Schneider-Teitelbaum duality for locally profinite groups

We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider–Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.


Introduction
Let k denote a non-Archimedean local field, and O k ⊂ k the valuation ring of k. The paper is devoted to two topics. One topic is to give monoidal structures on several categories of linear topological O k -modules. We are interested mainly in the closed symmetric monoidal category C cg of CG linear topological O k -modules. A CG linear topological O k -module is a linear topological O k -module given as the colimit of totally bounded O ksubmodules. By the definition, it is a module theoretic analogue of a compactly generated topological space. We show that every Banach k-vector space and every compact linear topological O k -module are CG. Therefore C cg contains both of the categories of Banach k-vector spaces and compact Hausdorff flat linear topological O k -modules, which play the roles of the foundation in Schneider-Teitelbaum duality (cf. [13] The other topic is to define a generalised Iwasawa algebra O k [[G]] associated to a locally profinite group G, and to extend Schneider-Teitelbaum duality, which is applicable to a profinite group, to duality applicable to G by using module theory over O k [[G]]. We note that O k [[G]] is defined as a monoid in C cg , and does not necessarily form a topological O k -algebra. This is one of the main reasons why we need monoidal structures. As the classical Iwasawa algebra associated to a profinite group is naturally identified with the O k -algebra of O k -valued measures, O k [[G]] is naturally identified with the O k -algebra of O k -valued measures on G satisfying a certain property called the normality. As the original Schneider-Teitelbaum duality is given by a module theoretic interpretation of a Banach k-linear representations through the integration of the action along measures (cf. [13] Corollary 2.2), the generalised Schneider-Teitelbaum duality is give by a module theoretic interpretation through the integration of the action of G by normal measures.
As applications, we establish a criterion of the irreducibility of a unitary Banach k-linear representation of G, and give a description of the continuous induction of a unitary Banach k-linear representation of a closed subgroup P ⊂ G such that the homogeneous space P \G is compact. In particular, we give an explicit description of the continuous parabolic induction for the case G is an algebraic group over a local field so that the representation space of the continuous parabolic induction is independent of the choice of the action of P .
We explain the contents of this paper. In §2.1, we study several categories of linear topological O k -modules. In §2.2, we introduce a notion of the normality of an O k -valued measure on a topological space. In §3.1, we define monoidal structures on several categories of linear topological O kmodules. In §3.2, we define a notion of a CGLT O k -algebra as a monoid in C cg , which is a counterpart of a topological O k -algebra, and define O k [[G]] as a CGLT O k -algebra. In §3.3, we define a notion of a CGLT module over a CGLT O k -algebra, which is a counterpart of a topological left module over a topological O k -algebra. In §4.1, we recall a unitary Banach k-linear representation of G and interpret it in terms of a CGLT O k [[G]]-module. In §4.2, we interpret a continuous action of G on a compact Hausdorff flat linear topological O k -module in terms of a CGLT O k [[G]]-module. In §4.3, we define a notion of the dual of a unitary Banach k-linear representation of G, and extend Schneider-Teitelbaum duality to duality applicable to G. In §5.1, we study the dual of several operations on Banach k-linear representations such as the continuous induction. In §5.2, we give an explicit description of the continuous parabolic induction in the case where G is an algebraic group.

Preliminaries
Let k denote a local field, that is, a complete discrete valuation field with finite residue field, O k ⊂ k the valuation ring of k, and G a locally profinite group. We denote by ω the set of natural numbers. For a set X, we denote by P <ω (X) the set of finite subsets of X. Since we deal with many pairs, we abbreviate (• i ) 1 i=0 to (• i ), 1 i=0 • i to • i , and 1 i=0 • i to • i . Let Θ be a category. We say that Θ is ω-cocomplete (respectively, cocomplete, complete) if it admits all small filtered colimits (respectively, colimits, limits), and is bicomplete if it is cocomplete and complete. Let F be a functor. We say that F is ω-cocontinuous (respectively, cocontinuous, continuous) if it commutes with all small filtered colimits (respectively, colimits, limits), and is bicontinuous if it is cocontinuous and continuous. We denote by Set the bicomplete category of sets and maps, and by Top the bicomplete category of topological spaces and continuous maps. We abbreviate Hom Top to C.

Linear topological modules
Let M be a topological O k -module, and C ⊂ M a subset. We say that C is pre-compact (respectively, complete) if C is totally bounded (respectively, complete) with respect to the restriction of the uniform structure on M associated to the structure as a topological Abelian group to C. By the definition of the uniformity on M , C is totally bounded if and only if for any open neighbourhood U ⊂ M of 0 ∈ M , there exists a finite subset C 0 ⊂ C such that C ⊂ {m 0 + m 1 | (m 0 , m 1 ) ∈ U × C 0 }. The following are well-known facts (cf. [3] 8.3.2 Theorem, [4], and [3] 8.3.16 Theorem, respectively) on the pre-compactness: It implies m∈C O k m ∈ K (M ).
We denote by C the category of O k -modules and O k -linear homomorphisms. We denote by U : C → Top and F : C → C the forgetful functors.
Proof. The completeness of C and the continuity of U and F follow from the definition of the limits in Top and C . The ω-cocomleteness of C and the ω-cocontinuity of U and F follow from [6] Proposition 1.3. For any small family (M s ) s∈S in C , s∈S F (M s ) forms a linear topological O kmodule with respect to the topology generated by {m + s∈S F (L s ) | (m, (L s ) s∈S ) ∈ ( s∈S F (M s ))× s∈S O(M s )}, and satisfies the universality of the direct sum of (M s ) s∈S in C . Thus C is cocomplete, and F is cocontinuous.
Since we will introduce several full subcategories of C , we prepare a convention for colimits (respectively, limits) in order to avoid the ambiguity of categories in which we consider the universality. Let (M s ) s∈S be a small diagram in a full subcategory Θ ⊂ C . We always denote by lim − →s∈S M s (respectively, lim ← −s∈S M s ) the colimit (respectively, limit) of (M s ) s∈S in C but not in Θ. As an immediate consequence of Proposition 2.6, we obtain the following: We denote by C c ⊂ C the full subcategory of pre-compact linear topological O k -modules and by I c the inclusion C c → C . We put U c := U •I c and F c := F • I c . Proof. The functoriality of (•) and the assertion (ii) immediately follow from the definition. The assertion (iii) immediately follows from the assertion (i) and Proposition 2.6. We show the assertion (i). We consider two functors F, G : C op × C c → Set given as F := L (• 0 , • 1 ) and G := which is a natural equivalence by the assertion (ii). For a K ∈ ob(C c ), we consider maps T M,K : T M,K , T M,K give natural transformations T : F ⇒ G and T : G ⇒ F satisfying T • T = id G and T • T = id F by the bijectivity of of values of π c . We obtain adjunction data ((•), I c , T, π c , (π c I c ) −1 ) between C c and C . It implies that (•) is left adjoint to I c .
Suppose that M is linear in the following in this subsection. Then K (M ) forms a small filtered diagram in C by Proposition 2.4. We put M K := lim − →K∈K (M ) K. By the universality of the colimit, the system of inclusions induces a continuous injective O k -linear homomorphism ι cg M : M K → M . By Corollary 2.5, ι cg M is bijective. We show that ι cg M preserves the precompactness of O k -submodules.
The pre-compactness of K is equivalent to that of K .
Proof. The assertion (ii) follows from Proposition 2.3 and the assertion (i). We show the assertion (i). By K ∈ K (M ), we have ι cg We denote by C cg ⊂ C the full subcategory of CG linear topological O k -modules and by I cg the inclusion C cg → C . We put U cg := U • I cg and F cg := F • I cg . We study properties of C cg analogous to those of the category of compactly generated topological spaces. Corollary 2.10. (i) The correspondence M M K gives a functor (•) K : C → C cg right adjoint to I cg such that the counit is given as a natural equivalence.
(ii) The category C cg is bicomplete, and the colimit of a small diagram Proof. To begin with, we show that C cg is closed under small colimits in C . Let (M s ) s∈S be a small diagram in C cg . Put M := lim − →s∈S I cg (M s ). In order to verify that M is pre-compactly generated, it suffices to show ι cg M (L) ∈ O(M ) for any L ∈ O(M K ). Let s ∈ S. We denote by L s the preimage of ι cg M (L) in M s . Let K 0 ∈ K (M s ). We denote by K ⊂ M the image of K 0 . By Proposition 2.3 and Proposition 2.9 (ii), we have (ι cg . By Proposition 2.9 (i), we obtain by Corollary 2.7. We show the assertion (i). Since C cg is closed under small colimits in C , the correspondence M M K gives a functor (•) K : C → C cg by Proposition 2.3 and Proposition 2.9 (i). We consider two functors F, G : (C cg ) op × C → Set given as F := L (I cg (• 0 ), • 1 ) and G := L (• 0 , (• 1 ) K ). The correspondence M ι cg M gives a unit ι cg : I cg • (•) K ⇒ id C , and we also have a counit (ι cg which is a natural equivalence by definition. For an (M i ) ∈ ob(C cg × C ), we consider maps give natural transformations T : F ⇒ G and T : G ⇒ F satisfying T • T = id G and T • T = id F by the bijectivity of values of ι. We obtain adjunction data (I cg , (•) K , T, ι cg , (ι cg I cg ) −1 ) between C and C cg . It implies that (•) K is right adjoint to I cg .
We show the assertion (ii). By the assertion (i), (•) K is continuous and I cg is cocontinuous. Since the counit (ι cg I cg ) −1 is a natural equivalence, C cg is complete by Proposition 2.6. Since we have already verified that C cg is closed under small colimits in C , it implies the assertion (ii) by Proposition 2.6 We have three criteria of CG linear topological O k -modules.
Proof. The assertion (ii) follows from Proposition 2.1 (ii) and Proposition 2.9 (i), because M is locally compact if and only if M admits a compact clopen O k -submodule. We verify the assertion (i). [6] Lemma 2.23. By Corollary 2.7, we obtain an isomorphism lim − →K∈K (M ) (K ∩ M 0 ) → M 0 . By Proposition 2.1 (i), K ∩ M 0 lies in K (M 0 ) for any K ∈ K (M ). It implies that M 0 is CG by Corollary 2.10 (i).
We verify the assertion (iii). Let L ∈ O(M K ). We show ι cg . Take an decreasing sequence (L r ) r∈ω ∈ O(M ) ω such that {L r | r ∈ ω} forms a fundamental system of neighbourhoods of 0 ∈ M . By the assumption, we have L r \ ι cg M (L) = ∅ for any r ∈ ω. Take an (m r ) r∈ω ∈ r∈ω (L r \ ι cg M (L)). Put C := {m r | r ∈ ω}. We have C = r h=0 (m h + L r ) for any r ∈ ω, and hence C is pre-compact. We denote by C ch f ⊂ C the full subcategory of compact Hausdorff flat linear topological O k -modules, by Ban(k) the k-linear category of Banach k-vector spaces and bounded k-linear homomorphisms, by Ban ≤ (k) ⊂ Ban(k) the O k -linear subcategory of submetric k-linear homomorphisms, and by Ban ur ≤ (k) ⊂ Ban ≤ (k) the full subcategory of unramified Banach k-vector spaces. By Proposition 2.1 (ii), C ch f is a full subcategory of C c . For a (V i ) ∈ ob(Ban ur ≤ (k)) 2 , we denote by S ((V i )) the O k -module Hom Ban ur ≤ (k) ((V i )) equipped with the topology of pointwise convergence. For a V ∈ ob(Ban ur ≤ (k)), we put V D d := S (V, k). For a K ∈ ob(C ch f ), we denote by K Dc the k-vector space L (K, k) equipped with the supremum norm. The correspondence V V D d gives a functor D d : Ban ur ≤ (k) op → C ch f , and the correspondence K K Dc gives a functor D c : C ch f → Ban ur ≤ (k) op . Theorem 2.12 (Schikhof duality). The pair (D d , D c ) is an O k -linear equivalence between Ban ur ≤ (k) op and C ch f .

Normal Measures
We study a non-Archimedean analogue of the normality of a measure. For this purpose, we introduce a convention of infinite sums. Let S be a set. For an f ∈ k S , we denote by s∈S f (s) the limit of the net ( s∈S 0 f (s)) S 0 ∈P<ω(S) , where P <ω (S) is directed by inclusions. It is elementary to show the following: Let X be a topological space. We denote by CO(X) the set of clopen subsets of X, and by P(X) the set of subsets P ⊂ CO(X) satisfying X = An O k -valued measure µ on X is said to be normal if U ∈P µ(U ) converges to µ(U ) for any U ∈ CO(X) and P ∈ P(U ).
Let P ∈ P(X). For a subset U ⊂ X, we put P | U := {U ∈ P | U ⊂ U }. We define a partial order P 0 ≤ P 1 on (P i ) ∈ P(X) 2 as P i } ∈ P(X) forms the least upper bound of {P 0 , P 1 } with respect to ≤. In particular, P(X) is directed with respect to ≤. Suppose P 0 ≤ P 1 . Let f ∈ C 0 (P 0 , O k ) and U ∈ P 1 . By P 0 | U ⊂ P 0 and Proposition 2.13, f (U ) : is a converging sum. For any ∈ (0, ∞), there is a P 0 ∈ P <ω (P 0 ) such that |f (U )| < for any U ∈ P 0 \ P 0 , and hence P 1 := {U ∈ P 1 | P 0 ∩ (P 0 | U ) = ∅} is a finite set satisfying | f (U )| < for any U ∈ P 1 \ P 1 . It implies that the map f : We  , µ → (µ(U )) U ∈CO(X) is injective. We identify F (M(X)) with the O k -module of normal O k -valued measures on X through the evaluation map. For a U ∈ CO(X), we denote by 1 U : X → k the characteristic function of U .
Proof. By the compactness of X, every O k -valued measure on X is normal, and hence the map in the assertion gives an O k -linear homomorphism C(X, k) D d → M(X), which is continuous by the finiteness of pairwise disjoint clopen coverings of X. On the other hand, again by the compactness of X, every continuous k-valued function is uniformly approximated by a finite k-linear combination of characteristic functions of clopen subsets. Therefore we obtain the inverse M(X) → C(X, k) D d , which is continuous because C(X, k) D d is compact and M(X) is Hausdorff.
We denote by δ X,x ∈ M(X) the normal O k -valued measure which assigns 1 if x ∈ U and 0 otherwise to each U ∈ CO(X) for an x ∈ X, by δ X : X → M(X) the map given by setting δ X (x) := δ X,x for an x ∈ X, and by (ii) If X is zero-dimensional, that is, CO(X) generates the topology of X, and Hausdorff, then O ⊕δ X k is injective.
By the assumption, there is a (µ 0 , U 0 ) ∈ C × CO(G \ K) such that |µ 0 (U 0 )| ≥ . By the normality of µ 0 , we have µ 0 (U 0 ) = C∈G/K µ 0 (U 0 ∩ C), and hence |µ 0 By the normality of µ r , we may assume that U r is contained in a C r ∈ G/K satisfying C r = K. By induction on r ∈ ω, we obtain a desired family (µ r , U r , C r ) r∈ω .
Since (C r ) r∈ω is a system of pairwise disjoint subsets of G, It contradicts that the inequality |µ r (U r )| ≥ holds for any r ∈ ω. This completes the proof of the assertion.
For an increasing sequence (X r ) r∈ω of compact clopen subsets of X and a decreasing sequence . Then (G r ) r∈ω forms an increasing sequence of compact clopen subsets of G satisfying C ⊂ M(G; (G r ) r∈ω , ( r ) r∈ω ).
On the other hand, suppose that C is contained in M(G; (G r ) r∈ω , ( r ) r∈ω ) for an increasing sequence (G r ) r∈ω of compact clopen subsets of G. Let L ∈ O(M(G)). By Corollary 2.7, there is a (P, ) ∈ P(X) × (0, 1] such that M(G; P, ) ⊂ L. By ∈ (0, 1), there is an r ∈ ω such that r ≤ . By the compactness of G r , there is a P 0 ∈ P <ω (P ) such that Take an open profinite subgroup K ⊂ G. The direct implication follows from the continuity of ι cg is an open subset of f (gK) by Proposition 2.1 (ii) and Corollary 2.5. By the continuity of f , Proof of Proposition 2.17. Take an open profinite subgroup K ⊂ G. Then G/K gives an element {gK | g ∈ G} of P(G). For any g ∈ G, δ G | gK is a closed continuous map by Proposition 2.16 (i) because gK is compact and M(G) is Hausdorff, and its image is contained in δ G,g + M(G; G/K, 1). Therefore δ G is an injective local homeomorphism onto the image by Proposition 2.16 (ii), because {δ G,g + M(G; G/K, 1) | g ∈ G} forms a covering of the image of δ G consisting of pairwise disjoint clopen subsets of M(G). It implies that δ G is a homeomorphism onto the image, and so is d G by Lemma 2.20. Let We

Topological tensor products
We define symmetric monoidal structures on C , C c , and C cg . First, we study C . Let (M i ) ∈ ob(C 2 ). We denote by . Let (M s ) s∈S be a small diagram in C . By the functoriality of ⊗ and the universality of the colimit, the . We note that ⊗ seems not to be cocontinuous.
Proof. We denote by (A, L, R, B) the data of the associator, the left unitor, the right unitor, and the braiding of (C , By the symmetry of the sub-base of the topology of every value of ⊗ , Since C c is a full subcategory of C , we obtain the following by Proposition 3.1: We have a comparison of the endomorphism algebras, which corresponds to [13] Lemma 1.6 in the case ch(k) = 0.
Proposition 3.6. The natural transformation T c is a natural equivalence.
By Proposition 3.6, we obtain an adjoint property between ⊗ c and H om c . It does not ensure that ⊗ c is cocontinuous, because we used I c in the description of the adjoint property. On the other hand, we have a commutativity between ⊗ c and colimits in C c in a special case. Let (K s ) s∈S be a small diagram in C c . We put M := lim − →s∈S I c (K s ). We recall that the colimit of (K s ) s∈S in C c is given as M by Proposition 2.8 (i). Therefore if M is pre-compact, then S (I c (Ks)) s∈S ,I c (K) gives a morphism S c (Ks) s∈S ,K : lim Finally, we study C cg . We put . By Proposition 2.9 (i) and Corollary 2.10, M 0 ⊗ cg M 1 forms a CG linear topological O k -module. By Corollary 2.7 and the naturality of ∇ c , the system (∇ c Suppose (M i ) ∈ ob((C cg ) 2 ) in the following in this subsection. By the universality of the colimit, the system of the inclusions gives a functor ( . We construct an adjunction By the continuity of f , f (K i ) is continuous. By Proposition 3.5, we have L ∩ K 0 ∈ O(K 0 ). By Corollary 2.7, we obtain L ∈ O(M 0 ). Thus f R is continuous.
given as the composite (ι cg H om cg ((M i+1 )) ) −1 • f R is continuous by Corollary 2.10 (i) and Lemma 3.11. We obtain a map Proof of Theorem 3.8. We denote by (A, L, R, B) the data of the associator, the left unitor, the right unitor, and the braiding of (C , ⊗ , O k ).
. We denote by K 0,1 ⊂ M 0 ⊗ cg M 1 the closure of K 0,1,0 , which is pre-compact by Proposition 2.1 (i). Let (K i ) ∈ K (M i ). We denote by (K i ) K 0,1 ⊂ K 0 ⊗ c K 1 the preimage of K 0,1 , and by By the universality of the colimit, the system ( . Similarly, we obtain a natural formation of the opposite direction, which is the inverse ofÃ. By the construction, the data (Ã,L,R,B, T cg ) is sent to the data of the associator, the left unitor, the right unitor, the braiding, and the Currying of (C , ⊗ O k , O k ) through F cg and ι cg . Since F cg is faithful, it ensures the coherence so that (Ã,L,R,B) forms data of an associator, a left unitor, a right unitor, a braiding, and an injective Currying of (C cg , ⊗ cg , O k ). We have only to verify that T cg As a consequence of Theorem 3.8, we obtain the following: Corollary 3.12. The functor ⊗ cg is cocontinuous.

CGLT algebras
We will verify that O k [[G]] forms a CGLT O k -algebra. Before that, we give examples of CGLT O k -algebras. For this purpose, we compare ⊗ , the tensor product⊗ k of Banach k-vector spaces (cf. [1] p. 12), and the tensor product of compact Hausdorff flat linear topological O k -modules given as the inverse limit of the algebraic tensor product of finite quotients. For this purpose, we recall an elementary property of⊗ k .
Proposition 3.13. For any (X, V ) ∈ ob(Top × Ban ur ≤ (k)), the multiplication C(X, k)×V → C(X, V ) extends to a unique isomorphism C(X, k)⊗ k V → C(X, V ) in Ban ur ≤ (k). Proof. The assertion immediately follows from the orthonormalisability of an unramified Banach k-vector space (cf. The underlying linear topological O k -module of any Banach k-vector space is CG by Proposition 2.11 (iii). We denote by I k : Ban(k) → C cg the forgetful functor. Let (V i ) ∈ ob(Ban(k) 2 ). By the definition of ⊗ , I cg (I k (V 0 )) ⊗ I cg (I k (V 1 )) is first countable. The natural embedding I cg (I k (V 0 )) ⊗ I cg (I k (V 1 )) → I cg (I k (V 0⊗k V 1 )) is a homeomorphism onto the dense image by the definition of ⊗ and⊗ k , and hence induces a homeomorphism T⊗ k ,⊗ cg gives a natural transformation T⊗ k ,⊗ cg : I k (• 0 ) ⊗ cg I k (• 1 ) → I k (• 0⊗k • 1 ). As a consequence, we obtain the following: Proposition 3.14. Every Banach k-algebra, that is, monoid in (Ban(k),⊗ k , k), forms a CGLT O k -algebra through I k and T⊗ k ,⊗ cg .

By [7] Corollary 2.8 (i)
, if G is a profinite group, then C(G, k) admits a unique Hopf monoid structure in (Ban(O k ),⊗ k , k) extending the pointwise k-algebra structure. Therefore by Proposition 3.14, we obtain the following: Corollary 3.15. If G is a profinite group, then C(G, k) admits a unique structure of a commutative CGLT O k -algebra such that the multiplication is a continuous O k -linear extension of the pointwise multiplication.

Every compact topological O k -module is CG by Proposition 2.1 (ii) and Proposition 2.11 (ii). We denote by
is a homeomorphism onto the dense image by the definition of ⊗ c and ⊗ O k , and it induces a homeomorphism T⊗ gives a natural transformation As a consequence, we obtain the following:

By Proposition 2.21 and [7] Proposition 2.7, if G is a profinite group, then O k [[G]] admits a unique Hopf monoid structure in (C
. Therefore by Proposition 3.16, we obtain the following: We note that Corollary 3.17 will be extended to the case where G is not necessarily a profinite group, as we mentioned in the beginning of this subsection. Another simple example of a CGLT O k -algebra is given by a topological O k -algebra. The rest of this subsection is devoted to the following extension of Corollary 3.17: In order to verify Theorem 3.19, we define a convolution product on M(G). Let (µ i ) ∈ M(G) 2 . We define elements µ i ∈ M(G 2 ) and µ 0 * µ 1 ∈ M(G). Let U ∈ CO(G 2 ). To begin with, suppose that U is compact. Take an S ∈ P <ω (CO(G) 2 By the finite additivity of µ 0 and µ 1 , ( µ i )(U ) depends only on U . In particular, the equality ( µ i )( U i ) = µ i (U i ) holds for any compact clopen subsets U 0 and U 1 of G.
Next, we consider the case where U is not necessarily compact. Take a compact clopen subgroup K ⊂ G. Then (G/K) 2 = {(g i K) | (g i ) ∈ G 2 } gives an element of P(G 2 ) consisting of compact clopen subsets. Put ( µ i )(U ) := (C i )∈(G/K) 2 ( µ i )(U ∩ C i ). By the normality of µ 0 and µ 1 , the infinite sum in the right hand side actually converges, and ( µ i )(U ) is independent of the choice of K. We obtain a normal O k -valued measure µ i on G 2 . For a U ∈ CO(G), we denote by U ⊂ G 2 the preimage of U by the multiplication G 2 → G. Set (µ 0 * µ 1 )(U ) := ( µ i )( U ). Since µ i is a normal O k -valued measure on G 2 , so is µ 0 * µ 1 on G. We have constructed an element µ 0 * µ 1 ∈ M(G). By the construction, the convolution product . We note that * G is not necessarily continuous.
, there is an increasing sequence (G i,r ) r∈ω of compact clopen subsets such that Lemma 2.19. For an r ∈ ω, put G r := r h=0 {g 0 g 1 | (g i ) ∈ G r−h,0 × G h,1 }. Then (G r ) r∈ω forms an increasing sequence of compact clopen subsets of G satisfying K ⊂ M(G, (G r ) r∈ω , (2 −r ) r∈ω ) by the definition of * G . Therefore K is pre-compact by Lemma 2.19.
By the bijectivity of ι cg M(G) and ∇ cg× Lemma 3.21. The convolution product * cg G is continuous.
] be an open neighbourhood of µ 0 * µ 1 for a (µ i ) ∈ M(G) 2 . It suffices to show that for any (  (iii) If G admits a closed subgroup H ⊂ G and a compact subset C ⊂ G such that the multiplication H × C → G is bijective, then the multiplication is actually a homeomorphism by [ As an application of Example 3.22 (ii) and (vi), we immediately obtain the following:

CGLT modules Let
We give three examples of CGLT modules as immediate consequences of Proposition 3.14, Proposition 3.16, and Proposition 3.18, respectively: Proposition 3.24. Let A be a Banach k-algebra. Then every Banach left A -module, that is, left A -module in (Ban(k),⊗ k , k), forms a CGLT I k (A )-module through I k and T⊗ k ,⊗ cg .
Proposition 3.26. Let A be a topological O k -algebra whose underlying topological O k -module is linear and CG. Then every topological left Amodule whose underlying topological O k -module is linear and CG forms a CGLT A -module through ∇ cg⊗ • (∇ cg× ) −1 .
A BT A-module is a CGLT A-module V whose underlying O k -module structure extends to a k-vector space structure equipped with a complete non-Archimedean norm on the underlying k-vector space of V giving its original topology. Let V be a BT A-module. Then V forms a topological k-vector space because it forms a Banach k-vector space. We say that such that ρ(f , m ) ∈ U for any (f , m ) ∈ ((f + L 0 ) × K) ∪ (K 0 × (m + L 1 )). In particular, we have L 0 ⊂ L ∩ K 0 and hence L ∩ K 0 ∈ O(K 0 ). It implies L ∈ O(A) by Corollary 2.7. By L × K ⊂ ρ −1 (U ), ρ −1 (U ) is an open neighbourhood of (f, m). Thus ρ is continuous. We give an example of a CGHLT A-module. Let K ∈ ob(Mod ch f (A)). We denote by K k the left F cg (A)-module k ⊗ O k F c (K) equipped with the strongest topology for which K k forms a topological k-vector space and the natural embedding ι c K : K → K k is continuous. We identify F c (K) with its image in k ⊗ O k F c (K). The following is an analogue of [13] Lemma 1.4: Proposition 3.28. The linear topological O k -module K k forms a CGHLT A-module, and ι c K is a homeomorphism onto a core. In order to verify Proposition 3.28, we characterise the topology of K k . Proof of Proposition 3.28. Take a uniformiser ∈ O k . Put K r := K for an r ∈ ω, and denote by K ω the colimit in C of (K r ) r∈ω with respect to the transition maps K r → K r+1 , m → cm indexed by r ∈ ω. Then K ω forms a CG linear topological O k -module by Proposition 2.1 (ii), Proposition 2.9 (i), and Corollary 2.10. It is Hausdorff by the same computation as that in the proof of [6] Proposition 1.27 using Corollary 2.7 and a well-known property of T 1 normal topological spaces. By Corollary 3.12 and the functoriality of the colimit, the scalar multiplication A ⊗ cg K → K induces a continuous O k -linear homomorphism A ⊗ cg K ω → K ω , for which K ω forms a CGLT A-module.
By the universality of the colimit and the flatness of K, ι c K induces a continuous bijective O k -linear homomorphism kι c K : K ω → K k . By Corollary 2.7, the map K ω → K ω , m → m is an isomorphism in C , and hence K ω forms a topological k-vector space. We show that kι c K is an open map.
) coincides with the preimage of cL in K 0 , and hence is open by the continuity of the canonical embedding K 0 → K ω . It ensures (kι c K )(L) ∈ O(K k ) by Lemma 3.29. Therefore kι c K is an isomorphism in C , and K k forms a Hausdorff CGLT Amodule. Since K is compact and K k is Hausdorff, ι c K is a homeomorphism onto the image, which is a core of K k .
We obtain a characterisation of a CGHLT A-module.

Modules over Iwasawa algebras
We study relation between module theory over O k [[G]] and representation theory of G. As a main result, we generalise Schneider-Teitelbaum duality to duality applicable to G, and give a criterion of the irreducibility of unitary Banach k-linear representations of G.

Unitary Banach representations A Banach k-linear represen-
tation of G is a pair (V, ρ) of a V ∈ ob(Ban(k)) and a continuous map ρ : G × V → V giving a k-linear action of G on V . Let (V, ρ) be a Banach k-linear representation of G. We say that (V, ρ) is unitarisable if there is an R ∈ (0, ∞) such that ρ(g, v) ≤ R v for any (g, v) ∈ G × V , is isometric if ρ(g, v) = v for any (g, v) ∈ G × V , and is said to be unitary if V is unramified and (V, ρ) is isometric. A map between Banach k-linear representations is said to be a k[G]-linear homomorphism if it is a G-equivariant k-linear homomorphism. We denote by Rep G (Ban(k)) the k-linear category of unitarisable Banach k-linear representations of G and bounded k[G]linear homomorphisms, by Rep G (Ban ≤ (k)) ⊂ Rep G (Ban(k)) the O k -linear subcategory of isometric Banach k-linear representations of G and submetric k[G]-linear homomorphisms, and by Rep G (Ban ur ≤ (k)) ⊂ Rep G (Ban ≤ (k)) the full subcategory of unitary Banach k-linear representations of G.
We compare the notion of a BT O k [[G]]-module and the notion of a Banach k-linear representation of G. For this purpose, we consider a partial generalisation of Banach-Steinhaus theorem (cf. [11] Corollary 6.16). Let (X 0 , (V i )) ∈ ob(Top × Ban(k) 2 ).
Proof. We denote by ρ : X 0 × V 1 → V 2 the induced map. The direct implication follows from the continuity of the map . It implies that ρ is continuous.
is an isomorphism in C .
Proof. Denote by i the map in the assertion. By Corollary 2.5, i is continu- We also consider a similar comparison without the assumption of the submetric condition. Let (V, ρ) ∈ ob(Rep G (Ban(k))). Take a c ∈ k × satisfying ρ(g, v) ≤ |c| v for any (g, v) ∈ G×V . By Proposition 4.1, the map   For this purpose, we consider a compact analogue of Banach-Steinhaus theorem (cf. [11] Corollary 6.16). We denote by Unf the category of compact uniform spaces and uniformly continuous maps. Let (X 0 , (C i+1 )) ∈ ob(Top × Unf 2 ). We equip Hom Unf ((C i+1 )) the topology of uniform convergence.

CHFLT modules
Proof. If C 1 = ∅, then the assertion is obvious. We assume C 1 = ∅. We denote by ρ : X 0 ×C 1 → C 2 the induced map. Suppose that ϕ is continuous.
By the uniform continuity of ϕ(x 0 ), there is an entourage E 2 ⊂ C 2 1 such that every (m i ) ∈ E 2 satisfies (ϕ(x 0 )(m i )) ∈ E 1 . By the continuity of ϕ, there exists an open neighbourhood U 0 ⊂ X of x 0 such that (ϕ( Then the collection of subsets of the form U E forms a fundamental system of neighbourhoods of ϕ(x 0 ). Take entourages E 0 , E 1 ⊂ C 2 2 satisfying U E 0 ⊂ U and that for any there are open neighbourhoods U 0 ⊂ X and U 1 ⊂ C 1 of x 0 and m 0 , respectively, such that (ρ(x i , m i )) ∈ E 1 for any (x 1 , m 1 ) ∈ U i by the continuity of ρ. We denote by S the set of such an (m 0 , (U i )) satisfying m 0 ∈ C 1 . Since C 1 is compact and non-empty, there is an S 0 ∈ P <ω (S)\{∅} such that 1 by the choice of m 1 and U 1 . Therefore we obtain (ϕ(x i )(m 1 )) = (ρ(x i , m 1 )) ∈ E 0 by the choice of E 1 . It ensures ϕ(x 1 ) ∈ U E 0 . It implies that V 0 ⊂ ϕ −1 (U E 0 ). Thus ϕ is continuous.
Let (K, ρ) ∈ ob(Rep G (C ch f )). The monoid homomorphism ϕ ρ : G → H om c (K, K) × induced by ρ is continuous by Proposition 4.7. In order to obtain a CHFLT O k [[G]]-module structure on K associated to ρ, we prepare a partial generalisation of [13] Lemma 2.1 for the compact side. By Proposition 3.3 and Proposition 4.2, we obtain the following: By Theorem 3.19 and Proposition 4.8, we obtain a locally profinite counterpart of [13] Corollary 2.2 for the compact side.  Let (M, ρ) ∈ ob(Rep G (C cg )). A G-stable O k -submodule K ⊂ M is said to be a core of (M, ρ) if K is compact, the inclusion K → M induces an isomorphism k ⊗ O k F c (K) → F cg (M ) in C , and every O k -submodule L ⊂ M satisfying cL ∩ K ∈ O(K) for any c ∈ O k \ {0} is open. We say that (M, ρ) is a CGHLT k-linear representation of G if M is Hausdorff and (M, ρ) admits a core. If (M, ρ) is a CGHLT k-linear representation of G, then M forms a topological k-vector space because O(M ) is closed under the action of k × . We denote by Rep G (kC ch f ) ⊂ Rep G (C cg ) the full subcategory of CGHLT k-linear representations of G. We give an example of a CGHLT k-linear representation of G. We denote by (K, ρ) k the pair of K k ∈ ob(Mod cgh (O k )) and the k-linear extension of ρ.
Proposition 4.11. The pair (K, ρ) k forms a CGHLT k-linear representation of G, and ι c K is a homeomorphic O k [G]-linear isomorphism onto a core.
] ⊗ cg M induced by the inclusion K 1 → M , which is continuous by the functoriality of ⊗ cg , and the scalar Let U 2 ⊂ M 0 be an open subset. Take an open profinite subgroup H ⊂ G. For a g ∈ G, put U g,1 := gH and U g,2 := {m ∈ M 0 | ∀g ∈ U g,1 , ρ M (g , m) ∈ U 2 }. Then we have ρ −1 M (U 2 ) = g∈G U g,i . Therefore in order to show that ρ −1 M (U 2 ) is open, it suffices to show U g,2 is open for any g ∈ G. Let g ∈ G. We show that cU g,2 ∩ K 1 is open in K 1 for any Let m ∈ cU g,2 ∩ K 1 . By Proposition 2.1 (ii), Corollary 2.5, and Proposition 2.17 (ii), we have . By Proposition 2.1 (ii) and the continuity of the scalar mul-
Proof. By Proposition 4.1, ρ induces a continuous monoid homomorphism ) is a continuous by Proposition 3.3. Therefore ρ D d is continuous by Proposition 4.7.
for any m ∈ V D d by Theorem 2.12. It ensures m(v) = v(0) = 0 for any m ∈ M 0 . We obtain v ∈ V 0 and hence V 0 = {0}. Since V 0 is a closed G-stable k-vector subspace of (V, ρ) and (V, ρ) is irreducible, we obtain V 0 = V . It ensures M 0 = {0}. It implies that ((V, ρ) D d ) k is simple.

Applications
As applications of the module theory in the monoidal structure, we give an explicit description of a continuous parabolic induction of unitary Banach k-linear representations.
Let K 0 ∈ ob(Mod ch f (O k [[P ]])). We describe Ind G P (K Dc 0 ) D d explicitly by G and K 0 . Since the underlying topological space of G is a disjoint union of compact clopen subspaces, a map ϕ : G → K Dc 0 is continuous if and only if the induced map G × K 0 → k : (g, m) → ϕ(g)(m) is continuous by Proposition 4.7. Therefore we obtain an isometric k-linear homomorphism C bd (G, K Dc 0 ) → C bd (G×K 0 , k) onto the closed image. We consider the map ρ : G × C bd (G × K 0 , k) → C bd (G × K 0 , k) given by setting ρ(g, f )(g , m) := f (g g, m) for a (g, f, g , m) ∈ G × C bd (G × K 0 , k) × G × K 0 , which is not necessarily continuous. The inclusion Ind G P (K Dc 0 ) → C bd (G, K Dc 0 ) ⊂ C bd (G × K 0 , k) is an isometric G-equivariant k-linear homomorphism, and its image is the closed G-stable k-vector subspace consisting of functions f : G × K 0 → k satisfying the following: (I) The equality f (g, cm) = cf (g, m) holds for any (g, c, m) ∈ G×O k ×K 0 .
The inclusion Ind G P (K Dc 0 ) → C bd (G × K 0 , k) induces a continuous surjective G-equivariant O k -linear homomorphism ϕ G,P : C bd (G × K 0 , k) D d Ind G P (K Dc 0 ) D d by Hahn-Banach theorem (cf. [5] Theorem 3 and [11] Proposition 9.2). Since the target and the source of ϕ G,P are compact and Hausdorff, the target is homeomorphic to the coimage. We determine ker(ϕ G,P ) in order to describe the target. We denote by e g,m the submetric k-linear homomorphism C bd (G × K 0 , k) → k, f → f (g, m) for a (g, m) ∈ G × K 0 . We put µ I g,c,m := ce g,m − e g,cm for a (g, c, m) ∈ G × O k × K 0 , µ II g,(m i ) := e g, m i − e g,m i for a (g, (m i )) ∈ G × K 2 0 , and µ III g,h,m := e hg,m − e g,d G,h −1 m for a (g, h, m) ∈ G × P × K 0 . We denote by µ I +µ II +µ III ⊂ C bd (G×K 0 , k) D d the closed O k -submodule generated by the union of {µ I g,c,m | (g, c, m) ∈ G × O k × K 0 }, {µ II g,(m i ) | (g, (m i )) ∈ G × K 2 0 }, and {µ III g,h,m | (g, h, m) ∈ G × P × K 0 }.

We set Ind
O k [[P ]] (K 0 ) := C bd (G × K 0 , k) D d /(µ I + µ II + µ III ). By Proposition 5.1, we obtain the following: . To begin with, we prepare a compact complete representative C ⊂ G of P \G. We denote by Σ the set of open subsets U ⊂ P \G admitting a continuous section U → G of the canonical projection G P \G. Take an open profinite subgroup G 0 ⊂ G. Since G is a topological group, the canonical projection G P \G is an open map. Therefore the image G 0 g ⊂ P \G of G 0 g is an open subset, and the map G 0 → G, h → hg induces a homeomorphism (P ∩ G 0 )\G 0 → G 0 g for any g ∈ G. It implies that Σ forms an open covering of P \G by [9] Theorem 2. Take a Σ 0 ∈ P <ω (Σ) satisfying P \G = U ∈Σ 0 U . Gluing continuous sections on each U ∈ Σ 0 , we obtain a continuous section P \G → G, whose image forms a compact subset C ⊂ G such that the multiplication P × C → G is a continuous bijective map. Conversely, let C ⊂ G be an arbitrary compact subset such that the multiplication P × C → G is a continuous bijective map. As is mentioned in Example 3.22 (iii), the multiplication P × C → G is a homeomorphism, and induces a homeomorphic We denote by π 0 : G P (respectively, π 1 : G C) the composite of the inverse G → P × C of the multiplication and the canonical projection P × C P (respectively, P × C P ). As a result, C is obtained as the image of the continuous section P \ G → G induced by π 1 .
Let F be a local field, G an algebraic group over Spec(F ), and P ⊂ G a parabolic subgroup. Then G(F ) forms a locally profinite group with respect to the topology induced by the valuation of F , and P(F ) is naturally identified with a closed subgroup of G(F ). Since P\G forms a proper algebraic variety over Spec(F ), P(F )\G(F ) forms a totally disconnected compact Hausdorff topological space. Henceforth, we consider the case G = G(F ) and P = P(F ).
Let (V 0 , ρ 0 ) ∈ ob(Rep P (Ban ur ≤ (k))). We consider the composite r C,V 0 : Ind G P (V 0 ) → C(C, V 0 ) of the inclusion Ind G P (V 0 ) → C bd (G, V 0 ) and the restriction map C bd (C, V 0 ) → C(C, V 0 ). Then r C,V 0 is injective by the conditions (III) in §5.1 and P C = G. The quotient norm on the source of r C,V 0 coincides with the norm restricted to the image of r C,V 0 because P acts isometrically on V 0 . Therefore r C,V 0 is isometric. For any f ∈ C(C, V 0 ), the mapf : G → V 0 , g → ρ 0 (π 0 (g), (f • π 1 (g))) lies in Ind G P (K 0 ). We obtain an isometric section C(C, V 0 ) → Ind G P (V 0 ), f →f , and hence r C,V 0 is an isomorphism in Ban ur ≤ (k). Pulling back Ind G P (ρ 0 ) by r C,V 0 and the isomorphism