton, and rather than reproducing theintricate proofs in full, we have simply indicated how the original arguments needto be modified in order to apply to the metaplectic case.2.
Classical cohomology of metaplectic groups
In this section, we shall recall some standard results on metaplectic groups,and recall the construction of admissible smooth representations arising from thecohomology of arithmetic quotients of these groups.2.1.
Metaplectic groups.
As before, we let G be a connected reductive groupover an algebraic number field k . It is shown in [Del96] that if k contains a primi-tive m -th root of unity, then there is a canonical non-trivial metaplectic extensionof G by the group µ m of all m -th roots of unity in k , and also a canonical liftˆ G ( k ). If G is absolutely simple and algebraically simply connected then there is auniversal metaplectic extension, whose kernel is the group of all roots of unity in k . The universal metaplectic cover coincides with Deligne’s cover. However for ourpurposes, it is sufficient to choose a metaplectic cover, together with a lift ˆ G ( k ). -ADIC INTERPOLATION OF METAPLECTIC FORMS 7 Let K ◦∞ ⊂ G ◦∞ be a maximal compact subgroup. We have a topological centralextension of compact Lie groups:(4) 1 → µ → ˜ K ◦∞ → K ◦∞ → . Recall that G ◦∞ has an Iwasawa decomposition as the product of K ◦∞ and auniquely divisible, topologically contractible group H . This group H thereforehas a unique lift to a subgroup ˆ H of ˜ G ◦∞ , so we have a corresponding Iwasawadecomposition ˜ G ◦∞ = ˜ K ◦∞ ˆ H . Thus the inclusions K ◦∞ (cid:44) → G ◦∞ and ˜ K ∞ (cid:44) → ˜ G ◦∞ arehomotopy equivalences. It follows that the extension (1) splits if and only if (4)splits. Example.
Let G = SL n / Q . There is a unique non-trivial metaplectic double cover (cid:102) SL n ( A ). We may take K ∞ = SO( n ), and then ˜ K ∞ = Spin( n ). The extension is oftype 2, since Spin( n ) is connected.The following proposition gives an ample supply of type 1 metaplectic covers. Proposition 2.1.1.
Let G be semi-simple and algebraically simply connected, andsuppose that for every real place v of k , the group G v is compact. Then everymetaplectic extension of G is of type 1. In particular this holds if k is totallycomplex.Proof. We will show that under these hypotheses, G v is topologically simply con-nected for each infinite place v .If v is a complex place of k then G v is a complex algebraically simply connectedgroup. By the Iwasawa decomposition, G v is homotopy equivalent to a maximalcompact subgroup. On the other hand, if k v is real and G v is compact, then G v is itself a maximal compact subgroup of the complexification G ( C ). So it sufficesto show that if G is an algebraically simply connected semi-simple Lie group over C , then any maximal compact subgroup of G ( C ) is topologically simply connected.This follows readily from Theorem 1.1 of [Ada04].The group G ∞ is therefore a product of simply connected spaces, so is simplyconnected. It follows that the extension splits over G ∞ . (cid:3) Remark . It is a widely held misconception that when k is totally complexevery metaplectic extension is of type 1. By the above proposition, this holdswhen G is semi-simple and simply connected, but it is false in general. Indeed itis even false for double covers of GL (the extension constructed in [Hil10a] is acounterexample).2.2. Arithmetic quotients.
We assume for the remainder of this paper that ˜ G is of type 1.Recall that for a compact open subgroup K f of G ( A f ) we have defined anarithmetic quotient Y ( K f ). Assuming that K f has a lift ˆ K f to ˜ G ( A ), we havedefined a µ -covering space ˜ Y ( ˆ K f ) of Y ( K f ). The topological spaces Y ( K f ) and˜ Y ( ˆ K f ) are homotopic to finite simplicial complexes (see [BS73]). If K f is sufficientlysmall, then these are topological manifolds.As described in the introduction, for any finite-dimensional algebraic represen-tation W of G , over some field E containing Q p , and any character ε : µ → E × , wehave a locally constant sheaf of E -vector spaces on Y ( K f ), V W ⊗ ε = ˜ G ( k ) \ (cid:16) ( ˜ G ( A ) / ˆ K ◦∞ ˆ K f ) × ( W ⊗ ε ) (cid:17) . Since W is finite-dimensional and Y ( K f ) is homotopic to a finite simplicial complex,the cohomology groups H • ( Y ( K f ) , V W ⊗ ε ) are finite-dimensional E -vector spaces. RICHARD HILL AND DAVID LOEFFLER
The formation of V W ⊗ ε is compatible with pullback via the natural maps Y ( K (cid:48) f ) → Y ( K f ), for K (cid:48) f ⊆ K f . Moreover, for g ∈ ˜ G ( A f ), right translation defines anisomorphism [ g ] : Y ( K f ) → Y ( g − K f g ), and an isomorphism of local systems[ g ] ∗ V W ⊗ ε ∼ = V W ⊗ ε ; when g ∈ µ , we have Y ( g − K f g ) = Y ( K f ), and the map issimply multiplication by ε ( g ). These compatibilities imply that if K p is a tamelevel, then the spaces H • cl ,ε ( ˆ K p , W ) := lim −→ U ⊂ K f H • ( Y ( U K p ) , V W ⊗ ε ) . are smooth representations of ˜ G p , on which µ acts via the character ε . Since the ˆ K p -invariants of the representation H • cl ,ε ( ˆ K p , W ) can be identified with H • ( Y ( K p K p ) , V W ⊗ ε ),the representations H • cl ,ε ( ˆ K p , W ) are admissible smooth representations of ˜ G p .Note that we also have a local system on ˜ Y ( ˆ K f ) defined by V W = ˆ G ( k ) \ (cid:16) ( ˜ G ( A ) / ˆ K ◦∞ ˆ K f ) × W (cid:17) . If we write pr : ˜ Y ( ˆ K f ) → Y ( K f ) for the projection map, then we have an isomor-phism of local systems: pr ∗ ( V W ) = (cid:77) η : µ → E × V W ⊗ η , and hence by Shapiro’s lemma (or the spectral sequence of the map pr), H • ( ˜ Y ( ˆ K f ) , V W ) = (cid:77) η : µ → E × H • ( Y ( K f ) , V W ⊗ η ) . Connection with the Kubota symbol.
We give an alternative descriptionof the cohomology groups in the special case that G is absolutely simple, simplyconnected, and has positive real rank. In this situation there is a universal meta-plectic cover ˜ G ( A ) by the group µ m of all roots of unity in k . The groups G ∞ and K ∞ are connected, and their quotient X = G ∞ /K ∞ is a symmetric space. Thecompact open subgroup K f determines an arithmetic subgroupΓ = G ( k ) ∩ ( K f × G ∞ ) . Furthermore the arithmetic quotient Y ( K f ) is connected, and may be identifiedwith a quotient of X as follows: Y ( K f ) = Γ \ X. Recall that we have lifts τ : G ( k ) → ˆ G ( k ), and τ : K f × G ∞ → ˆ K f × ˆ G ∞ . Theselifts are both defined on the arithmetic subgroup Γ, but they are not equal on thatsubgroup. The Kubota symbol is defined to be the ratio of these two lifts: κ ( γ ) = τ ( γ ) τ ( γ ) − , γ ∈ Γ . We recall that the Kubota symbol is a surjective homomorphism Γ → µ m . Its kernelis a non-congruence subgroup of Γ. Indeed in many cases κ gives an isomorphismbetween the congruence kernel and µ m (see for example Theorem 2.9 of [PR83]).Suppose again that W is an algebraic representation of G over E . By restriction,we obtain an action of Γ on W , and we may twist this action by the character ε ◦ κ to get a new action. We can form the local system on Y ( K f ): V (cid:48) W ⊗ ε = Γ \ (cid:16) X × ( W ⊗ ( ε ◦ κ )) (cid:17) . One can check that V (cid:48) W ⊗ ε is an isomorphic local system to V W ⊗ ε . In particular, wemay express our cohomology groups in terms of group cohomology: H • ( Y ( K f ) , V W ⊗ ε ) = H • group (Γ , W ⊗ ( ε ◦ κ )) . -ADIC INTERPOLATION OF METAPLECTIC FORMS 9 Non-vanishing of metaplectic cohomology.
We next show in some simplecases that the spaces H • cl ,ε ( ˆ K p , W ) are non-zero. Proposition 2.4.1.
Suppose that G ∞ is compact. Then for any K f sufficientlysmall and any W , the vector space H ,ε ( ˆ K f , W ) is non-zero.Proof. Since G ∞ is compact, the double quotient˜ G ( k ) \ G ( A ) / ˆ K ◦∞ ˆ K f is a finite set. Moreover, if µ , . . . , µ r is a set of coset representatives, the groupsΓ j = ˜ G ( k ) ∩ µ j ˆ K ◦∞ ˆ K f µ − j are finite. Then, for any W , the space H ( Y ( K f ) , V W ⊗ ε ) can be identified with thespace of maps from the finite set µ , . . . , µ r to W ⊗ ε for which f ( µ j ) ∈ ( W ⊗ ε ) Γ j .By shrinking K f if necessary, we may assume that ˆ K f is torsion-free, and hence allthe groups Γ j are trivial. Therefore H ( Y ( K f ) , V W ⊗ ε ) is non-zero. (cid:3) Proposition 2.4.2.
Let k be an imaginary quadratic field containing an m -th rootof unity; let G = SL /k and let ˜ G be the canonical metaplectic extension of G by µ m . Then for K f sufficiently small there is a non-trivial character ε : µ m → C × such that the space H ( Y ( K f ) , C ⊗ ε ) is non-zero.Proof. In the case of SL , the Kubota symbol has been determined on Γ = Γ( m ).It is given by (see Proposition 1 in § κ (cid:18) a bc d (cid:19) = (cid:40)(cid:0) cd (cid:1) k,m if c (cid:54) = 0,1 otherwise.Here the notation (cid:0) cd (cid:1) k,m means the m -th power residue symbol in the field k . LetΓ = ker( κ ). Then we have a decomposition H (Γ , C ) = (cid:77) η : µ m → C × H (Γ , η ◦ κ ) . We shall suppose that each of the spaces H (Γ , η ◦ κ ) is zero apart from the spacewhere η is trivial, and so we are assuming H (Γ , C ) = H (Γ , C ).We shall write Y for the arithmetic quotient Γ \ X and ˜ Y for the µ m -cover Γ \ X .We shall also write ∂Y and ∂ ˜ Y for the boundaries of the Borel–Serre compactifica-tions of Y and ˜ Y respectively. We have a commutative diagram, in which the rowsare exact: H ( Y, C ) → H ( ∂Y, C ) → H ( Y, C ) → ↓ ↓ ↓ H ( ˜ Y , C ) → H ( ∂ ˜ Y , C ) → H ( ˜ Y , C ) → Y and ˜ Y are both connected topological 3-manifolds, it follows that H ( Y, C ) and H ( ˜ Y , C )are both one-dimensional, and the third vertical map is also an isomorphism. Adiagram chase shows that the middle vertical map is surjective. However, themiddle vertical arrow is known to be injective, since the composition H ( ∂Y, C ) pr ∗ → H ( ∂ ˜ Y , C ) pr ∗ → H ( ∂Y, C )is known to be scalar multiplication by the degree of the cover ∂ ˜ Y → ∂Y . We havetherefore shown that the map H ( ∂ ˜ Y , C ) → H ( ∂Y, C ) is bijective. By examiningthe map ∂ ˜ Y → ∂Y , will show that this is not the case.Recall that the connected components of ∂Y correspond to Γ-conjugacy classesof Borel subgroups B = TN defined over k . For each such Borel subgroup we let Γ B = Γ ∩ B ( k ). As Γ is torsion-free, we have Γ B ⊂ N ( k ), and the correspondingboundary component is defined by ∂Y ( B ) = Γ B \ N ( C ). Since Γ B is a lattice in N ( C ) ∼ = C , we see that each boundary component is a 2-torus. Hence the dimensionof H ( ∂Y, C ) is exactly the number of Γ-conjugacy classes of Borel subgroups.Each Γ-conjugacy class of Borel subgroups is a finite union of Γ -conjugacy classes.However, since we are assuming that H ( ∂Y, C ) = H ( ∂ ˜ Y , C ), we conclude thatthe Γ-conjugacy class of each Borel subgroup is equal to its Γ -conjugacy class.Recall that a Borel subgroup B is called essential if the restriction of κ to Γ B istrivial, or equivalently if Γ B ⊂ Γ . Not every Borel subgroup is essential; howeverthe standard Borel subgroup of upper triangular matrices is clearly essential by (5).Suppose B is any essential Borel subgroup. Since we are assuming that the Γ- andΓ -conjugacy classes of B are equal, it follows that the inclusion Γ → Γ gives us abijection Γ / Γ B ∼ = Γ / Γ B . From this we conclude that Γ = Γ , which gives us the desired contradiction. (cid:3) Remark . The argument in the proof of Proposition 2.4.2 only shows that H ( Y ( K f ) , C ⊗ ε ) contains some non-trivial Eisenstein cohomology classes. In fact,one can show that there are also metaplectic cusp forms of cohomological type on˜SL . This follows by examining the Shimura correspondence for the group ˜GL /k (see [Fli80]). In particular, it is shown that if ˜ π is an automorphic representationof ˜GL , then there is a corresponding automorphic representation π of GL . If π iscuspidal, then so is ˜ π . The image of the map ˜ π (cid:55)→ π is also calculated. In particularif π has level 1, then it has a preimage ˜ π . Finally, one can check that if π is ofcohomological type, then a certain twist of ˜ π will be of cohomological type.3. Background on p -adic representation theory Continuous cohomology.
Throughout this section, we let G be a locallycompact, totally disconnected group. By a continuous G -module, we shall mean anabelian topological group V , together with an action of G by endomorphisms of V ,such that the map G × V → V is continuous. Suppose V and W are continuous G -modules. We shall write H • cts ( G , V ) and Ext • G ( V, W ) for the continuous cohomologygroups (see for example [CW74]). We shall sometimes consider continuous G -modules V , which are locally convex topological vector spaces over a field E . Inthis case, the field E will always be a complete discretely valued subfield of C p .We begin by recalling a rather technical aspect of continuous cohomology from[CW74]. Let V be a continuous representation of G and let C n ( G , V ) be the abeliangroup of continuous maps G n +1 → V . We regard C n ( G , V ) as a topological groupwith the compact-open topology. We have an exact sequence of G -modules(6) 0 → V → C ( G , V ) → C ( G , V ) → · · · . Recall that H • cts ( G , V ) is the cohomology of the cochain complex C n ( G , V ) G . Definition 3.1.1. (see § [CW74] ) The group Z n ( G , V ) = ker( C n ( G , V ) → C n +1 ( G , V )) is given the subspace topology; the group B n ( G , V ) = Im( C n − ( G , V ) → C n ( G , V )) is given the quotient topology as C n − ( G , V ) /Z n − ( G , V ) . Since (6) is exact, thegroups B n ( G , V ) and Z n ( G , V ) are identical for n > , and the identity map givesa continuous bijective homomorphism B n ( G , V ) → Z n ( G , V ) . We say that thecohomology H • cts ( G , V ) is strongly Hausdorff if the maps B n ( G , V ) → Z n ( G , V ) are open, i.e. if the two topologies are the same. -ADIC INTERPOLATION OF METAPLECTIC FORMS 11 Lemma 3.1.2.
Suppose G is a union of countably many compact subsets and V isa Fr´echet space over E with a continuous action of G . Then the cohomology groups H • cts ( G , V ) are strongly Hausdorff.Proof. The spaces C n ( G , V ) are Fr´echet spaces. Hence the closed subspaces Z n ( G , V )are also Fr´echet spaces, and the quotient spaces B n +1 = C n /Z n are Fr´echet spaces.The map B n ( G , V ) → Z n ( G , V ) is a continuous linear bijection of Fr´echet spaces.By the open mapping theorem [Sch02, Proposition 8.6], this map is an isomorphismof topological vector spaces. (cid:3) Theorem 3.1.3 (Standard facts about continuous cohomology) . Let G be a locallycompact, totally disconnected topological group and H a closed subgroup of G . Let E be a field as described above. (1) The vector space C ( G , E ) of continuous functions from G to E is continu-ously injective as a module over H . In particular H r cts ( H , C ( G , E )) is zerofor r > . (2) Suppose H is normal in G and V is a continuous G -module. If the groups H • cts ( H , V ) are strongly Hausdorff then there is a natural continuous actionof G / H on H • cts ( H , V ) , and there is a spectral sequence E r,s = H r cts ( G / H , H s cts ( H , V )) which converges to H r + s cts ( G , V ) . (3) Suppose G is a profinite group. Then for each r > there is a short exactsequence → lim ←− (1) t H r − ( G , Z /p t ) → H r cts ( G , Z p ) → lim ←− t H r cts ( G , Z /p t ) → . The notation lim ←− (1) means the first derived functor of the projective limitfunctor (see for example [Wei94, section 3.5] ).Proof. Part (1) is a special case of Proposition 4(a) of [CW74]. Part (2) is propo-sition 5 of [CW74]. Part (3) is a special case of Theorem 2.3.4 of [NSW00]. (cid:3)
Some functional analysis.
We shall consider continuous representations ofa topological group on locally convex topological vector spaces over a coefficientfield E containing Q p . We again suppose that E is a complete discretely valuedsubfield of C p , so in particular E is spherically complete [Sch02, Lemma 1.6].For two topological vector spaces V, W , we shall write L ( V, W ) for the vectorspace of continuous linear maps from V to W . We shall always regard L ( V, W ) asa topological vector space with the strong topology, and sometimes write L b ( V, W )to emphasise this. The notation V (cid:48) will mean the strong dual of V .Recall that a Fr´echet space V may be written as the projective limit of a sequence V ← V ← · · · , where each V i is a Banach space and each transition map is surjective. If eachtransition map is nuclear, then V is called a nuclear Fr´echet space . A topologicalvector space V over E is said to be of compact type if it is the locally convexinductive limit of a sequence V → V → V → · · · , where each V i is a Banach space over E , and each transition map is compact andinjective. Compact type spaces are Hausdorff, complete, bornological, reflexive (andhence barrelled). The strong dual of a compact type space is a nuclear Fr´echet spaceand vice versa. More precisely, the functor which takes a compact type space toits strong dual is an antiequivalence of categories, between the category of compacttype spaces and the category of nuclear Fr´echet spaces. Suppose V and W are Fr´echet spaces. There is a canonical topology on V ⊗ W ,in which the continuous bilinear maps V × W → X correspond to the continuouslinear maps V ⊗ W → X . This topology is not, in general, Hausdorff. We shallwrite V ˆ ⊗ W for the Hausdorff completion of V ⊗ W with respect to this topology.3.3. Continuous admissible representations.
In this section, we shall supposethat we have a connected reductive group G defined over Q p , and we write G forthe group of Q p -valued points. We shall suppose also that we have a topologicalcentral extension 1 → µ → ˜ G pr → G → , where µ is a finite abelian group.Let K be a compact open subgroup of ˜ G and let C ( K ) be the vector space ofcontinuous functions f : K → E . The supremum norm on functions makes C ( K )into a Banach space over E . We shall write D ( K ) for its strong dual. The space D ( K ) is naturally a Banach algebra over E , with multiplication given by convolutionof distributions. This algebra is known to be Noetherian [Eme04, Theorem 6.2.8].If V is a continuous representation of ˜ G , and V is also a Banach space, then thereis a natural action of D ( K ) on the dual space V (cid:48) (see [Eme04, Proposition 5.1.7]). Definition 3.3.1 ([Eme04, Proposition-Definition 6.2.3]) . Let V be a continuousrepresentation of ˜ G on a Banach space. We call this representation admissible con-tinuous if the dual space V (cid:48) is finitely generated as a D ( K ) -module. This conditiondoes not depend on the choice of compact open subgroup K . If the coefficient field E is a finite extension of Q p , then the above definitionis equivalent to the definition of an admissible continuous representation given in[ST02a, § Locally analytic representations.
We shall write g for the Lie algebra of G over Q p . By a Lie sublattice h in g , we shall mean a finitely generated Z p -submodule,which spans g over Q p , and which is closed under the Lie bracket operation. Sucha sublattice defines a norm on g , with respect to which h is the unit ball. Hencethere is an affinoid H , such that h = H ( Q p ). If h is sufficiently small, then theBaker–Campbell–Hausdorff formula converges on H , and gives H the structure of arigid analytic group. Furthermore the exponential map converges on H , and givesa bijection exp : H ( Q p ) → H for some compact open subgroup H of G . A subgroup H which arises in this way is called a good analytic open subgroup of G . Recall thatthere is a compact open subgroup K of G , which lifts to a subgroup ˆ K of ˜ G . Weshall fix such a subgroup, together with its lift. Definition 3.4.1.
By a good analytic open subgroup of ˜ G , we shall mean a compactopen subgroup ˆ H of ˆ K , such that the image of ˆ H in G is a good analytic opensubgroup of G . Given a good analytic open subgroup H of ˜ G and a Banach space V , we write C an ( H , V ) for the vector space of V -valued functions in H , which are given by apower series which converges on the whole of H . We regard C an ( H , V ) as a Banachspace in which the topology is given by the supremum norm on H . More generally,if V is a Hausdorff locally compact topological vector space, then we define C an ( H , V ) = lim −→ W → V C an ( H , W ) , where W → V runs through Banach spaces which map continuously and injectivelyinto V .Suppose that V is also a continuous representation of ˜ G . We call a vector v ∈ V H -analytic if the orbit map H → V given by h (cid:55)→ hv is represented by an element -ADIC INTERPOLATION OF METAPLECTIC FORMS 13 of C an ( H , V ). We shall write V H − an for the subspace of H -analytic vectors in V .The topology on V H − an is defined to be that given by the supremum norm on C an ( H , V ).A vector v ∈ V is said to be locally analytic if it is H -analytic for a suitable goodanalytic open subgroup H . The subspace of locally analytic vectors in V will bewritten V la . We have an isomorphism of vector spaces: V la = lim −→ H V H − an . We shall regard V la as a topological vector space with the direct limit topology. If H is a proper subgroup of H then the map V H − an → V H − an is compact, andso V la is a compact type space.There is a natural continuous map V la → V . We call V a locally analytic repre-sentation if this map is an isomorphism of topological vector spaces.3.5. Fr´echet–Stein algebras.
Let A be a locally convex topological E -algebra. AFr´echet–Stein structure [ST03] on A is an isomorphism of locally convex topological E -algebras, A = lim ←− n A n , such that(1) each A n is a left-Noetherian Banach algebra;(2) each map A n +1 → A n is a continuous homomorphism, and is right-flat.An algebra A with such a structure is called a Fr´echet–Stein algebra. Suppose A = lim ←− n A n is a Fr´echet–Stein algebra and M is an A -module. We say that M is coadmissible if there is an isomorphism of A -modules M = lim ←− n M n , such that(1) each M n is a finitely generated locally convex topological A n -module;(2) each map M n +1 → M n of A n +1 -modules induces an isomorphism M n +1 ⊗ A A n → M n of A n -modules.If M = lim ←− n M n is coadmissible, then it automatically follows that M n = M ⊗ A A n . The category of coadmissible modules over a Fr´echet–Stein algebra has manyof the same good properties as the category of finitely–generated modules over aNoetherian Banach algebra (which is a special case); in particular, it is an abeliancategory.3.6. Locally analytic distributions.
Let K be a compact open subgroup of ˜ G .Fix a good analytic open subgroup H of K . A function K → E is said to be H - analytic if its restriction to every H -coset can be written as a power seriesexpansion on the corresponding Lie sublattice in g . The H -analytic functions on K form a Banach space with respect to the supremum norm on the functions. Weshall call this space C H − an ( K ).If H ⊂ H is a proper subgroup, and is also a good analytic open subgroup,then every H -analytic function is H -analytic, and so we have an inclusion C H − an ( K ) (cid:44) → C H − an ( K ) . A function on K is said to be locally analytic if there exists a good analytic opensubgroup H , such that the function is H -analytic. We shall write C la ( K ) for thespace of such functions. We clearly have C la ( K ) = lim −→ n C H n − an ( K ) , where H n is a basis of neighbourhoods of the identity in ˜ G consisting of good analyticopen subgroups. By construction, C la ( K ) has compact type. We shall write D la ( K )for its strong dual, which is therefore a nuclear Fr´echet space. Furthermore, thereis a convolution multiplication on D la ( K ) induced by the group law on K . We canwrite D la ( K ) as a projective limit of Banach algebras: D la ( K ) = lim ←− n D H n − an ( K ) , D H n − an ( K ) = C H n − an ( K ) (cid:48) . This gives D la ( K ) the structure of a Fr´echet–Stein algebra [ST03, Theorem 5.1].3.7. Admissible locally analytic representations.
We now recall the definitionof an admissible locally analytic representation, introduced in [ST03]. If V is alocally analytic representation of ˜ G , and K is a compact open subgroup of ˜ G , thenthe action of K on V (cid:48) extends to a continuous D la ( K )-module structure (see [ST02b,Proposition 3.2] or [Eme04, Proposition 5.1.9(ii)]). We say V is admissible locallyanalytic if V (cid:48) is coadmissible for one, or equivalently every, open compact K . Onthe other hand, V is said to be strongly admissible if V (cid:48) is finitely generated over D la ( K ). Since every finitely generated module is coadmissible, it follows that everystrongly admissible locally analytic representation is an admissible locally analyticrepresentation. Theorem 3.7.1.
Let V be an admissible continuous representation of ˜ G . Then thesubspace V la of locally analytic vectors in V has the structure of a locally analyticrepresentation of ˜ G . Furthermore, V la is strongly admissible.Proof. This is shown in [ST03, Theorem 7.1] under the mild additional hypothesisthat E is a finite extension of Q p , and for general complete discretely valued E in[Eme04, 6.2.4] (which is stated for locally analytic groups). (cid:3) If V and W are locally analytic representations of ˜ G , then they have an actionof the Lie algebra g of G over Q p , and we shall write H • Lie ( g , V ) and Ext • g ( V, W ) forthe Lie algebra cohomology. There is a smooth action of ˜ G on Ext • g ( V, W ). Theorem 3.7.2 (Emerton) . Let V be an admissible continuous representation of K and let W be a finite dimensional algebraic representation of K . Then there arecanonical isomorphisms Ext • K ( W, V ) ∼ = Ext • g ( W, V la ) K , H • cts ( K , V ) ∼ = H • Lie ( g , V la ) K . In particular we have
Ext • g ( W, V la ) ∼ = lim −→ U Ext • U ( W, V ) , H • Lie ( g , V la ) ∼ = lim −→ U H • cts ( U , V ) , where the limits are taken over subgroups U of finite index in K .Proof. The formulae of the left column follow from those of the right column appliedto V ⊗ W (cid:48) , so it suffices to prove the latter. These follow from [Eme06b, Prop.1.1.12(ii) and Theorem 1.1.13]. (cid:3) Essentially admissible locally analytic representations.
The definitionof essentially admissible locally analytic representations, introduced in § G is locally Q p -analytic, and its centre Z = Z ˜ G is topologically finitely generated. Following the construction loc.cit. , we may con-struct a rigid space (cid:98) Z over E , together with a “universal character” Z (cid:44) → C an ( (cid:98) Z ).This space parametrises the locally Q p -analytic characters of Z , in the sense thatif E (cid:48) /E is any finite extension, there is a canonical bijection between the E (cid:48) -pointsof (cid:98) Z and the E (cid:48) -valued characters of Z , where a point x ∈ (cid:98) Z ( E (cid:48) ) corresponds tothe composition of the universal character with the evaluation map at x . -ADIC INTERPOLATION OF METAPLECTIC FORMS 15 If V is a locally analytic representation of ˜ G , then the dual space V (cid:48) has anaction of D la ( K ). Suppose also that the Z ˜ G -action on V extends to a separatelycontinuous action C an ( (cid:98) Z ) × V → V. Then by [Eme04, Proposition 6.4.7], V (cid:48) is also a topological C an ( (cid:98) Z )-module. Theactions of C an ( (cid:98) Z ) and D la ( K ) on V (cid:48) commute, and so V (cid:48) has an action of C an ( (cid:98) Z ) ˆ ⊗D la ( K ), which is a Fr´echet–Stein algebra. Definition 3.8.1.
Let V be a locally analytic representation of ˜ G . We say that V is essentially admissible if it satisfies the following two conditions: (1) the Z -action on V extends to a separately continuous action of C an ( (cid:98) Z ) , (2) for one, and hence for every compact open subgroup K of ˜ G , the dual space V (cid:48) is co-admissible as a module over the Fr´echet–Stein algebra C an ( (cid:98) Z ) ˆ ⊗D la ( K ) . Theorem 3.8.2.
Every admissible locally analytic representation of ˜ G is an essen-tially admissible locally analytic representation.Proof. This is [Eme04, Proposition 6.4.10], and is stated there for arbitrary locallyanalytic groups. (cid:3)
Suppose that the centre Z has finite index in ˜ G , and let V be an essentiallyadmissible locally analytic representation of ˜ G . Then V is by restriction an es-sentially admissible representation of Z . For any affinoid U ⊂ (cid:98) Z , the module V (cid:48) ⊗ C an ( (cid:98) Z ) C an ( U ) is finitely generated over C an ( U ), and so we may regard V (cid:48) asa coherent sheaf on (cid:98) Z . The functor which takes V to V (cid:48) is an anti-equivalence ofcategories between the category of essentially admissible locally analytic represen-tations of Z and the category of coherent sheaves on (cid:98) Z (see the discussion followingProposition 6.4.10 in [Eme04]).4. Completed cohomology of metaplectic groups
In this section, we adapt some definitions and results of Emerton [Eme06b] tothe metaplectic case. Suppose again that we have a connected reductive group G defined over a number field k , and that we have a type 1 metaplectic cover of G by µ . Given a representation W of ˜ K p , we may define a corresponding local system on Y ( K p K p ) as follows: V p ( W ) = (cid:16) ( ˆ G ( k ) \ ˜ G ( A ) / ˆ K p ˆ K ◦∞ ) × W (cid:17) / ˜ K p . Suppose as before that our coefficient field E is an extension of Q p . Again, let W be an algebraic representation of G over E and let ε : µ → E × be an injectivecharacter. We then have an action of G p on W . This gives rise to an action of˜ K p = ˆ K p ⊕ µ , in which ˆ K p acts through its isomorphism with K p and µ acts byscalar multiplication by ε . We shall call this representation W ⊗ ε . We thereforehave a local system V p ( W ⊗ ε ) on Y ( K p K p ).As in Lemma 2.2.4 of Emerton [Eme06b], we note that V p ( W ⊗ ε ) is canonicallyisomorphic to the local system V W ⊗ ε defined in the introduction. In particular,these two local systems have the same cohomology groups. It is important tosee these cohomology groups from both points of view: when regarding them asthe cohomology of V W ⊗ ε , it is clear that these groups are defined over any field ofdefinition of W . In particular, it follows that the eigenvalues of the Hecke operatorson these spaces are algebraic. When regarding them as cohomology groups of V p ( W ⊗ ε ), we shall see that we are able to p -adically interpolate the systems ofeigenvalues.4.1. The representations ¯ H s . Let C ( ˜ K p ) be the vector space of continuous func-tions f : ˜ K p → E . The vector space C ( ˜ K p ) is a continuous representation of˜ K p × ˜ K p , where the first ˜ K p acts on function by left-translation and the secondby right-translation. Using one of these ˜ K p actions, we can define a local system V ( ˜ K p ) of ˜ K p -modules on Y ( ˆ K p ˆ K p ) by V ( ˜ K p ) = (cid:16) ( ˆ G ( k ) \ ˜ G ( A ) / ˆ K ◦∞ ˆ K p ) × C ( ˜ K p ) (cid:17) / ˜ K p , and we shall be interested in the cohomology groups of this local system:¯ H • ( ˆ K p , E ) = H • ( Y ( ˆ K p ˆ K p ) , V ( ˜ K p )) . The vector spaces ¯ H • ( ˆ K p , E ) have an action of ˜ K p , together with a commutingactions of H p . To avoid sign errors, we state now that we have used the right-translation action of ˜ K p to form the local system V ( ˜ K p ); the action of ˜ K p on¯ H • ( ˆ K p , E ) is given by the left-translation action.Note that the vector space C ( ˜ K p ) decomposes as a direct sum of µ -isotypicsubspaces: C ( ˜ K p ) = (cid:77) η : µ → E × C ( ˜ K p ) η , where C ( ˜ K p ) η consists of continuous functions f : ˜ K p → E , such that f ( ζx ) = η ( ζ ) · f ( x ) for all ζ ∈ µ . As a consequence, we have a similar decomposition¯ H • ( ˆ K p , E ) = (cid:77) η : µ → E × ¯ H • η ( ˆ K p , E ) . Note that as the left and right translation actions of µ differ by a sign, we have thefollowing: Lemma 4.1.1.
The subspace ¯ H • η ( ˆ K p , E ) of ¯ H • ( ˆ K p , E ) is the η (cid:48) -eigenspace for theaction of µ . Lemma 4.1.2.
As representations of ˜ K p , the spaces ¯ H • ( ˆ K p , E ) are admissiblecontinuous representations.Proof. Recall that Y ( K p K p ) is homotopic to a finite simplicial complex Y . Let Y ( d )be the set of simplices of Y of dimension d . Then ¯ H • ( ˜ K p , E ) is the cohomology ofthe chain complex(7) 0 → C ( ˜ K p ) Y (0) → C ( ˜ K p ) Y (1) → C ( ˜ K p ) Y (2) → · · · . Hence each cohomology group is a subquotient of finitely many copies of C ( ˜ K p ). (cid:3) Theorem 4.1.3.
There is a canonical isomorphism: ¯ H • ( ˆ K p , E ) ∼ = (cid:32) lim ←− n lim −→ K p H • ( ˜ Y ( ˆ K p ˆ K p ) , O E /p n ) (cid:33) ⊗ O E E. Proof.
This is a special case of [Hil10b, Theorems 2.5 and 2.10]. (cid:3)
Corollary 4.1.4.
The action of ˜ K p on ¯ H • ( ˆ K p , E ) extends to a canonical actionof ˜ G p . -ADIC INTERPOLATION OF METAPLECTIC FORMS 17 Proof.
This is immediate from the previous theorem, since ˜ G p already acts on thespace lim −→ K p H • ( ˜ Y ( ˆ K p ˆ K p ) , O E /p n ) . (cid:3) Corollary 4.1.5.
The group ¯ H • ( ˆ K p , E ) is independent (up to a canonical isomor-phism) on the choice of K p .Proof. This is immediate from Theorem 4.1.3. (cid:3)
Some spectral sequences.Theorem 4.2.1.
Let W be any continuous representation of ˜ K p over E , and let W (cid:48) be the contragredient representation on the continuous dual space. There is aspectral sequence Ext r ˜ K p ( W, ¯ H s ( ˆ K p , E )) ⇒ H r + s ( Y ( K p K p ) , V p ( W (cid:48) )) . Proof.
This is a special case of Theorem 3.5 of [Hil10b]. We note that for algebraic W , Emerton’s original proof in [Eme06b] extends to the metaplectic case withoutmodification. (cid:3) In what follows we let G = Rest k p Q p ( G × k k p ). Thus G is a reductive group over Q p and we may canonically identify G ( Q p ) with G ( k p ). Suppose that W is analgebraic representation of G over E . For a character ε : µ → E × , we let W ⊗ ε bethe representation of ˜ K p in which ˆ K p acts through K p and µ acts by ε . Corollary 4.2.2.
There is a spectral sequence
Ext r ˆ K p ( W (cid:48) , ¯ H sε ( ˆ K p , E )) ⇒ H r + s ( Y ( K p K p ) , V W ⊗ ε ) . Proof.
We shall apply the previous theorem to the representation ( W ⊗ ε ) (cid:48) . Tosimplify notation, we shall write ¯ H s in place of ¯ H s ( ˆ K p , E ). By the theorem, wehave a spectral sequence whose E r,s term is Ext r ˜ K p ( W (cid:48) ⊗ ε (cid:48) , ¯ H s ). The corollary isproved by the following calculation:Ext r ˜ K p ( W (cid:48) ⊗ ε (cid:48) , ¯ H s ) = Ext r ˆ K p ( W (cid:48) , Hom µ ( ε (cid:48) , ¯ H s ))= Ext r ˆ K p ( W (cid:48) , ¯ H sε ) . In the first line above we have used the Hochschild–Serre spectral sequence, whichdegenerates because µ is finite and E has characteristic zero. The second line isimmediate from Lemma 4.1.1. (cid:3) Corollary 4.2.3.
There is a spectral sequence
Ext r g ( W (cid:48) , ¯ H sε ( ˆ K p , E ) la ) ⇒ H r + s cl ,ε ( ˆ K p , W ) . Proof.
This follows by taking the direct limit of the formula of the previous corol-lary, over levels ˆ K p , and applying Theorem 3.7.2. (cid:3) Definition 4.2.4.
We shall say that the triple ( ˜ G , ˆ K p , ε ) satisfies the edge mapcriterion in dimension n if for every finite dimensional algebraic representation W of G , the edge map in the spectral sequence above gives an isomorphism Hom g ( W (cid:48) , ¯ H nε ( ˆ K p , E ) la ) ∼ = H n cl ,ε ( ˆ K p , W ) . Calculation of certain spaces ¯ H nε . In this section we let G be absolutelysimple, simply connected, and of positive real rank. We recall if k contains aprimitive m -th root of unity, then there is a canonical metaplectic extension of G by µ m , and this extension is the universal metaplectic extension if µ m is thegroup of all roots of unity in k . We shall assume that ˜ G is one of these canonicalextensions.Recall that such a group G satisfies strong approximation, and so we may iden-tify the arithmetic quotient Y ( K p K p ) with Γ \ X , where X is the symmetric space G ∞ /K ∞ , and Γ is the congruence subgroup of level K p K p . The local system V ( K p ) may by identified with Γ \ ( X × C ( ˜ K p )), where the action of Γ on C ( ˜ K p ) isby right-translation. As a consequence, we have¯ H • ( ˆ K p , E ) = H • group (Γ , C ( ˜ K p )) . Here we are regarding Γ as a subgroup of ˜ K p through the mapsΓ ∼ = { g ∈ ˆ G ( k ) : pr( g ) ∈ Γ } (cid:44) → ˜ K f = ˜ K p × ˆ K p → ˜ K p . Theorem 4.3.1.
Let ˜ G be as described above. If ε is non-trivial then ¯ H ε ( ˆ K p , E ) =0 . If ε is trivial then ¯ H ε ( ˆ K p , E ) ∼ = E .Proof. By strong approximation, the arithmetic quotient Y ( K p K p ) is connected,and so ¯ H ( ˆ K p ) is the space of elements of C ( ˜ K p ), which are right- ˜ K p -invariant.This is simply the space of constant functions. The action of µ by left translationsis trivial on the constant functions. This proves the result. (cid:3) By the strong approximation theorem, the closure of Γ in G ( A f ) is K f . Weshall write K f for the profinite completion of Γ. There is a canonical surjectivehomomorphism K f → K f . The congruence kernel Cong( G ) is defined to be thekernel of this map. We therefore have a short exact sequence of profinite groups:1 → Cong( G ) → K f → K f → . The congruence kernel measures the extent to which Γ has non-congruence sub-groups of finite index. The Kubota symbol gives a map Cong( G ) → µ m , and we letCong( G ) be the kernel of this map. It is conjectured (and in many cases proved)that when µ m is the group of all roots of unity in k , the Kubota symbol is anisomorphism whenever G has real rank at least 2. When G has real rank 1, it isconjectured that Cong( G ) is infinite. Theorem 4.3.2.
Let ˜ G be as described above. There is a canonical isomorphism ¯ H ( ˆ K p , E ) = Hom cts , ˆ K p (Cong , E ) . Proof.
Recall that K f is a compact open subgroup of G ( A f ), which lifts to asubgroup ˆ K f of ˜ G ( A f ). We shall choose such a K f of the form K p K p . As before,we write ˜ K f for the preimage of K f in ˜ G ( A f ). Furthermore let ¯ K f be the profinitecompletion of Γ( K f ). The congruence kernel is defined to be the kernel of thecanonical map ¯ K f → K f . This map factors through ˜ K f , and the kernel of the map¯ K f → ˜ K f is Cong . We therefore have an extension of groups:1 → Cong → ¯ K f → ˜ K f → . Let C ( ¯ K f ) be the vector space of continuous functions from ¯ K f to E . We shallregard C ( ¯ K f ) as a Γ × Cong -module. By Theorem 3.1.3 there are two spectralsequences: H r (Γ , H s cts (Cong , C ( ¯ K f ))) ⇒ H r + s cts (Γ × Cong , C ( ¯ K f )) ,H r cts (Cong , H s (Γ , C ( ¯ K f ))) ⇒ H r + s cts (Γ × Cong , C ( ¯ K f )) . -ADIC INTERPOLATION OF METAPLECTIC FORMS 19 (Here we are regarding Γ × Cong as a topological group, in which the profinitegroup Cong is an open subgroup.) Again by Theorem 3.1.3, the first of thesespectral sequences degenerates, as we have H s cts (Cong , C ( ¯ K p )) = (cid:40) C ( ˜ K p ) if s = 0 , s > . From this, it follows that H r cts (Γ × Cong , C ( ¯ K f )) = H r (Γ , C ( ˜ K f ))From the second spectral sequence, we have an inflation-restriction sequence:0 → H (Cong , E ) → H (Γ , C ( ˜ K f )) → H (Γ , C ( ¯ K f )) Cong . By Theorems 5 and 6 of [Hil10b], we have H • (Γ , C ( ¯ K f )) = E ⊗ Z p lim ←− t lim −→ Υ H • (Υ , Z /p t ) , where Υ runs through the subgroups of finite index in Γ. In particular, we have H (Γ , C ( ¯ K f )) = 0, and so(8) H (Cong , E ) ∼ = H (Γ , C ( ˜ K f )) . Next, we consider C ( ˜ K f ) as a Γ × ˆ K p -module. As before, we have H s cts ( ˆ K p , C ( ˜ K f )) = (cid:40) C ( ˜ K p ) s = 0 , s > . This implies H • (Γ × ˆ K p , C ( ˜ K f )) = H • (Γ , C ( ˜ K p )) = ¯ H • ( ˆ K p , E ) . Hence there is a spectral sequence H r cts ( ˆ K p , H s (Γ , C ( ˜ K f ))) ⇒ ¯ H r + s ( ˆ K p , E ) . The sequence of low degree terms gives: H ( K p , E ) → ¯ H ( ˆ K p , E ) → H (Γ , C ( ˜ K f )) ˆ K p → H ( K p , E ) . The theorem will be proved by (8) when we have shown that the middle mapin the sequence above is an isomorphism. We shall show that the first and lastterms above are zero. As E is a field of characteristic zero, it follows that for anynormal subgroup U of finite index in K p , we have H • cts ( K p , E ) = H • cts ( U, E ) K p .It is therefore sufficient to prove that H ( U, E ) and H ( U, E ) are trivial for asuitable subgroup U of finite index in K p . Let S be finite set of finite places of k , distinct from p , such that for every prime q outside S ∪ { p } ∪ ∞ , the subgroup K q = K p ∩ G ( k q ) is a hyperspecial maximal compact subgroup, and is perfect.For primes q in S , we choose a subgroup K q contained in K p ∩ G ( k q ). We shalltake U to be the subgroup (cid:81) q (cid:54) = p K q . This can be written as U = U S ⊕ U S , where U S = (cid:81) p (cid:54) = q / ∈ S K q and U S = (cid:81) q ∈ S K q . Since G is semisimple, it follows that forevery prime q , the commutator subgroup [ K q , K q ] has finite index in K q , and henceHom( K q , E ) = 0. We therefore have H ( U, E ) = 0.To calculate H ( U S , E ), recall the short exact sequence of Theorem 3.1.3:0 → lim ←− (1) t H ( U S , Z /p t ) → H ( U S , Z p ) → lim ←− t H ( U S , Z /p t ) → . By the same argument as above, the groups H ( U S , Z /p t ) are zero, and so the firstterm in this short exact sequence is zero. We have from [Ser73, § H • ( U S , Z /p t ) = lim −→ T finite H • ( K T , Z /p t ) , where T runs over the finite sets of places of k which do not intersect S ∪ { p } ∪ ∞ and K T = (cid:81) q ∈ T K q . In particular, since K T is a perfect group it has a universaltopological central extension, whose kernel we shall denote π ( K T ). It follows that H ( U S , Z /p t ) = lim −→ T finite Hom( π ( K T ) , Z /p t ) = Hom (cid:0) Π , Z /p t (cid:1) , where Π = (cid:89) q / ∈ S ∪{ p }∪∞ π ( K q ) . The group Π is a product of finite groups, and so we havelim ←− t Hom (cid:0) Π , Z /p t (cid:1) = 0 . This shows that H ( U S , Z p ) = 0, and in particular H ( U S , E ) = 0.Next let q be a prime in S which does not lie above p . In this case there is an E -valued Haar measure on K q , so it follows that H r ( K q , E ) = 0 for all r > r = 2.Finally, suppose q is a prime in S which lies above p . In this case there is anisomorphism [Laz65, Theorem 2.4.10 of Chapter V] H • cts ( K q , E ) = ( H • Lie ( g , Q p ) ⊗ E ) K q , where the Lie algebra g of K q is regarded as a Lie algebra over Q p . By Whitehead’sSecond Lemma [Wei94] we have H ( g , Q p ) = 0, and so in this case we also have H ( K q , E ) = 0.As a consequence, we deduce that H ( U, E ) = 0, which finishes the proof ofthe theorem. (cid:3)
In particular, if the congruence kernel is finite then ¯ H = 0. As a result, wehave: Theorem 4.3.3.
For any metaplectic group the edge map criterion holds in di-mension . If G is semi-simple, simply connected and has positive real rank, and ε is non-trivial then the edge map criterion holds in dimension . If in addition G has finite congruence kernel, then the edge map criterion holds in dimension 2.Proof. In dimension 0 the edge map is clearly an isomorphism, since it is in thebottom left corner of the spectral sequence. If G is semi-simple, simply connectedand has positive real rank, and ε is non-trivial then we’ve seen that ¯ H ε = 0, andso the edge map is an isomorphism in dimension 1. When the congruence kernel isfinite we also have ¯ H ε = 0, and so the edge map is an isomorphism in dimension 2. (cid:3) The p -adic metaplectic Jacquet functor The results of this section are of a local nature, so we shall alter our notation.We now suppose that we have a connected reductive group G defined over Q p , andwe write G for the group of Q p -valued points. We shall suppose also that we havea topological central extension1 → µ → ˜ G pr → G → , where µ is a finite abelian group. There is a compact open subgroup K of G whichlifts to a subgroup ˆ K of ˜ G .Let P be a parabolic subgroup of G defined over Q p with unipotent radical N ,and choose a Levi component M . We shall also write P , M and N for the groups P ( Q p ), M ( Q p ) and N ( Q p ) respectively. We write ˜ P , ˜ M and ˜ N for the preimagesof P , M and N in ˜ G . -ADIC INTERPOLATION OF METAPLECTIC FORMS 21 Lemma 5.0.4.
Let N , M and P be as above. (a) There is a unique subgroup ˆ N of ˜ N , such that ˆ N projects bijectively onto N . (b) The subgroup ˆ N is open (and hence closed) in ˜ N and normal in ˜ P . Fur-thermore we have ˜ P = ˜ M (cid:110) ˆ N . (c) Let τ : N → ˆ N be the unique splitting of pr : ˜ N → N . For any ˜ m ∈ ˜ M andany n ∈ N , we have ˜ m − τ ( n ) ˜ m = τ ( m − nm ) , where m = pr( ˜ m ) .Proof. For the moment we shall regard Q p as a discrete additive group. As such, Q p is uniquely divisible, and so we have H • group ( Q p , µ ) = µ . By this we mean that H ( Q p , µ ) = µ and H n group ( Q p , µ ) = 0 for n >
0. Suppose we have a centralextension of discrete groups 1 → Q p → N → N → . It follows by the Hochschild–Serre spectral sequence that H • group ( N , µ ) = H • group ( N , µ ).The group N may be constructed from a sequence of central extensions by Q p ofthe trivial group. Therefore H • group ( N , µ ) = µ , and in particular H ( N , µ ) = 0.This shows that the extension splits on N , and hence shows the existence of ˆ N .For uniqueness, suppose that τ , τ : N → ˜ N are two splittings. It follows that n (cid:55)→ τ ( n ) τ ( n ) − is a homomorphism from N to µ . Since N is divisible, thishomomorphism must be trivial, so τ = τ .To see that ˆ N is open and closed in ˜ N , we note that the same calculation asabove proves for the continuous cohomology N that H • cts ( N , µ ) = µ . In particular,the section τ : N → ˆ N is continuous (and hence a homeomorphism).As N is normal in P , it follows that ˜ N is normal in ˜ P . The uniqueness propertyof ˆ N shows that ˆ N is normal in ˜ P .Part (c) also follows from the uniqueness of ˆ N . (cid:3) Now let N be a compact open subgroup of N , and let ˆ N = τ ( N ). Following § M + = { m ∈ M : m N m − ⊂ N } , and ˜ M + = { ˜ m ∈ ˜ M : ˜ m ˆ N ˜ m − ⊂ ˆ N } . Lemma 5.0.5.
With the notation described above, ˜ M + is the preimage of M + in ˜ M .Proof. This is immediate from the fact that ˆ N is a normal subgroup of ˜ P . (cid:3) Let Z M and Z ˜ M be the centres of M and ˜ M respectively. Lemma 5.0.6.
The image of Z ˜ M in G is a subgroup of Z M of finite index.Proof. Let ˜ z ∈ pr − ( Z M ) and ˜ m ∈ ˜ M . Furthermore let z = pr(˜ z ) and m = pr( ˜ m ).Since Z M is central in M , we have [ z, m ] = 1, and therefore [˜ z, ˜ m ] ∈ µ . Since ourextension is central, it follows that the commutator [˜ z, ˜ m ] depends only on z and m . Furthermore, one easily checks that the map Z M × M → µ given by( z, m ) (cid:55)→ [˜ z, ˜ m ]is bimultiplicative. In particular, if z is an | µ | -th power, then [˜ z, ˜ m ] = 1 for all˜ m . This shows that the projection of Z ˜ M contains Z | µ | M . Since Z M is topologicallyfinitely generated, it follows that Z | µ | M has finite index in Z M . This proves thelemma. (cid:3) Remark . The projection of Z ˜ M is typically not equal to Z M . For example,suppose G = GL ( Q p ). Assume Q p contains an m -th root of unity. Then Kubotahas defined an m -fold cover ˜ G of G . We obviously have Z G = Q × p . On the otherhand, the image in Z ˜ G of Z G is { x ∈ Q × p : ∀ y ∈ Q × p , ( x, y ) p ,m = ( y, x ) p ,m } . This is the set of x such that x is an m -th power in Q × p . Lemma 5.0.8.
The group ˜ M is generated as a semigroup by ˜ M + and Z ˜ M .Proof. Choose any ˜ m ∈ ˜ M and let m = pr( ˜ m ). Since pr( Z ˜ M ) has finite index in Z M , Lemma 3.3.1 of [Eme06a] shows that there is an element z ∈ pr( Z ˜ M ), suchthat zm N m − z − ⊂ N . Hence zm ∈ M + . If ˜ z is any preimage of z in Z ˜ M , thenby Lemma 5.0.5 we have ˜ z ˜ m ∈ ˜ M + . (cid:3) Definition of the Jacquet functor.
We shall consider representations ofthe group ˜ G over a coefficient field E containing Q p . Let ( V, π ) be a locally analyticrepresentation of ˜ P . In this section we shall define the Jacquet functor J P ( V ).Fix a compact open subgroup P of P and let ˜ P be the preimage of P in ˜ G . Letˆ N and ˜ M be the intersections of ˜ P with ˜ M and ˆ N respectively. We also definetwo semigroups:˜ M + = { m ∈ ˜ M : m ˆ N m − ⊂ ˆ N } , ˜ Z + = ˜ M + ∩ Z ˜ M . Recall that the subspace V ˆ N has a natural action of ˜ M + . This action is definedas follows. For m ∈ ˜ M + and v ∈ V ˆ N , the vector π ( m ) v will be in V m ˆ N m − , andwe define (cid:16) π ˆ N ( m ) (cid:17) ( v ) = (cid:90) ˆ N π ( nm ) v dn, where the Haar measure on ˆ N is normalized to have total measure 1. For elements m ∈ ˜ M we have π ˆ N ( m ) v = π ( m ) v , and so the action π ˆ N of ˜ M + on V ˆ N is locallyanalytic.Let (cid:98) Z ˜ M be the rigid analytic space of locally analytic characters of Z ˜ M , andwrite C an ( (cid:98) Z ˜ M , E ) for the ring of E -valued analytic functions on (cid:98) Z ˜ M . Definition 5.1.1. If V is a locally analytic representation of ˜ M + then we definethe finite slope part of V by V fs = L b, ˜ Z + ( C an ( (cid:98) Z ˜ M , E ) , V ) . There is a natural map Z ˜ M → C an ( (cid:98) Z ˜ M , E ), which makes V fs into a Z ˜ M -module.Furthermore, the action of ˜ M + on V gives rise to an action of ˜ M + on V fs . Theactions of Z ˜ M and ˜ M + coincide on their intersection Z +˜ M , and so generate an actionof the group ˜ M = ˜ M + Z ˜ M on V fs . Definition 5.1.2. If V is a locally analytic representation of ˜ P , then we define theJacquet functor of V by J P ( V ) = (cid:16) V ˆ N (cid:17) fs . The Jacquet functor preserves essential admissibility.Theorem 5.2.1. If V is an essentially admissible locally analytic representationof ˜ G then J P ( V ) is an essentially admissible locally analytic representation of ˜ M . -ADIC INTERPOLATION OF METAPLECTIC FORMS 23 Theorem 5.2.1 was proved by Emerton in the algebraic case. The proof is ratherlong, and most of it carries through word for word to the metaplectic case. Theonly difference is a technical lemma on the structure of the group G . We quote thislemma below, and we prove its generalization to the metaplectic case.Before stating these lemmata, we must recall the definition of a rigid analyticIwahori decomposition, and generalize this concept to the metaplectic case. Let P and ¯ P be parabolic subgroups of G defined over Q p , so that M = P ∩ ¯ P is a Levicomponent of both P and ¯ P . We shall write N and ¯ N for the unipotent radicalsof P and ¯ P respectively. We shall write n , ¯ n and m for the Lie algebras of N , ¯ N and M over Q p , and N , ¯ N and M for their groups of Q p -valued points. Suppose H is a good analytic open subgroup of G , which arises from the Lie sublattice h of g ,with underlying rigid analytic group H . Furthermore define M = M ∩ H , N = N ∩ H , ¯ N = ¯ N ∩ H . Finally, we let M , N and ¯ N denote the rigid analytic closures of M , N and ¯ N in H . The subgroup H is said to admit a rigid analytic Iwahori decomposition ifthe following conditions are satisfied:(1) The groups M , N and ¯ N are good analytic open subgroups of M , N and¯ N corresponding to the Lie sublattices m ∩ h , n ∩ h and ¯ n ∩ h , and withunderlying rigid analytic groups M , N and ¯ N .(2) The rigid analytic map ¯ N × M × N → H given by multiplication in H is an isomorphism of rigid analytic spaces.We next give a corresponding definition for subgroups of ˜ G . Note that we have acompact open subgroup K of G , which lifts to a subgroup ˆ K of ˜ G , and we also havea unique lifts ˆ N and ˆ¯ N of N and ¯ N to ˜ G . These lifts do not necessarily coincide on K ∩ N and K ∩ ¯ N . However, by reducing the size of K is necessary, we may assumethat ˆ K ∩ ˆ N pr ∼ = K ∩ N , ˆ K ∩ ˆ¯ N pr ∼ = K ∩ ¯ N . Definition 5.2.2.
Suppose ˆ K and K are chosen to satisfy these conditions above.We say that a good analytic subgroup H of ˜ G has a rigid analytic Iwahori decom-position with respect to P and ¯ P if (i) H is contained in ˆ K and (ii) the image of H in G has a rigid analytic Iwahori decomposition with respect to P and ¯ P . Recall the following technical result of Emerton:
Proposition 5.2.3. [Eme06a, Prop. 4.1.6]
We may find a decreasing sequence { H n } n ≥ of good analytic open subgroups of G , cofinal in the directed set of allanalytic open subgroups of G , and satisfying the following conditions: (i) For each n ≥ , the inclusion H n +1 ⊂ H n extends to a relatively compactrigid analytic map H n +1 ⊂ H n . (ii) For each n ≥ , the subgroup H n of H is normal.Let P ∅ be a minimal parabolic subgroup of G defined over Q p . The remainingproperties refer to any pair P and ¯ P of opposite parabolic subgroups of G , chosenso that P contains P ∅ and ¯ P contains ¯ P ∅ . (iii) Each H n admits a rigid analytic Iwahori decomposition with respect to P and ¯ P . (iv) If z ∈ Z M is such that z − ¯ N z ⊂ ¯ N , then z − ¯ N n z ⊂ ¯ N n for each n ≥ . (v) If z ∈ Z M is such that z N z − ⊂ N , then z N n z − ⊂ N n for each n ≥ . (vi) We may find z ∈ Z M such that z − ¯ N z ⊂ ¯ N and z N z − ⊂ N , andsuch that, for each n ≥ , the embedding of part (iv) factors through theinclusion ¯ N n +1 ⊂ ¯ N n . In order to prove Theorem 5.2.1, it is sufficient to prove the following resultanalogous to Proposition 5.2.3. The rest of the proof of the theorem is word forword the same as in [Eme06a].
Proposition 5.2.4.
We may find a decreasing sequence { H n } n ≥ of good analyticopen subgroups of ˜ G , cofinal in the directed set of all analytic open subgroups of ˜ G ,and satisfying the following conditions: (i) For each n ≥ , the inclusion H n +1 ⊂ H n extends to a relatively compactrigid analytic map H n +1 ⊂ H n . (ii) For each n ≥ , the subgroup H n of H is normal.Let P ∅ be a minimal parabolic subgroup of G defined over Q p . The remainingproperties refer to any pair P and ¯ P of opposite parabolic subgroups of G , chosenso that P contains P ∅ and ¯ P contains ¯ P ∅ . (iii) Each H n admits a rigid analytic Iwahori decomposition with respect to P and ¯ P . (iv) If ˜ z ∈ Z ˜ M is such that ˜ z − ˆ¯ N ˜ z ⊂ ˆ¯ N , then ˜ z − ˆ¯ N n ˜ z ⊂ ˆ¯ N n for each n ≥ . (v) If ˜ z ∈ Z ˜ M is such that ˜ z ˆ N ˜ z − ⊂ ˆ N , then ˜ z ˆ N n ˜ z − ⊂ N n for each n ≥ . (vi) We may find ˜ z ∈ Z ˜ M such that ˜ z − ˆ¯ N ˜ z ⊂ ˆ¯ N and ˜ z ˆ N ˜ z − ⊂ ˆ N , andsuch that, for each n ≥ , the embedding of part (iv) factors through theinclusion ˆ¯ N n +1 ⊂ ˆ¯ N n .Proof. Recall that we have a compact open subgroup K of G , which lifts to asubgroup ˆ K of ˜ G . Furthermore, K and ˆ K are chosen small enough so that for eachstandard parabolic subgroup P = MN , the lift K → ˆ K coincides with the lift N → ˆ N (resp. ¯ N → ˆ¯ N ) on K ∩ N (resp. K ∩ ¯ N ). Emerton’s proof of Proposition5.2.3 actually shows a little bit more than is stated. He in fact shows that we mayin addition take H to be arbitrarily small. We may therefore take a sequenceof subgroups H n satisfying Proposition 5.2.3 with H contained in K . We thendefine a new sequence of subgroups ˆ H n in ˜ G , where each ˆ H n is the lift of H n to ˆ K .We claim that the sequence ˆ H n satisfies Proposition 5.2.4. Properties (i), (ii) and(iii) for ˆ H n are clear, since they only depend on the original groups H n . We nextconsider property (iv). For an element ˜ z ∈ Z ˜ M , we shall write z for the image of z in Z M . Lemma 5.0.4 (c) shows that the equation ˜ z − ˆ N n ˜ z ⊂ ˆ N n is equivalent to z − N n z ⊂ N n . Hence property (iv) of Proposition 5.2.4 is a consequence of property(iv) of Proposition 5.2.3. Similarly, property (v) of Proposition 5.2.4 follows fromthe corresponding property in Proposition 5.2.3. Suppose that z ∈ Z M is chosento satisfy property (vi) of Proposition 5.2.3. It follows that every power z r ( r > z r which is also in pr( Z ˜ M ). We replace z by such a power and let ˜ z be a pre-imagein Z ˜ M of z . Again using Lemma 5.0.4 (c), we deduce that ˜ z has property (vi) ofProposition 5.2.4. (cid:3) Exactly as in [HL11], we may strengthen Theorem 5.2.1 as follows. Let D be thederived subgroup of M (a semisimple algebraic group over Q p ), and D = D ( Q p ).Then if D is an open compact subgroup of D which lifts to a subgroup ˆ D of ˜ D ,and W a finite-dimensional continuous representation of ˆ D , we may consider therepresentation ( J P ( V ) ⊗ W ) ˆ D -ADIC INTERPOLATION OF METAPLECTIC FORMS 25 of Z ˜ M . This representation is essentially admissible, by proposition 3.3 of [HL11].Since Z ˜ M is commutative, this space corresponds to a coherent sheaf on the char-acter space (cid:100) Z ˜ M . Let Σ be the support of this sheaf. Differentiation of charactersgives a map (cid:100) Z ˜ M → ˇ z , where z is the Lie algebra of Z M over E and ˇ z its dual space. Theorem 5.2.5. If V is admissible, then the map Σ → ˇ z has discrete fibres. If P = B is a Borel subgroup, so D is trivial, then this is a metaplectic analogueof [Eme06a, Proposition 4.2.23], and the proof also carries over identically usingthe family of subgroups H n constructed above. In the general case, one need onlyadd the requirement that the subgroups M n have rigid-analytic decompositions asproducts of subgroups of D and Z M ; then the proof proceeds exactly as in [HL11, § The Jacquet functor of an admissible smooth representation.
In thissection, we consider a smooth admissible representation V of ˜ G . The classicaltheory of the Jacquet functor for smooth representations of algebraic groups appliesequally to metaplectic covers such as ˜ G (see [McN10, § N -coinvariants V ˆ N . This is the largest quotient of V on which ˆ N acts trivially, andis a smooth representation of ˜ M . We may however regard V as a locally analyticrepresentation, so we also have the locally analytic Jacquet functor J P ( V ) definedabove. In this section we show that J P ( V ) is canonically isomorphic to V ˆ N .For a smooth representation V of ˜ G , we shall always regard the vector space V asa topological vector space with the finest locally convex topology. In this topology, every vector subspace V (cid:48) ⊆ V is closed, and the subspace topology on V (cid:48) is againthe finest locally convex topology. This has the following consequence: if U is aFr´echet space, and f : U → V is a continuous linear map, then f must have finiterank. This is because the space Coim( f ) = U/ ker( f ), with its quotient topology, isa Fr´echet space; but it maps continuously and bijectively to Im( f ), which has thefinest locally convex topology. Hence this map is a topological isomorphism, andwe see that the finest locally convex topology on Im( f ) is Fr´echet, which can onlyhappen if Im( f ) is finite-dimensional. Theorem 5.3.1.
Let V be an admissible smooth representation of ˜ G . Then thereis a canonical isomorphism J P ( V ) ∼ = V ˆ N , where V ˆ N is the space of ˆ N -coinvariants of V .Proof. The proof of this theorem is exactly the same as in the algebraic case, whichis dealt with in [Eme06a]. We shall merely recall the main steps. Composingthe inclusion V ˆ N → V with the projection V → V ˆ N we get a canonical mapΦ : V ˆ N → V ˆ N , and this map is ˜ M + -equivariant. Using the smoothness of V , wecan show that Φ is also surjective: indeed for any v ∈ V we may define v (cid:48) = π ˆ N ( v ) = 1 | ˆ N | (cid:90) ˆ N π ( n ) v dn. Clearly v (cid:48) is in V ˆ N , and has the same image in V ˆ N as v . The kernel of Φ consists ofthose vectors v ∈ V ˆ N for which there is a sufficiently large compact open subgroupˆ N ⊂ ˆ N , for which π ˆ N ( v ) = 0.We call a vector v ∈ V ˆ N null if there is a z ∈ Z +˜ M , such that π ˆ N ( z )( v ) = 0. Oneeasily checks that the kernel of Φ consists of the null vectors in V ˆ N . To completethe proof, it suffices to show that V ˆ N = ( V ˆ N ) null ⊕ ( V ˆ N ) fs . Using the fact that V is an admissible smooth representation, one can show thateach v ∈ V ˆ N is contained in a finite dimensional Z +˜ M -invariant subspace W . Wetherefore have V ˆ N = lim −→ W with W ranging over such subspaces. Since C an ( (cid:98) Z ˜ M )is a Fr´echet space, the remark preceding the theorem shows that ( V ˆ N ) fs = lim −→ W fs .Furthermore it is clear that ( V ˆ N ) null = lim −→ W null . It follows by elementary linearalgebra, that W = W fs ⊕ W null for any finite dimensional representation W of Z +˜ M ,and so the result follows. (cid:3) More generally, we have the following result applying to locally algebraic repre-sentations of ˜ G : Theorem 5.3.2.
Let V be an admissible smooth representation of ˜ G and let W bea finite dimensional irreducible algebraic representation of G , which we shall regardas a representation of ˜ G , trivial on µ . Then there is a canonical isomorphism J P ( V ⊗ W ) ∼ = V ˆ N ⊗ W N . Proof.
The proof of this result in the algebraic case (Proposition 4.3.6 of [Eme06a])works in the metaplectic case. We shall recall some details here.Let n be the Lie algebra of ˆ N . The action of n on V is trivial, and so we have( V ⊗ W ) n = V ⊗ W n . On the other hand, since the action of ˆ N on W is algebraic(and ˆ N is connected) we have W n = W ˆ N . In particular, the action of ˆ N on W n istrivial, so we have ( V ⊗ W n ) ˆ N = V ˆ N ⊗ W ˆ N . This implies ( V ⊗ W ) ˆ N = V ˆ N ⊗ W N .The action of ˜ M + on W ˆ N is the restriction of the usual action of M . Hence byproposition 3.2.9 of [Eme06a] (which is stated in sufficient generality for our needs),it follows that J P ( V ⊗ W ) = ( V ˆ N ) fs ⊗ W N . The result now follows from Theorem5.3.1. (cid:3) Corollary 5.3.3.
Suppose that G is quasi-split over Q p and let B = MN be aBorel subgroup defined over Q p . Assume also that G splits over E . Let V be a ad-missible smooth representation of ˜ G and let W ψ be the finite dimensional irreduciblealgebraic representation of G with highest weight ψ with respect to B . Then thereis a canonical isomorphism of representations of ˜ M : J B ( V ⊗ W ψ ) ∼ = V ˆ N ⊗ ψ. Proof.
This is just a special case of the previous result. (cid:3)
Small slope vectors.
In this section, we’ll assume for simplicity that G issplit over the coefficient field E , so every irreducible algebraic representation of G over E is absolutely irreducible. We shall write Z M for the centre of M ; the group Z M is a torus, and we write S for the maximal subtorus of Z M which splits over Q p . Let ord denote the valuation on Q p mapping p to 1.Let χ : Z ˜ M → E × be a continuous (hence locally Q p -analytic) character. Thehomomorphism Z ˜ M → Q given by t (cid:55)→ ord( χ ( t )) clearly factors through the pro-jection to Z M . Since the image of Z ˜ M has finite index in Z M (and Q is uniquelydivisible) this extends uniquely to a linear functional on Q ⊗ Z ( Z M / ( Z M ) ), where( Z M ) is the maximal compact subgroup of Z M .As in [Eme06a, § Q ⊗ Z ( Z M / ( Z M ) ) with Q ⊗ Z Y • , where Y • is the cocharacter group of S . The linear functional constructed above thusdefines an element of Q ⊗ Z Y • , where Y • is the character group of the maximalsplit subtorus; and as in op.cit. we may define slope( χ ) ∈ Q ⊗ Z Y • to be thiselement.Let us write ∆( G , S ) for the set of positive restricted roots of S in G (that is,the set of characters of S appearing in the adjoint action on Lie( N )). We write R -ADIC INTERPOLATION OF METAPLECTIC FORMS 27 for the sublattice of Y • generated by ∆( G , S ), which is not necessarily of full rank,and ( Q ⊗ Z R ) ≥ for the Q ≥ -cone in Q ⊗ Z Y • generated by ∆( G , S ). Finally, we let ρ denote the weighted half-sum of ∆( G , S ), i.e. half the sum of the characters of S appearing in the adjoint action on N weighted by their multiplicities.The usefulness of these definitions arises from the following two lemmas, gener-alising Lemmas 4.4.1 and 4.4.2 of [Eme06a] to the metaplectic case. Recall that wedefined Z +˜ M = Z ˜ M ∩ ˜ M + . Lemma 5.4.1.
We have | χ ( a ) | ≤ for all a ∈ Z +˜ M if and only if slope( χ ) ∈ ( Q ⊗ Z R ) ≥ .Proof. It suffices to note that the projection of Z ˜ M has finite index in Z M ; thusthe projection of Z +˜ M is cofinal with Z + M , and hence the proof given in [Eme06a]extends to the metaplectic case also. (cid:3) Using this in place of Lemma 4.4.1 of [Eme06a], we deduce the following analogueof Lemma 4.4.2 of op.cit. : Lemma 5.4.2.
If a locally analytic representation V of ˜ P admits a norm which is ˜ P -invariant, and χ ∈ (cid:100) Z ˜ M is such that ( V ˆ N )[ Z +˜ M = χ ] (cid:54) = 0 , then ρ + slope( χ ) ∈ ( Q ⊗ Z R ) ≥ . We now recall what is meant by an element of Q ⊗ Z Y • being of non-criticalslope . We write ∆( G , Z M ) for the set of positive restricted roots of Z M (that is,the set of characters α of Z M appearing in the adjoint action on Lie N ).By hypothesis, G is split over E , so we may choose a Borel subgroup B of G defined over E . We can and do assume that the unipotent radical N (cid:48) of B contains N E , and we choose a Levi factor T of B such that Z M ⊆ T ⊆ M . We define∆( G , T ) as the set of characters of T appearing in the adjoint action on n (cid:48) = Lie N (cid:48) .As shown in [Eme06a, § α ∈ ∆( G , Z M ),there is a unique simple positive root ˜ α ∈ ∆( G , T ) with ˜ α | Z M = α . To ˜ α is attachedan element s ˜ α of the Weyl group W ( G , T ). We define ρ to be the weighted half-sumof ∆( G , S ), i.e. half the sum of the characters of S appearing in the adjoint actionon N weighted by their multiplicities; and ˜ ρ the half-sum of ∆( G , T ), so ˜ ρ | S = ρ .Let W be an irreducible algebraic representation of G over E . Then W N is anirreducible algebraic representation of M , and in particular Z M acts on W N viaa character ψ . Let ˜ ψ be the highest weight of W N with respect to M ∩ B ; then˜ ψ | Z M = ψ . It is shown in op.cit. that the element s ˜ α ( ˜ ψ + ˜ ρ ) | S ∈ Y • is independent of the choice of Borel subgroup B . Let χ be a character of Z ˜ M whichis locally ψ -algebraic (that is, we may write χ = θψ , where θ is locally constant). Definition 5.4.3 ([Eme06a, Definition 4.4.3]) . We say χ = θψ is of criticalslope with respect to the representation W N if, for some simple positive root α ∈ ∆( G , Z M ) , the element s ˜ α ( ˜ ψ + ˜ ρ ) | S + ρ + slope( θ ) ∈ ( Q ⊗ Z Y • ) lies in ( Q ⊗ Z R ) ≥ . Otherwise, we say χ is of non-critical slope. Exactly as in [Eme06a, § Theorem 5.4.4.
Let V be a locally analytic representation of ˜ G , and suppose that V admits a ˜ G -invariant norm. Let W be an irreducible algebraic representation of G , and let ψ : Z M → G m be the central character of W N . Then for any character χ : Z ˜ M → E × which is locally ψ -algebraic and of non-critical slope with respect to W N , the map (9) J P ( V W − loc . alg . )[ Z ˜ M = χ ] → J P ( V ) W N − loc . alg . [ Z ˜ M = χ ] is an isomorphism. This result is proved for algebraic groups in [Eme06a, section 4.4], and the sameproof works for representations of metaplectic groups, using the key lemma 5.4.2.
Proof.
Injectivity of (9) follows from the fact that J P is left-exact. We must provesurjectivity. Note that by [Eme06a, Prop 3.2.12] we have isomorphisms J P ( V )[ Z +˜ M = χ ] ∼ = ( V ˆ N )[ Z +˜ M = χ ] , J P ( V W − loc . alg . )[ Z +˜ M = χ ] ∼ = ( V ˆ N W − loc . alg . )[ Z +˜ M = χ ] . We must therefore show that every vector in ( V ˆ N ) W N − loc . alg . [ Z +˜ M = χ ] is locally W -algebraic.Suppose that v is in ( V ˆ N ) W N − loc . alg . [ Z +˜ M = χ ] and is not locally W -algebraic.Since v generates a direct sum of copies of W N under the action of Lie( P ), we mayassume that v is annihilated by n (cid:48) = Lie( N (cid:48) ); that is, v is a highest weight vectorfor g with respect to n (cid:48) , of weight ˜ ψ .Hence the U ( g )-submodule ( U g ) · v ⊆ V is a quotient of a Verma module withhighest weight ψ . Since by assumption v is not locally algebraic, this quotient mustbe infinite-dimensional. It follows from the Bernstein-Gelfand-Gelfand resolutionof W in terms of Verma modules that there must exist a positive simple root α such that X m +1 − α v (cid:54) = 0, where X − α is in the − α root space in g ; m = (cid:104) ψ, α ∨ (cid:105) ; and α ∨ is the corresponding coroot. The calculation [Eme06a, Prop. 4.4.4] shows that X m +1 − α v is in V ˆ N [ Z +˜ M = γ ], where γ = α − m − χ . Applying Lemma 5.4.2 to γ showsthat if V has a ˜ P -invariant norm, and ( V ˆ N )[ Z +˜ M = γ ] (cid:54) = 0, then ρ + slope( γ ) mustlie in ( Q ⊗ R • ) ≥ . From this, we deduce that χ has critical slope, so we have acontradiction. (cid:3) Emerton’s eigenvariety machine.
In this paragraph, our discussion be-comes global again. We therefore begin with a connected reductive group G definedover an algebraic number field k and a fixed prime p of k over p . We define G = Rest k p Q p ( G × k k p ) . Thus G is an algebraic group over Q p and we let G be the group of Q p -valued pointsof G . There is an isomorphism of groups G = G p . As before, we assume thatwe have a metaplectic extension ˜ G of G by µ , and we define ˜ G to be the centralextension of G obtained by identifying G with G p .The local theory of the preceding paragraphs was, for a few technical reasons,described only over Q p . We shall apply this theory to the group ˜ G .We make the assumption that G is quasi-split over Q p ; note that this holds ifand only if G is quasi-split over k p , which is well known to be true for all butfinitely many primes p . Let B be a Borel subgroup of G , and T a Levi factor of B .Following our general notational conventions, we let B and T denote the Q p -pointsof these, and ˜ B and ˜ T their preimages in ˜ G = ˜ G p .Let V be an essentially admissible locally analytic ˜ G -representation, equippedwith a commuting action of H p . Let H sph be the spherical part of the Heckealgebra. By functoriality, there is an action of H p on J B ( V ), which is an essentially -ADIC INTERPOLATION OF METAPLECTIC FORMS 29 admissible representation of ˜ T . Let Z = Z ˜ T be the centre of ˜ T . Then J B ( V ) restrictsto give an essentially admissible representation of Z , and so there is a correspondingcoherent rigid analytic sheaf E on (cid:98) Z . There is an action of H sph on the sheaf E ,and we let A be the image of H sph in the sheaf of endomorphisms of E . We thendefine the Eigenvariety of V to be the following a rigid analytic spaceEig( V ) = Spec( A ) ⊂ (cid:98) Z × Spec( H sph ) . A point ( χ, λ ) ∈ (cid:98) Z × Spec( H sph ) is in Eig( V ) if and only if the ( Z = χ, H sph = λ )-eigenspace in J B ( V ) is non-zero. By construction, the sheaf E is the push-forwardto (cid:98) Z of a sheaf on Eig( V ), which we also denote by E . This sheaf E is a sheaf ofright H p -modules, and the fibre of E over a point ( χ, λ ) is isomorphic as a right H p -module to the dual of the ( Z = χ, H sph = λ )-eigenspace in V . Theorem 5.5.1. (i)
The map
Eig( V ) → (cid:98) Z is finite. (ii) If V is admissible as a representation of ˜ G , then the map Eig( V ) → ˇ t has discrete fibres; in particular the dimension of Eig( V ) is at most thedimension of T over Q p .Proof. Part (i) is true by construction, since Eig( V ) is defined as the relative spec-trum of a coherent sheaf of algebras on (cid:98) Z . Part (ii) follows from the correspondingstatement for the support of the coherent sheaf E on (cid:98) Z , which is Theorem 5.2.5above. (cid:3) Let us now take V = ¯ H nε ( ˆ K p , E ) Q p − la . This is admissible, so the precedingtheorem applies to Eig( V ). We suppose for the remainder of this section that theedge map criterion (Definition 1.4.1) holds for ( ˜ G , p , ˆ K p , ε, n ). Then for all algebraicrepresentations W of G we have an isomorphism of smooth ˜ G × H ( K p )-modules: H n cl ,ε ( ˆ K p , W (cid:48) ) = Hom g ( W, V ) . We shall show that the eigenvariety Eig( V ) interpolates the finite slope represen-tations in H n cl ,ε ( ˆ K p , W (cid:48) ).Suppose π is an absolutely irreducible representation of ˜ G ×H p , which appears asa subquotient of H n cl ,ε ( K p , W (cid:48) ) for some irreducible algebraic representation W of G ,and suppose that π p embeds in ind ˜ G Z ˆ N ( θ ) for some smooth character θ of the centre Z of ˜ T . The Hecke algebra H sph acts on π by a character λ ∈ Spec( H sph )( ¯ Q p ). Let ψ be the highest weight character of W with respect to B , regarded as a characterof Z p . Then the point ( θψ, λ ) ∈ (cid:98) Z × Spec( H sph ) is called a classical point. Theorem 5.5.2.
Every classical point is in
Eig( V ) .Proof. We first note that J B is exact on the subcategory of admissible smoothrepresentations of ˜ G . This is because (i) it is constructed as a composition oftwo left-exact functors, and is therefore left-exact, and (ii) by theorem 5.3.1, theJacquet functor coincides on smooth representations with the coinvariants, whichis right-exact.By exactness, there is a sub-quotient of J B ( H n ( ˆ K p , W (cid:48) )) on which Z × H sph acts by ( θ, λ ). The vector space J B ( H n ( ˆ K p , W (cid:48) )) ⊗ E ¯ Q p is an admissible smoothrepresentation of Z , and is therefore a direct limit of finite dimensional Z × H sph -modules. Therefore the ( θ, λ )-eigenspace in J B ( H n ( ˆ K p , W (cid:48) )) ⊗ E ¯ Q p is non-zero.By Corollary 5.3.3 we deduce that the ( θψ, λ )-eigenspace in J B ( H n ( ˆ K p , W (cid:48) ) ⊗ W )is non-zero. By the edge map criterion, the representation H n ( ˆ K p , W (cid:48) ) ⊗ W is isomorphic to the closed subspace of W -locally algebraic vectors in V . By left-exactness, we deduce that the ( θψ, λ )-eigenspace in J B ( V ) is non-zero. This impliesthat ( θψ, λ ) is in Eig( V ). (cid:3) Theorem 5.5.3.
Let ( θψ, λ ) be a point of Eig( V ) , where θ is locally constant and ψ is the highest weight of some algebraic representation W of G . If χ = θψ hasnon-critical slope, then ( θψ, λ ) is a classical point.Proof. This is immediate from Theorem 5.4.4, since V = ¯ H nε ( ˆ K p , E ) Q p − la admits a˜ G -invariant norm given by the gauge of the lattice ¯ H nε ( ˆ K p , O E ) ⊂ ¯ H nε ( ˆ K p , E ). (cid:3) A p -adic analytic Stone–von Neumann theorem If the machinery of the previous section is applied in the case where the parabolicsubgroup P is a Borel subgroup, the Jacquet module is a representation of thepreimage in ˜ G of a maximal torus in G . This is a topological central extension of acommutative group, but need not itself be commutative. We therefore turn to thequestion of classifying locally analytic representations of metaplectic tori.In this section, we shall let T be any topologically finitely generated abelian Q p -analytic group. We define the rank of T to be the rank of the finitely-generatedabelian group T / T , for any compact open subgroup T ⊆ T . Let ˜ T be a topologicalcentral extension of T by µ , so there is an exact sequence of topological groups1 → µ → ˜ T → T → . Let us fix a character ε : µ → E × , where E is (as above) a discretely valued closedsubfield of C p . We shall restrict our attention to representations of ˜ T on which µ acts via ε . Without loss of generality, we suppose that ε is injective, and inparticular µ is cyclic.Let Z be the centre of ˜ T . This contains µ , and therefore is the preimage of asubgroup Z ⊆ T . The group ˜ T /Z ∼ = T / Z is a finite abelian group, of exponentdividing the order of µ . This quotient group is equipped with a non-degenerate,alternating bilinear form Λ (˜ T /Z ) → µ given by t ∧ u (cid:55)→ [ t, u ]; in particular, the index [ T : Z ] is a square. If A is asubgroup of T containing Z , then the preimage ˜ A is abelian if and only if A / Z isan isotropic subspace of T / Z . We shall abuse notation and call such subgroups isotropic subgroups of T .For a locally p -adic analytic group G whose centre is topologically finitely gen-erated (such as all of the groups considered above), we let Rep ess ( G ) denote thecategory of essentially admissible locally analytic representations of G . Note thatif G is commutative, then Rep ess ( G ) is the opposite category of the category ofcoherent sheaves on the rigid-analytic space (cid:98) G . For G a subgroup of ˜ T containing µ , we let Rep ess ( G ) ε denote the subcategory of representations on which µ acts viathe character ε . If G is commutative, then this category is anti-equivalent to thecoherent sheaves on a subspace (cid:98) G ε ⊆ (cid:98) G , which is a union of components of (cid:98) G .6.1. Irreducible representations.
We begin with a weak form of the Stone–vonNeumann theorem, classifying the irreducible objects of Rep ess (˜ T ) ε . Let us choosea maximal isotropic subgroup A ⊆ T , as above. Lemma 6.1.1.
Let χ be a continuous (hence locally Q p -analytic) character of Z restricting to ε , I χ the set of E -valued characters of ˜ A extending χ , and ψ ∈ I χ .Then the map ˜ T / ˜ A → I χ mapping t to the character a (cid:55)→ ψ ( t − at ) is a bijection.(In other words, I χ is a torsor for ˜ T / ˜ A ). -ADIC INTERPOLATION OF METAPLECTIC FORMS 31 (Note that any ψ ∈ I χ is defined over a finite, and hence complete and discretelyvalued, extension of E . Moreover, since Z is open in ˜ T , any such ψ is locally Q p -analytic.) Proof.
Let ψ t be the character defined by ψ t ( a ) = ψ ( t − at ). We shall first showthat the map t (cid:55)→ ψ t is injective on ˜ T / ˜ A . Suppose ψ t = ψ u for t, u ∈ ˜ T . We needto show that t − u ∈ ˜ A . By definition we have for all a ∈ ˜ A : ψ ( t − at ) = ψ ( u − au ) . Hence the element t − atu − a − u is in the kernel of ψ . Since T is abelian, theimage of this element in T is the identity, so t − atu − a − u is also in µ m . Sincethe restriction of ψ to µ m is ε , which is assumed to be injective, we deduce that t − at = u − au . In other words ut − commutes with every element a ∈ ˜ A . As ˜ A isa maximal abelian subgroup, we must therefore have ut − ∈ ˜ A .To prove surjectivity, we’ll show that ˜ T / ˜ A and I χ have the same number ofelements. The number of elements in I χ is | ˜ A /Z | , which is the same as | A / Z | . Since A / Z is a maximal isotropic subspace of T / Z with respect to the skew-symmetricform above, T / A is identified with the Pontryagin dual of A / Z and hence | T / A | = | A / Z | . (cid:3) Proposition 6.1.2.
Let ˜ T be a topological central extension by µ of a topologicallyfinitely generated abelian locally Q p -analytic group T , and let Z be the centre of ˜ T .Fix an injective character ε of µ .Then for every locally Q p -analytic E -valued character χ of Z extending ε , thereis a unique irreducible finite-dimensional locally Q p -analytic representation V χ of ˜ T on an E -vector space having central character χ .Proof. Let A χ be the algebra E [˜ T ] / (cid:104) z − χ ( z ) : z ∈ Z (cid:105) . This is a finite-dimensional E -algebra of dimension d , where d = | T / A | = | A / Z | ,and it is clear that any representation of ˜ T on which Z acts via χ is a module over A χ . Note that since Z is open and has finite index in ˜ T , such a representation isessentially admissible if and only if it is finite-dimensional over E , or equivalentlyfinitely-generated over A χ ; thus the essentially admissible representations of ˜ T withcentral character χ are precisely the finitely-generated A χ -modules.We claim that A χ is a central simple E -algebra. It suffices to check this afterany finite base extension, so let us choose a finite extension E (cid:48) /E sufficiently largethat all characters ψ ∈ I χ are defined over E (cid:48) .We consider A χ as a representation of ˜ A × ˜ A via right and left translation. If E (cid:48) /E is a finite extension sufficiently large that all ψ ∈ I χ have values in E (cid:48) , wemay decompose A (cid:48) χ = A χ ⊗ E E (cid:48) as a direct sum of isotypical components A ( ψ ,ψ ) χ for the action of ˜ A × ˜ A , indexed by pairs ( ψ , ψ ) ∈ I χ × I χ , and any ˜ A × ˜ A -stablesubspace of A (cid:48) χ is equal to the direct sum of its intersections with these isotypicalsubspaces.So let S be any two-sided ideal in A (cid:48) χ . It follows that S is ˜ A × ˜ A -invariant, andthe action of ˜ A × ˜ A on S is obviously diagonalizable. Hence if S is non-zero, thenit must have nontrivial intersection with A ( ψ ,ψ ) χ for some pair ( ψ , ψ ). However,since S is a two-sided ideal, and conjugation by ˜ T permutes I χ transitively, wededuce that it must have nontrivial intersection with all of the subspaces A ( ψ ,ψ ) χ .Hence its dimension is at least d , which is the dimension of A (cid:48) χ ; thus S = A (cid:48) χ . Sowe have shown that A (cid:48) χ is simple, from which it follows that A χ is simple. Since A χ is a central simple algebra, up to isomorphism there is a unique simpleleft A χ -module; and we define V χ to be this representation. (cid:3) Remark . Note that the dimension of V χ is equal to de , where e is the orderof A χ in the Brauer group of E . In particular, V χ is absolutely irreducible if andonly if A χ is isomorphic to a matrix algebra, so e = 1. Proposition 6.1.4.
Let W be any essentially admissible locally Q p -analytic repre-sentation of ˜ T on which µ acts via ε , and let V χ be the representation constructedabove. Then for any character χ : Z → E × , we have Hom Z ( χ, W ) (cid:54) = 0 ⇔ Hom ˜ T ( V χ , W ) (cid:54) = 0 . Proof.
By replacing W with the closed ˜ T -stable subspace W Z = χ , it suffices toshow that if Z acts on W via χ , then there is a nonzero homomorphism of ˜ T -representations V χ → W . But since W is essentially admissible as a representationof Z , it must be finite-dimensional, and thus finitely-generated as an A -modulewhere A is the central simple algebra constructed above. Since every finitely-generated module over a central simple algebra is a direct sum of copies of the uniquesimple module, it follows that Hom A ( V χ , W ) = Hom ˜ T ( V χ , W ) is nonzero. (cid:3) We now extend the definition of V χ slightly. The continuous characters χ : Z → E × extending ε are precisely the absolutely irreducible objects of the categoryRep ess ( Z ) ε . If F/E is a finite extension and χ : Z → F × is a continuous character,we regard χ as an object of Rep ess ( Z ) ε via restriction of scalars. If the values of χ generate F over E , this representation is irreducible (but not absolutely so, ofcourse, unless F = E ). This gives us a bijection between each of the following sets: • irreducible objects of Rep ess ( Z ) ε , • points of the rigid space (cid:98) Z ε (in the sense of rigid geometry, i.e. maximalideals of its structure sheaf), • Gal(
E/E )-orbits of characters χ : Z → E × extending ε .For each Galois orbit of characters χ , applying Proposition 6.1.2 with the coeffi-cient field E replaced by the finite extension F/E generated by the values of χ , wesee that there is a unique irreducible F -linear representation V χ of ˜ T with centralcharacter χ ; and since the values of χ generate F over E , this is still irreduciblewhen regarded as an E -linear representation via restriction of scalars. We define V χ to be this representation, which clearly depends only on the Gal( E/E )-orbit of χ . As a representation of Z , V χ is isomorphic to a direct sum of copies of the objectof Rep ess ( Z ) ε constructed from χ above; we shall abuse notation slightly by writing“ V χ has central character χ ” even when χ is not defined over E . Corollary 6.1.5. (a)
The map χ (cid:55)→ V χ is a bijection between the set of irreducible objects of thecategories Rep ess ( Z ) ε and Rep ess (˜ T ) ε , which is uniquely characterised bythe fact that V χ has central character χ . (b) For any W ∈ Rep ess (˜ T ) ε , there is a closed rigid-analytic subvariety X ofthe character space (cid:98) Z ε such that the irreducible subrepresentations of W are precisely the V χ for χ ∈ X . (c) The irreducible representation V χ is absolutely irreducible if and only if χ is defined over E and the central simple algebra A χ of Proposition 6.1.2 istrivial in the Brauer group of E .Proof. For part (a), the fact that the map exists and is injective is clear; so itsuffices to check that it is surjective. Let W be an irreducible object in Rep ess (˜ T ) ε . -ADIC INTERPOLATION OF METAPLECTIC FORMS 33 Applying Proposition 6.1.4 to Rest ˜ T Z ( W ), we see that there is some χ such thatHom ˜ T ( V χ , W ) (cid:54) = 0. Since W is irreducible, we must have W ∼ = V χ .For part (b), we simply take X to be the support of the sheaf on (cid:98) Z correspondingto the locally analytic representation Rest ˜ T Z ( W ) of Z . By construction, the pointsof X are precisely the χ such that Hom Z ( χ, W ) (cid:54) = 0, and Proposition 6.1.4 showsthat these are precisely the irreducible ˜ T -subrepresentations of W .Part (c) is immediate from the construction of V χ and the remark on its dimen-sion above. (cid:3) Tame isotropic subgroups.
We now show that the results of the previoussubsection can be strengthened, under a mild additional hypothesis on ˜ T . Definition 6.2.1.
Let A ⊃ Z be a maximal isotropic subgroup. We say A is tame if ZA tors = A , where A tors is the torsion subgroup of A . We have used the word “tame” to indicate the analogy to the tame symbol, asthe following example illustrates.
Example.
Let k p be a finite extension of Q p containing a primitive m -th root ofunity. Let T = k × p and let µ m be the group of all m -th roots of unity in k p . Thenwe may define an extension ˜ T of T by µ m by setting˜ T = { ( x, ζ ) : x ∈ k × p , y ∈ µ m } with the group law ( x, ζ )( y, ξ ) = ( xy, ζξ ( x, y ) m ) , where ( x, y ) m is the m -th power Hilbert symbol in k p .One finds that Z = { x ∈ k × p : x ∈ k × m p } . Let us define n = m if m is odd, and n = m/ m is even. Then, since we clearlyhave µ n ⊆ k × p and hence − ∈ k × n p , we see that Z = k × n p .If n is not a multiple of p , then we have [ T : Z ] = [ k × p : k × n p ] = n . If we take A = π Z O × n , where O is the ring of integers of k p and π is a uniformizer, then A / Z is isotropic (as any lift of π to ˜ T clearly commutes with itself). As [ A : Z ] = n , A is maximal.Since we have assumed p does not divide n , we have 1 + π O ⊆ O × n ; hence everyelement of A / Z ∼ = O × / O × n has a representative that is a Teichmuller lift of anelement of the residue field, and thus in particular lies in A tors . Hence A is tame.On the other hand, the (possibly more natural) choice A (cid:48) = π n Z O × is a maximalisotropic subgroup which is not tame.If on the other hand p is a factor of n , then there is no tame maximal isotropicsubgroup.It is clear that if p (cid:45) m and the rank of T is 0 or 1, there must exist a tamemaximal isotropic subgroup. On the other hand, if T is the discrete group Z and ˜ T is the unique non-split extension of T by ±
1, then there are precisely threemaximal isotropic subgroups of ˜ T and none of these are tame.Let R denote the finite ´etale cover (cid:98) ˜ A → (cid:98) Z given by restriction of characters. Proposition 6.2.2. If A is tame, then R maps every component of (cid:98) ˜ A isomorphi-cally to a component of (cid:98) Z .Proof. Recall that if H is an abelian locally analytic group, the geometrically con-nected components of (cid:98) H correspond bijectively with the characters of H tors . By [Eme04, Proposition 6.4.1], we can (non-uniquely) decompose ˜ A and Z asproducts ˜ A = ˜ A tors × ˜ A ∞ and Z = Z tors × Z ∞ . Moreover, it is clear that we maydo this in such a way that Z ∞ ⊆ ˜ A ∞ . The assumption that A is tame, so that˜ A = Z ˜ A tors , implies that in fact Z ∞ = ˜ A ∞ .Since the contravariant functor (cid:100) ( − ) takes direct products of groups to fibre prod-ucts of rigid spaces, the map (cid:98) ˜ A → (cid:98) Z is obtained by taking the fibre product of themap of finite rigid spaces (cid:91) ˜ A tors → (cid:91) Z tors with the connected rigid space (cid:100) Z ∞ , fromwhich the result is clear. (cid:3) Corollary 6.2.3.
For all sufficiently large E , there exists a map S : (cid:98) Z → (cid:98) ˜ A ofrigid spaces over E which is a section of R .Proof. By the same argument as in the previous proposition, it suffices to showthat the map (cid:91) ˜ A tors → (cid:91) Z tors admits a section; this is clear, since both spaces arefinite unions of points. (cid:3) Remark . It suffices to assume that ζ r ∈ E , where r is the exponent of ˜ A tors .If E does not contain enough roots of unity then such a section may well not exist.Note also that the choice of the section S is highly non-canonical. Theorem 6.2.5. If ˜ T contains a tame maximal commutative subgroup, and E issufficiently large, then there is an equivalence of categories Rep ess ( Z ) ε ∼ → Rep ess (˜ T ) ε . Proof.
Let V be an object of Rep ess ( Z ). We decompose V in the form V = (cid:77) χ V χ where the sum is over the characters of Z tors . We let F A ( V ) be the representationof A which is isomorphic to V as a representation of Z ∞ , but with ˜ A tors acting onthe summand V χ by the extension of χ to a character of ˜ A tors determined by thesection S above.It is clear that F A is a functor Rep ess ( Z ) ε → Rep ess ( ˜ A ) ε . Composing thiswith the functor Ind ˜ T ˜ A : Rep ess ( ˜ A ) → Rep ess (˜ T ) gives a functor F : Rep ess ( Z ) ε → Rep ess (˜ T ) ε .We construct an inverse functor G : Rep ess (˜ T ) → Rep ess ( Z ) as follows. Restric-tion to ˜ A gives a functor Rep ess (˜ T ) → Rep ess ( ˜ A ). We may decompose Rep ess ( ˜ A )as a direct sum of subcategories corresponding to the characters of ˜ A tors .The choice of section S above determines a subset I ⊂ (cid:91) ˜ A tors . The functorpr I : Rep ess ( ˜ A ) → Rep ess ( ˜ A ) given by V (cid:55)→ (cid:76) ψ ∈ I V ψ is well-defined. We define G = Res Z ˜ A ◦ pr I ◦ Res ˜ A ˜ T , which clearly defines a functor Rep ess (˜ T ) ε → Rep ess ( Z ) ε .We claim that these functors are inverse to each other. Let us first consider thecomposition G ◦ F : Rep ess ( Z ) ε → Rep ess ( Z ) ε . We note that there is a naturalisomorphism of functors Rep ess ( ˜ A ) → Rep ess ( ˜ A ) between the functors Res ˜ A ˜ T ◦ Ind ˜ T ˜ A and the functor mapping V to (cid:76) t ∈ ˜ T / ˜ A V t . Using lemma 6.1.1, we deduce that G ◦ F is naturally isomorphic to the identity functor.On the other hand, let W be an object of Rep ess (˜ T ) ε . Since all our functorscommute with direct sums, we may as well assume that Z tors acts on W via acharacter χ (extending ε ). Then ψ = S ( χ ) is a choice of extension of χ to ˜ A tors .We find that F ( G ( W )) = Ind ˜ T ˜ A (cid:16) W [ ˜ A tors = ψ ] (cid:17) . -ADIC INTERPOLATION OF METAPLECTIC FORMS 35 We must construct a natural transformation between this and the identity func-tor. Let ψ , . . . , ψ s be the conjugates of ψ , and fix t , . . . , t s ∈ ˜ T such that ψ ( t − i at i ) = ψ i ( a ). Then the map f (cid:55)→ (cid:80) t i f ( t − i ) : F ( G ( W )) → W gives anatural transformation between F ◦ G and the identity functor. (cid:3) Remark . If T has no tame maximal isotropic subgroup, then the categoryRep ess (˜ T ) ε may genuinely fail to be isomorphic to Rep ess ( Z ) ε , even if the scalarfield E is large.For instance, let ˜ T be the extension of Z by ± ε thenontrivial character. If we set D = E [˜ T ] ⊗ E [ Z ] C an ( (cid:98) Z ε , E ), then D is a Fr´echet–Steinalgebra, and Rep ess (˜ T ) is precisely the opposite category of coadmissible D -modules;but D is a non-trivial central simple algebra of rank 4 over C an ( (cid:98) Z ε , E ), and hencethe categories of coadmissible modules over these rings are not equivalent.Nonetheless, restriction of representations certainly gives a functor Rep ess (˜ T ) ε → Rep ess ( Z ) ε , and one deduces that for any V ∈ Rep ess (˜ T ) ε there is a coherent sheafon (cid:98) Z ε whose support consists precisely of the characters of Z appearing in V ; thedisadvantage is that since the restriction functor is not full, morphisms betweensuch sheaves do not necessarily correspond to morphisms of ˜ T -representations.We may summarise the discussion as follows: Theorem 6.2.7.
Let ˜ T be a topological central extension by µ of a topologicallyfinitely generated abelian locally Q p -analytic group T , and let Z be the centre of ˜ T .Fix an injective character ε of µ . (1) For every
Gal(
E/E ) -orbit of continuous characters χ : Z → E × extending ε , there is a unique irreducible locally analytic representation V χ of ˜ T havingcentral character χ . (2) If W is an essentially admissible locally analytic representation of ˜ T onwhich µ acts via ε , then there is a closed rigid-analytic subvariety Supp( W ) of (cid:98) Z having the property that Hom ˜ T ( V χ , W ) (cid:54) = 0 if and only if χ ∈ Supp( W ) . (3) If, moreover, T has a tame maximal isotropic subgroup, then the categoryof locally analytic representations of ˜ T is anti-equivalent to the categoryof rigid-analytic sheaves on (cid:98) Z ε , and Supp( W ) is the support of the sheafcorresponding to W . Theorem 6.2.8.
Let G be semi-simple, simply connected and split over k p and let ˜ G be the canonical metaplectic extension of G by µ m . Let T be a maximal splittorus in G , and let T and ˜ T be as before. If p does not divide m , then there is atame maximal isotropic subgroup of T .Proof. The torus T is split, so we have an isomorphism T ∼ = G nm for some n . Weshall regard elements of T as vectors t = ( t i ) ni =1 with t i ∈ k × p . By [Mat69, LemmeII.5.4 and Lemme II.5.8] we know that the commutator has the form[ t, u ] = (cid:89) i,j ( t i , u j ) m ( i,j ) p ,m , where m ( i, j ) are certain integers. One easily shows that Z is the set of elements t satisfying for j = 1 , . . . , n the following relation: n (cid:89) i =1 t m ( i,j ) i ∈ k × m p . Clearly, Z contains T m . Define a subgroup A of T by the relations n (cid:89) i =1 t m ( i,j ) i ∈ O × · k × m p , j = 1 , . . . , n. It follows that A = Z · ( O × ) n . We claim that A is a tame maximal isotropicsubgroup of T .We first show that A is isotropic. The elements of A are, up to an element of Z ,in the subgroup ( O × ) n , and so it suffices to show that ( O × ) n is isotropic. This inturn follows from the fact that ( − , − ) p ,m is the tame symbol (as p does not divide m ) and so is trivial on O × × O × .We next show that A is maximal. Assume that u satisfies [ u, t ] = 1 for all t ∈ A .This implies for each j and every element t ∈ O × the relation n (cid:89) i =1 ( t, u j ) m ( i,j ) p ,m = 1 . Hence (cid:81) ni =1 u m ( i,j ) j is an element of O × · k × m p , and so u is in A .Finally, we show that A is tame. To see this, it is sufficient to note that thecosets of O × / O × m have representatives which are roots of unity (Teichm¨uller rep-resentatives). Again, we have used the fact that p does not divide m , and so everyelement in O × which is congruent to 1 modulo p is an m -th power. (cid:3) Implications for eigenvarieties.
We now return to the situation of section5.5. The group ˜ T is an extension of the abelian, topologically finitely generated p -adic analytic group T by µ , so we may describe its essentially admissible rep-resentations by means of Theorem 6.2.7. As before, we write Z for the centre of˜ T . Recall that we have shown that, for any essentially admissible ˜ G -representation V equipped with an action of H p , there exists a rigid-analytic subspace Eig( V ) ⊆ (cid:98) Z × Spec( H sph ), and a coherent sheaf of right H p -modules E on Eig( V ), such thatthe fibre of E at a point ( χ, λ ) ∈ (cid:98) Z is canonically isomorphic as an H p -module tothe dual of the ( Z = χ, H sph = λ ) eigenspace of V .As before, let ε be an injective character of µ . We write Eig( V ) ε for the inter-section of Eig( V ) with (cid:98) Z ε × Spec( H sph ) ⊂ (cid:98) Z × Spec( H sph ). From Theorem 6.2.7,we deduce: Corollary 6.3.1. (1)
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