From category O ∞ to locally analytic representations
aa r X i v : . [ m a t h . R T ] J a n FROM CATEGORY O TO LOCALLY ANALYTICREPRESENTATIONS
SHISHIR AGRAWAL AND MATTHIAS STRAUCH
Abstract.
Suppose G Ě P Ě T are a p -adic split reductive group, a parabolicsubgroup, and a maximal torus, respectively, and let g , p , and t denote their Liealgebras. A functor from O p alg into the category of admissible locally analyticrepresentations of G “ G p F q is constructed in [OS15]. We construct a family ofextensions of this functor defined on the extension closure O p , alg of the category O p alg inside the category of all g -modules. Each extension is determined by achoice of logarithm on T “ T p F q . We prove that these functors are exact, andwe establish a relationship between these functors as the parabolic subgroup P changes. Contents
Introduction 2Notation 41. Preliminaries 51.1. Locally finite dimensional representations 51.2. Modules over an abelian Lie algebra 51.3. Rational points of a reductive group 82. Lifting Lie algebra actions 82.1. Lifting to actions of unipotent groups 82.2. Logarithms on tori 92.3. Lifting to actions of tori 102.4. Lifting to actions of general groups 122.5. Changing the logarithm 163. Categories O p , and O p , alg O p and O p alg O p , alg D p H q -modules 28 A.1. Some functional analysis 29A.2. Constructions and functoriality 29A.3. Adjunction 35A.4. Finite generation 37Appendix B. Modules over D p g , H q D p g , H q D p g , H q -modules 45B.5. The closure D p G q H of D p g , H q D p G q H D p g , H q Introduction
Let F be a finite field extension of Q p , and let G Ě P Ě B Ě T be a splitreductive group over F , a parabolic subgroup, a Borel subgroup, and a maximaltorus, respectively. Denote by g , p , b , and t their respective Lie algebras, andlet G, P, B , and T be their groups of F -points, regarded as locally F -analyticgroups. In analogy with the Bernstein-Gelfand-Gelfand category O [BGG76],we denote by O p alg the category of finitely generated g -modules which are locallyfinite dimensional over p and on which t acts semisimply with algebraic weights.The paper [OS15] introduced exact functors F GP from O p alg to the category ofadmissible locally analytic representations of G .The category O p alg is not stable under extensions in the category of all g -modules, and we denote by O p , alg its extension closure. Objects in this categoryare still finitely generated over g and the action of p is still locally finite, but theaction of t is not necessarily semisimple (cf. section 3). This category has beenstudied in more detail elsewhere (eg, [Soe85, CM15]) and is sometimes called the“thick category O .” The object of this paper is to extend the functors F GP from O p alg to its extension closure O p , alg . In fact, we construct a family of extensions,indexed by “logarithms” on T (i.e., homomorphisms T Ñ t which invert theexponential exp : t T , cf. definition 2.2.1)In [OS15], the strategy behind defining the functor F GP on category O p alg is asfollows. We first set ˇ F GP p M q “ D p G q b D p g ,P q M, where D p G q is the locally analytic distribution algebra [ST02b, ST03] and D p g , P q is the subring of D p G q generated by U p g q and the distribution algebra D p P q of P .The key point here is that the action of p on M lifts functorially to an action of ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 3 P , and thus gives rise to a structure of a D p P q -module on M which is compatiblewith the structure of a g -module. It turns out that ˇ F GP p M q is a coadmissible D p G q -module, and we define F GP p M q to be its continuous dual.Our strategy for extending this functor to O p , alg is similar, with one key differ-ence. To construct a functorial lift of the action of p to an action of P , we mustnow make a choice of a logarithm log : T Ñ t . Having made this choice, we canlift a nilpotent action ϕ : t Ñ End p V q on a finite dimensional vector space V toan action of T that is given by the composite exp ˝ ϕ ˝ log (cf. lemma 2.3.1). Bytwisting this action appropriately for other generalized weight spaces, we can liftarbitrary algebraic actions of t on finite dimensional vector spaces to actions of T . It turns out that this lifted action is suitably natural and yields an action of P (cf. section 2.4).We thus obtain a D p g , P q -module structure on any M P O p , alg which dependson the choice of logarithm log : T Ñ t . Denoting M equipped with this D p g , P q -module structure by Lift p M, log q , we proceed as described above and setˇ F GP p M q “ ˇ F GP, log p M q “ D p G q b D p g ,P q Lift p M, log q , where we usually drop the subscript log for convenience. We show that thisfunctor is exact (cf. theorem 4.2.4) and that ˇ F GP p M q is coadmissible (cf. theo-rem 4.2.3). Defining F GP p M q to be the continuous dual of ˇ F GP p M q thus definesan exact functor F GP from O p , alg into admissible locally analytic representations.In fact, as in [OS15], we actually construct a bifunctor p M, V q ÞÑ F GP p M, V q where M P O p , alg , V is a smooth strongly admissible representation of P . Theoutput F GP p M, V q is an admissible locally analytic representation of G . Thisbifunctor is contravariant in the first argument, covariant in the second, and exactin both. Finally, as in [OS15, proposition 4.9], we establish a relationship between F GP and F GQ where Q is a parabolic subgroup containing P (cf. theorem 4.3.3).Locally analytic representations associated to objects in the category O p , alg viaour functor appear naturally in the p -adic Langlands program (cf. section 5).In [Bre04] and [Sch11], the authors construct locally analytic representationsof GL p Q p q and GL p Q p q , respectively, which are supposed to be related tosemistable Galois representations of dimension 2 and 3, respectively, which arenot potentially crystalline. Indeed, if V is such a 2-dimensional Galois represen-tation, the representation Σ p k, L q of [Bre04] appears as a subrepresentation ofthe space of locally analytic vectors Π p V q an of Colmez’ p -adic Banach space rep-resentation Π p V q . This Σ p k, L q is in turn a subquotient of a representation whichis in the image of our functor F GL p Q p q B, log , where the logarithm log used dependson the L -invariant of V . Similarly, the representation Σ p λ, L , L q of [Sch11] isa subquotient of a representation in the image of our functor F GL p Q p q B, log , where,again, the logarithm log used depends on the pair of L -inariants p L , L q of acertain semistable Galois representation. SHISHIR AGRAWAL AND MATTHIAS STRAUCH
Extending the functors F GP from category O p alg to its extension closure O p , alg isalso convenient, even necessary, if one wants to work with these categories andfunctors in the setting of derived categories. On the side of locally analytic repre-sentations, the paper [ST05] considers the bounded derived category D b C G p D p G qq of complexes of D p G q -modules whose cohomology objects are coadmissible. Thenatural pendant to this category on the side of Lie algebra representations is thenthe bounded derived category D b O p , alg p U p g qq of complexes of U p g q -modules whosecohomology objects are in category O p , alg . Here, in order for this category to bea triangulated category, it is indeed necessary to work with O p , alg , and not onlywith O p alg , as the latter category is not closed under extensions. Moreover, by[CM15], the natural functor D b p O p , alg q Ñ D b O p , alg p U p g qq is an equivalence of cate-gories, which makes it possible to define the functors ˇ F GP on D b O p , alg p U p g qq . Thestudy of the functors ˇ F GP in the setting of derived categories will be taken up inforthcoming work. Notation.
Modules are assumed to be left modules unless otherwise specified.Let F denote a finite extension of Q p , and E a finite extension of F . If X is anobject over F (vector space, algebra, etc), we denote by X E its base change to E . Vector spaces, tensor products, homs, etc, are assumed to be over E unlessotherwise specified. If X is an object of a category with a faithful functor intovector spaces, we say that X is locally finite dimensional if it is a colimit of itsfinite dimensional subobjects.If g is a Lie algebra over F , we write U p g q in place of U p g E q . Also, a g -module is an E -vector space M equipped with a Lie algebra homomorphism ϕ M : g E Ñ End E p M q (in other words, we use the term g -module to refer towhat might more accurately be called a g E -module). For m P M and x P g E , wesometimes write x . m in place of ϕ M p x qp m q . Also, when M is clear from context,we sometimes drop the subscript from ϕ M and simply write ϕ instead. We letMod g denote the E -linear category of g -modules. We write Mod ˚ g for various ˚ to denote various full subcategories of Mod g , as introduced below.We use upper case letters ( G, P , etc) to denote locally analytic groups over F ,and we use corresponding lower case fraktur letters ( g , p , etc) to denote their Liealgebras. If G is a locally analytic group, we write Rep an G for the category of locallyanalytic representations of G with coefficients in E . We write C an p G q , C p G q ,and D p G q for the algebras of analytic functions, smooth functions, and analyticdistributions on G with coefficients in E .We denote blackboard bold letters ( G , P , etc) to denote algebraic groups over F . In this case, we then use the corresponding upper case letter ( G, P , etc) todenote the group of F -points of the algebraic group regarded as a locally analyticgroup, and the corresponding lower case fraktur letter ( g , p , etc) to denote its Liealgebra. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 5 Preliminaries
Locally finite dimensional representations.
Let g be a Lie algebra over F . We write Mod fd g for the category of finite dimensional g -modules, and Mod lfd g for the category of locally finite dimensional g -modules (ie, the ones for whichevery finitely generated g -submodule is finite dimensional). Lemma 1.1.1.
Mod fd g and Mod lfd g are both Serre subcategories of Mod g .Proof. Suppose 0 M M M g -modules. It is clear that M is finite dimensional if andonly if M and M are. It is also clear that if M is locally finite dimensional, then M and M are both locally finite dimensional as well. Conversely, suppose M and M are locally finite dimensional. Let N be a finitely generated submoduleof M . Then we have an exact sequence as follows.0 M X N N N {p M X N q N {p M X N q is a finitely generated submodule of M , it is finite dimen-sional. Since U p g q is noetherian and N is finitely generated, M X N is also finitelygenerated; so, since M is locally finite dimensional, M X N must be finite di-mensional. Thus, since M X N and N {p M X N q are both finite dimensional, N must be as well. (cid:3) Modules over an abelian Lie algebra.
Let t be an abelian Lie algebraover F . Let E r X s be the polynomial ring in an indeterminate X over E . Weidentify points of MaxSpec p E r X sq with their monic generators. Note that, if ¯ E isan algebraic closure of E , then there is a surjective function ¯ E Ñ MaxSpec p E r X sq which maps elements of ¯ E to their minimal polynomials. Definition 1.2.1.
Let t E denote the set of functions π : t E Ñ MaxSpec p E r X sq which factor through an E -linear map t E Ñ ¯ E for any algebraic closure ¯ E of E .Note that if this is true for one choice of algebraic closure, it is true for any otherchoice as well. Definition 1.2.2.
Let M be a locally finite dimensional t -module. If π P t E , the primary component M π of M associated to π is M π “ t m P M : p π p x qp ϕ p x qqq n p m q “ n ě u , where x P t and π p x qp ϕ p x qq is the endomorphism of M obtained by plugging ϕ p x q P End p M q into the monic irreducible polynomial π p x q P MaxSpec p E r X sq . Theorem 1.2.3.
For any locally finite dimensional t -module V , we have M “ à π P t E M π . SHISHIR AGRAWAL AND MATTHIAS STRAUCH
This is called the primary component decomposition of M .Proof. When M is finite dimensional, this is the statement of [Jac79, chapter II,section 4, theorem 5; p. 40–41]. In general, observe that we have N “ à π P t E N π for any finite dimensional submodule of M . Since M is the colimit over all suchfinite dimensional submodules N , and since colimits commute with direct sums,we have M “ colim N “ colim ¨˝ à π P t E N π ˛‚ “ à π P t E colim N π “ à π P t E M π . (cid:3) Proposition 1.2.4. M ÞÑ M π defines an exact functor on Mod lfd t for any π P t E .Proof. One can check directly from the definition of the primary component that,if σ : M Ñ N is a homomorphism of t -modules, then σ p M π q Ď N π , which showsthat M ÞÑ M π is in fact functorial. For exactness, suppose we have an exactsequence of t -modules M M M σ τ and suppose m P ker p τ q X M π . Since m P ker p τ q , there exists m P M such that σ p m q “ m . Since M has a primary component decomposition by theorem 1.2.3,we can write m “ m ` ¨ ¨ ¨ ` m n where m i P M π i for distinct π i P t E . Then m “ σ p m q “ σ p m q ` ¨ ¨ ¨ ` σ p m n q and we have σ p m i q P M π i . Since m P M π , there must exist a single k such that m “ σ p m k q and π “ π k and σ p m i q “ i ‰ k . Thus m P σ p M π q , provingthat M π M π M πσ τ is exact. (cid:3) Definition 1.2.5.
There is a natural embedding t ˚ E “ Hom p t E , E q ã Ñ t E whichcarries λ P t ˚ E to the function π λ p x q “ T ´ λ p x q P MaxSpec p E r X sq . For a locallyfinite dimensional t -module M , we write M λ instead of M π λ and call M λ the generalized weight space of M associated to λ . Explicitly, we have M λ “ t m P M : p ϕ p x q ´ λ p x qq n p m q “ n ě u . Definition 1.2.6. A t -module M is split if it is locally finite dimensional and M π “ π “ π λ for some λ P t ˚ E . In other words, M is split if it is locallyfinite dimensional and there exists a t -module decomposition(1.2.7) M “ à λ P t ˚ E M λ . ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 7
This decomposition is called the generalized weight space decomposition of M .We write Mod s t for the category of split t -modules.Note that every finite dimensional t -module becomes split after a finite fieldextension [Jac79, chapter II, section 4]. Lemma 1.2.8.
Mod s t is a Serre subcategory of Mod t .Proof. Suppose 0 M M M t -modules. We know from lemma 1.1.1 that M is locallyfinite dimensional if and only if M and M are. Now if π P t E and π ‰ π λ forany λ P t ˚ E , then we know that0 M π M π M π M π “ M π “ M π “ (cid:3) Definition 1.2.9.
Let M be a split t -module. Let ϕ s : t Ñ End p M q be themap given by ϕ s p x q : m ÞÑ λ p x q m on M λ for each λ . Then ϕ s is a Lie algebrahomomorphism, and it is called the semisimple part of ϕ . The nilpotent part of ϕ is ϕ n “ ϕ ´ ϕ s .For the remainder of this subsection, we regard t as the Lie algebra of analgebraic torus T . Definition 1.2.10.
We say that λ P t ˚ E is algebraic if there exists an algebraiccharacter χ λ : T E Ñ G m,E which differentiates to λ , in the sense that λ p x q “ ddt χ λ p exp p tx qq ˇˇˇˇ t “ . Such a character χ , if it exists, is uniquely determined by λ . Definition 1.2.11.
A split t -module M has algebraic weights if M λ ‰ λ P t ˚ E only if λ is algebraic. We write Mod alg t for the category of split t -modules which have algebraic weights, and Mod fd , alg t for the subcategory of finitedimensional ones. Lemma 1.2.12.
Mod alg t and Mod fd , alg t are both Serre subcategories of Mod t .Proof. This follows from lemma 1.1.1 and proposition 1.2.4, similar to the proofof lemma 1.2.8. (cid:3)
SHISHIR AGRAWAL AND MATTHIAS STRAUCH
Rational points of a reductive group.
Let p G , T q be a split reductive al-gebraic group over F . Let Φ p G , T q denote the corresponding root system. Recallour standing convention that when an algebraic group is denoted in blackboardbold, we use a corresponding upper case letter to denote its group of F -points. Lemma 1.3.1.
The group G is generated by T and U α for all α P Φ p G , T q , where U α is the root group of G corresponding to α .Proof. Fix a Borel subgroup B containing T and let U be its unipotent radical.The multiplication map ś α ą U α Ñ U is an isomorphism of F -schemes [BT65,2.3], so on F -points, we see that U is generated by U α for all α ą
0. Let N be thenormalizer of T in G . Then G “ U N U [BT65, 2.11], so it suffices to show thatevery element n P N is in the subgroup of G generated by T and U α for all α .The quotient N { T is the Weyl group of G , so it is generated by the reflectionsacross hyperplanes perpendicular to α for all α . Thus it suffices to fix α P Φ p G , T q and assume that n induces reflection across the hyperplane perpendicular to α .Fix a nontrivial u P U α . Then N X U ´ α uU ´ α “ t m p u qu for some m p u q which also induces reflection across the hyperplane perpendicularto α [Lan96, lemma 0.19]. Since both m p u q and n induce the same element of theWeyl group, there exists t P T such that n “ m p u q t . Since m p u q P U ´ α uU α , whichin turn is contained in the subgroup generated by U α and U ´ α , this concludes theproof. (cid:3) Corollary 1.3.2.
The group G is generated by T and G , where G is the derivedsubgroup of G .Proof. After lemma 1.3.1, it is sufficient to show that each root group U α of G is contained in G . Since g “ Lie p G q is the derived subalgebra of g , it contains g α . Let U α be the root group of G corresponding to g α . Then U α and U α areboth closed connected subgroups of G with Lie algebra g α , so U α “ U α [Hum75,theorem 13.1]. Thus U α is in G . (cid:3) Lifting Lie algebra actions
In this section, we describe ways of lifting actions of Lie algebras over F toactions of locally analytic groups.2.1. Lifting to actions of unipotent groups.
The easiest situation is liftingactions of nilpotent Lie algebras to actions of corresponding unipotent locallyanalytic groups.
Let U be a unipotent algebraic group. Then u is nilpotent, the exponentialmap exp : u Ñ U is bijective, and we write log : U Ñ u to denote its inverse.Suppose M is a finite dimensional u -module. Since u is nilpotent, its image ϕ p u q ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 9 inside End p M q is also nilpotent. By Engel’s theorem, ϕ p u q Ď Nil p M q , whereNil p M q denotes the set of nilpotent endomorphisms of M . We let ˜ ϕ M denote thecomposite U u Nil p M q GL p M q . log ϕ exp Then ˜ ϕ M defines a locally analytic action of U on M (in fact, the action iseven algebraic). We write Lift p M q to denote M regarded as a locally analyticrepresentation of U . When M is clear from context, we drop the subscript andsimply write ˜ ϕ instead of ˜ ϕ M . Lemma 2.1.2 below shows that ˜ ϕ does in fact liftthe original action of u on M . Lemma 2.1.2.
Suppose M is a finite dimensional u -module. Then ϕ M p x q “ ddt ˜ ϕ M p exp p tx qq ˇˇˇˇ t “ for all x P u .Proof. Observe that˜ ϕ p exp p tx qq “ exp p ϕ p log p exp p tx qqqq “ exp p ϕ p tx qq “ exp p tϕ p x qq which means that ddt ˜ ϕ p exp p tx qq ˇˇˇˇ t “ “ ddt exp p tϕ p x qq ˇˇˇˇ t “ “ exp p tϕ p x qq ϕ p x q ˇˇˇˇ t “ “ ϕ p x q . (cid:3) This construction is evidently natural and defines a functorLift : Mod fd u Ñ Rep an U on the category Mod fd u of finite dimensional u -modules.2.2. Logarithms on tori.
Let T be a split algebraic torus and let T be themaximal compact subgroup of T “ T p F q . Definition 2.2.1. A logarithm on T is a locally analytic group homomorphism T Ñ t E such that, on a neighborhood of 0 where exp : t T is defined, thecomposite log ˝ exp equals the natural map t Ñ t E (we will sometimes abusivelywrite log ˝ exp “ id for brevity). We write Logs p T q for the set of all logarithmson T . Lemma 2.2.2.
Logs p T q is a torsor for Hom p T { T , t E q .Proof. Let T f be the set of x P T such that x n Ñ n , as in [Bou72, III.6, proposition 10]. Then T f is the unionof all compact subgroups of T [Bou72, III.6, corollary to proposition 13], so T f “ T since T is the unique maximal compact subgroup of T . By [Bou72,III.6, propositions 10–11], there exists a unique group homomorphism λ : T Ñ t E which inverts the exponential map on a neighborhood of 0 in t E . In other words, Logs p T q is precisely the set of locally analytic homomorphisms T Ñ t E whichrestrict to λ on T .Since T { T is a discrete, finite free abelian group, the exact sequence1 T T T { T p´ , t E q yields anexact sequence(2.2.3)0 Hom p T { T , t E q Hom p T, t E q Hom p T , t E q . Then Logs p T q Ď Hom p T, t E q is precisely the preimage of λ P Hom p T , t E q , so itis a torsor over the kernel Hom p T { T , t E q . (cid:3) Lifting to actions of tori.
Let T be a split algebraic torus and fix log P Logs p T q . Lemma 2.3.1.
Suppose M is a finite dimensional t -module such that ϕ M p t q Ď Nil p M q , where Nil p M q is the set of nilpotent endomorphisms of M as a vectorspace. Then the composite T t E Nil p M q GL p M q log ϕ M exp defines a locally analytic homomorphism ˜ ϕ M : T Ñ GL p M q and ϕ M p x q “ ddt ˜ ϕ M p exp p tx qq ˇˇˇˇ t “ for all x P t . Thus M equipped with ˜ ϕ M is a representation of T which lifts theoriginal action of t . Moreover, if N is another finite dimensional t -module with ϕ N p t q Ď Nil p N q and σ : M Ñ N is a homomorphism of t -modules, then σ is alsoa homomorphism of representations of T .Proof. It is clear that ˜ ϕ is a locally analytic homomorphism, so we only need tocheck that that it differentiates to ϕ , but this is more or less identical to the proofof lemma 2.1.2. Fix x P t . If t is small enough that exp is defined on tx , we havelog p exp p tx qq “ tx , which means that˜ ϕ p exp p tx qq “ exp p ϕ p log p exp p tx qqqq “ exp p ϕ p tx qq “ exp p tϕ p x qq , so ddt ˜ ϕ p exp p tx qq ˇˇˇˇ t “ “ ddt exp p tϕ p x qq ˇˇˇˇ t “ “ exp p tϕ p x qq ϕ p x q ˇˇˇˇ t “ “ ϕ p x q . ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 11
Finally, if σ : M Ñ N is a homomorphism of finite dimensional t -modules with ϕ N p t q Ď Nil p N q , then for t P T and m P M , we have σ p ˜ ϕ M p t qp m qq “ σ p exp p ϕ M p log p t qqqp m qq“ σ p m ` ϕ M p log p t qqp m q ` ¨ ¨ ¨ q“ σ p m q ` σ p ϕ M p log p t qqp m qq ` ¨ ¨ ¨“ σ p m q ` ϕ N p log p t qqp σ p m qq ` ¨ ¨ ¨“ exp p ϕ N p log p t qqp σ p m qq“ ˜ ϕ N p t qp σ p m qq which proves that σ is also a homomorphism of representations of T . (cid:3) We now generalize the above lifting procedure. If M “ M λ for a single algebraic λ P t ˚ , we define ˜ ϕ M : T Ñ GL p M q by˜ ϕ M p t q “ χ λ p t q exp p ϕ n p log p t qqq . This defines a locally analytic action of T on M lifting the original action of t . More generally, if M is a split finite dimensional t -module with algebraicweights, we can lift the action of t one generalized weight space at a time. In otherwords, let χ : T Ñ GL p M q be the map where χ p t q acts on M λ by multiplicationby χ λ p t q . Then defining ˜ ϕ M : T Ñ GL p M q by˜ ϕ M p t q “ χ p t q exp p ϕ n p log p t qqq gives a lift of the action of t . We write Lift p M, log q to denote M regarded as arepresentation of T via this procedure. Since the generalized weight space decomposition is functorial, this liftingconstruction is also functorial. In other words, M ÞÑ Lift p M, log q defines a functorLift p´ , log q : Mod fd , alg t Ñ Rep an T on the category Mod fd , alg t of finite dimensional t -modules with algebraic weights.We conclude this subsection with the following observation about how thislifting construction interacts with passage to subtori. Lemma 2.3.5.
Let S Ď T be split algebraic tori and suppose log P Logs p T q .Then log | S P Logs p S q , and if M is a finite dimensional t -module with algebraicweights, then Lift p M, log q| S “ Lift p M | s , log | S q . Proof.
Let S be the maximal compact subgroup of S . Then log | S must be theunique logarithm map on S in the sense of [Bou72, III.6], so log p S q Ď s . Then,for any s P S , there exists a positive integer n such that s n P S , which meansthat n log p s q “ log p s n q P s . Since s is a divisible abelian group, we conclude that log p s q P s as well. This shows that log p S q Ď s , from which it follows thatlog | S P Logs p S q . The latter statement follows immediately. (cid:3) Lifting to actions of general groups.
Suppose P is a connected algebraicgroup and T is a split maximal torus of P . Then t is a Cartan subalgebra of p ,and we have the root space decomposition p “ à α P t ˚ E p α . Lemma 2.4.1.
Suppose M is a finite dimensional p -module. Then the root space p α maps the generalized weight space M λ into M α ` λ for all α, λ P t ˚ E .Proof. Suppose x P g E , y P t E and m P M . Then p ϕ p y q ´ α p y q ´ λ p y qqp x . m q “ y . p x . m q ´ α p y qp x . m q ´ λ p y qp x . m q“ r y, x s . m ` x . p y . m q ´ α p y qp x . m q ´ λ p y qp x . m q“ p ad p y q ´ α p y qqp x q . m ` x . p ϕ p y q ´ λ p y qqp m q By induction, we have p ϕ p y q ´ α p y q ´ λ p y qq n p x . m q “ n ÿ k “ ˆ nk ˙ p ad p y q ´ α p y qq k p x q . p ϕ p y q ´ λ p y qq n ´ k p m q . Suppose now that x P p α,E and m P M λ . Let n ě dim M λ , so that, for all0 ď k ď n , either k ě n ´ k ě dim M λ . If k ě
1, then p ad p y q ´ α p y qq k p x q “ x is in the root space corresponding to α . If n ´ k ě dim M λ , then p ϕ p y q ´ λ p y qq n ´ k p m q “ m is in the generalized weight space correspondingto λ . This means that p ϕ p y q ´ α p y q ´ λ p y qq n p x . m q “ , so x . m is a generalized weight vector with weight α ` λ . (cid:3) Lemma 2.4.2.
Let M be a finite dimensional p -module and let ϕ n denote thenilpotent part of ϕ | t . Then ϕ p x q ˝ ϕ n p y q “ ϕ n p y q ˝ ϕ p x q for all x P p E and y P t E .Proof. Since p “ À p α , we may assume without loss of generality that x P p α,E .Furthermore, since M “ À M λ , it is sufficient to show that ϕ p x q and ϕ n p y q commute when the domain is restricted to M λ . In other words, it is sufficient toshow that(2.4.3) x . ϕ n p y qp m q “ ϕ n p x . m q ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 13 for all m P M λ . Let ϕ s denote the semisimple part of ϕ | t E . Since x P p α,E , wehave r y, x s “ ad p y qp x q “ α p y q x , so α p y q x . m “ r y, x s . m “ ϕ p y qp x . m q ´ x . ϕ p y qp m q“ ϕ s p y qp x . m q ` ϕ n p y qp x . m q ´ x . ϕ s p y qp m q ´ x . ϕ n p y qp m q“ p α ` λ qp y qp x . m q ´ x . λ p y q m ` ϕ n p y qp x . m q ´ x . ϕ n p y qp m q“ α p y q x . m ` ϕ n p y qp x . m q ´ x . ϕ n p y qp m q . Here, we have used lemma 2.4.1 for the equality ϕ s p x . m q “ p α ` λ qp x . m q .Subtracting α p y q x . m from both sides of the above equation α p y q x . m “ α p y q x . m ` ϕ n p y qp x . m q ´ x . ϕ n p y qp m q yields precisely equation (2.4.3). (cid:3) Definition 2.4.4.
Let Mod alg p denote the category of locally finite dimensional p -modules M such that M | t is split with algebraic weights. Let Mod fd , alg p denotethe subcategory of finite dimensional p -modules. Lemma 2.4.5.
Suppose M P Mod fd , alg p . Fix log P Logs p T q and let θ “ ˜ ϕ M | t : T Ñ GL p M q denote the action of T on Lift p M | t , log q . Then ϕ p Ad p t qp x qq “ θ p t q ˝ ϕ p x q ˝ θ p t q ´ for all t P T and x P p E .Proof. It is evidently equivalent to prove that(2.4.6) Ad p t qp x q . θ p t qp m q “ θ p t qp x . m q for all t P T , x P p E , and m P M . Since p “ À p α and M “ À M λ , it is enoughto prove equation (2.4.6) for x P p α,E and m P M λ . First of all, notice that, since m P M λ , we have θ p t qp m q “ χ λ p t q exp p ϕ n p log p t qqqp m q where ϕ n is the nilpotent part of ϕ | t . Then we have Ad p t qp x q “ χ α p t q x since x P p α , so Ad p t qp x q . θ p t qp m q “ p χ α p t q x q . p χ λ p t q exp p ϕ n p log p t qqqp m qq“ χ α p t q χ λ p t q x . exp p ϕ n p log p t qqqp m q On the other hand, note that x . m P M α ` λ by lemma 2.4.1, so θ p t qp x . m q “ χ α ` λ p t q exp p ϕ n p log p t qqqp x . m q . Thus equation (2.4.6) is equivalent to the assertion that ϕ p x q ˝ exp p ϕ n p log p t qqq “ exp p ϕ n p log p t qqq ˝ ϕ p x q , but this follows immediately from lemma 2.4.2. (cid:3) The upshot of lemma 2.4.5 is that Lift p M | t , log q is naturally a “locallyanalytic p p , T q -module” (cf. definition B.4.1). We use this below to extend theaction of T to an action of all of P . (Semisimple groups) . Suppose P is semisimple. The Cartan subalgebra t consists of semisimple elements in the semisimple Lie algebra p , and representa-tions of semisimple Lie algebras preserve Jordan decompositions [Hum80, section6.4]; thus, t acts semisimply on any p -module M . This means that, when M hasalgebraic weights, the action of T induced by the construction of section 2.3 isalgebraic; in other words, M is a p p , T q -module in the sense of [Jan03, part II,section 1.20], so the discussion in loc. cit. implies that M naturally has an actionof the entire group P which lifts the original action of p . We write Lift p M q todenote this representation of P . (Reductive groups) . Suppose P is reductive. Let ρ : T Ñ GL p M q denotethe action of T on Lift p M | t , log q . Let P be the derived subgroup of P , whichis a semisimple group [Spr98, 8.1.6]. Then T “ P X T is a maximal torus in P [Hum75, exercise 27.9] [Mil17, theorem 17.82], so we can lift the action of P on M as in paragraph 2.4.8. Let σ : P Ñ GL p M q denote the action of P onLift p M | p q . It follows from lemma 2.3.5 that σ T “ ρ | T . Note that P is normalin P , and we have(2.4.10) ρ p t q ˝ σ p h q “ σ p h q ˝ ρ p t q for all t P T and h P P (see proof below). This means that ρ ˆ σ definesa homomorphism T ˙ P Ñ GL p M q , where T ˙ P is the external semidirectproduct of T and P in which T acts on P by conjugation. But P is generatedby T and P by corollary 1.3.2, so the multiplication map T ˙ P Ñ P is surjectiveand we have an exact sequence as follows.1 T T ˙ P P p M q ρ ˆ σ Since ρ and σ agree on T , it follows that ρ ˆ σ factors through ˜ ϕ : P Ñ GL p M q as indicated above. Let Lift p M, log q denote this representation of P . Proof of equation (2.4.10) . Let ϕ n denote the nilpotent part of ϕ | t and let χ : T Ñ GL p M q represent the action of T where T acts on a generalized weightspace M λ by χ λ , as in paragraph 2.3.3. Then ρ p t q “ χ p t q exp p ϕ n p log p t qq . By lemma 1.3.1, we know that the group of F -points P is generated by T and its root subgroups U α for all α P Φ p P , T q . Thus, it is sufficient to showequation (2.4.10) for h P T and h P U α . Suppose first that h P T . Then ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 15 σ p h q “ χ p h q is just a scalar on each generalized weight space, so equation (2.4.10)is clear. Next, suppose that h is in a root subgroup of P . Then σ p h q “ exp p ϕ p log p h qqq , so, to show equation (2.4.10) in this case, it is sufficient to show that ϕ n p log p t qq and ϕ p log p h qq commute. This follows from lemma 2.4.2. (cid:3) (General groups) . Suppose P is any connected algebraic group. By atheorem of Mostow’s, P has a Levi decomposition U ¸ L , where U is the unipotentradical of P and L is a reductive subgroup [Hoc81, VIII.4.3]. Note that themaximal torus T of P is contained in L . Let log P Logs p T q . Let ρ : U Ñ GL p M q be the action induced by the construction of section 2.1, and let σ : L Ñ GL p M q be the action of L on Lift p M | l , log q induced by the construction of paragraph 2.4.9above. To show that ρ and σ induce an action of the entire group P , it is sufficientto show that(2.4.12) ρ p huh ´ q “ σ p h q ˝ ρ p u q ˝ σ p h q ´ for all u P U and h P L . Proof of equation (2.4.12) . The group L is generated by T and U α for all α P Φ p L , T q by lemma 1.3.1. Thus it suffices to assume that h is in T or in U α forsome α .Suppose first that h P U α for some α . Let U ¸ U α be the subgroup of P generated by U and U α . Since the class of unipotent groups is stable underextension, we know that U ¸ U α is also unipotent. Let ρ α : U ¸ U α Ñ GL p M q denote the action of this unipotent group on M furnished by section 2.1. Since ρ α | U and ρ are both actions of the unipotent group U on M which lift ϕ | u , wemust have ρ α | U “ ρ . Similarly, since σ | U α and ρ α | U α are both actions of theunipotent group U α on M which lift ϕ | u α , we must have σ | U α “ ρ α | U α . Thus ρ p huh ´ q “ ρ α p huh ´ q “ ρ α p h q ˝ ρ α p u q ˝ ρ α p h q ´ “ σ p h q ˝ ρ p u q ˝ σ p h q ´ , proving equation (2.4.12) for h P U α .Next, suppose h P T . Note that ρ p u q “ exp p ϕ p log p u qqq for all u P U . Setting x “ log p u q , we see that it is sufficient to show that ϕ p Ad p h qp x qq “ σ p h q ˝ ϕ p x q ˝ σ p h q ´ for all x P u . This is precisely lemma 2.4.5. (cid:3) We let Lift p M, log q denote this locally analytic representation of P . Wehave thus defined a functor Lift p´ , log q : Mod fd , alg p Ñ Rep an P . (Extension to locally finite dimensional modules) . Finally, we extend thelifting functor Lift p´ , log q from Mod fd , alg p to all of Mod alg p in the universal way,using a left Kan extension. Let us tentatively write Lift ˚ p´ , log q for the left Kan extension of Lift p´ , log q along the inclusion Mod fd , alg p ã Ñ Mod lfd , alg p (we willultimately drop the asterisk from this notation). If M P Mod alg p , then by definition(2.4.15) Lift ˚ p M, log q “ colim N P I p M q Lift p N, log q where I p M q is the category of p -module homomorphisms N Ñ M where N P Mod fd , alg p . Observe that the subcategory of finite dimensional p -submodules of M is cofinal in I p M q , since any p -module homomorphism N Ñ M in I p M q factors through the finite dimensional submodule f p N q . Thus we may replace I p M q with this cofinal subcategory without affecting equation (2.4.15). Now, if M P Mod fd , alg p , it is clear that there is a natural isomorphism Lift ˚ p M, log q “ Lift p M, log q . Thus, we may remove the asterisk and defineLift p´ , log q : Mod alg p Ñ Rep an P to be the left Kan extension of the functor on Mod fd , alg p defined in paragraph 2.4.11without introducing any conflict of notation.2.5. Changing the logarithm.
Suppose log , log P Logs p T q . The exact se-quence (2.2.3) shows that the difference ǫ “ log ´ log is a homomorphism T Ñ t which annihilates T . In other words, it is locally constant.Let M be a finite dimensional t -module such that ϕ p t q Ď Nil p M q , and let ˜ ϕ and ˜ ϕ denote the homomorphisms T Ñ GL p M q representing the actions of T onLift p M, log q and Lift p M, log q , respectively. Then˜ ϕ p t q “ exp p ϕ p log p t qqq “ exp p ϕ p log p t q ` ǫ p t qqq “ exp p ϕ p log p t qq ` ϕ p ǫ p t qqq . Observe that r ϕ p log p t qq , ϕ p ǫ p t qqs “ ϕ pr log p t q , ǫ p t qsq “ t is abelian, so ˜ ϕ p t q “ exp p ϕ p log p t qqq ˝ exp p ϕ p ǫ p t qqq . We set ˜ ǫ p t q “ exp p ϕ p ǫ p t qqq . This is a locally constant, i.e., smooth, representation T Ñ GL p M q , and ˜ ϕ p t q “ ˜ ϕ p t q ˝ ˜ ǫ p t q .3. Categories O p , and O p , alg Let p G , T q be a split reductive algebraic group over F and let P be a parabolicsubgroup containing T .3.1. Definition and basic properties.Definition 3.1.1.
Let O p , be the full subcategory of Mod g consisting of finitelygenerated g -modules M such that M | p is locally finite dimensional and M | t issplit (cf. definition 1.2.6). Proposition 3.1.2. O p , is a Serre subcategory of Mod g . ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 17
Proof.
Suppose 0 M M M g -modules. It is clear that if M and M are finitelygenerated, then M is as well; it is also clear that if M is finitely generated, sois M . Since U p g q is noetherian, we also know that M being finitely generatedimplies that M is. Also, it follows from lemma 1.1.1 that M | p is locally finitedimensional if and only if M | p and M | p are. Finally, it follows from lemma 1.2.8that M | t is split if and only if M | t and M | t are. (cid:3) Definition 3.1.3.
Let O p , alg be the full subcategory of O p , consisting of modules M such that M | t P Mod alg t . Proposition 3.1.4. O p , alg is a Serre subcategory of Mod g .Proof. This follows from proposition 3.1.2 and lemma 1.2.12. (cid:3)
Relation to categories O p and O p alg .Definition 3.2.1. For any positive integer n , let O p ,n be the full subcategory ofMod g consisting of finitely generated g -modules M such that M | p is locally finitedimensional and M “ À λ P t ˚ E M λ,n , where M λ,n “ t m P M : p ϕ p x q ´ λ p x qq n p m q “ x P t E u . Observe that M λ,n Ď M λ,n ` Ď M λ and that O p ,n Ď O p ,n ` Ď O p , for all n . Wedefine O p : “ O p , . Lemma 3.2.2. If M P O p , , then g α . M λ,n Ď M α ` λ,n for all α, λ P t ˚ E .Proof. The proof is identical to the proof of lemma 2.4.1. (cid:3)
Lemma 3.2.3. O p , “ Ť n ě O p ,n .Proof. Suppose M P O p , . Since M is a finitely generated g -module and since M “ À M λ , we can choose a finite set of generators m , . . . , m r such that m i P M λ i for all i . Since m i P M λ i , there exists an integer n i such that p ϕ p x q ´ λ p x qq n i p m i q “ x P t E . Set n “ max n i , so that m i P M λ i ,n for all i . It then follows fromlemma 3.2.2 that M “ À λ P t ˚ E M λ,n , proving that M P O p ,n . (cid:3) Proposition 3.2.4 ([Soe85, 2.1.2.5]) . A g -module M is in category O p , if andonly if it has a finite filtration whose successive quotients are in O p .Proof. If M has a finite filtration whose successive quotients are in O p , the factthat M P O p , follows by induction on the length of the filtration using propo-sition 3.1.2. Conversely, suppose M P O p , . Then there exists an n such that M P O p ,n by lemma 3.2.3. Let M k “ à λ P t ˚ E M λ,k , and observe that each M k is a g -submodule of M by lemma 3.2.2. In other words,we have a filtration M “ M n Ě M n ´ Ě ¨ ¨ ¨ Ě M Ě M “ M by g -submodules, and it is clear from definitions that Q i : “ M i { M i ´ P O p . (cid:3) Corollary 3.2.5.
Simple objects in O p , are in O p . (cid:3) Corollary 3.2.6.
Every object in O p , has finite length.Proof. This follows from proposition 3.2.4 plus the fact that objects in category O p have finite length. (cid:3) Definition 3.2.7.
Let O p alg “ O p X O p , alg . Proposition 3.2.8. A g -module M is in the category O p , alg if and only if it hasa finite filtration whose successive quotients are in O p alg .Proof. One direction follows from proposition 3.1.4. Conversely, if M is in cate-gory O p , alg , we know from proposition 3.2.4 that M has a filtration M “ M n Ě M n ´ Ě ¨ ¨ ¨ Ě M “ Q i : “ M i { M i ´ P O p . Since M P O p , alg , we know that M i P O p , alg for all i by proposition 3.1.4. Furthermore, we have exact sequences0 M i ´ M i Q i , so Q i P O p , alg again by proposition 3.1.4. Thus Q i P O p , alg X O p “ O p . (cid:3) Lifting from category O p , alg . Suppose M P O p , alg and fix log P Logs p T q . Note that M | p P Mod alg p , so we can form the liftLift p M, log q : “ Lift p M | p , log q . This is a locally analytic representation of P . Also, since M “ Lift p M, log q as sets,so Lift p M, log q is also a g -module. Since the action of P on Lift p M, log q lifts theoriginal action of p , the two actions of p agree. The proposition below shows thatLift p M, log q is a locally analytic p g , P q -module in the sense of definition B.4.1.Thus it is naturally a D p g , P q -module by lemma B.4.2. Proposition 3.3.2. If m P Lift p M, log q , x P g , and h P P , then h . p x . p h ´ . m qq “ Ad p h qp x q . m. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 19
Proof.
We remark that it suffices to show this identity for a generating system of P . Now, P is generated by T and the subgroups exp U α p g α q , where α P Φ p l p , t q Y Φ p g , t q ` . When h P T , the proof is identical to the proof of lemma 2.4.5. For y P g α , where α P Φ p l p , t q Y Φ p g , t q ` , the map ad p y q : g Ñ g is nilpotent, henceexp p ad p y qq : g Ñ g is well-defined. It is easy to show by induction that for everyinteger n ě U p g q holds: y n .x “ n ÿ i “ ad p y q i p x q y n ´ i . This implies rather straightforwardly that as endomorphisms of Lift p M, log q wehave exp p y q ˝ x “ exp p ad p y qqp x q ˝ exp p y q , or equivalently(3.3.3) exp p ad p y qqp x q “ exp p y q ˝ x ˝ exp p y q ´ . We choose an embedding of algebraic groups G ã Ñ GL n and use this to regard G as a subgroup of GL n p F q and g as a Lie subalgebra of gl n . Then we apply 3.3.3to the representation of G on F n (furnished by the inclusion G Ď GL n p F q ). Then y , considered as an element of g α Ă gl n is nilpotent, and the right-hand side of3.3.3 is Ad GL n p F q p exp U α p y qqp x q . By functoriality of the adjoint representation Adand the exponential map we haveAd GL n p F q p exp p y qqp x q “ Ad G p exp U α p y qqp x q . We have thus shown that in g we have the identity(3.3.4) exp p ad p y qqp x q “ Ad G p exp U α p y qqp x q Substituting the right-hand side of 3.3.4 into the left-hand side of 3.3.3 givesAd G p exp U α p y qqp x q “ exp p y q ˝ x ˝ exp p y q ´ , as endomorphisms of Lift p M, log q . This proves the assertion. (cid:3) Globalization functor
Let p G , T q be a split reductive algebraic group over F and let P be a parabolicsubgroup containing T . Also, fix a logarithm log P Logs p T q .4.1. Setup and preliminaries.
Let M P O p , alg and let V be a smooth stronglyadmissible representation of P over E . We endow V with its finest locally convex topology. Then the D p P q -module V is finitely generated as a D p P q -module [ST02b, p. 453], where P isa compact open subgroup of P , and it is annihilated by p [ST01b, proposition For example, if V is a smooth admissible representation of P of finite length, it is stronglyadmissible. Indeed, the unipotent radical of P acts trivially on V [Boy99, lemme 13.2.3], so V is a smooth representation of the Levi subgroup L P of finite length, which must be stronglyadmissible [ST01b, proposition 2.2]. V is a finitely generated module over the quotient D p P q of D p P q by the two-sided ideal generated by p . In fact, V is even finitely presented over D p P q (or, equivalently, over D p P q ), where P is any compact open subgroup of P . Note that smooth stronglyadmissibile representations of P can be characterized as those whose restrictionsto P are subrepresentations of C p P q ‘ m for some m [ST01a, section 2]. If V is a smooth strongly admissible representation of P , we can set up a short exactsequence 0 V C p P q ‘ m W W is the cokernel of the inclusion. But C p P q ‘ m is semisimple, so W isitself a subrepresentation of C p P q ‘ m , so it is also strongly admissible. Dualizingthe above sequence yields an exact sequence0 W D p P q ‘ m V . Since W is a smooth strongly admissible representation of P , its dual W is alsofinitely generated over D p P q , so we conclude that V is finitely presented over D p P q . Since V is a D p P q -module, we can use the map from lemma B.3.8to regard V as a D p g , P q -module on which g acts trivially. Then, applyingthe construction of lemma B.4.3, we see that Lift p M, log q b E V is naturally a D p g , P q -module. Since the action of g is trivial on V , the action of x P g is givenexplicitly by x . p m b λ q “ p x . m q b λ. Lemma 4.1.4. If W is any p -module, and if X is a trivial p -module, then thereis a natural g -linear isomorphism p U p g q b U p p q W q b E X “ U p g q b U p p q p W b E X q , where we regard the left-hand side as a g -module by extending the trivial actionof p on X to a trivial action of g .Proof. There is a natural p -linear map W b E X p U p g q b U p p q W q b E X given by w b x ÞÑ p b w q b x , and this induces a natural g -linear map U p g q b U p p q p W b E X q p U p g q b U p p q W q b E X. We know that U p g q is free as a left U p p q -module with basis U p u ´ q , so both sidesare compatibly isomorphic to U p u ´ qb E W b E X as vector spaces. Thus the abovemap is an isomorphism. (cid:3) ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 21
Proposition 4.1.5.
Suppose M P O p , alg and V is a smooth strongly admissiblerepresentation of P . If P is a compact open subgroup of P , then Lift p M, log q b E V is finitely presented as a D p g , P q -module.Proof. To ease notation, set ˜ M “ Lift p M, log q . Let W be a finite dimensional p -submodule of M which generates M as a g -module, and set ˜ W “ Lift p W, log q .The locally analytic P -representation ˜ W is naturally a D p P q -module [ST02b,proposition 3.2 and the sentence before lemma 3.1]. Moreover, since W generates M as a g -module, the natural map D p g , P q b D p P q ˜ W Ñ ˜ M is surjective. Let Z be its kernel, so that we have an exact sequence(4.1.6) 0 Z D p g , P q b D p P q ˜ W ˜ M . Observe that D p g , P q b D p P q ˜ W “ p U p g q b U p p q D p P qq b D p P q ˜ W “ U p g q b U p p q W where we have used the fact that D p g , P q “ U p g q b U p p q D p P q [SS16, sentenceafter lemma 4.1] and the fact that ˜ W “ W as a p -module. Since W is finitedimensional, we see that U p g q b U p p q W is a finitely generated g -module. Since˜ M “ M is also a finitely generated g -module, and since U p g q is noetherian, wesee that Z is also finitely generated as a g -module.Let us tensor (4.1.6) with V over E .(4.1.7)0 Z b V p D p g , P q b D p P q ˜ W q b E V ˜ M b E V D p g , P q -module insuch a way that the map on the right is D p g , P q -linear. Using [SS16, sentenceafter lemma 4.1] as well as lemma 4.1.4, note that we have a natural isomorphism p D p g , P q b D p P q ˜ W q b E V “ p U p g q b U p p q W q b V “ U p g q b U p p q p W b V q“ D p g , P q b D p P q p ˜ W b E V q , which shows that the middle term is in fact a D p g , P q -module. Using this naturalidentification, the fact that the map on the right of (4.1.7) is D p g , P q -linearfollows from the fact that the map D p g , P q b D p P q ˜ W Ñ ˜ M is D p g , P q -linear. Itfollows that Z b V is naturally a D p g , P q -module as well.Note that V is a finitely generated D p P q -module (cf. paragraph 4.1.1), and˜ W is a finite dimensional locally analytic representation of P , so ˜ W b E V is afinitely generated D p P q -module by proposition A.4.1. Thus the middle term of(4.1.7) is a finitely generated D p g , P q -module.We now claim that Z b E V is also finitely generated. Because Z too is in O p , alg , there is a finite-dimensional p -submodule Z Ă Z which generates Z as U p g q -module. Then, given any λ P V and w P Z there are w , . . . , w n P Z and u , . . . , u n P U p g q such that w “ ř i u i .w i , and hence w b λ “ ř i u i . p w i b λ q . Hence Z b E V is generated by Z b E V as a U p g q -module. The map D p g , P q b D p P q p Z b V q Ñ Z b E V is hence surjective, because Z b E V is finitely generatedas a D p P q -module by proposition A.4.1. (cid:3) Definition and properties.
We continue to assume that M P O p , alg andthat V is a smooth strongly admissible representation of P over E . As we notedin paragraph 4.1.3, Lift p M, log q b E V is naturally a D p g , P q -module. Definition 4.2.1.
We defineˇ F GP p M, V q “ D p G q b D p g ,P q p Lift p M, log q b E V q . When G and P can be understood from context, we drop the superscript andsubscript and simply write ˇ F p M, V q instead. Also, when V “ E is the trivialrepresentation of P , we simply write ˇ F p M q . Let G be a maximal compact subgroup of G , and let P “ G X P .Observe that ˇ F p M, V q “ D p G q b D p g ,P q p Lift p M, log q b E V q , as D p G q -modules [SS16, lemma 4.2]. We use this observation repeatedly below. Theorem 4.2.3. If G is a maximal compact subgroup of G , then ˇ F p M, V q is afinitely presented D p G q -module. In particular, it is a coadmissible D p G q -module.Proof. Let P “ G X P . Since Lift p M, log q b E V is finitely presented as a D p g , P q -module by proposition 4.1.5, the result follows immediately. (cid:3) Theorem 4.2.4.
The functor p M, V q ÞÑ ˇ F p M, V q is exact in each argument.Proof. Certainly p M, V q ÞÑ
Lift p M, log q b E V is exact in each argument. Let G be a maximal compact subgroup of G , and let P “ G X P . Since Lift p M, log qb E V is a finitely presented D p g , P q -module, it follows from corollary B.8.7 that p M, V q ÞÑ D p G q b D p g ,P q p Lift p M, log q b E V q “ ˇ F p M, V q is also exact in each argument. (cid:3) Change of parabolic.
Let Q Ě P be a parabolic subgroup, and let V be asmooth strongly admissible representation of P . We let i p V q “ ind QP p V q denotethe smooth induction of V to Q . Lemma 4.3.1. i p V q is a smooth strongly admissible representation of Q .Proof. Let Q be a compact open subgroup of Q and let P Ď Q X P be acompact open subgroup. By the characterization of [ST01a, section 2], we see ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 23 that V restricted to P is a subrepresentation of C p P q ‘ m for some m [ST01a,section 2]. Since smooth induction is exact, ind Q P p V q is a subrepresentation ofind Q P p C p P q ‘ m q “ C p Q q ‘ m , and ind Q P p V q is thus strongly admissible again by the characterization of [ST01a,section 2]. But i p V q| Q “ à g P Q z Q { P ind Q Q X gP g ´ p g V q . Because V is a strongly admissible as representation of the compact open sub-group g ´ Q g X P , so is g V as a representation of Q X gP g ´ . Therefore,ind Q Q X gP g ´ p g V q is strongly admissible by the argument above. Since P is par-abolic, the double quotient Q z Q { P is finite, so i p V q is a finite direct sum ofstrongly admissible representations and is therefore itself smooth strongly admis-sible. (cid:3) Lemma 4.3.2.
Let D p Q q denote the quotient of D p Q q by the two-sided ideal I p q q generated by q . There is an isomorphism of D p Q q -modules D p Q q b D p P q V “ i p V q which is natural in the smooth strongly admissible representation V .Proof. We first claim that D p Q q b D p P q V “ Ind QP p V q where Ind QP p V q is the locally analytic induction. To see this, observe first that D p Q q b D p P q V is coadmissible. Indeed, suppose Q is a compact open subgroupof Q and P “ Q X P . Then D p Q q b D p P q V “ D p Q q b D p P q V as D p P q -modules [ST05, lemma 6.1(ii)], and V is a finitely presented D p P q -module byparagraph 4.1.2, so D p Q q b D p P q V is a finitely presented D p Q q -module, whichin turn means that D p Q q b D p P q V is coadmissible.We now relate D p Q qb D p P q V to what is denoted D p Q q r b D p P q V in the notationof [Koh11, section 2]. Note that, by [Koh11, eq. (53)], there an isomorphism oftopological vector spaces D p Q q r b D p P q V » D p Q { P q ˆ b E,i V which depends on the choice of a locally analytic section Q { P Ñ Q of the pro-jection map Q Ñ Q { P . Since P is parabolic, the quotient Q { P is compact, so D p Q { P q is Fr´echet; thus D p Q q r b D p P q V is Fr´echet as well. Since D p Q q b D p P q V is coadmissible and therefore complete, there is a natural continuous map α : D p Q q r b D p P q V Ñ D p Q q b D p P q V . By universal property of D p Q q b D p P q V , thereis also a natural map β : D p Q q b D p P q V Ñ D p Q q r b D p P q V , and it is clear that α ˝ β “ id. This means that α is surjective; since its domain D p Q q r b D p P q V isFr´echet, we see that α is an open map by the open mapping theorem [Sch02, proposition 8.6]. But then β must continuous as well, and it is clear that β ˝ α is the identity when restricted to im p α q , which is dense in D p Q q r b D p P q V ,which means that β ˝ α “ D p Q q b D p P q V “ D p Q q r b D p P q V .Observe that p D p Q q b D p P q V q “ Ind QP p V q using [Koh11, proposition 5.3 and remark 5.4] together with reflexivity of V . Nownote that D p Q q b D p P q V is also reflexive [Koh11, proposition 5.3 and theorem3.1], so we can dualize both sides of the above isomorphism to get D p Q q b D p P q V “ Ind QP p V q . Finally, observe that the smooth induction i p V q “ ind QP p V q is the subspace ofvectors in the locally analytic induction Ind QP p V q that are annihilated by q . Thus i p V q “ p D p Q q b D p P q V q{ I p q qp D p Q q b D p P q V q “ D p Q q b D p P q V . (cid:3) Theorem 4.3.3.
Suppose M P O , q alg and V is a strongly admissible smoothrepresentation of P . Then there is an isomorphism of D p G q -modules ˇ F GP p M, V q “ ˇ F GQ p M, i p V qq which is natural in both M and V .Proof. To ease notation, let us write ˜ M “ Lift p M, log q . It is sufficient to provethat(4.3.4) D p g , Q q b D p g ,P q ´ ˜ M b E V ¯ “ ˜ M b E i p V q since then applying D p G q b D p g ,Q q ´ yields the desired isomorphism. Observe that i p V q “ D p Q q b D p P q V “ D p Q q b D p P q V “ D p g , Q q{ J p g q b D p g ,P q{ J p g q V “ D p g , Q q{ J p g q b D p g ,P q V “ D p g , Q q b D p g ,P q V , where the first isomorphism is lemma 4.3.2, the third is lemma B.3.8, and thelast is because g acts trivially on V . Thus (4.3.4) is equivalent to(4.3.5) D p g , Q q b D p g ,P q ´ ˜ M b E V ¯ “ ˜ M b E ` D p g , Q q b D p g ,P q V ˘ . ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 25
Observe that we have natural isomorphismsHom D p g ,Q q ´ D p g , Q q b D p g ,P q ´ ˜ M b E V ¯ , ´ ¯ “ Hom D p g ,P q ´ ˜ M b E V , ´ ¯ “ Hom D p g ,P q ´ V , Hom E p ˜ M , ´q ¯ “ Hom D p g ,Q q ´ D p g , Q q b D p g ,P q V , Hom E p ˜ M , ´q ¯ “ Hom D p g ,Q q ´ ˜ M b E ` D p g , Q q b D p g ,P q V ˘ , ´ ¯ where we have used the adjunction of theorem B.4.5 twice (for the second andfinal steps). The fully faithfulness of the Yoneda embedding thus implies (4.3.5).By unwinding Yoneda’s lemma and the above isomorphisms, we find that theisomorphism (4.3.5) is given explicitly by 1 b m b v ÞÑ m b b v . (cid:3) Two Examples
As usual, let p G , T q be a split reductive group over F and let P be a para-bolic subgroup of G containing T . Fix log P Logs p T q . We begin with a generalcalculation. Lemma 5.1.1.
Suppose M is a finite dimensional p -module and α : P Ñ E ˆ isa smooth character. Then ˇ F GP p Ind gp p M q , α q “ Ind GP p Lift p M, log q b α q . Proof.
The p -linear map M Ñ Ind gp p M q “ U p g q b U p p q M induces a morphismof locally analytic representations Lift p M, log q Ñ Lift p Ind gp p M q , log q , which weregard as a homomorphism of D p P q -modules. Tensoring with α yields another D p P q -module homomorphism whose codomain is naturally a D p g , P q -module (cf.lemma B.4.3), so there is a natural D p g , P q -linear map(5.1.2) D p g , P q b D p P q p Lift p M, log q b α q Lift p Ind gp p M q , log q b α . We claim that (5.1.2) is an isomorphism. To see this, observe that D p g , P q “ U p g q b U p p q D p P q as left g -modules, so the domain is D p g , P q b D p P q p Lift p M, log q b α q “ U p g q b U p p q p Lift p M, log q b α q “ U p g q b U p p q M as g -modules, since Lift p M, log q “ M and α “ E as p -modules. On the otherhand, we also have Lift p Ind gp p M q , log q b α “ Ind gp p M q as g -modules, and under these g -linear identifications, the map (5.1.2) becomesthe identity map. To conclude, observe that the isomorphism (5.1.2) produces the first isomor-phism in the following sequence of isomorphisms which prove the desired result.ˇ F p Ind gp p M q , α q “ D p G q b D p P q p Lift p M, log q b α q“ D p G q ˜ b D p P q p Lift p M, log q b α q“ Ind GP p Lift p M, log q b α q Here, we use arguments similar to those in the proof of lemma 4.3.2 (now usingfinite dimensionality) for the second and third isomorphisms. (cid:3)
This brings us to our examples.
Example 5.1.3 (Breuil’s representations) . Let F “ Q p and let E be a finiteextension. Let G “ GL . Let P Ď G be the Borel of upper triangular matricesand T Ď G the maximal torus of diagonal matrices.Let M “ Em ‘ Em be the 2-dimensional p -module where ϕ M ˆ x ˚ y ˙ : m ÞÑ m ÞÑ p x ´ y q m . Identifying End p M q “ M p E q , we can also write ϕ M : ˆ x ˚ y ˙ ÞÑ ˆ x ´ y ˙ . Observe that M is the inflation of a t -module and its only weight is 0.If we choose L P E , there is a unique logarithm map log L : Q ˆ p Ñ E such thatlog L p p q “ L . We can use this to construct a logarithm log P Logs p T q bylog : ˆ a d ˙ ÞÑ ˆ log L p a q
00 log L p d q ˙ . Then Lift p M, log q is precisely the representation of P that is denoted σ p L q in[Bre04, section 2.1], and lemma 5.1.1 implies thatˇ F GP p Ind gp p M qq “ Ind GP p σ p L qq . More generally, for an integer k ě
2, let N p k q “ En denote the 1-dimensional p -module given by ϕ N p k q : ˆ x ˚ y ˙ ÞÑ p k ´ q y, and let α k : P Ñ E ˆ be the smooth character α k : ˆ a ˚ b ˙ ÞÑ | ab | ´p k ´ q{ . Note that if k is odd, we must assume that E is large enough that ? p P E inorder for this character to be defined. Then Lift p M b N p k q , log q b α k is precisely ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 27 the representation of P denoted σ p k, L q in [Bre04, section 2.1], and lemma 5.1.1implies that ˇ F GP p Ind gp p M b N p k qq , α k q “ Ind GP p σ p k, L qq . This representation has the representation Σ p k, L q of [Bre04, section 2] as a sub-quotient. Remark 5.1.4.
Here’s another perspective on the choice of logarithms in the ex-ample above. We can choose coordinates on T using the “standard” isomorphism T Ñ G m given by diag p a, b q ÞÑ p a, b q , but we can also use other isomorphismsto choose coordinates. Suppose we choose coordinates using the isomorphism ζ : T Ñ G m given by diag p a, b q ÞÑ p ab ´ , b q . If we choose L , L P E , we get alogarithm log : p Q ˆ p q Ñ E on G m given by log p a, b q “ p log L p a q , log L p b qq . Nowconsider the following commutative diagram. t E E t F T p F ˆ q dζdζ exp exp ζ log We can use this to define a logarithm log : T Ñ t E given bylog “ p dζ q ´ ˝ log ˝ ζ , and this is in fact a logarithm sincelog ˝ exp “ p dζ ´ ˝ log ˝ ζ q ˝ exp “ dζ ´ ˝ log ˝ exp ˝ dζ “ dζ ´ ˝ id ˝ dζ “ id , where we abusively write id to denote the natural maps t Ñ t E and F Ñ E .Explicitly, this logarithm is given bylog ˆ a b ˙ “ ˆ log L p ab ´ q ` log L p b q
00 log L p b q ˙ “ ˆ log L p a q ` γ p a, b q
00 log L p b q ˙ where γ is the smooth function γ p a, b q “ log L p b q ´ log L p b q . If we fix L “ L , the above construction of a logarithm on T applied withany choice of L P E will furnish us with a logarithm log P Logs p T q with theproperty that Lift p M b N p k q , log q b α k is Breuil’s representation σ p k, L q . Also,in preparation for example 5.1.5 below, we remark that it is not necessary for ζ to be an isomorphism; it is sufficient for it to be an ´etale homomorphism of groupschemes (since this will imply that dζ is invertible). Example 5.1.5 (Schraen’s representations) . Suppose F “ Q p and E is a finiteextension. Let G “ GL . Let P Ď G be the Borel of upper triangular matricesand T Ď G the maximal torus of diagonal matrices.Let M “ Em ‘ Em ‘ Em be the 3-dimensional representation of p givenby ϕ M : ¨˝ x ˚ ˚ y ˚ z ˛‚ ÞÑ ¨˝ ´ x ` y ` z ´ x ´ y ` z ˛‚ , where we have identified End p M q “ M p E q using the basis m , m , m . Again, M is the inflation of a representation of t and its only weight is 0. We can use afinite ´etale homomorphism ζ : T Ñ G m of the form ¨˝ a b
00 0 c ˛‚ ÞÑ p a ´ bc, a ´ b ´ c , ˚q to choose a logarithm on T (eg, take ˚ “ c ). More precisely, given L , L P E ,choose a logarithm on G m which is given by log L on the first coordinate and log L on the second (there are infinitely many such logarithms). Via ζ , this inducesa logarithm log P Logs p T q as in remark 5.1.4. This logarithm has the propertythat Lift p M, log q is precisely the representation of P that is denoted σ p L , L q in[Sch11, remarque 5.14]. For example, we can take the logarithm log : T Ñ t E given bylog ¨˝ a b
00 0 c ˛‚ “ ¨˝ log L p a q ` γ p a, b, c q L p b q ` γ p a, b, c q
00 0 log L p c q ˛‚ where γ is the smooth function γ p a, b, c q “ ´ p log L p a ´ b ´ c q ´ log L p a ´ b ´ c qq . Thus lemma 5.1.1 implies thatˇ F GP p Ind gp p M qq “ Ind GP p σ p L , L qq . This representation has the representation Σ p k, L , L q of [Sch11, 5.12] as a sub-quotient. Appendix A. Tensor-hom adjunction for D p H q -modules Let H be a locally analytic group. In the following, whenever we considerseparately or jointly continuous D p H q -modules, we assume that the underlyingtopological vector space is locally convex. If M is a locally analytic representation,there exists a separately continuous D p H q -module structure on M which has theproperty that δ h . m “ h . m for all h P H and m P M [ST02b, proposition 3.2]. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 29
We will assume in this section that M is locally finite dimensional (ie, that itis the colimit of its finite dimensional subrepresentations). Our goal is to showthat, if X is any D p H q -module, then we can endow M b X and Hom p M, X q with natural D p H q -module structures in such a way that M b ´ is left adjointto Hom p M, ´q . This would be clear if D p H q were a Hopf algebra, but it is notquite a Hopf algebra [ST05, appendix to section 3].A.1. Some functional analysis.Lemma A.1.1.
Suppose V is a locally convex vector space and W is a finitedimensional vector space. Then the inductive and projective topologies on V b W coincide. In other words, if U is a locally convex vector space and σ : V ˆ W Ñ U is bilinear and separately continuous, then it is automatically jointly continuous.Proof. Let N be an open lattice in U . Choose a basis w , . . . , w n of W . Then σ p´ , w i q : V Ñ U is continuous for all i , so there exist open lattices L i Ď V suchthat σ p L i , w i q Ď N . Then let L “ X L i , so that σ p L, w i q Ď N . Let M be the O E -submodule of W generated by w , . . . , w n . Then M is an open lattice in W .If v P L and if a , . . . , a n P O E , then σ ´ v, ÿ a i w i ¯ “ ÿ i a i σ p v, w i q . We know that σ p v, w i q P N , and N is an O E -submodule of U , so σ p v, ř a i w i q P N as well. Thus σ p L, M q Ď N , proving that σ is continuous. (cid:3) Remark A.1.2.
We frequently use lemma A.1.1 tacitly. Whenever V is a lo-cally convex vector space and W is finite dimensional, we regard V b W as alocally convex vector space with the “inductive = projective” topology withoutspecifying a subscript i or π . Remark A.1.3. If M is a finite dimensional locally analytic representation of H , the separately continuous D p H q -module structure D p H q ˆ M Ñ M is auto-matically continuous by lemma A.1.1. Taking colimits, we find the same is truewhen M is only locally finite dimensional.A.2. Constructions and functoriality.
We begin by constructing a D p H q -module structure on M b X , where M is a locally finite dimensional locallyanalytic representation of H and X is a D p H q -module. Lemma A.2.1 does thiswhen M is finite dimensional, and lemma A.2.3 upgrades this to the general case.Intermediate to this we have lemma A.2.2 which proves that the construction ofthe D p H q -module structure on M b X is functorial in M when restricted to thecategory of finite dimensional locally analytic repersentations of H (and again,lemma A.2.4 upgrades this statement to the general case). Lemma A.2.1.
Suppose M is a finite dimensional locally analytic representationof H . For any D p H q -module X , there exists a natural D p H q -module structure on the tensor product M b X with the property that δ h . p m b x q “ p h . m q b p δ h . x q for all h P H , m P M , and x P X . Moreover, if X is a separately (resp. jointly)continuous D p H q -module, then so is M b X .Proof. The tensor product M b X naturally has the structure of a module over D p H q b D p H q , given by p µ b µ qp m b x q “ p µ . m q b p µ . x q . Observe that the structure map D p H q b D p H q Ñ End p M b X q factors as D p H q b D p H q End p M q b D p H q End p M b X q , where the first map is the structure map D p H q Ñ End p M q tensored with D p H q ,and the second map is given by σ b µ ÞÑ r m b x ÞÑ σ p m q b µ . x s . Since M is a separately continuous D p H q -module, the map D p H q Ñ End p M q iscontinuous, so the map D p H q b D p H q Ñ End p M q b D p H q is also continuous.Since End p M q is finite dimensional and D p H q is complete, the tensor productEnd p M qb D p H q is also complete. Thus D p H qb D p H q Ñ End p M qb D p H q factorsnaturally through D p H q ˆ b i D p H q . D p H q ˆ b i D p H q End p M q b D p H q End p M b X q . Now note that D p H ˆ H q “ D p H q ˆ b i D p H q [ST05, proposition A.3], and thatthe diagonal map ∆ : H Ñ H ˆ H induces a continuous homomorphism ∆ ˚ : D p H q Ñ D p H ˆ H q of algberas. The composite D p H q D p H ˆ H q “ D p H q ˆ b i D p H q End p M q b D p H q End p M b X q ∆ ˚ then defines the structure of a D p H q -module on M b X . It is easy to see thatthis D p H q -module structure is given by the stated formula on delta distributions.Note that the formula is not so clear on other distributions (ie, ones that are notdelta distributions) precisely because we cannot explicitly say how to transfersuch distributions through the isomorphism D p H ˆ H q “ D p H q ˆ b i D p H q .If X happens to be a separately continuous D p H q -module, we would like toshow that the resulting D p H q -module structure on M b X is also separatelycontinuous. In other words, we need to show that the maps D p H q Ñ End p M b X q and M b X Ñ Hom p D p H q , M b X q land in the spaces End cts p M b X q andHom cts p D p H q , M b X q of continuous linear maps, respectively. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 31
To show that D p H q Ñ End p M b X q lands inside End cts p M b X q , we note that,by using the above factorization, it is sufficient to show that End p M q b D p H q Ñ End p M b X q lands inside End cts p M b X q . In other words, we have to show that,for fixed σ P End p M q and µ P D p H q , the map M b X Ñ M b X given by m b x ÞÑ σ p m q b µ . x is continuous. But this map factors as M b X M b X M b X σ b X M b µ and each of these maps is continuous (the former since M is finite dimensional,the latter since X is separately continuous). Thus the composite is continuousas well.Next, we need to show that the image of M b X Ñ Hom p D p H q , M b X q is inside Hom cts p D p H q , M b X q . Fix m b x . We want to show that the map D p H q Ñ M b X given by µ ÞÑ µ . p m b x q is continuous. Observe that this mapfactors as follows. D p H q D p H ˆ H q “ D p H q ˆ b i D p H q M b D p H q M b X ∆ ˚ m b D p H q M b x The first two maps are continuous, and the last map is continuous by separatecontinuity of X . To see that D p H q ˆ b i D p H q Ñ M b D p H q is continuous, notethat the map D p H q Ñ M given by µ ÞÑ µ . x is continuous by separate continuityof M , so D p H q b i D p H q Ñ M b D p H q is continuous as well. Since M is finitedimensional, we know that M b D p H q is complete, so this continuous map factorscontinuously through D p H q ˆ b i D p H q , as desired.Now suppose X is jointly continuous. Then D p H q b π X Ñ X is continuous.Since M is finite dimensional, separate continuity of D p H q ˆ M Ñ M impliescontinuity, so D p H q b π M Ñ M is also continuous. Thus the map p D p H q b π D p H qq b π p M b π X q “ p D p H q b π M q b π p D p H q b π X q M b π X is continuous. Since the inductive topology is finer than the projective topology,this implies that p D p H q b i D p H qq b π p M b X q M b X is continuous. We noted above that the action of D p H q b D p H q on M b X factorsthrough D p H q ˆ b i D p H q , so the above map factors through a continuous map p D p H q ˆ b i D p H qq b π p M b X q M b X. Now note that D p H q Ñ D p H q ˆ b i D p H q is continuous, so the first map in thecomposite D p H q b π p M b X q p D p H q ˆ b i D p H qq b π p M b X q M b X. is also continuous. This proves that the D p H q -module structure on M b X isjointly continuous. (cid:3) Lemma A.2.2.
Suppose f : M Ñ N is a homomorphism of finite dimensionallocally analytic representations of H . If X is a D p H q -module, then f b X : M b X Ñ N b X is also D p H q -linear.Proof. We want to show that the following diagram commutes. D p H q End p M b X q End p N b X q Hom p M b X, N b X q f ˚ f ˚ Note that D p H q Ñ End p M b X q factors through End p M q b D p H q and D p H q Ñ End p N b X q factors through End p N q b X , and the black part of the followingdiagram clearly commutes. D p H q End p M q b D p H q End p N q b D p H q Hom p M, N q b D p H q End p M b X q End p N b X q Hom p M b X, N b X q f ˚ f ˚ f ˚ f ˚ Thus it is sufficient to show that the following diagram commutes. D p H q End p M q b D p H q End p N q b D p H q Hom p M, N q b D p H q f ˚ f ˚ These are all continuous maps. Thus it sufficies to check commutativity afterrestricting to E r H s , where this is clear. (cid:3) Lemma A.2.3.
Suppose M is a locally finite dimensional locally analytic repre-sentation of H . For any D p H q -module X , there exists a natural D p H q -modulestructure on the tensor product M b X with the property that δ h . p m b x q “ p h . m q b p δ h . x q for all h P H , m P M , and x P X . Moreover, if X is a separately (resp. jointly)continuous D p H q -module, then so is M b X .Proof. We know that M is the colimit of its finite dimensional subrepresentations N . Note that M b X “ p colim N q b X “ colim p N b X q . Each N b X has a D p H q -module structure as in lemma A.2.1, and the transitionmaps are all D p H q -linear by lemma A.2.2. Thus M b X acquires a D p H q -module ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 33 structure from each N b X . It too is given by the same formula on delta dis-tributions. Moreover, if X is separately (resp. jointly) continuous, then M b X inherits the same continuity property from each N b X . (cid:3) Lemma A.2.4.
Suppose f : M Ñ N is a homomorphism of locally finite di-mensional locally analytic representations of H . If X is a D p H q -module, then f b X : M b X Ñ N b X is also D p H q -linear.Proof. Suppose M Ă M and N Ă N are finite dimensional subrepresentationssuch that f p M q Ă N . By lemma A.2.2, we know that M b X Ñ N b X is D p H q -linear. As M and N vary, the transition maps are also D p H q -linear bylemma A.2.2. Taking a colimit over all such M and N thus yields the result. (cid:3) Lemma A.2.5.
The following diagram commutes. U p h q U p h q b U p h q D p H q D p H ˆ H q D p H q ˆ b i D p H q Here, U p h q Ñ U p h qb U p h q is the comultiplication on U p h q , D p H q Ñ D p H ˆ H q isthe continuous algebra homomorphism induced by the diagonal map H Ñ H ˆ H ,and D p H ˆ H q “ D p H q ˆ b i D p H q is the isomorphism of [ST05, proposition A.3].Proof. The comultiplication on U p h q factors as U p h q Ñ U p h ‘ h q induced by thediagonal map h Ñ h ‘ h and a natural isomorphism U p h ‘ h q “ U p h q b U p h q . U p h q U p h ‘ h q U p h q b U p h q D p H q D p H ˆ H q D p H q ˆ b i D p H q Since the inclusion U p Lie p´qq Ñ D p´q is functorial in its argument (a locallyanalytic group), we can apply this functoriality to the diagonal map H Ñ H ˆ H and conclude that the above diagram commutes. (cid:3) Corollary A.2.6.
Suppose M is a locally finite dimensional locally analytic rep-resentation of H . The D p H q -module structure on M b X from A.2.3 restricts tothe usual U p h q -module structure on M b X . Lemma A.2.7.
Suppose M and N are both finite dimensional locally analyticrepresentations of H . For any D p H q -module X , the associativity isomorphism M b p N b X q “ p M b N q b X is D p H q -linear.Proof. Let α : M b p N b X q Ñ p M b N q b X be the associativity isomorphism m b p n b x q ÞÑ p m b n q b x. We want to show that the following diagram commutes. D p H q End p M b p N b X qq End pp M b N q b X q Hom p M b p N b X q , p M b N q b X q α ˚ α ˚ Note that the structure map D p H q Ñ End p M b p N b X qq factors throughEnd p M qbp End p N qb D p H qq , while the structure map D p H q Ñ End pp M b N qb X q factors through End p M b N q b D p H q . It is clear that the black part of the followdiagram commutes. D p H q End p M q b p End p N q b D p H qq End p M b N q b D p H q End p M b p N b X qq End pp M b N q b X q Hom pp M b N q b X, M b p N b X qq α ˚ α ˚ Thus it is sufficient to show that the following diagram commutes. D p H q End p M q b p End p N q b D p H qq End p M b N q b D p H q These are all continuous maps. Thus it sufficies to check commutativity afterrestricting to E r H s , where this is clear. (cid:3) Lemma A.2.8.
Suppose M is a locally finite dimensional locally analytic rep-resentation of H . If f : X Ñ Y is a homomorphism of D p H q -modules, then M b f : M b X Ñ M b Y is also D p H q -linear.Proof. It is sufficient to consider the case when M is finite dimensional. We wantto show that the following diagram commutes. D p H q End p M b X q End p M b Y q Hom p M b X, M b Y q f ˚ f ˚ By construction, the two maps coming out of D p H q factor through End p M q b D p H q in the same way, so it is sufficient to show that the following diagram ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 35 commutes. End p M q b D p H q End p M b X q End p M b Y q Hom p M b X, M b Y q f ˚ f ˚ This follows from D p H q -linearity of f . (cid:3) Remark A.2.9.
Suppose M is a finite dimensional locally analytic representationof H and X is a D p H q -module. Then the contragredient M is also a finitedimensional locally analytic representation of H . If we transfer the D p H q -modulestructure on M b X through the natural isomorphism M b X “ Hom p M, X q ,it is given by p δ h . σ qp m q “ δ h . σ p h ´ . m q for h P H, m P M , and σ P Hom p M, X q . To see this, observe that the naturalisomorphism ψ : M b X Ñ Hom p M, X q is given by ψ p λ b x qp m q “ λ p m q x. If ψ p λ b x q “ σ , then p δ h . σ qp m q “ ψ p δ h . ψ ´ p σ qqp m q“ ψ p δ h . p λ b x qqp m q“ ψ pp h . λ q b p δ h . x qqp m q“ p h . λ qp m qp δ h . x q“ δ h . p h . λ qp m q x “ δ h . λ p h ´ . m q x “ δ h . σ p h ´ . m q . If X is a separately (resp. jointly) D p H q -module, then so is Hom p M, X q , simplybecause M b X “ Hom p M, X q is a topological isomorphism. If M is only locally finite dimensional, it is the colimit of its finite dimensional subrepresentations N ,so Hom p M, X q “
Hom p colim N, X q “ lim Hom p N, X q has a natural structure of a D p H q -module, induced by the D p H q -module structuredescribed above on each Hom p N, X q . And again, if X is a separately (resp.jointly) continuous D p H q -module, then Hom p M, X q inherits the same continuityproperty from all of the Hom p N, X q .A.3. Adjunction.Theorem A.3.1.
Suppose M is a locally finite dimensional locally analytic rep-resentation of H . Then there is an adjunction of D p H q -module-valued functors p M b ´q % Hom p M, ´q . Proof.
Suppose first that M is finite dimensional. We know that there is anadjunction p M b ´q % Hom p M, ´q of vector-space-valued functors, and we wantto upgrade this to an adjunction of D p H q -module valued functors. Suppose X is a D p H q -module. Since adjunctions are determined by their counit and unitmaps, it is sufficient to show that the counit ǫ : M b Hom p M, X q Ñ X and theunit η : X Ñ Hom p M, M b X q of the vector-space-valued adjunction are both D p H q -linear. First let us show that ǫ is D p H q -linear. M b Hom p M, X q “ M b p M b X q “ p M b M q b X as D p H q -modules by remark A.2.9 and lemma A.2.7. Let ev : M b M Ñ E bethe evaluation map. Under the isomorphism M b Hom p M, X q “ p M b M q b X ,the counit ǫ corresponds to the map ev b X : p M b M q b X Ñ X . Since ev isa homomorphism of representations, it follows from lemma A.2.4 that ev b X is D p H q -linear. Thus ǫ is also D p H q -linear.Next we show that η is D p H q -linear in the same way. Note thatHom p M, M b X q “ M b p M b X q “ p M b M q b X as D p H q -modules again by remark A.2.9 and lemma A.2.7. Let ι : E Ñ M b M be the coevaluation. Under the isomorphism Hom p M, M b X q “ p M b M q b X ,the unit η corresponds to the map ι b X : X Ñ p M b M q b X . Since ι isa homomorphism of representations, it follows from lemma A.2.4 that ι b X is D p H q -linear. Thus η is also D p H q -linear.This completes the proof when M is finite dimensional. If M is locally finitedimensional, it is the colimit of its finite dimensional subrepresentations N . Since M b ´ is functorial in M by lemma A.2.4, and since Hom p M, ´q is consequentlyalso functorial in M , we have thatHom p M b ´ , ´q “ lim Hom p N b ´ , ´q“ lim Hom p´ , Hom p N, ´qq“ Hom p´ , lim Hom p N, ´qq“ Hom p´ , Hom p M, ´qq . (cid:3) Remark A.3.2. If X happens to be a separately continuous D p H q -module, then ǫ and η are continuous, so it is sufficient to show that ǫ and η are both E r H s -linear. This can be done in a more “hands-on” way than the general case discussed Suppose X and Y are D p H q -modules and f : M b X Ñ Y is D p H q -linear. Under theisomorphism Hom p M b X, Y q “
Hom p X, Hom p M, Y qq , the map f on the left correspondson the right to the composite f ˚ ˝ η , where f ˚ : Hom p M, M b X q Ñ Hom p M, Y q . Then D p H q -linearity of η and f implies D p H q -linearity of f ˚ ˝ η , proving that the isomorphismHom p M b X, Y q “
Hom p X, Hom p M, Y qq restricts to a well-defined map Hom D p H q p M b X, Y q Ñ
Hom D p H q p X, Hom p M, Y qq . We use the counit ǫ to go the other way. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 37 above. Notice that ǫ p m b σ q “ σ p m q , so ǫ p δ h . p m b σ qq “ ǫ p h . m b δ h . σ q“ p δ h . σ qp h . m q“ δ h . σ p h ´ . h . m q“ δ h . σ p m q“ δ h . ǫ p m b σ q , proving that ǫ is E r H s -linear. Similarly, since η p x qp m q “ m b x , we have η p δ h . x qp m q “ m b δ h . x “ δ h . p h ´ . m b x q“ δ h . η p x qp h ´ . m q“ p δ h . η p x qqp m q so η is also E r H s -linear.A.4. Finite generation.
Our goal in this subsection is to prove that the con-struction that the module produced by lemma A.2.3 is sometimes finitely gener-ated. This statement is analogous to [ST01a, lemma 3.3], and its proof largelyfollows the proof of that result. The precise statement is the following.
Proposition A.4.1.
Suppose H is compact. Let M be a finite dimensional locallyanalytic representation of H , and let X be finitely generated module over D p H q which is annihilated by h . Then M b X is a finitely generated D p H q -module. Before proceeding with the proof, let us make a preliminary observation.
Remark A.4.2.
Suppose M is a finite dimensional locally analytic representationof H (and H need not be compact for the purposes of this remark). There aretwo ways to endow M with the structure of a D p H q -module:(i) We can follow the construction of [ST02a, proposition 3.2 and the sentencebefore lemma 3.1].(ii) We can note that the contragredient representation M is also locally ana-lytic representation, so its dual p M q is a D p H q -module [ST02a, corollary3.3]. We can then transfer this structure through the canonical isomorphism M “ p M q .These two D p H q -module structures on M coincide. Indeed, both D p H q -modulestructures are separately continuous, so it is sufficient to check that the E r H s -module structures coincide, but this follows from the fact that M “ p M q is anisomorphism of representationsof H . Proof of proposition A.4.1.
We remark that the assertion is true for H if it is truefor any open subgroup H Ď H (which is hence itself compact), since there is a canonical injective homomorphism D p H q Ñ D p H q . We may thus replace H byan open subgroup in the course of the proof.After possibly shrinking H , we may assume that there is a good analytic opensubgroup H of H (in the sense of [Eme17, sec. 5.2]) with the property that H “ H ˝ p F q (cf. [Eme17, pp. 101-102] for notation). We claim that the canonicalmap(A.4.3) O p H ˝ q b F C p H q C an p H q given by ψ b f ÞÑ ψ | H ¨ f is injective. Proof of injectivity of (A.4.3) . Suppose that for linearly independent elements ψ , . . . , ψ r P O p H ˝ q and functions f , . . . , f r P C p H q one has ř ri “ ψ i | G ¨ f i “ H Ă H ˝ such that f i | H h is constantfor all i P t , . . . , r u and h P H , where H “ H p F q . If c i,h is the value of f i onthe coset H h , then this implies that ř ri “ c i,h ψ i | G h “
0. Replacing ψ i by thefunction x ÞÑ ψ i p xh q , we may assume h “
1. After choosing coordinates for H (coming from the O F -Lie lattice in h which gives rise to H , cf. [Eme17, p.100]), we may assume that H is, as a rigid analytic space, isomorphic to a d -dimensional polydisc of polyradius p , . . . , q and H is a d -dimensional polydiscof polyradius p| p | n , . . . , | p | n q for some n ą
0. Then O p H ˝ q is the ring of powerseries ř ν P N d a ν X ν P F rr X , . . . , X d ss with lim | ν |Ñ8 | a ν | r | ν | “ r ă
1. Ifthe restriction of such a power series to p p n o F q d vanishes, then all coefficientsmust vanish. But this implies that ř ri “ c i,g ψ i “
0, and hence c i,g “ i P t , . . . , r u and g P H . (cid:3) It suffices to prove that M b E X is finitely generated when X “ D p H q . Write D an p H ˝ q “ p O p H ˝ q E q b for the analytic distribution algebra of H ˝ with coefficientsin E (cf. [Eme17, 2.2.2] for the definition of D an p H ˝ q , and the discussion of in[Eme17, p. 95-102] for some properties of this algebra). This is a topologicalalgebra whose underlying topological vector space is of compact type [Eme17, p.102], and hence it is reflexive [Sch02, 16.10]. The canonical map O p H ˝ q b F E Ñ C an p H q gives rise to a continuous homorphism of topological algebras D p H q Ñ D an p H ˝ q .Since M is finite dimensional and locally analytic, the same is true for thedual space M with its contragredient action. There is thus a small enough group This can be seen by induction on d . We may assume without loss of generality that f p X , . . . , X d q is a power series in F x X , . . . , X d y which vanishes on o dF . If d “ f “ f “ ř k “ f k p X , . . . , X d q X k is as above, then, for all fixed p a , . . . , a d q P o d ´ F , one has f p X , a , . . . , a d q “
0, by the case d “
1. Hence f k p a , . . . , a d q “ k ě p a , . . . , a d q P o d ´ F . By induction, this impliesthat all f k vanish identically. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 39 of the form H ˝ such that M is an analytic representation of H ˝ in the sense of[Eme17, 3.6.1]. As a consequence, M “ p M q is naturally a module over D an p H ˝ q ,i.e., the map D p H q ˆ M Ñ M which gives M the structure of a D p H q -modulefactors as D p H q ˆ M Ñ D an p H ˝ q ˆ M . Since M is finite dimensional, we mayargue by induction on the length of M as a D an p H ˝ q -module. It thus sufficesto prove the case when M is simple as D an p H ˝ q -module. Choosing a generatorfor M as a D an p H ˝ q -module we obtain a surjection D an p H ˝ q Ñ M of D an p H ˝ q -modules. This induces an H -equivariant injection M ã Ñ D an p H ˝ q b “ O p H ˝ q E ,by reflexivity. This map gives rise to an H -equivariant injection M b C p G q O p H ˝ q E b C p H q “ O p H ˝ q b F C p H q which we compose with the injection (A.4.3) to obtain an injective H -equivariantmap(A.4.4) M b C p H q C an p H q . The group action of H on the domain and target of A.4.4 induces a derived actionof h and hence the structure of a U p h q -module on these representations. Since M is finite-dimensional, the kernel J of the map U p h q Ñ End p M q is a two-sidedideal of finite codimension. Because the action of h on C p H q is trivial, the ideal J also acts trivially on the domain of A.4.4. This implies that the image of A.4.4is contained in C an p H q J “ “ t f P C an p H q | z . f “ z P J u . By [ST01a, 3.1], the subspace topology on C an p H q J “ induced from C an p H q isthe finest locally convex topology. Because the natural topology on M b C p H q is also the finest locally convex topology, we see that the map A.4.4 is strict. Asa consequence, the map D p H q Ñ M b D p H q obtained from A.4.4 by passingto continuous dual spaces is surjective, cf. [Sch02, 9.4]. (cid:3) Appendix B. Modules over D p g , H q Let G be a locally analytic group and H a closed subgroup. In this section,we establish some foundational facts about the module theory of D p g , H q . Recallthat this is defined to be the subring of D p G q generated by g and D p H q [OS15,section 3.4] [SS16, section4].B.1. Some functional analysis.Lemma B.1.1.
Suppose β : V ˆ W Ñ Z is separately (resp. jointly) contin-uous bilinear map of locally convex vector spaces. Let β ˚ : V Ñ Hom p W, Z q be the adjoint map. Then β ˚ p V q Ď Hom cts p W, Z q and β ˚ is continuous when Hom cts p W, Z q is given the topology of pointwise (resp. bounded) convergence. Proof.
For fixed v P V , note that w ÞÑ β p v, w q is continuous by separate continu-ity of β . Fix an open lattice N in Z . If w P W , continuity of v ÞÑ β p v, w q impliesthat there exists an open lattice L in V such that β p L, w q Ď N . In other words, β ˚ p L q Ď L p w, N q , so β ˚ is continuous for the topology of pointwise convergence(ie, the weak topology [Sch02, p. 30]).Now suppose β is jointly continuous and B is a bounded subset of W . Thenthere exist open lattices L Ď V and M Ď W such that β p L, M q Ď N . Since B isbounded, there exists an a P E such that B Ď aM . Then β p a ´ L, B q Ď β p a ´ L, aM q “ β p L, M q Ď N, so β ˚ p a ´ L q Ď L p B, M q . This shows that β ˚ is continuous for the topology ofbounded convergence (ie, the strong topology [Sch02, p. 30]). (cid:3) The following is a version of [Eme17, proposition 1.2.28] in which the space isnot necessarily compact, but the vector space is finite dimensional.
Lemma B.1.2. If V is a finite dimensional vector space, then the natural map C an p H q b V Ñ C an p H, V q given by f b v ÞÑ r h ÞÑ f p h q v s is a topological isomor-phism.Proof. The bilinear map p f, v q ÞÑ r h ÞÑ f p h q v s is separately continuous, so theinduced map C an p H q b V Ñ C an p H, V q is continuous. If v , . . . , v n is a basis for V , and if f P C an p H, V q , then we can define functions f i P C an p H q by f p h q “ ÿ i f i p h q v i . The map C an p H, V q Ñ C an p H q b V given by f ÞÑ ř f i b v i defines a continuousinverse to the map in the statement. (cid:3) Definition B.1.3.
Any function f P C an p H q defines a continuous multiplica-tion map C an p H q Ñ C an p H q . It thus induces a continuous map on dual spaces D p H q Ñ D p H q , which we denote by µ ÞÑ µ f . Explicitly, for g P C an p H q , we have µ f p g q “ µ p f g q . Note that the map p f, µ q ÞÑ µ f is bilinear and separately continuous (see below).Thus the induced map C an p H q b i D p H q Ñ D p H q is also continuous. Proof of separate continuity.
We have already noted that p f, µ q ÞÑ µ f is continu-ous for fixed f . For fixed µ , let M be an open lattice in E , and let B is a boundedsubset of C an p H q , so that L p B, M q “ t µ P D p H q : µ p B q Ď M u is an open lattice in D p H q . We want to construct an open lattice L in C an p H q such that µ f P L p B, M q for all f P L .Since the multiplication map C an p H q b π C an p H q Ñ C an p H q and µ : C an p H q Ñ E are both continuous, there exist lattices L and L in C an p H q such that ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 41 µ p L L q Ď M . Since B is bounded, there exists an a P E such that B Ď aL .Let L “ a ´ L and note that this an open lattice in C an p H q . Then any f P L can be written as a ´ f for f P L . Similarly, since B Ď aL , any g P B can bewritten as ag where g P L . Then µ f p g q “ µ p f g q “ µ p a ´ f ag q “ µ p f f q P µ p L L q Ď M which shows that µ f P L p B, M q for all f P L . (cid:3) B.2.
The adjoint-twisted braiding.Definition B.2.1.
Let ζ denote the composite p D p H q b g q b i C an p H, End p g qq “ p D p H q b g q b i p C an p H q b End p g qq g b D p H q where the first map is the tensor product of D p H q b g with the topologicalisomorphism from lemma B.1.2 (applied with V “ End p g q ), and the second isgiven by p µ b x q b p f b σ q ÞÑ σ p x q b µ f . The evaluation map g b End p g q Ñ g is automatically continuous (as it is amap between finite dimensional vector spaces), and we have already noted that C an p H q b i D p H q Ñ D p H q is also continuous in definition B.1.3. It follows that ζ is continuous. Definition B.2.2.
Let Ad P C an p H, End p g qq denote the adjoint action of H on g . We define D p H q b g g b D p H q ζ Ad to be the map given by µ b x ÞÑ ζ p µ b x b Ad q . Continuity of ζ implies that ζ Ad is continuous. We call this map the adjoint-twisted braiding of D p H q and g . Lemma B.2.3.
Following [OS15, proof of lemma 3.5], suppose x , . . . , x n is abasis for g , and fix x P g . Let c i P C an p H q be defined by Ad p h qp x q “ ř i c i p h q x i and let µ i “ µ c i . Then ζ Ad p µ b x q “ ÿ i x i b µ i . Proof.
This is a straightforward calculation. Let σ i,j P End p g q be the map thatsends x j to x i and annihilates all x k for k ‰ j . Under the isomorphism oflemma B.1.2, Ad P C an p H, End p g qq corresponds to ř i,j c i,j b σ i,j where c i,j P C an p H q is defined by Ad p h q “ ÿ i,j c i,j p h q σ i,j . Then ζ Ad p µ b x q “ ÿ i,j σ i,j p x q b µ c i,j . Suppose x “ ř j b j x j for b j P E . Then σ i,j p x q “ b j x i , so ζ Ad p µ b x q “ ÿ i,j b j x i b µ c i,j “ ÿ i x i b ˜ÿ j b j µ c i,j ¸ . Now note that ÿ i c i p h q x i “ Ad p h qp x q “ ÿ i ˜ÿ j c i,j p h q b j ¸ x i so c i “ ř b j c i,j . Thus ´ÿ b j µ c i,j ¯ p g q “ ÿ j µ p b j c i,j g q “ µ p c i g q “ µ c i p g q “ µ i p g q , so ζ Ad p µ b x q “ ř i x i b µ i . (cid:3) Corollary B.2.4.
Suppose x P g . If h P H and δ h P D p H q is the correspondingdelta distribution, then ζ Ad p δ h b x q “ Ad p h qp x q b δ h . Proof.
As in lemma B.2.3, let x , . . . , x n be a basis for g and let c i P C an p H q bedefined by Ad p h qp x q “ ř i c i p h q x i . For a distribution µ P D p H q , set µ i “ µ c i .Then Ad p h qp x q b δ h “ ÿ i x i b c i p h q δ h , but p c i p h q δ h qp g q “ c i p h q g p h q “ δ h,i p g q and then lemma B.2.3 allows us to conclude thatAd p h qp x q “ ÿ x i b δ h,i “ ζ Ad p δ h b x q . (cid:3) Corollary B.2.5.
The following diagram of continuous maps commutes. D p H q b g g b D p H q D p G q ζ Ad Here, both maps into D p G q are restrictions of the multiplication map on D p G q .Proof. The commutativity follows from lemma B.2.3 and [OS15, proof of lemma3.5]. The fact that the maps into D p G q are continuous is a consequence of the factthat the multiplication map on D p G q is separately continuous [ST02b, proposition2.3]. (cid:3) ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 43
B.3.
Universal property of D p g , H q .Definition B.3.1. Let C denote the category of triples p A, α, β q where A is a E -algebra, α : D p H q Ñ A is a homomorphism of E -algebras, and β : g Ñ A is ahomomorphism of Lie algebras, such that α | h “ β | h and such that the followingdiagram commutes, where ζ Ad is the adjoint-twisted braiding. D p H q b g g b D p H q A ζ Ad α b β β b α Morphisms p A, α, β q Ñ p A , α , β q in C are E -algebra homomorphisms A Ñ A commuting with the maps from D p H q and g . D p H q A g AA A αα ββ Remark B.3.2.
Suppose p A, α, β q P C . If A is the E -subalgebra of A generatedby the images of α and β , then evidently p A , α, β q is again an object of C . Theorem B.3.3. D p g , H q is an initial object of C .Proof. It follows from corollary B.2.5 that D p G q is naturally in C , and thenremark B.3.2 implies that D p g , H q is also naturally in C . Let p R, α, β q be aninitial object of C (which exists, for instance by [Ber15, theorem 9.4.14]) andconsider the induced map σ : R Ñ D p g , H q in C . We will show that σ is anisomorphism.Since β is a homomorphism of Lie algebras and since α | h “ β | h , the map β b α induces a well-defined U p g q - D p H q -bimodule homomorphism γ : U p g qb U p h q Ñ R such that the following diagram commutes. U p g q b U p h q D p H q RD p g , H q γ „ σ We know that U p g q b U p h q D p H q Ñ D p g , H q is an isomorphism [SS16, section 4],so σ is surjective. To show that σ is injective, it is sufficient to show that γ issurjective.Observe that, since R is initial in C , it must be generated as an algebra bythe images of α and β (cf. remark B.3.2). Using the fact that α is an algebrahomomorphism, this implies that R is generated as a right D p H q -module by expressions of the form(B.3.4) α p µ q β p x q α p µ q β p x q ¨ ¨ ¨ α p µ n q β p x n q where n ě µ , . . . , µ n P D p H q , and x , . . . , x n P g . Since γ is right D p H q -linear,it is sufficient to show that every element of the form (B.3.4) is in the image of γ . To show this, we induct on n and use the fact that the following diagramcommutes. D p H q b g g b D p H q R ζ Ad α b β β b α This tells us that α p µ q β p x q “ ÿ i β p x ,i q α p µ ,i q for some x ,i P g and µ ,i P D p H q , so(B.3.4) “ ÿ i β p x ,i q α p µ ,i µ q β p x q ¨ ¨ ¨ α p µ n q β p x n q , where we have again used the fact that α is a homomorphism of E -algebras. Byinduction, we know that α p µ ,i µ q β p x q ¨ ¨ ¨ α p µ n q β p x n q is in the image of γ . Since γ is left U p g q -linear, it follows that (B.3.4) is also in the image of γ . (cid:3) Corollary B.3.5.
The data of a D p g , H q -module structure on a vector space M is precisely the data of a D p H q -module structure and a g -module structure suchthat the two h -module structures agree and such that, for any µ P D p H q and x P g and m P M , ζ Ad p µ b x q “ ÿ x i b µ i ùñ µ . p x . m q “ ÿ i x i . p µ i . m q . Moreover, a map between two D p g , H q -modules is D p g , H q -linear if and only if itis both D p H q -linear and g -linear. Remark B.3.6.
The group ring E r H s is dense in D p H q , so E r H s b g is densein D p H q b g . Suppose p A, α, β q is a triple consisting of a separately continuous E -algebra, a continuous E -algebra homomorphism D p H q Ñ A , and a continuousLie algebra homomorphism g Ñ A such that α | h “ β | h . Since multiplication on A is separately continuous, the two maps D p H q b g Ñ A and g b D p H q Ñ A arecontinuous. Thus p β b α q ˝ ζ Ad “ α b β if and only if this is true when restrictedto E r H s b g . E r H s b g D p H q b g g b D p H q A ζ Ad α b β β b α ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 45
In light of corollary B.2.4, this means that p A, α, β q P C if and only if α p δ h q β p x q “ β p Ad p h qp x qq α p δ h q for all h P H and x P g . Lemma B.3.7.
Suppose M is simultaneously a separately continuous g -moduleand a separately continuous D p H q -module such that the two actions of h agree.Then M is a D p g , H q -module if and only if δ h . p x . m q “ Ad p h qp x q . p δ h . m q for all h P H , x P g , and m P M .Proof. Giving End cts p M q the topology of pointwise convergence, we have a contin-uous map D p H q Ñ End cts p M q by lemma B.1.1. Thus we can apply remark B.3.6with A “ End cts p M q . (cid:3) Lemma B.3.8.
Let D p H q denote the quotient of D p H q by the two-sided idealgenerated by h . There exists a natural surjective homomorphism D p g , H q Ñ D p H q of E -algebras whose kernel is the two-sided ideal of D p g , H q generated by g .Proof. Observe that D p H q becomes an object of C when equipped with thequotient map D p H q Ñ D p H q and the zero map g Ñ D p H q . Thus thereexists a natural E -algebra homomorphism σ : D p g , H q Ñ D p H q , which isclearly surjective. It is also clear that its kernel contains g . Letting J p g q de-note the two-sided ideal generated by g , we see that there is an induced map¯ σ : D p g , H q{ J p g q Ñ D p H q . Showing that ker p σ q “ J p g q is equivalent to show-ing that ¯ σ is injective.We have a map τ : D p H q Ñ D p g , H q{ J p g q which annihilates h , so it induces amap ¯ τ : D p H q Ñ D p g , H q{ J p g q . It is clear that ¯ σ ˝ ¯ τ is the identity on D p H q .Thus ¯ σ is injective if and only if ¯ τ is surjective. And ¯ τ is surjective if and only if τ is surjective, so we show that τ is surjective.As we noted in the proof of theorem B.3.3, every element of D p g , H q can bewritten as a finite sum of expressions of the form µ x µ x ¨ ¨ ¨ µ n x n µ n ` for n ě µ , . . . , µ n ` P D p H q , and x , . . . , x n P g . But all such expressionsare in J p g q if n ě
1, and the above expression is just an element of D p H q when n “
0. Thus any element of D p g , H q is congruent to an element of D p H q modulo J p g q . This shows that τ is surjective. (cid:3) B.4.
Tensor-hom adjunction for D p g , H q -modules. The following is a locallyanalytic analog of a definition made in [Jan03, part II, section 1.20].
Definition B.4.1. A locally analytic p g , H q -module M is a locally analytic rep-resentation of H which is also simultaneously a separately continuous g -module,such that these two actions satisfy two compatibility conditions: (C1) The two induced actions of h agree, and(C2) δ h . p x . m q “ Ad p h qp x q . p h . m q for all h P H, x P g and m P M . Lemma B.4.2.
A locally analytic p g , H q module is naturally a D p g , H q -module.Proof. Any locally analytic representation of H can be given the structure ofa separately continuous D p H q -module [ST02b, proposition 3.2], so this followsfrom lemma B.3.7 (cid:3) Lemma B.4.3.
Suppose M is a locally analytic p g , H q -module which is locallyfinite dimensional as a representation of H . If X is a D p g , H q -module, then M b X is also a D p g , H q -module. This construction is functorial in both M and X .Proof. Suppose that M is finite dimensional. Then M b X is naturally a g -module, and it is also a D p H q -module by lemma A.2.3. Corollary A.2.6 impliesthat the two h -module structures coincide. We want to show that the followingcommutes. D p H q b g g b D p H q End p M b X q ζ Ad For brevity, let us define L “ ` p D p H q b g q ˆ b i D p H q ˘ ‘ ` D p H q ˆ b i p D p H q b g q ˘ R “ ` g b p D p H qq ˆ b i D p H q ˘ ‘ ` D p H q ˆ b i p g b D p H qq ˘ and observe that there is a map ζ : L Ñ R given by ζ “ p ζ Ad b D p H qq ‘ p D p H q ‘ ζ Ad q . Note that we can decompose the above diagram as follows. D p H q b g g b D p H q D p H ˆ H q b p g ‘ g q p g ‘ g q b D p H ˆ H q ` D p H q ˆ b i D p H q ˘ b p g ‘ g q p g ‘ g q b ` D p H q ˆ b i D p H q ˘ L R
End p M b X q ζ Ad ζ Ad ζ ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 47
It is straightforward to verify that the triangle at the bottom commutes because X and M are both individually D p g , H q -modules.The square at the very top commutes since ζ Ad is functorial in the pair p G, H q and we can apply this functoriality to the diagonal map p G, H q Ñ p G ˆ G, H ˆ H q ,but we won’t need to use this commutativity. We only note that the maps g Ñ p g ‘ g q involved in this square are the diagonal map x ÞÑ p x, x q . We willshow that the following square commutes. D p H q b g g b D p H q L R ζ Ad ζ All of these maps are continuous, so we can restrict to check commutativity on E r H s b g . If h P H and δ h is the corresponding delta distribution and x P g , notethat the map into L carries δ h b x to δ h b x b δ h ` δ h b δ h b x, and similarly with the map into R . Using the formula for ζ Ad from corollary B.2.4,commutativity follows. (cid:3) Corollary B.4.4.
Suppose M is a locally analytic p g , H q -module which is locallyfinite dimensional as a representation of H . Then Hom p M, X q is naturally a D p g , H q -module for any D p g , H q -module X .Proof. We proceed just as in remark A.2.9. (cid:3)
Theorem B.4.5.
Suppose M is a locally analytic p g , H q -module which is locallyfinite dimensional as a representation of H . Then there is an adjunction of D p g , H q -module-valued functors p M b ´q % Hom p M, ´q .Proof. It is sufficient to show that the unit and counit of the adjunction are D p g , H q -linear. But we know that p M b ´q % Hom p M, ´q as g -module-valuedfunctors and as D p H q -module functors (by theorem A.3.1), so the counit andunit are both g and D p H q -linear. This implies D p g , H q -linearity. (cid:3) B.5.
The closure D p G q H of D p g , H q . Let us now consider Kohlhaase’s ring D p G q H of distributions supported in H [Koh07, 1.2.1–6]. This is precisely thetopological closure of D p g , H q inside D p G q [Koh07, 1.2.10]. In this section, weprove that, if G is compact, then D p G q H is a Fr´echet-Stein algebra. Let D r p G q H denote the closure of D p G q H inside D r p G q [Koh07, p. 30]. In other words, D r p G q H is the completion of D p G q H for the topology defined by the r -norm. Lemma B.5.1. If G is compact, then D r p G q H is left noetherian.Proof. We know that U r p g q is noetherian, and D r p G q H is finite free as a moduleover U r p g q [Koh07, theorem 1.4.2], so D r p G q H is also noetherian. (cid:3) Lemma B.5.2. If G is compact, then D r p G q is finite free as a right module over D r p G q H .Proof. Choose topological generators a , . . . , a d of G such that a k ` , . . . , a d aretopological generators for H . Set b i “ a i ´
1. Then there exists an ℓ i such that D r p G q is finite free as a right module over U r p g q with basis given by monomials b α “ b α ¨ ¨ ¨ b α d d where α i ď ℓ i for all i [Koh07, theorem 1.4.2]. Furthermore, thesubring D r p G q H is also finite free as a right module over U r p g q , with basis givenby monomials of the form b α k ` k ` ¨ ¨ ¨ b α d d where α i ď ℓ i for all i [Koh07, corollary1.4.3].This implies that D r p G q is free over D r p G q H with basis given by monomialsof the form b α ¨ ¨ ¨ b α k k where α i ď ℓ i for all i . To see this, first observe that anyelement of D r p G q can be written in the form ř α b α c α where c α P U r p g q . But ÿ α b α c α “ ÿ α b α ¨ ¨ ¨ b α d d c α “ ÿ α ,...,α k b α ¨ ¨ ¨ b α k k ˜ ÿ α k ` ,...,α d b α k ` k ` ¨ ¨ ¨ b α d d c α ¸ and ÿ α k ` ,...,α d b α k ` k ` ¨ ¨ ¨ b α d d c α is an element of D r p G q H , which shows that D r p G q is generated as a right moduleover D r p G q H by monomials of the form b α ¨ ¨ ¨ b α k k .Moreover, these monomials are linearly independent over D r p G q H . Indeed,suppose we have a dependence relation0 “ ÿ α ,...,α k b α ¨ ¨ ¨ b α k k s α ,...,α k where s α ,...,α k P D r p G q H . Each s α ,...,α k can be written as s α ,...,α k “ ÿ α k ` ,...,α d b α k ` k ` ¨ ¨ ¨ b α d d c α where c α P U r p g q . Thus we can rewrite the dependence relation as0 “ ÿ α ,...,α k b α ¨ ¨ ¨ b α k k s α ,...,α k “ ÿ α ,...,α k b α ¨ ¨ ¨ b α k k ˜ ÿ α k ` ,...,α d b α k ` k ` ¨ ¨ ¨ b α d d c α ¸ “ ÿ α b α ¨ ¨ ¨ b α d d c α “ ÿ α b α c α . Since the monomials b α are linearly independent over U r p g q , this means that c α “ α , so s α ,...,α k “ α , . . . , α k as well. (cid:3) ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 49
Proposition B.5.3. If G is compact, then D p G q H is a Fr´echet-Stein algebra.Proof. Certainly D p G q H , as a closed subalgebra of D p G q , is Fr´echet, and wealready know that D r p G q H is noetherian by lemma B.5.1, so we only need tocheck that the transition maps D r p G q P Ñ D r p G q H are flat. Observe that wehave a commutative square as follows. D r p G q H D r p G q H D r p G q D r p G q Since D p G q is Fr´echet-Stein, we know that the horizontal map D r p G q Ñ D r p G q is flat. We also saw above in lemma B.5.2 that both vertical maps are faithfullyflat. This means that D r p G q is flat as a D r p G q H -module and faithfully flat asa D r p G q H -module, so it follows that D r p G q P Ñ D r p G q H is flat [Sta20, tag039V]. (cid:3) B.6. c-Flatness.
Recall that, if A Ñ B is a continuous homomorphism ofFr´echet-Stein algebras, there is a right-exact functor B > b A ´ from coadmissibleleft A -modules to coadmissible left B -modules [AW19, section 7]. One says that B is right c-flat if this functor is exact, and right faithfully c-flat if this functor isfaithfully exact. If M is a finitely presented A -module, then B > b A M “ B b A M . Lemma B.6.1.
Suppose A Ñ B is a continuous homomorphism of Fr´echet-Steinalgebras, and let p q n q n and p p n q n be sequences of norms defining the topologies on A and B , respectively, such that p A, q n q Ñ p B, p n q is continuous for all n (cf.[ST03, proof of lemma 3.8]).(a) If A q n Ñ B p n is right flat for all n , then A Ñ B is right c-flat.(b) If A q n Ñ B p n is right faithfully flat for all n , then A Ñ B is right faithfullyc-flat.Proof. Suppose M Ñ N is an injective map of coadmissible left A -modules. Since A q n is right flat over A [ST03, remark 3.2], and B p n is right flat over A q n ,we seethat B p n b A M Ñ B p n b A N is also injective. Since limits are left exact, weconclude that B > b A M “ lim B p n b A M lim B p n b A N “ B > b A N is injective, proving that B is right c-flat.Now suppose A q n Ñ B p n is right faithfully flat for all n and that M is acoadmissible left A -module such that B > b A M “
0. Since the image of B > b A M isdense in B p n b A M [ST03, theorem A and corollary 3.1], this means that B p n b A qn p A q n b A M q “ B p n b A M “ for all n . Since B p n is faithfully flat over A q n , we see that A q n b A M “ n , so M “ lim A q n b A M “ (cid:3) Lemma B.6.2.
Suppose G is compact. Then D p G q is faithfully c-flat as a rightmodule over D p G q H .Proof. This follows from lemmas B.5.2 and B.6.1 and proposition B.5.3. (cid:3)
Remark B.6.3. If D p G q H is a left coherent ring, it would also be true that D p G q is right flat over D p G q H . Indeed, then any finitely generated left ideal I in D p G q H would also be finitely presented, so D p G q b D p G q H I “ D p G q > b D p G q H I. Furthermore, since finite presentation implies coadmissibility [ST03, corollary3.4(v)], we would also know that D p G q b D p G q H I “ D p G q > b D p G q H I Ñ D p G q isinjective by lemma B.6.2. This would prove that D p G q is right flat over D p G q H [Lam99, 4.12]. However, we do not know if D p G q H is a coherent ring.B.7. Coadmissible modules over D p G q H .Lemma B.7.1 ([Koh07, equation (1.7)]) . Suppose G is a compact open subgroupof G and H “ H X G . Then D p G q H “ D p G q H b D p H z H q as left D p G q H -modules. Corollary B.7.2.
Suppose G Ď G are two compact open subgroups of G , andlet H i “ H X G i . Then D p G q H is finite free as a left module over D p G q H .Proof. Lemma B.7.1 tells us that D p G q H “ D p G q H b D p H z H q . Since H is compact and H is open, we see that H z H is finite, so D p H z H q is finitedimensional. (cid:3) Definition B.7.3. A D p G q H -module M is coadmissible if there exists a compactopen subgroup G of G such that M is coadmissible as a module over the Fr´echet-Stein algebra D p G q H , where H “ H X G . By corollary B.7.2 and [ST03, lemma3.8], it follows that the same is then true for any other compact open subgroup.B.8. Pre-coadmissibile modules over D p g , H q .Lemma B.8.1. If G is a compact open subgroup of G and H “ H X G , then D p G q H b D p g ,H q D p g , H q “ D p G q H as D p G q H - D p g , H q -bimodules. ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 51
Proof.
Observe that D p g , H q “ U p g q b U p h q D p H q“ U p g q b U p h q p D p H q b D p H z H qq“ ` U p g q b U p h q D p H q ˘ b D p H z H q“ D p g , H q b D p H z H q . so D p G q H b D p g ,H q D p g , H q “ D p G q H b D p H z H q “ D p G q H , using lemma B.7.1 for the last step. (cid:3) Corollary B.8.2.
Suppose M is a D p g , H q -module. For any compact open sub-group G of G , D p G q H b D p g ,H q M “ D p G q H b D p g ,H q M as D p G q H -modules. (cid:3) Definition B.8.3. A D p g , H q -module M is pre-coadmissible if D p G q H b D p g ,H q M is coadmissible.Suppose M is a D p g , H q -module. If G is a compact open subgroup of G , H “ H X G , and M is finitely presented over D p g , H q , then M is pre-coadmissible.Also, if M is finitely generated over U p g q , it is pre-coadmissible (by the sameproof as in [SS16, lemma 4.3], mutatis mutandis). Lemma B.8.4 ([SS16, lemma 4.6]) . For a D p g , H q -module M , the natural map U r p g q b U p g q M D r p G q H b D p g ,H q M is an isomorphism of left U r p g q -modules. Proposition B.8.5. If (B.8.6) 0 M M M is an exact sequence of pre-coadmissible D p g , H q -modules, then D p G q H b D p g ,H q (B.8.6) is an exact sequence of coadmissible D p G q H -modules.Proof. Using corollary B.8.2, we can assume that G is compact. Then D p G q H b D p g,H q (B.8.6)is a sequence of coadmissible modules over the Fr´echet-Stein algebra D p G q H , soit is exact if and only if D r p G q H b D p G q H ` D p G q H b D p g ,H q (B.8.6) ˘ “ D r p G q H b D p g ,H q (B.8.6)is exact for a cofinal set of values of r . But D r p G q H b D p g ,H q (B.8.6) “ U r p g q b U p g q (B.8.6) by lemma B.8.4. Furthermore, we know that U r p g q is flat over U p g q , since ˆ U p g q is flat over U p g q [SS16, theorem 3.13] and U r p g q is flat over ˆ U p g q [ST03, remark3.2]. Thus U r p g q b U p g q (B.8.6) is in fact exact for all r . (cid:3) Corollary B.8.7.
Suppose G is compact. If (B.8.6) is an exact sequence ofpre-coadmissible D p g , H q -modules. Then D p G q > b D p G q H ` D p G q H b D p g ,H q (B.8.6) ˘ is an exact sequence of coadmissible D p G q -modules. In particular, if (B.8.6) isan exact sequence of finitely presented D p g , H q -modules, then D p G q b D p g ,H q (B.8.6) is an exact sequence of coadmissible D p G q -modules.Proof. The first part follows from lemma B.6.2 and proposition B.8.5. For the sec-ond part, suppose M is a finitely presented D p g , H q -module. Then is D p G q H b D p g ,H q M is also finitely presented, and D p G q > b D p G q H ´ agrees with D p G q b D p G q H ´ forfinitely presented D p G q H -modules. (cid:3) References [AW19] K. Ardakov and S. Wadsley. u D -modules on rigid analytic spaces I. J.Reine u. Angew. Math. , 747:221–276, 2019.[Ber15] George M. Bergman.
An invitation to general algebra and universalconstructions . Universitext. Springer, Cham, second edition, 2015. doi:10.1007/978-3-319-11478-1 .[BGG76] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand. A certain category of g -modules. Funkcional. Anal. i Priloˇzen. , 10(2):1–8, 1976.[Bou72] N. Bourbaki. ´El´ements de math´ematique. Fasc. XXXVII. Groupes etalg`ebres de Lie. Chap. II/III . Hermann, Paris, 1972. Act.Sci. et Ind.,No. 1349.[Boy99] P. Boyer. Mauvaise r´eduction des vari´et´es de Drinfeld et correspon-dance de Langlands locale.
Invent. Math. , 138(3):573–629, 1999. doi:10.1007/s002220050354 .[Bre04] Christophe Breuil. Invariant L et s´erie sp´eciale p -adique. Ann. Sci. ´Ecole Norm. Sup. (4) , 37(4):559–610, 2004.URL: http://dx.doi.org/10.1016/j.ansens.2004.02.001 , doi:10.1016/j.ansens.2004.02.001 .[BT65] A. Borel and J. Tits. Groupes r´eductifs. Inst. Hautes ´Etudes Sci. Publ.Math. , (27):55–150, 1965.[CM15] Kevin Coulembier and Volodymyr Mazorchuk. Some homologi-cal properties of category O . III. Adv. Math. , 283:204–231, 2015. doi:10.1016/j.aim.2015.06.019 . ROM CATEGORY O TO LOCALLY ANALYTIC REPRESENTATIONS 53 [Eme17] Matthew Emerton. Locally analytic vectors in representations of locally p -adic analytic groups. Mem. Amer. Math. Soc. , 248(1175):iv+158,2017. doi:10.1090/memo/1175 .[Hoc81] Gerhard P. Hochschild.
Basic theory of algebraic groups and Lie alge-bras , volume 75 of
Graduate Texts in Mathematics . Springer-Verlag,New York-Berlin, 1981.[Hum75] James E. Humphreys.
Linear algebraic groups . Springer-Verlag, NewYork-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21.[Hum80] James E. Humphreys.
Introduction to Lie algebras and representationtheory , volume 9 of
Graduate Texts in Mathematics . Springer-Verlag,New York-Berlin, 1980. Third printing, revised.[Jac79] Nathan Jacobson.
Lie algebras . Dover Publications, Inc., New York,1979. Republication of the 1962 original.[Jan03] Jens Carsten Jantzen.
Representations of algebraic groups , volume 107of
Mathematical Surveys and Monographs . American Mathematical So-ciety, Providence, RI, second edition, 2003.[Koh07] J. Kohlhaase. Invariant distributions on p -adic analytic groups. DukeMath. J. , 137(1):19–62, 2007. doi:10.1215/S0012-7094-07-13712-8 .[Koh11] J. Kohlhaase. The cohomology of locally analytic representations.
J.reine angew. Math. (Crelle) , 651:187–240, 2011.[Lam99] T. Y. Lam.
Lectures on modules and rings , volume 189 of
Grad-uate Texts in Mathematics . Springer-Verlag, New York, 1999. doi:10.1007/978-1-4612-0525-8 .[Lan96] Erasmus Landvogt.
A compactification of the Bruhat-Tits building , vol-ume 1619 of
Lecture Notes in Mathematics . Springer-Verlag, Berlin,1996. doi:10.1007/BFb0094594 .[Mil17] J. S. Milne.
Algebraic groups , volume 170 of
Cambridge Studiesin Advanced Mathematics . Cambridge University Press, Cambridge,2017. The theory of group schemes of finite type over a field. doi:10.1017/9781316711736 .[OS15] S. Orlik and M. Strauch. On Jordan-H¨older series of some locally ana-lytic representations.
Journal of the AMS , 28(1):99–157, 2015.[Sch02] P. Schneider.
Nonarchimedean functional analysis . Springer Mono-graphs in Mathematics. Springer-Verlag, Berlin, 2002.[Sch11] Benjamin Schraen. Repr´esentations localement analytiques deGL p Q p q . Ann. Sci. ´Ec. Norm. Sup´er. (4) , 44(1):43–145, 2011. doi:10.24033/asens.2140 .[Soe85] Wolfgang Soergel. ¨Uber den “Erweiterungs-Abschluß” der KategorieO in der Kategorie aller Moduln ¨uber einer halb-einfachen Liealgebra,1985. Diplomarbeit im Fach Mathematik an der Rheinische Friedrich-Wilhelms-Universit¨at Bonn. [Spr98] T. A. Springer.
Linear algebraic groups , volume 9 of
Progress in Math-ematics . Birkh¨auser Boston, Inc., Boston, MA, second edition, 1998. doi:10.1007/978-0-8176-4840-4 .[SS16] T. Schmidt and M. Strauch. Dimensions of certain locally analyticrepresentations.
Representation Theory , 20:14–38, 2016.[ST01a] P. Schneider and J. Teitelbaum. p -adic Fourier theory. Doc. Math. ,6:447–481 (electronic), 2001.[ST01b] P. Schneider and J. Teitelbaum. U p g q -finite locally ana-lytic representations. Represent. Theory , 5:111–128 (elec-tronic), 2001. With an appendix by D. Prasad. URL: http://dx.doi.org/10.1090/S1088-4165-01-00109-1 , doi:10.1090/S1088-4165-01-00109-1 .[ST02a] P. Schneider and J. Teitelbaum. Banach space representations andIwasawa theory. Israel J. Math. , 127:359–380, 2002.[ST02b] P. Schneider and J. Teitelbaum. Locally analytic distributions and p -adic representation theory, with applications to GL . J. Amer. Math.Soc. , 15(2):443–468 (electronic), 2002.[ST03] P. Schneider and J. Teitelbaum. Algebras of p -adic distributions andadmissible representations. Invent. Math. , 153(1):145–196, 2003.[ST05] P. Schneider and J. Teitelbaum. Duality for admissible locally analyticrepresentations.
Represent. Theory , 9:297–326 (electronic), 2005.[Sta20] The Stacks Project Authors. Stacks Project. http://stacks.math.columbia.edu , 2020.
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