Duality theory of p-adic Hopf algebras

We show the monoidal functoriality of Schikhof duality, and cultivate new duality theory of p-adic Hopf algebras. Through the duality, we introduce two sorts of p-adic Pontryagin dualities. One is a duality between discrete Abelian groups and affine formal group schemes of specific type, and the other one is a duality between profinite Abelian groups and analytic groups of specific type. We extend Amice transform to a p-adic Fourier transform compatible with the second p-adic Pontryagin duality. As applications, we give explicit presentations of a universal family of irreducible p-adic unitary Banach representations of the open unit disc of the general linear group and its q-deformation in the case of dimension 2.


Introduction
Let k be a non-Archimedean local field. We denote by O k its valuation ring, and by p the characteristic of the residue field of O k . As a p-adic analogue of the reflexivity of Hilbert spaces, Schikhof duality (cf. [6] Theorem 4.6) gives a contravariant categorical equivalence between Banach k-vector spaces and localisations of compact Hausdorff flat linear topological O k -modules. We show the monoidal functoriality of Schikhof duality in Theorem 3.2, and study a new duality between Hopf monoid objects in Theorem 3.6.
Through the duality restricted to commutative cocommutative Hopf monoid objects, we introduce two sorts of p-adic Pontryagin dualities. One is a contravariant categorical equivalence between discrete Abelian groups (discrete side) and affine formal group schemes over O k of specific type (compact side), and the other one is a contravariant categorical equivalence between profinite Abelian groups (compact side) and analytic groups over k of specific type (discrete side). Therefore they are variants of the duality between compactness and discreteness.
We explain the relation with classical results. Iwasawa isomorphism is a well-known isomorphism between the Iwasawa algebra O k [[Z p ]] and the formal power series algebra O k [[T ]], which represents the affine formal group scheme U 1/O k given as the completion of the multiplicative group scheme G m/O k . Through the identification of O k [[Z p ]] and the continuous dual of the continuous function algebra C(Z p , k), it gives an isomorphism called Amice transform from the continuous dual of C(Z p , k) to the algebra of global sections on U 1/O k . Therefore Amice transform gives a non-trivial connection between functions on the profinite Abelian group Z p and the affine formal group scheme U 1/O k . In addition, the second p-adic Pontryagin duality sends Z p to U 1/O k , and hence Amice transform can be regarded as a p-adic analogue of Fourier transform for the specific dual pair (Z p , U 1/O k ). The duality between Hopf monoid objects yields a Hopf monoid isomorphism in a wider case in Theorem 4.23, and hence can be regarded as a p-adic analogue of Fourier transform extending Amice transform.
The notion of a representation of an affine formal group scheme G of specific type can be formulated in terms of a comodule object. Through the duality between Hopf monoid objects, it can be naturally identified with the notion of a module object over the dual Hopf monoid A of the coordinate ring of G . By the functoriality of G , every A-module object can be regarded as a unitary Banach k-linear representation of the discrete group G (O k ). Then the regular A-module can be regarded as a universal family of irreducible unitary Banach k-linear representations of G (O k ) which can be obtained in this way.
As applications of the duality between Hopf monoid objects, we give explicit presentations of the universal family of irreducible unitary Banach k-linear representations of the open unit disc, that is, the discrete group of the O k -valued points of the affine formal group scheme given as the completion, of the general linear group and a q-deformation of the universal family in the case of dimension 2.
We explain contents of this paper. First, §2 consists of two subsections. In §2.1, we recall compact Hausdorff flat linear topological O k -modules. In §2.2, we recall Banach k-vector spaces. Secondly, §3 consists of two subsections. In §3.1, we show the monoidal functoriality of Schikhof duality. In §3.2, we study the associated duality between Hopf monoid objects. Thirdly, §4, consists of three subsections. In §4.1, we investigate the p-adic Pontryagin duality between discrete Abelian groups and affine formal groups schemes over O k of specific type. In §4.2, we investigate the p-adic Pontryagin duality between profinite Abelian groups and analytic groups over k of specific type. In §4.3, we study the p-adic Fourier transform. Finally, §5 consists of two subsections. In 5.1, we give an explicit description of the universal family of irreducible unitary Banach k-linear representations of the open unit disc of the general linear group. In 5.2, we give a similar description of a q-deformation of the universal family.

Preliminaries
We fix a Grothendieck universe U in order to avoid set-theoretic problems.
A set x is said to be U -small if x ∈ U . Throughout this paper, let k denote a fixed U -small local field, that is, a complete discrete valuation field with finite residue field, O k ⊂ k the valuation ring of k, and p the characteristic of the residue field of O k .
We introduce the convention and the terminology for categories, which is always assumed to be small but is not assumed to be U -small. We denote by C (respectively, C k ) the Abelian category of U -small O k -modules (respectively, k-vector spaces) and O k -linear homomorphisms. Then the triads (C , ⊗ O k , O k ) and (C k , ⊗ k , k) form symmetric monoidal categories. The correspondence M k ⊗ O k M restricted to U gives a symmetric monoidal functor (C , ⊗ O k , O k ) → (C k , ⊗ k , k). We abbreviate "(C , ⊗ O k , O k )-enriched" (respectively, "(C k , ⊗ k , k)-enriched") to "O k -linear" (respectively, "k-linear"). For an O k -linear category Θ, we denote by kΘ the k-linear category obtained as the localisation of Θ by the symmetric monoidal functor (C , ⊗ O k , O k ) → (C k , ⊗ k , k). For an O k -linear functor Φ, we denote by kΦ the k-linear extension of Φ.
We denote by Set the category of U -small sets and maps, by Top the category of U -small topological spaces and continuous maps, by PTop ⊂ Top the full subcategory of U -small totally disconnected compact Hausdorff topological spaces, by Grp the category of U -small discrete groups and group homomorphisms, by Ab ⊂ Grp the full subcategory of U -small discrete Abelian groups, by PGrp the category of U -small profinite groups and continuous group homomorphisms, and by PAb ⊂ PGrp the full subcategory of U -small profinite Abelian groups. For topological spaces X and Y , we denote by C(X, Y ) the set of continuous maps X → Y .

Topological module
the discrete topology so that it again forms a compact Hausdorff linear topological O k -module by the finiteness of its underlying set. We denote by with respect to canonical projections equipped with the inverse limit topology. We give an explicit example of the completed tensor product. Proposition 2.2. Let I 0 and I 1 be sets. Then there is a natural homeo- We denote by f L 0 ,L 1 the composite of the canonical projection O I 0 ×I 1 k O J 0 ×J 1 k and the following O k -linear homomorphism, which is continuous by the continuity of the O k -module structure of M L 0 ,L 1 : By the definition, (f L 0 ,L 1 ) (L 0 ,L 1 ) forms a compatible system of surjective continuous O k -linear homomorphisms, and hence induces a continuous O klinear homomorphism f : , whose image is dense by the surjectivity of the composite with any canonical projection. Let c = ( By the definition, the completed tensor product of compact Hausdorff linear topological O k -modules is again a compact Hausdorff linear topological O k -module. Moreover, we show that it also preserves the flatness.

Banach space
called the norm satisfying the following: Example 2.4. Let X be a topological space, and V a Banach k-vector space, e.g. k. We denote by C bd (X, V ) ⊂ C(X, V ) the k-vector subspace of bounded continuous maps. Then C bd (X, V ) forms a Banach k-vector space with respect to the supremum norm • : by the discreteness of the valuation. We denote by C 0 (X, V ) ⊂ C bd (X, V ) the k-vector subspace of functions f : X → V such that for any ∈ (0, ∞), there is a compact subset C ⊂ X such that f (x) < for any x ∈ X \ C. Then C 0 (X, V ) is closed in C bd (X, V ), and hence forms a Banach k-vector space with respect to the restriction of the supremum norm. Suppose that X is compact. Then we have C 0 (X, V ) = C bd (X, V ) = C(X, V ) by the maximal modulus principle, and hence regard C(X, V ) as a Banach k-vector space. Let V 0 and V 1 be Banach k-vector spaces, and f : Since the valuation of k is not trivial, a k-linear homomorphism between Banach kvector spaces is bounded if and only if it is continuous by [2] 2.1.8 Corollary 3.
Let V 0 and V 1 be Banach k-vector spaces. We denote by V 1⊗k V 2 the completed non-Archimedean tensor product of V 1 and V 2 (cf. [1] p. 12). We give an explicit example of the completed tensor product analogous to the one in Proposition 2.2.
Proof. The following proof is essentially the same as the argument in the proof of [9] Theorem 1.2, in which k is assumed to be of characteristic 0. Denote by F the functor in the assertion. The fullness of F follows from the definition of a bounded homomorphism and non-triviality of the valuation of k. The faithfulness of F follows from the flatness of hom modules. For a Banach k-vector space V , denote by V ur the underlying k-vector space of Then V ur forms a Banach k-vector space, and the identity map I V : V → V ur is an isomorphism in Ban(k). The correspondence V V ur restricted to U gives an endofunctor (•) ur of Ban(k), and the correspondence V I V restricted to U gives a natural equivalence I : id Ban(k) ⇒ (•) ur . Moreover, I V is the identity for any unramified Banach k-vector space V . Therefore (•) ur induces a quasi-inverse of F , because F is fully faithful.
A Banach k-algebra is a Banach k-vector space A equipped with a kalgebra structure satisfying f 0 f 1 ≤ f 0 f 1 for any (f 0 , f 1 ) ∈ A 2 and 1 = 1 as long as A is not a zero ring. It is elementary to show that the notion of a U -small Banach k-algebra is equivalent to that of a monoid object in (Ban ur ≤ ,⊗ k , k). We note that there are several distinct formulations of the notion of a Banach k-algebra, and such an equivalence does not necessarily hold if one applies another formulation.
Let X be a topological space. We equip C bd (X, k) with the pointwise multiplication C bd (X, k) × C bd (X, k) → C bd (X, k), for which it forms a commutative Banach k-algebra and hence a commutative monoid object in (Ban ur ≤ ,⊗ k , k) as long as X is U -small. In order to introduce another example of a monoid object in (Ban ur ≤ ,⊗ k , k), we recall the notion of a (possibly uncountable) infinite sum. Let I be a discrete set, and f : I → k a map. We denote by i∈I f (i) the limit of the net ( i∈J f (i)) J⊂I,#J<∞ indexed by the set of finite subsets J ⊂ I directed with respect to the inclusion relation. When I = {i ∈ N | i < n} for some n ∈ N (respectively, I = N), then i∈I f (i) coincides with the usual finite Let G be a discrete group. We equip C 0 (G, k) with the convolution product where the sum in the definition makes sense by the argument above, for which it forms a Banach k-algebra and hence a monoid object in (Ban ur ≤ ,⊗ k , k) as long as G is U -small. The canonical embedding ι d G : k[G] → C 0 (G, k) is an k-algebra homomorphism whose image is dense, and hence C 0 (G, k) is an analogue of k[G].
We denote by Alg(k) the category of monoid objects in (Ban ur ≤ (k),⊗ k , k) and monoid homomorphisms, and by CAlg(k) ⊂ Alg(k) the full subcategory of commutative monoid objects in (Ban ur We recall non-Archimedean Gel'fand-Naimark theorem. Proposition 2.8. The functor C bd (•, k) restricted to PTop is fully faithful.

Monoidal Structure
We introduce symmetric monoidal structures on Ban ur ≤ (k), Ban(k), C ch fl , and kC ch fl , and verify the monoidal functoriality of Schikhof duality (cf. [6] Theorem 4.6 and [9] Theorem 1.2). As a corollary, we obtain a duality between Hopf monoid objects.

Monoidal functoriality
. In order to verify Theorem 3.2, we prepare several lemmas.
Proof. The functoriality is obvious. We show that the functors are compatible with the symmetric monoidal structures. For U -small sets I 0 and ) explicitly constructed in the proof of Proposition 2.2 (Proposition 2.6). The correspondence (I 0 , I 1 ) T(I 0 , I 1 ) gives a natural equivalence T : ) with the desired coherence by the construction.
We recall classical results on classifications of unramified Banach kvector spaces and compact Hausdorff flat linear topological O k -modules.
Proof. The first assertion follows from [5]  We construct a natural equivalence ( Ban with the desired coherence. For this purpose, it suffices to show that the Ban in C ch fl for any unramified Banach k-vector spacesV 0 and V 1 . By Lemma 3.4 (i), it is reduced to the case V 0 = C 0 (I, k) and V 1 = C 0 (J, k) for sets I and J.
Ban extending the given homomorphism. Therefore D ur Ban forms a symmetric monoidal equivalence. We construct a natural equivalence with the desired coherence. For this purpose, it suffices to show that the fl extending the given homomorphism. Therefore D O k and D ch fl are symmetric monoidal functors.
We also have a relation to Cartesian products of topological spaces.
(ii) For any topological spaces X 0 and X 1 , the k-algebra homomor- Proof. The assertion (i) immediately follows from Theorem 3.2 and the assertion (ii) applied to the underlying topological spaces of G 0 and G 1 . We show the assertion (ii). In the case where X 0 and X 1 are totally disconnected compact Hausdorff topological spaces, the assertion immediately follows from Proposition 2.8, because⊗ k satisfies the universality of the coproduct. In the general case, the assertion follows from the fact that the inclusion PTop → Top is a right adjoint functor (cf. [4] Corollary 2.3). We note that the universality and the adjoint property are formalisable without using categories, and hence we do not have to assume the U -smallness of X 0 and X 1 . We give an explicit example of a Hopf monoid object in (C ch fl ,⊗ O k , O k ). We recall one of the simplest example of a Hopf O k -algebra, that is, a Hopf monoid object in (C , ⊗ O k , O k ), is the group algebra over O k . Similarly, we construct two sorts of Hopf monoid objects in (C ch fl ,⊗ O k , O k ) by using a topological group.

Hopf monoid
such that ι c G preserves the comultiplication, the counit, and the antipode.
(ii) Let G be a U -small discrete group. Then the monoid object structure of C 0 (G, k) in (Ban ur ≤ ,⊗ k , k) extends to a unique Hopf monoid object structure in (Ban ur ≤ (k),⊗ k , k) such that ι d G preserves the comultiplication, the counit, and the antipode. Moreover, C 0 (G, k) is cocommutative.
Proof. (i) The uniqueness and the cocommutativity follow from the fact (ii) The uniqueness and the cocommutativity follow from the fact that and an antipode C 0 (G, k) • → C 0 (G, k) • , which extend to a comultiplication, a counit, and an antipode on C 0 (G, k) respectively with respect to (⊗ k , k), for which C 0 (G, k) forms a Hopf monoid object in (Ban ur ≤ (k),⊗ k , k) satisfying the desired conditions again because the image of ι d G is dense in C 0 (G, k).
Corollary 3.8. (i) Let G be a U -small profinite group. Then the monoid object structure of C(G, k) extends to a unique Hopf monoid object structure in (Ban ur ≤ (k),⊗ k , k) such that the composite of the comultiplication and the isometric k-linear isomorphism ι : C(G, k)⊗ k C(G, k) → C(G × G, k) in Corollary 3.5 (ii) coincides with the composition to the multiplication Ban , and the antipode is the involution C(G, k) → C(G, k) given as the composition to the map G → G, g → g −1 .
(ii) Let G be a U -small discrete group. Then the monoid object structure of O G k extends to a unique Hopf monoid object structure in (C ch fl ,⊗ O k , O k ) such that the composite of the comultiplication and the homeomorphic O k - Proof. The assertions immediately follow from Example 3.1, Lemma 3.3, Theorem 3.6, and Proposition 3.7 by the fact that the dual of the comultiplication of O k [[G]] (respectively, C 0 (G, k)) coincides with the pointwise multiplication of C(G, k) (respectively, O G k ).

Pontryagin duality
We introduce two sorts of p-adic Pontryagin dualities in terms of functors of points. Through the duality in Theorem 3.6, we establish p-adic analogues of Fourier transform and Plancherel's theorem extending Amice transform

Discrete Abelian group
For a Banach k-algebra A, we denote by A G m/k the discrete group which shares the underlying group with (A • ) × . The correspondence A A G m/k restricted to U gives a functor G m/k : CAlg(k) → Ab, which is an analytic geometric counterpart of S 1 .
For any U -small set I, the correspondence A (A G m/k ) I restricted to U gives a functor G I m/k : CAlg(k) → Ab. A functor G : CAlg(k) → Ab is said to be a torus if it is naturally isomorphic to G I m/k for some U -small set I, and is said to be a multiplicative analytic group over k if it is naturally isomorphic to the kernel of a natural transformation between tori. We establish a p-adic Pontryagin duality between discrete Abelian groups and multiplicative analytic groups over k.
Example 4.1. (i) Let n ∈ N \ {0}. For a Banach k-algebra A, we denote by A µ n/k ⊂ A G m/k the subgroup {f ∈ A G m/k | f n = 1}. The correspondence A A µ n/k restricted to U gives a functor µ n/k : CAlg(k) → Ab, which is a multiplicative analytic group over k because it is the kernel of the natural transformation G m/k → G m/k given by the n-th power.
(ii) Let D N denote the set {(n 0 , n 1 ) ∈ (N \ {0}) 2 | ∃d ∈ N, dn 0 = n 1 }. For a Banach k-algebra A, we denote by A Z(1) /k the inverse limit of (A µ n/k ) n∈N\{0} with respect to the system of n −1 0 n 1 -th powers A µ n 1 /k → A µ n 0 /k indexed by (n 0 , n 1 ) ∈ D N equipped with the discrete topology. The correspondence A A Z(1) /k restricted to U gives a functor which is a multiplicative analytic group over k because it is the kernel of the natural transformation For discrete Abelian groups G and H, we denote by H om Ab (G, H) the discrete set of group homomorphisms G → H equipped with the pointwise multiplication. Let G be a U -small discrete Abelian group. The correspondence A H om Ab (G, A G m/k ) restricted to U gives a functor H om Ab (G, G m/k ) : CAlg(k) → Ab. Since G m/k is a counterpart of S 1 , Hom Ab (•, G m/k ) is a p-adic analogue of the Pontryagin dual. In order to verify Proposition 4.2, we prepare several lemmata.
Proof. By Proposition 3.7 (ii), C 0 (G, k) represents a functor G : CAlg(k) → Ab. Denote by u ∈ H om Ab (G, C 0 (G, k) G m/k ) the composite of the canonical embedding G → k[G] and ι d G , and by F the natural transformation G ⇒ H om Ab (G, G m/k ) determined by the equality F (C 0 (G, k))(id C 0 (G,k) ) = u by Yoneda's lemma because u is a group homomorphism. We show that F is a natural equivalence. Let A be a commutative monoid object in (Ban ur ≤ (k),⊗ k , k). Then F (A) is the group homomorphism Since the image of u generates a dense k-subalgebra of C 0 (G, k), F (A) is injective. Let ψ ∈ H om Ab (G, A G m/k ). For any g ∈ G, we have ψ(g) = 1 by ψ(g) ∈ (A • ) × . Therefore the k-algebra homomorphism ϕ : k[G] → A associated to ψ by the universality of the group algebra satisfies ϕ . It implies that ϕ extends to a unique submetric k-algebra homomorphism ϕ : It implies the surjectivity of F (A). Thus F is a natural equivalence.
Corollary 4.4. For any U -small set I, G I m/k is representable by C 0 (Z ⊕I , k). Proof. The assertion immediately follows from Lemma 4.3 applied to the case G = Z ⊕I by the universality of the direct sum, because the identity functor Ab → Ab is represented by Z.
It implies that f is a group homomorphism G → A G m/k , and hence is an element of the image of F 0 (A). Therefore (Hom Ab (G, A G m/k ), F 0 (A)) satisfies the universality of the kernel of F 1 (A). It implies that (H om Ab (G, G m/k ), F 0 ) satisfies the universality of the kernel of F 1 .
We denote by PAb k the category of multiplicative analytic groups over k and natural transformations. By Proposition 4.2, the correspondence G H om Ab (G, G m/k ) gives a functor H om Ab (•, G m/k ) : Ab op → PAb k . Now we state a p-adic analogue of the Pontryagin duality. By the axiom of choice and the smallness of Ab, Theorem 4.5 implies that Ab op is naturally equivalent to PAb k . However, there seems to be no natural construction of an inverse of H om Ab (•, G m/k ). Although the fullness and the faithfulness immediately follows from the proof of Corollary 3.8 (ii) because the maximal spectrum of O G k modulo the maximal ideal of O k is canonically homeomorphic to the Stone-Čech compactification of G, we give an alternative proof constructing an "inverse correspondence". In order to verify Theorem 4.5, we prepare several notions and lemmata.
Let G be a multiplicative analytic group over k. We denote by H om PAb k (G , G m/k ) the discrete set Hom PAb k (G , G m/k ) equipped with the pointwise multiplication. We note that H om PAb k (G , G m/k ) is not U -small in our context, and hence is not an object of Ab. Therefore the correspondence G H om PAb k (G , G m/k ) does not give a functor Nevertheless, it will essentially play a role of an inverse of H om Ab (•, G m/k ). Indeed, it is functorial in the sense that for any natural transformation F between multiplicative analytic groups G 0 and G 1 over k, the map is a group homomorphism, and the correspondence F H om PAb k (F, G m/k ) preserves identities and compositions.
Let G be a U -small discrete Abelian group. For any g ∈ G, ι d G (g) is a group-like element of C 0 (G, k) because ι d G preserves the comultiplication, and hence ι d G (g) ∈ G m/k (C 0 (G, k)) yields a natural transformation η Ab (G)(g) : H om Ab (G, G m/k ) ⇒ G m/k by Lemma 4.3 and Yoneda's lemma.
Lemma 4.6. For any U -small discrete Abelian group G, the map is a group isomorphism.
Proof. The map η Ab (G) is a group homomorphism because the evaluation at C 0 (G, k) preserves the multiplication by the definition of H om PAb k (G, G m/k ). For a Hopf monoid homomorphism ϕ : C 0 (k) → C 0 (G, k), we denote by is bijective by Yoneda's lemma. Put H := F −1 • η Ab (G). It suffices to verify the bijectivity of H.
Put G := coker(ϕ). Since G is a quotient of Z ⊕I , G is U -small. We denote by π the canonical projection Z ⊕I G. By the left exactness of the functor H om Ab (•, A G m/k ) : Ab op → Ab for any commutative monoid object A in (Ban ur ≤ (k),⊗ k , k), (H om Ab (G, G m/k ), H om Ab (π, G m/k )) satisfies the universality of the kernel of F identified with H om PAb k (ϕ, G m/k ) through the natural isomorphisms H om Ab (Z ⊕I , G m/k ) ⇒ G I m/k and Thus G is naturally isomorphic to H om Ab (G, G m/k ).
By Lemma 4.3, Lemma 4.6, and Lemma 4.7, we obtain the following: Corollary 4.8. Every multiplicative analytic group G over k is representable by C 0 (G, k) for a U -small Abelian group G isomorphic to H om PAb k (G , G m/k ).
Proof of Theorem 4.5. The fullness and faithfulness immediately follow from Lemma 4.6. The essential surjectivity precisely coincides with the assertion of Lemma 4.7.
By Theorem 4.5, we obtain the following: Corollary 4.9. The category PAb k forms an Abelian category with respect to a natural Ab-enrichment, and admits all U -small colimits and all Usmall limits.
Remark 4.10. The functor C 0 (•, k) : Ab → Hopf(Ban ur ≤ (k),⊗ k , k) does not preserve U -small limits. For example, the limit of (Z/nZ) n∈N\{0} in Ab with respect to the canonical projections is Z equipped with the discrete topology, while the colimit of (µ n/k ) n∈N\{0} ∼ = (H om Ab (Z/nZ, G m/k )) n∈N\{0} in PAb k is the functor µ ∞/k : CAlg(k) → Ab which assigns the torsion group of A G m/k to each commutative monoid object A in (Ban ur ≤ (k),⊗ k , k). The natural transformation µ ∞/k ⇒ H om Ab ( Z, G m/k ) given by the universality of the colimit of (µ n/k ) n∈N\{0} is not a natural isomorphism, because ι d Z : Z → C 0 ( Z, k) G m/k assigns to 1 ∈ Z a Z-torsionfree element. It implies that µ ∞/k is not a multiplicative analytic group over k and does not admit a maximal multiplicative analytic subgroup over k.

Profinite Abelian group Let A be a compact Hausdorff flat
linear topological O k -algebra. Then A × ⊂ A is the intersection of the preimages of the multiplicative groups of finite quotients, and hence is a closed subset. Therefore it forms a profinite group. We denote by A G m/O k the discrete group which shares the underlying group with A × .
The correspondence A A G m/O k restricted to U gives a functor G m/O k : CAlg(O k ) → Ab, which is a formal geometric counterpart of S 1 .
For any U -small set I, the correspondence Ab is said to be a torus if it is naturally isomorphic to G I m/O k for some U -small set I, and is said to be a multiplicative formal group over O k if it is naturally isomorphic to the kernel of a natural transformation between tori . We establish a p-adic Pontryagin duality between profinite Abelian groups and multiplicative formal groups over O k in a parallel way to the p-adic Pontryagin duality in §4.1.  For profinite Abelian groups G and H, we denote by H om PAb (G, H) the discrete Abelian group whose underlying set is the set of continuous group homomorphisms G → H and whose operation is the pointwise multiplication. Let G be a U -small profinite Abelian group. The correspondence A H om PAb (G, A × ) restricted to U gives a functor We note that this convention is misleading because H om PAb (G, G m/O k ) is not a functor which assigns the discrete Abelian group of continuous group homomorphism G → A G m/O k to each commutative monoid object A in (C ch fl ,⊗ O k , O k ), but employ it in order to formulate a p-adic Pontryagin duality in a parallel way to the p-adic Pontryagin duality in §4.1. Since G m/O k is a counterpart of S 1 , Hom PAb (•, G m/O k ) is a p-adic analogue of the Pontryagin dual.
forms a multiplicative formal group over O k .
In order to verify Proposition 4.12, we prepare several lemmata.
which is bijective by the universality of the Iwasawa algebra. Therefore F is a natural equivalence.
Let I be a set. We denote by Z⊕ I the profinite completion of the direct sum Z ⊕I of I-copies of the underlying discrete group of Z. For each i ∈ I, we denote by ι I,i : Z → Z⊕ I the composite of the i-th canonical embedding Z → Z ⊕I and the canonical embedding Z ⊕I → Z⊕ I . Lemma 4.14. Let I be a set. For any profinite group G, the map H om PAb ( Z⊕ I , G) → G I , χ → (χ(ι I,i (1))) i∈I is a group isomorphism, where G I is equipped with the pointwise multiplication.
Proof. Since every subgroup of Z of finite index is closed, ι I,i is continuous for any i ∈ I. Therefore the assertion immediately follows from the universality of the completion and the direct sum, because the forgetful functor PAb → Ab is represented by Z. Proof. The assertion immediately follows from Lemma 4.13 applied to the case G = Z⊕ I and Lemma 4.14.
Proof of Proposition 4.12. For a set I and a profinite group H, denote by ϕ I,H the group isomorphism H om PAb ( Z⊕ I , H) → H I in Lemma 4.14. Denote by ϕ 0 the group homomorphism ϕ −1 G,G (id G ) : Z⊕ G → G, and by ϕ 1 the composite of the group homomorphism ϕ ker(ϕ),ker(ϕ) : Z⊕ ker(ϕ 0 ) → ker(ϕ 0 ) and the inclusion ker(ϕ 0 ) → Z⊕ G . By Proposition 3.7 (i), form Hopf monoid homomorphisms. By Lemma 4.13 and Corollary 4.15, they induce natural trans- Since ϕ 0 is a continuous surjective group homomorphism between compact Hausdorff topological groups, ker(ϕ 0 ) ⊂ Z⊕ G is closed and the group isomorphism Z⊕ G / ker(ϕ 0 ) → G induced by ϕ 0 is a homeomorphism. Therefore (H om PAb (G, G m/O k ), F 0 ) satisfies the universality of the kernel of F 1 by a completely similar argument in the second paragraph in the proof of Proposition 4.2.
We denote by Ab O k the category of multiplicative formal groups over O k and natural transformations. By Proposition 4.12, the correspondence  By the axiom of choice and the smallness of PAb, Theorem 4.16 implies that PAb op is naturally equivalent to Ab O k , but there is an issue on an inverse similar to that for H om Ab (•, G m/k ). Although the fullness and the faithfulness immediately follows from the proof of Corollary 3.8 (i) because the Berkovich spectrum of C(G, k) is canonically homeomorphic to G by [1] 9.2.7 Corollary, we give an alternative proof constructing an "inverse correspondence". For this purpose, we prepare several notions and lemmata.
Let G be a multiplicative formal group over O k . We denote by , and an open subset U ⊂ A. It actually forms a topological group by the definition of the pointwise multiplication. By a reason similar to that for but is functorial in the sense that for any natural transformation F between multiplicative formal groups G 0 and G 1 over O k , the map is bijective by Yoneda's lemma. Put H := F −1 •η PAb (G). It suffices to verify the bijectivity of H. We have H(g)(ι c Z (n)) = ι c G (g) n for any (g, n) ∈ G × Z by the construction. In particular, we obtain H(g)(ι c Then π G 0 (ϕ(ι c Z (1))) is a group-like element of O k [G/G 0 ], and hence there is a unique g G 0 ∈ G/G 0 such that π G 0 (ϕ(ι c Z (1))) = ι c G/G 0 (g G 0 ). By the uniqueness, (g G 0 ) G 0 forms an element of the profinite completion of G, and hence corresponds to a unique g ∈ G. It implies ϕ(ι c Z (1)) = ι c G (g). For any n ∈ Z, we have For any open subgroup G 0 ⊂ G, the composite of ι c G and π G 0 gives a continuous group homomorphism ϕ :    Following the traditional convention of a representation of an algebraic group, we define the notion of a representation of formal group schemes of a certain type. Let G be a functor CAlg(O k ) → Grp represented by a commutative Hopf monoid object A in (C ch fl ,⊗ O k , O k ). A representation of G is a right A-comodule object in (C ch fl ,⊗ O k , O k ). A representation of G is said to be irreducible if it admits precisely two quotients as representations of G . By Theorem 3.2, the irreducibility of a representation of G is equivalent to the irreducibility of its dual as a Banach left A D ch fl -module in the sense that it admits precisely two closed A D ch fl -submodules. The left regular A D ch fl -module forms a "universal family" of irreducible unitary Banach k-linear representations of the discrete group G (O k ) in the sense that for any irreducible unitary Banach k-linear representation V of G (O k ) which is "analytic" in the sense that it admits a continuous fl . Let G be a multiplicative formal group over O k . By Corollary 4.19, G is represented by a commutative Hopf monoid object A in (C ch fl ,⊗ O k , O k ). Therefore the notion of a representation of G makes sense as long as we fix A, which is unique up to Hopf monoid isomorphism. At least, the following holds for any choice of such an A: Theorem 4.22. Every representation of a multiplicative formal group G finitely generated over O k is completely reducible, that is, admits a homeomorphic O k -linear isomorphism to the direct product of finitely many irreducible representations preserving the action of G . suffices to show that every C(G, k)-module object V in (Ban ur ≤ (k),⊗ k , k) of finite dimension admits a C(G, k)-module isomorphism to the direct sum of finitely many irreducible C(G, k)-module objects in (Ban ur ≤ (k),⊗ k , k). It immediately follows from the fact that idempotents generates a dense k-subalgebra of C(G, k).

Fourier transform We denote by
is naturally isomorphic to the underlying O k -algebra of A. Therefore if X is an affine formal scheme over Spf(O k ) represented by a commutative compact Hausdorff flat linear topological O k -algebra, then H 0 (X , A 1 O k ) can be regarded as the underlying O k -algebra of global sections. Therefore H 0 (X , A 1 O k ) is a generalised notion of a ring of functions on X .
Let G be a U -small profinite Abelian group. We denote by G the composite of H om PAb (G, G m/O k ) and the forgetful functor Ab → Set. Let f ∈ C(G, k) D ur Ban . For a commutative monoid object A in (C ch fl ,⊗ O k , O k ), we denote byf (A) : G(A) → A the group homomorphism which assigns to each It gives a nontrivial conversion connecting two function rings C(G, k) and H 0 ( G, A 1 O k ), and is a p-adic analogue of Fourier transform. We note that we do not need to take the first dual in the Archimedean setting, because every Hilbert space admits a canonical isomorphism to its first dual. We show a counterpart of Plancherel's theorem.
Theorem 4.23. For any U -small profinite Abelian group G, F G is a ring isomorphism.
Proof. We denote by δ : G → C(G, k) D ur Ban the map which assigns the delta function δ g : C(G, k) → k, f → f (g) concentrated at g to each g ∈ G. The assertion immediately follows from the fact that the map H 0 ( G, Ban which assigns to each natural transformation h : Zp , which is a homeomorphic isomorphism and is called Iwasawa's isomorphism. The evaluation map C(Z p , k) × Z p → k extends to a unique continuous O k -bilinear Ban . Combining these two homeomorphic O k -algebra isomorphisms, we obtain a homeomorphic O k -algebra isomor- , which is called Amice transform. We will observe in Theorem 4.24 that F is an extension of Amice transform. The ] forms a commutative cocommutative Hopf monoid ] is naturally equivalent to the unitary group of dimension 1, that is, the functor which assigns to each commutative monoid object A in (C ch fl , . The multiplicative formal group H om PAb (Z p , G m/O k ) is also naturally equivalent to the unitary group of dimension 1. Indeed, the evaluation at 1 ∈ Z p gives a natural isomorphism between them. Therefore F Zp also can be canonically regarded as an O k -algebra isomorphism C(Z p , k) D ur Ban →  3, and hence A n is canonically isomorphic to C(Z p , k) as a Hopf monoid object in (Ban ur ≤ (k),⊗ k , k). We study an explicit presentation of A n for the general case. Put ∆ n := {i ∈ N | i < n} and J n : is a homeomorphic isomorphism, the underlying Banach k-vector space of A n can be canonically identified with C 0 (N ∆n × N Jn , k). We study its O kalgebra structure through the presentation.
The projection O N ∆n ×N Jn k O N ∆n k , (c a,b ) (a,b)∈N ∆n ×N Jn → (c (a,0) ) a∈N ∆n induces an isometric k-linear homomorphism C 0 (N ∆n , k) → A n . The projection corresponds to the embedding U n 1/O k ⇒ U n/O k into the multiplicative formal subgroup of diagonal matrices. 21. Therefore C 0 (N ∆n , k) regarded as a closed k-subalgebra of A n is canonically identified with C(Z n p , k). Thus the embedding of the completion of the maximal torus corresponds to the embedding of the commutative O k -algebra of continuous functions on the weight space Z n p . We describe A n in terms of difference operators on C(Z n p , k). For each i ∈ ∆ n , we denote by δ i : ∆ n → Z p the characteristic function of {i}, and by ∂ ∂κ i the difference operator C(Z n p , k) → C(Z n p , k), f (κ) → f (κ + δ i ) − f (κ). For an (i, a) ∈ ∆ n × N, we abbreviate ( ∂ ∂κ i ) a to ∂ a ∂κ a i . For an a ∈ N ∆n , we abbreviate i∈∆n ∂ a(i) ∂κ a(i) i to ∂ a ∂κ a . For an element f of a Q-algebra A and an a ∈ N, we abbreviate the binomial coefficient f a ∈ A to B A (f, a).
For an (a, κ) ∈ (N ∆n ) 2 , we abbreviate i∈∆n (−1) a(i)−κ(i) B Q (a(i), κ(i)) ∈ O k to C(a, κ), where B Q (a(i), κ(i)) is an integer and hence gives an element of O k even if ch(k) = p. By the definition, the homeomorphic O k -linear isomorphism O N ∆n N ∆n × N Jn , we denote by E U ∈ A n ∼ = C 0 (N ∆n × N Jn , k) the characteristic function of U . For an (i, j) ∈ J n , we denote by δ (i,j) ∈ N ∆n × N Jn ∼ = N ∆n Jn the characteristic function ∆ n J n → N of {(i, j)}, and put E i,j := E {δ (i,j) } . For an (i, j) ∈ I n , we define X i,j as T i,j in the case (i, j) ∈ J n , and as 1 + T i,i in the case (i, j) / ∈ J n . Then the comultiplication of O N ∆n ×N Jn ] sends X i,j to n−1 h=0 X i,h ⊗ X h,j for any (i, j) ∈ I n . The descriptions of the correspondence C(Z n p , k) → C 0 (N ∆n , k) and the comultiplication of O N ∆n ×N Jn k allow us to compute the multiplication on A n in the following explicit way: (iv) For any (i, j) ∈ J n , E i,j E j,i = κ i + E {δ (i,j) +δ (j,i) } . where N ∆n is regarded as a submonoid of N ∆n × N Jn through the zero extension.
In particular, for any (f, (i, j)) ∈ C(Z n p , k) × J n , we have and E i,j f = a∈N ∆n ( ∂ a ∂κ a (f + ∂f ∂κ j ))(0)E {a+δ (i,j) } by Proposition 5.1 (v). By the results above, we obtain explicit presentations of the commutators. For any ((i, j), (j , i )) ∈ J 2 n satisfying j = j and i = i , we have E i,j E j ,i − E j ,i E i,j = 0. For any (i, j, i ) ∈ ∆ 3 n satisfying i = i , we have {T q,i,j | (i, j) ∈ I 2 } of non-commutative indeterminates by the two-sided closed ideal generated by (1 + T q,i,i )T q,i ,j − q (−1) i T q,i ,j (1 + T q,i,i ) (i, (i , j)) ∈ ∆ 2 × J 2 ∪ T q,0,0 T q,1,1 − T q,1,1 T q,0,0 − (q − q −1 )T q,0,1 T q,1,0 , T q,0,1 T q,1,0 − T q,1,0 T q,0,1 and whose comultiplication is characterised by the properties that it sends 1 + T q,i,i to ((1 + T q,i,i ) ⊗ (1 + T q,i,i )) + (T q,i,j ⊗ T q,j,i ) and T q,i,j to ((1 + T q,i,i ) ⊗ T q,i,j ) + (T q,i,j ⊗ (1 + T q,j,j )) for any (i, j) ∈ J 2 . We denote its dual by A 2,q . When q = 1, then A 2,q is canonically isomorphic to A 2 in Hopf(Ban ur ≤ (k),⊗ k , k), and hence A 2,q is a q-deformation of A 2 . Since T q,0,0 and T q,1,1 commute with each other modulo the two-sided ideal generated by {T q,1,2 , T q,2,1 }, A q,2 forms a C(Z 2 p , k)-algebra in a way similar to A 2 . We denote by O k [[T q,i,j | (i, j) ∈ ] q in C ch fl . Therefore it induces an isomorphism A 2,q → C 0 (N ∆ 2 × N J 2 , k) in Ban ur ≤ (k) in the same way as the isomorphism A 2 → C 0 (N ∆ 2 × N J 2 , k). For a finite subset U ⊂ N ∆ 2 × N J 2 , we denote by E q,U ∈ A 2,q ∼ = C 0 (N ∆ 2 × N J 2 , k) the characteristic function of U . For an (i, j) ∈ J 2 , we put E q,i,j := E q,{δ (i,j) } . Then we obtain the completely same results for A 2,q as Proposition 5.1, Lemma 5.3, and Lemma 5.6 under the careful computation of the vanishing of the contribution of T q,1,1 T q,0,0 − T q,0,0 T q,1,1 = (q − q −1 )T q,0,1 T q,1,0 . By a reasoning completely similar to Theorem 5.2, we obtain the following: We note that although A 2 and A 2,q admits the same presentation of topological generators of the closed unit discs, the O k -algebra structure heavily depends on q.