Frobenius Reciprocity for Topological Groups

We investigate the existence of left and right adjoints to the restriction functor in three categories of continuous representations of a topological group: discrete, linear complete and compact.

cardinality, provided that H is of finite index in G, Ind G H and Coind G H are isomorphic. However, in general this is not true. It is worth noting that in some of the literature the opposite convention is adopted -the left adjoint functor is called induction and the right -coinduction. This is always the case when studying the representation theory of rings. However, for p-adic groups the notions are the same as ours which is the reason for our terminology.
In this article, we wish to extend the idea of Frobenius reciprocity to continuous representations of topological groups and to investigate existence of the induction and coinduction functors. In particular, we fix a topological group G and a closed subgroup H 6 G. We look at continuous representations of such groups over an associative unitial ring R. These are topological modules ðV; T V Þ over R, such that the action map G Â V ! V is continuous with respect to the topology T V on V and the product topology on G Â V. Varying T V we obtain different categories of continuous representations for G. We are interested in three such: the category of discrete representations M d ðGÞ, where V is endowed with the discrete topology, the category of linearly topologized and complete representations M ltc ðGÞ, where the topology on V is linear and complete, and the category of compact representations M c ðGÞ, where V is given a linear, complete topology in which all quotients of open submodules are of finite length. In each of these categories we investigate the existence of a left and a right adjoint to the restriction functor Res G H . Our main tool is Freyd's Adjoint Functor Theorem. As opposed to the abstract groups case, in this setting we do not always have adjoints. Having topological, as well as algebraic conditions, creates some difficulties. As a start we need to construct products and coproducts in the aforementioned categories in a correct way. Furthermore, the functor Res G H has to commute with those, which is not always the case when taking a closed subgroup H 6 G. In the present paper we address all these problems in detail. More precisely, each section contains a main result which is a criterion for the existence of a left and a right adjoint to Res G H , i.e, we give variants of Frobenius reciprocity in each of the three categories described above. Putting those together, we establish the following: Theorem (Main). Let G be a topological group and H 6 G a closed subgroup. The restriction functor is also open. Now let us give a section-by-section outline of the present paper. Section 1 is a short introduction to the categories of interest. We set the notation and give a brief description of the objects and morphisms in each category. A more precise definition of a continuous representation of G is given.
In Section 2, we investigate Frobenius reciprocity in the category of discrete representations M d ðGÞ of G. Considering representations over modules endowed with the discrete topology is the standard approach to continuous representations. In particular, if G is a locally compact totally disconnected group, then M d ðGÞ is precisely the category of smooth representations of G [5,16,17]. This category is widely studied as examples of groups with such topology include padic groups [5,16], topological Kac-Moody groups and groups of Kac-Moody type [11]. The construction of the induction functor from a closed subgroup H 6 G in the case of smooth representations is well-known. Following Bushnell and Henniart in terminology, we generalize the construction of Ind G H to arbitrary topological groups [5]. Next, we move on to coinduction, which is more subtle. In the case of locally compact totally disconnected groups, the coinduction functor is called compact induction or induction with compact support [5,16]. In Theorem 2.3, we establish a sufficient condition for its existence in M d ðGÞ. In particular, we claim that H 6 G must also be an open subgroup. It is worth remarking that Bushnell and Henniart work over a field of characteristic zero. This is extended by Vign eras to positive characteristic. She also describes the induction and compact induction functors from a closed subgroup of a locally profinite group for modules over a commutative ring [16]. We, however, keep the generality of an associative ring with identity and arbitrary topological groups.
In Section 3, we move on to the category of linearly topologized and complete continuous Gmodules M ltc ðGÞ. First, we give a precise formulation of the continuity condition. We then begin our investigation of adjoints to Res G H . We wish to use Freyd's Theorem to establish whether Ind G H and Coind G H exist in M ltc ðGÞ. The main condition, which the category needs to satisfy to ensure such existence, is to be complete and cocomplete, i.e., to be closed under taking small limits and small colimits. Since M ltc ðGÞ is abelian, using the product-equalizer construction of limits and the coproduct-coequalizer construction of colimits, it is enough to show that the category is closed under taking products and coproducts. We show how to construct these in M ltc ðGÞ in Lemma 3.1 and Lemma 3.2, respectively. The main result of Section 3 is Theorem 3.3. It establishes the existence of the left adjoint to Res G H , i.e., Coind G H in the case when H 6 G is closed, and the existence of the right adjoint, i.e., Ind G H when H is also open. In Section 4, we describe the category of (linearly) compact continuous representations of G, denoted M c ðGÞ. The notion of linear compactness of vector spaces first appears in Lefschetz' "Algebraic Topology" [14]. He calls a vector space linearly compact if it is linearly topologized, Hausdorff, and satisfies the finite intersection property on cosets of closed subspaces. Such spaces are complete [14]. This leads to an alternative definition of linearly compact vector spaces given by Drinfeld as linearly topologized complete Hausdorff spaces with the property that open subspaces have finite codimension [8]. Dieudonn e unifies these definitions by showing they are all equivalent in the case of vector spaces [7]. Compact vector spaces are also topological duals of discrete ones [8,13]. Taking the duality viewpoint, we begin the section by constructing examples to show that if R is a field, then Res G H is cocontinuous in neither M c ðGÞ, nor M ltc ðGÞ, unless H is open. We then move on to the case of modules over an associative ring. In this setting the definitions for compactness given above are not equivalent. A linearly compact topological R-module V is linearly topologized, Hausdorff, and such that every family of closed cosets in V has the finite intersection property [18]. However, this is not equivalent to V being linearly topologized, complete, and such that open submodules have finite colength. We wish to take the point of view of the latter definition as it is closer to the Beilinson-Drinfeld approach to linearly compact topological vector spaces [1]. Modules defined as above are known in the literature as pseudocompact [3,9,12]. These come up in deformation theory, in particular, they are useful when describing lifts and deformations of representations of a profinite group over a perfect field of characteristic p [3].
After we have fixed the definition of the compact topology for a module over a ring, we follow our strategy from Section 3: we construct products (Lemma 4.5), coproducts (Lemma 4.6) and investigate the existence of Ind G H and Coind G H in M c ðGÞ. Theorem 4.7 is the main result of the section. It establishes the existence of the coinduction functor for H 6 G closed, and the existence of the induction functor, given that H is also open.
We finish the article with a brief discussion of the category of Tate representations M T ðGÞ. Tate spaces, or locally linearly compact spaces as defined by Lefschetz [14], are complete linearly topologized vector spaces, such that the basis at zero is given by mutually commensurable subspaces [2]. Equivalently, a Tate space is a vector space which splits as a topological direct sum of a discrete and a compact space [8]. The latter definition also generalizes to modules over a commutative ring [1]. Hence, for a topological group G one can define the category of Tate representations, as the category with objects Tate spaces, on which G acts continuously. These are an interesting object to study as they appear not only in the phenomenal work of Tate, but also in other areas of mathematics, such as the algebraic geometry of curves, the study of chiral algebras and infinite dimensional Lie algebras, as well as in Conformal Theory [1,2]. We do not fully investigate the analogue of Frobenius reciprocity in M T ðGÞ, but we pose some questions about it.

Introducing the categories
Throughout let R be an associative ring with 1 and G a topological group. We are interested in studying continuous representations of G over R. Let us explain precisely what we mean by this.
First, recall that ðV; T V Þ is called a topological R-module if T V makes ðV; þÞ into a topological group and the R-action map Á : R Â V ! V; ðr; vÞ 7 ! r Á v is continuous with respect to T V on the right and the product topology on the left (where R is endowed with the discrete topology). With this in mind, we make the following definition: Whenever we talk about topological R-modules, we always mean left modules, but of course the results remain true for right R-modules. From the definition above it is clear that the continuity condition depends on the topology we put on V. Hence, by changing this topology we obtain different categories of continuous representations. We are mainly interested in three such: M d ðGÞ -category of discrete representations of G. The objects are continuous representations ðp; VÞ of G, such that ðV; T V Þ is a topological R-module, endowed with the discrete topology. The morphisms between two objects ðp 1 ; V 1 Þ and ðp 2 ; V 2 Þ are given by R-linear maps f : In the next two categories of interest V is given a linear topology T V . More precisely, we say that a topology T V is linear, or that V is linearly topologized, if the open R-submodules of V form a fundamental system of neighborhoods at zero [18]. This gives rise to the following categories: M ltc ðGÞ -category of linearly topologized complete representations. The objects are pairs ðp; VÞ, where V is a continuous representation of G, endowed with a linear topology T V , such that ðV; T V Þ is a complete topological space. The morphisms between two objects ðp 1 ; V 1 Þ and ðp 2 ; V 2 Þ are given by continuous R-module homomorphisms f : We give further details on the topologies of the three categories defined above, as well as the explicit meaning of the continuity condition, in the sections to follow.

Category of discrete representations
Fix a topological group G and a closed subgroup H 6 G. We study the category M d ðGÞ of discrete representations of G and M d ðHÞ of discrete representations of H. These are connected by the restriction functor: where pj H : H ! Aut R ðVÞ is the restriction of p : G ! Aut R ðVÞ to H and H V denotes V as an H-module. Let ðp; VÞ be a representation of G and / : G Â V ! V, given by ðg; vÞ 7 ! pðgÞv, be the map induced by the action of G. A discrete representation ðp; Note that 1 G always satisfies pð1 G Þv ¼ v. Since the group topology is determined by the fundamental neighborhoods of identity, without loss of generality assume that K v is an open neighborhood of 1 G . We could go even further -for every v 2 V we can construct an open subgroup f K v 6 G generated by K v . Then clearly pðkÞv ¼ v, for every k 2 f K v . If the topology of G is locally compact and totally disconnected, then M d ðGÞ is the category of smooth representations of G. The smoothness condition there states that Stab G ðvÞ is open in G for every v 2 V, which is precisely our continuity condition.
The main goal of this section is to determine when the restriction functor Res G H has a left and a right adjoint in M d ðGÞ.
We start by investigating whether a right adjoint to Res G H exists. We claim that it exists and is given by the induction functor Ind G H : M d ðHÞ ! M d ðGÞ. We define Ind G H generalizing the construction of smooth induction for locally compact totally disconnected groups [5]: and b : Hom H ð H V; WÞ ! Hom G ðV; e WÞ given by: Similarly for e / 2 Hom G ðV; e W Þ. It is routine to check that a and b are inverse to each other. w Now we move on to the case of the left adjoint. This is more subtle. Let us lay out our conventions first. We use the following standard terminology: continuous if it preserves small limits, cocontinuous if it preserves small colimits. A category C is called: complete if all small diagrams have limits in C, cocomplete if all small diagrams have colimits in C.
Since limits can be constructed as equalizers of products, a category C is complete if all morphisms in C have equalizers and C is closed under arbitrary products [15]. Hence, to check continuity of a functor F : C ! D, it is sufficient to check that F preserves those (respectively coproduct and coequalizers for the cocontinuous case).
Recall the following criterion for existence of a left adjoint to a functor [15]: Dualise the statement to obtain a criterion for a right adjoint.
We wish to use Freyd's Theorem to determine whether the restriction functor has a left adjoint. First note that (SSC) holds in M d ðGÞ: it just says that every map in M d ðGÞ can be factored through a quotient. Also note that M d ðGÞ is abelian, so equalizers of all morphisms exist. Since Res G H does not change the morphisms between the objects, it commutes with equalizers. The next step is to check whether M d ðGÞ is closed under arbitrary products. Take a collection fV i g i2I 2 M d ðGÞ, for some arbitrary set I . Let V :¼ Q i2I V i denote the product of V i as R-modules. V remains a discrete space with respect to the box topology. It also has an obvious G-module structure -G acts componentwise: ; v 2 ; :::; v n ; :: ð Þ ¼ g Á v 1 ; g Á v 2 ; :::; g Á v n ; :: However, the action is not necessarily continuous: Fix v :¼ ðv 1 ; v 2 ; :::; v n ; :: But as I is chosen arbitrarily K v does not have to be open. Therefore, the representation is not continuous at v and V 6 2 M d ðVÞ. However, consider the continuous part of V, i.e., Clearly V sm is a continuous representation of G. We claim the following: It is also a G-map: Thus, f is a morphism in M d ðGÞ and the universal property of the product is satisfied.

Linearly topologized and complete G-modules
Let R be an associative ring with 1, V a topological R-module and G a topological group. In this section we investigate the category M ltc ðGÞ of linearly topologized and complete R-modules which admit a continuous action of G. We wish to investigate the existence of adjoints to this functor. We start with the left adjoint. Following the same strategy as in Section 2 we begin by constructing arbitrary products in M ltc ðGÞ. Proof. For an arbitrary collection fðp i ; V i Þg i2I of elements of M ltc ðGÞ, let V :¼ Q i2I V i denote their product in R-mod which is a topological R-module with respect to the product topology [4].
Following Lefschetz we show that the topology on V is linear [14]. Let fU Let z n ¼ ðz 1 n ; z 2 n ; ::; z i n ; :::Þ; z i n 2 V i and n 2 L for an ordinal number L, be a Cauchy net in V. Since all V i are complete, z i n is a convergent Cauchy net in V i . Let z i :¼ lim n2L z i n : Set z ¼ ðz 1 ; z 2 ; :::; z i ; :::Þ. Let U i be an open neighborhood of z i in V i and define Since each net z i n is convergent, there exists some l i , such that z i n 2 U i , for all nPl i in L. Pick the largest l i ; i 2 J , say l. Then for all nPl in L; z i n 2 U i for all i 2 I. Thus, z 2 U and z n is convergent in V with lim n2L z n ¼ z: The G-action on V is componentwise. We want to show it is continuous.
This is an open submodule of V. Thus, we found N G and and open submodule W 6 V, such that for g Á x 2 U, with x ¼ ðx 1 ; ::; x i ; ::Þ; Ng Á ðx þ WÞ 2 U. Hence, V 2 M ltc ðGÞ.
Let A 2 M ltc ðGÞ; p i : V ! V i be the projections in M ltc ðGÞ and f i : A ! V i be a family of morphisms in M ltc ðGÞ indexed by I . As V is the product of V i in R-mod, there exists a unique R-module homomorphism f : A ! V, making the following diagram commute: The map f has the following properties: Thus, f is a morphism in M ltc ðGÞ, finishing the proof.
w To continue our investigation of adjoint functors, we would also need existence of arbitrary coproducts in M ltc ðGÞ. We construct them explicitly. Let fV i g i2I be an arbitrary collection of elements in M ltc ðGÞ. Denote by V :¼ i2I V i their coproduct in R-mod and by a i : V i ! V the canonical injections. In this case they are just inclusion maps. We follow Higgins in defining the topology on V [10]: Consider pairs ðW; s W Þ, such that: 1. W 2 M ltc ðGÞ, such that there exists a surjective R-module homomorphism q W : V ! W, which is also G-linear, 2. s W is a topology on W in which the maps q i W : V i ! W that factor through q W are continuous.
is an embedding. We endow Q ðW;s W Þ W with the product topology and V with the topology induced by q. This is a group topology [10]. The map / : R Â Q ðW;s W Þ W ! Q ðW;s W Þ W is continuous, hence, the restriction /j qðVÞ : R Â qðVÞ ! qðVÞ is also continuous. Thus, the subspace topology on q(V), and respectively the induced one on V, is an R-module topology. By Lemma 3.1 Q ðW;s W Þ W lies in M ltc ðGÞ. Every subspace of a linearly topologized space is linearly topologized [14]. Thus, as V ffi qðVÞ, both as an R-module and as a topological space, the topology on it is linear. A priori V is not necessarily complete. However, its closure V is, as it is a closed subspace of a complete space [4]. Proof. By definition V is a linearly topologized and complete space. As V ¼ i2I V i and V i is a G-module for every i 2 I, then clearly so is V. Since Q ðW;s W Þ W 2 M ltc ðGÞ, the map G Â Q ðW;s W Þ W ! Q ðW;s W Þ W is continuous. Hence, its restriction to a subspace is also continuous. Therefore, V is a continuous G-module and, hence, so is its closure V . Thus, V 2 M ltc ðGÞ as required.
Let us check that V is indeed the coproduct of fV i g i2I . Let A be any module in M ltc ðGÞ and b i : V i ! A be morphisms in M ltc ðGÞ indexed by I . Since V is the coproduct of fV i g i2I in R-mod, there exists a unique R-linear homomorphism f : V ! A, such that for every i 2 I the diagram below commutes: The map f is G-linear: Lastly, let U A be open. By continuity where q i is given componentwise by the q i W defined above. By definition of the topology on V , it follows that f is continuous, finishing the proof.
w With notation as before, we have the following diagram: Since q i W is continuous for each i 2 I, then so is q i [4]. By definition of the topology on V, q is continuous. Hence, a i is continuous for each i. This means that the topology on V is contained in the final topology with respect to a i . However, the continuity of the a i implies that V appears as one of the W, thus, the coproduct topology defined above coincides with the final topology. Now we would like to give an explicit description of the basis of open neighborhoods of 0 in V. Chasco and Dom ınguez describe this basis with respect to the final topology for a coproduct of topological abelian groups [6]. We generalize their construction to topological R-modules: Let fU i g i2I be a sequence of neighborhoods of 0, with U i a neighborhood of 0 in V i . Let J I be finite. Then is a sequence of neighborhoods of 0 in V. Hence, the basis is given by can be written in the form of (2). This is a G-module via g Á f : x 7 ! f ðxgÞ. We claim that

Category of compact representations
Let us start by defining the category M c ðGÞ of compact representations of a topological group G.
As always the objects are pairs ðp; VÞ, where ðV; T V Þ is a topological module over an associative unitial ring R and p : G ! Aut R ðVÞ is a continuous representation of G. We need to describe the topology T V on V. Firstly, we look at the case when R ¼ F is a (skew-) field. As explained in the introduction, there are a few equivalent definitions of (linear) compactness for topological vector spaces. Following Beilinson-Drinfeld we view them as topological duals V ? of discrete vector spaces V [1,8]. By a topological dual we mean the space of all continuous linear functionals on V. The topology on V ? is given by orthogonal complements of finite dimensional subspaces of V with respect to the canonical pairing [1].
Define We move on to R-modules, where R is an associative ring with 1. We call an R-module V compact if V admits a linear complete topology with the additional property that if U V is an open submodule, then V/U is of finite length. Such modules are sometimes called pseudocompact [3,9,12]. We call a topological R-module V linearly compact if it is linearly topologized, Hausdorff, and such that every family of closed cosets in V has the finite intersection property [18]. Every compact module is linearly compact [12]. We denote by M c ðGÞ the category of all compact R-modules which admit a continuous action of the topological group G. We now construct products and coproducts in M c ðGÞ. Proof. A product of linearly compact R-modules is linearly compact with respect to the product topology [18]. Since every compact module is linearly compact, then the category of compact modules is closed under products.
By exactly the same argument as in Lemma 3.1 for an arbitrary collection fV i g i2I of elements of M c ðGÞ the product V :¼ Q i2I V i in R-mod, endowed with the product topology, is a continuous G-module with respect to the componentwise action of G. w We now wish to form coproducts in M c ðGÞ. For an arbitrary collection fV i g i2I 2 M c ðGÞ, we form the coproduct V :¼ i2I V i in R-mod. To define a topology T V on V we mimic the procedure from Section 3: Let W 2 M c ðGÞ. Suppose there exists a surjective R-linear map q W : V ! W which commutes with the G-action, such that the maps q i W : V i ! W, factoring through q W , are continuous. The topology T V on V is induced by the embedding Lemma 4.6. Let fV i g i2I be an arbitrary collection of elements of M c ðGÞ. Their coproduct is the module ðV; T V Þ described above. ðUÞ is open in V. But q À1 ðUÞ ¼ i2I q À1 ðUÞ \ V i . Since each q i W : V i ! W is continuous, it follows that q À1 ðUÞ \ V i is an open submodule of V i . Hence, the quotient is of finite length. This implies that V=q À1 ðUÞ is also of finite length, showing that the topology T V is compact. Every compact space is linearly compact. Thus, V is closed in Q ðW;s W Þ W [18]. In particular, V is complete [4]. The map G Â Q ðW;s W Þ W ! Q ðW;s W Þ W is continuous, and thus so is its restriction to a subspace, i.e., V 2 M c ðGÞ. As M c ðGÞ M ltc ðGÞ and the coproducts in the two categories are constructed in the same way, Lemma 3.2 implies that V satisfies the universal property of the coproduct in M c ðGÞ, finishing the proof.
we construct a free coproduct completion