On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations

In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $K_\infty$-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a ''smooth version'' of the space of automorphic forms, whose internal structure was the topic of a famous paper of Franke. We prove that the important decomposition along the parabolic support, and the even finer - and structurally more important - decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of a paper of Franke-Schwermer and of Moeglin-Waldspurger's book, III.2.6.


Introduction
Context.It has been almost 30 years since J. Franke wrote his epochal paper [Fra98] on the space of automorphic forms A J ([G]) attached to a connected reductive group G over a number field F (and a fixed, but arbitrary ideal J of finite codimension).We investigate a "smooth version" of Franke's space of automorphic forms, to be denoted A ∞ J ([G]) and to be called smooth-automorphic forms.This is to be understood as a certain topological completion of the original space A J ([G]) -by getting rid of the condition of K ∞ -finiteness of the elements of A J ([G]) -which provides us the benefit of now having the whole group G(A) act continuously by right translation and thus regaining the representation-theoretical symmetry between the archimedean and non-archimedean places of F .This idea of replacing classical K ∞ -finite automorphic forms by a "smooth" version of them (in the above sense) is not new, is well-known to the experts, and goes back to early ideas of Bernstein, Casselman, Wallach et al., which are in turn closely related to what one calls the Casselman-Wallach completion of Harish-Chandra modules 1 .striking early discoveries of Groethendieck that a closed subspace of an LF-space does not need to be an LF-space itself.However, the existence of an LF-space structure on V is indispensable, in order to obtain a (topologized) version of the restricted tensor product theorem in the smooth-automorphic context, i.e., the very key to a local-global principle for irreducible smooth-automorphic representations.Here, we solve both problems, providing a certain global analogue of a famous result of Casselman and Wallach, see Prop.3.8 and Thm.3.15.
We express our hope that this paper will become a useful reference for anyone, who seeks a clear and concise treatment of the internal, representation-theoretical structure of the space of smooth-automorphic forms in complete generality.

Notation and basic assumptions
1.1.Number fields.We let F be an algebraic number field.Its set of (non-trivial) places is denoted S and we will write S ∞ for the set of archimedean places.The ring of adeles of F is denoted A F or simply by A, the subring of finite adeles is denoted A f .If v ∈ S, F v stands for the topological completion of F with respect to the normalized absolute value, denoted | • | v , of F .We write • A := v∈S | • | v for the adelic norm.
1.2.Algebraic groups.In this paper, G is a connected, reductive linear algebraic group over F .
We assume to have fixed a minimal parabolic F -subgroup P 0 with Levi decomposition P 0 = L 0 N 0 over F and let A 0 be the maximal F -split torus in the center Z L0 of L 0 .This choice defines the set P of standard parabolic F -subgroups P with Levi decomposition P = L P N P , where L P ⊇ L 0 and N P ⊆ N 0 .We let A P be the maximal F -split torus in the center Z LP of L P , satisfying A P ⊆ A 0 .If it is clear from the context, we will also drop the subscript "P ".
We put ǎP := X * (A P ) ⊗ Z R and a P := X * (A P ) ⊗ Z R, where X * (resp.X * ) denotes the group of Frational characters (resp.co-characters).These real Lie algebras are in natural duality to each other.We denote by •, • the pairing between ǎP and a P and we let (•, •) be the standard euclidean inner product on ǎP ∼ = R dim aP .The inclusion A P ⊆ A 0 (resp.the restriction of characters to P 0 ) defines a P → a 0 (resp.ǎP → ǎ0 ), which gives rise to direct sum decompositions a 0 = a P ⊕ a P 0 and ǎ0 = ǎP ⊕ ǎP 0 .We let a Q P := a P ∩ a Q 0 and ǎQ P := ǎP ∩ ǎQ 0 for parabolic F -subgroups Q and P .Furthermore, we set ǎP,C := ǎP ⊗ R C and a P,C := a P ⊗ R C. Then the analogous assertions hold for these complex Lie algebras.
The group P acts on N P by the adjoint representation.The weights of this action with respect to the torus A P are denoted ∆(P, A P ) and ρ P denotes the half-sum of these weights, counted with multiplicity.We will not distinguish between ρ P and its derivative, so we may also view ρ P as an element of a P .In particular, ∆(P 0 , A 0 ) defines a choice of positive F -roots of G.With respect to this choice, we shall use the notation ǎG+ P for the open positive Weyl chambers in ǎG P .For P = LN ∈ P, let us write W L for the Weyl group of L with respect to A 0 , i.e., for the Weyl group attached to the (potentially non-reduced) root system given by the set of all ±α, where α ∈ ∆(P 0 , A 0 ) is a positive F -root of G, which is not in ∆(P, A P ).Following [Moe-Wal95], II.1.7,we will write W (L) for the set of representatives of W L \W G of minimal length, for which wLw −1 is again the Levi subgroup of a standard parabolic F -subgroup of G.

Locally compact groups.
1.3.1.Generalities.We put G ∞ := R F/Q (G)(R), where R F/Q denotes the restriction of scalars from F to Q.We shall also write G v := G(F v ), v ∈ S, whence G ∞ = v∈S∞ G v .The analogous notation is used for groups different from G. Lie algebras are denoted by the same but lower case gothic letter, e.g., g ∞ = Lie(G ∞ ) or a P,v = Lie(A P,v ).Moreover, we write g = g ∞ .For every real Lie algebra h, we denote by Z(h) the center of the universal enveloping algebra U(h) of the complex Lie algebra h C = h ⊗ R C. In this paper, J will always stand for an ideal of finite codimension in Z(g).
We fix a maximal compact subgroup K A of G(A), which is in good position with respect to our choice of standard parabolic subgroups, cf.[Moe-Wal95] I.1.4.It is of the form ∈S∞ K v is a maximal compact subgroup of G(A f ), which is hyperspecial at almost all places.For each v ∈ S, we choose the Haar measure dg v on G v v with respect to which vol(K v ) = 1.The product measures dg ∞ := v∈S∞ dg v , dg f := v / ∈S∞ dg v , and dg := dg ∞ • dg f are then Haar measures on the respective group.Once and for all, we will fix a cofinal sequence {K n } n∈N (subject to the conditions K n ⊃ K n+1 and n K n = {id}), forming a neighbourhood base of id ∈ G(A f ) of open compact subgroups We denote by H P : L P (A) → a P,C the standard Harish-Chandra height function, cf.[Moe-Wal95] I.1.4(4).The group L P (A) 1 := ker H P = χ∈X * (LP ) ker( χ A ) then admits a direct complement A R P ∼ = R dim aP + in L P (A) whose Lie algebra is isomorphic to a P .With respect to K A , we obtain an extension H P : G(A) → a P,C by setting H P (g) = H P (ℓ) for g = ℓnk.Recall that K A has trivial intersection with A R G (but may intersect non-trivially with A G,∞ ).The same is true for the image of G(F ) via the diagonal embedding into G(A).
The Lie algebra a P of the connected Lie group A R P is viewed as being diagonally embedded into a P,∞ .Moreover, we will denote by S(a P,C ) := n∈N Sym n (a P,C ) the symmetric algebra attached to a P,C , which we will think of as being identified with Z(a P ) as well as with the algebra of polynomials on ǎP,C , cf.
[Moe-Wal95], I.3.1.Analogously, S(ǎ G P,C ) will denote the symmetric algebra of ǎG P,C , which may be viewed as the space of differential operators with constant coefficients on ǎG P,C .1.3.2.Group norms.Once and for all we fix an embedding ι G : G → GL N defined over F and define the adelic group norm We note that g = g ∞ • g f for all g = (g ∞ , g f ) ∈ G(A) and that there exist c 0 , C 0 ∈ R >0 such that (1.1) g ≥ c 0 and gh ≤ C 0 g h for all g, h ∈ G(A), cf.[Moe-Wal95], I.2.2.1.4.Locally convex topological vector spaces, direct limits and LF-spaces.We will use the abbreviation LCTVS to refer to complex, Hausdorff, locally convex topological vector spaces.Moreover, we will use the notion of a strict inductive limit of an increasing sequence of LCTVSs as follows: Let (V n ) n∈N be a sequence of LCTVSs such that for each n, V n is a closed topological vector subspace of V n+1 (not necessarily a proper one).The strict inductive limit lim n→∞ V n of the sequence (V n ) n∈N is the space V := ∞ n=1 V n equipped with the finest locally convex topology such that the inclusion maps ι n : V n ֒→ V are continuous.Consequently, a linear map φ : V → V ′ into an LCTVS V ′ is continuous if and only if its restriction to each V n , n ∈ N, is, and a basis of neighbourhoods of 0 in V is given by the family of subsets where U n runs through a basis of neighbourhoods of 0 in each V n and "AConv" denotes the absolute convex hull in V .The space V induces on each step V n its original topology with which V n becomes a closed subspace of V .If each V n is complete (resp., barrelled), then so is V .If each V n is a Fréchet space, we say that V is an LF-space ("limit Fréchet") with a defining sequence (V n ) n∈N .In this case, V is complete, barrelled and bornological, but it is not Fréchet unless (V n ) n∈N becomes stationary (i.e., V = V n for some n), because it cannot be Baire (the closed steps V n have empty interior, as otherwise they would be absorbing and hence all of V ).We remark that, if each space V n is finite-dimensional, then V carries its finest locally convex topology.See [AGro75], Chp. 4, Part 1, Prop. 1, Cor. 1, Prop. 2 and Prop.3.
For a family {W n } n∈N of LCTVSs, we denote by n∈N W n the locally convex direct sum of the spaces W n , i.e., the strict inductive limit As it is well-known, a closed subspace W of an LF-space V does not need to be an LF-space, cf.[AGro54], p. 89.1.5.Representations.
1.5.1.A few necessary generalities.A word on our notions concerning representations (which are the standard ones, to be found, for instance in [Bor-Wal00]), mainly put here to explain their interplay with LF-spaces and in order to fix our specific use of notation, which is tailored to fit the needs when treating representations of adelic groups.To this end, let G be a second-countable, locally compact, Hausdorff topological group (e.g., then the latter is equivalent to (g, v) → π(g)v being separately continuous, i.e., that π(g) is a continuous linear operator V → V for all g ∈ G and that for each v ∈ V the orbit map is continuous, cf.[Bou04], VIII, §2, Sect. 1, Prop. 1.We let C G be the category of G-representations with morphisms the G-equivariant continuous linear maps, and irreducibility and equivalence of representations is henceforth meant within Hilbert space and π(g) preserves its Hermitian form for all g ∈ G.In particular, if G is compact, then any irreducible representation is finite-dimensional, cf.[Joh76], Thm.3.9, and admits an Hermitian form, with respect to which it is unitary.We will use the notions of smoothness of representations (π, V ) ∈ C G and vectors v ∈ V , as well as the one of It is worth noting that if G is totally disconnected, one obtains an equivalence of categories between the subcategory C sm,adm G of smooth admissible representations in C G and the category of abstract admissible Grepresentations, see [Bor-Wal00] X, §5.1 (i.e., using the notion of admissible representation as it is normally done in p-adic representation theory, cf.[Cas95], where V is given the discrete topology instead the one of an LCTVS): This equivalence is given by attaching to an abstract admissible G-representation the LF-topology defined by writing it as the countable increasing union of its invariant subspaces under a suitable cofinal sequence of open compact subgroups, see [Bor-Wal00] X, §1.3 & §5.1.As pointed out in §1.4,this LF-space topology is the finest locally convex topology.In particular, as any vector subspace of an LCTVS, which carries its finest locally convex topology, is closed, this equivalence respects irreducibles.1.5.2.Smooth representations.Remaining in the totally disconnected case, let {K n } n∈N be a decreasing cofinal sequence of compact open subgroups of G, and given a representation (π, V ) of G, let E Kn : V → V be the standard continuous projection onto V Kn : for some fixed Haar measure dk on K n .We define closed subspaces V n , n ∈ N, of V as follows: (1.2) , II, §4, Sect.5, Prop.6.We also record the following general lemma.

8.(i
= lim n U Kn , hence U is smooth.Finally, in order to prove that V /U smooth (if it is a G-representation at all, see Rem. 1.7 below), it suffices to note that there are the following isomorphisms of LCTVS: (1.6) Here, the first and second isomorphisms are the canonical ones (cf.[Bou03], II, §4, Sect.5, Prop.8.(ii)), while the last isomorphism is obtained from taking the inverse of the limit of the LCTVS-isomorphisms Remark 1.7.In this generality it is not automatic that the quotient of two (smooth) representations V /U as in Lem.1.4 is a representation.Indeed, it may fail both that the natural map U is continuous as well as that the quotient V /U is complete.However, suppose that each quotient V Kn /U Kn is complete and barrelled.Then, V /U is also complete and barrelled by (1.6) and [AGro75], Chp. 4, Part 1, Cor. 1 and Prop.3. Thus, the separate continuity of the map Likewise, if G is a real Lie group and (π, V ) a G-representation, then the C ∞ -topology on its space of smooth vectors, i.e., the subspace topology from C ∞ (G, V ), coincides with the locally convex topology given by the seminorms p ν,X (v) := ν(π(X)v), ν running through the continuous seminorms on V and X through U(Lie(G)), cf.[Cas89I], Lem.1.2.Hence, also for Lie groups G any subrepresentation of a smooth G-representation is smooth.Moreover, in the context of strict inductive limits, we obtain Lemma 1.8.Let V be a strict inductive limit of a sequence (V n ) n∈N of complete LCTVSs.Let (π, V ) be a representation of a real Lie group G such that for each n, the closed subspace V n ⊆ V is a smooth G-representation.Then, (π, V ) is a smooth G-representation.
Proof.Obviously, every v ∈ V is smooth, as the steps V n inherit from V their original topology.It therefore suffices to prove that for every continuous seminorm ν on V and X ∈ U(g), the seminorm p ν,X is continuous, i.e., by [Bou03], II, §4, Sect.4, Prop.5(ii), that the restrictions p ν,X Vn = ν Vn • π(X) Vn are continuous, which holds by the smoothness of V n .1.5.3.Representations of G(A).Let now (π, V ) be a representation of G(A).We will write and dense in V (by a classical argument of Gårding, see [Mui-Žun20], (10) and Cor.2(3) for an explicit proof in the current setup).We topologize V ∞ A as the inductive limit Kn , where V ∞ R is equipped with the C ∞ -topology.In this way, V ∞ A becomes a representation of G(A) carrying a (potentially strictly) finer topology than the subspace topology coming from V ∞ R (resp., V ).We say that (π, V ) is a smooth G(A)-representation, if V = V ∞ A holds topologically.One checks easily that (π, V ) is a smooth G(A)-representation if and only if its restrictions to G ∞ and G(A f ) are smooth representations.For every and hence V ∞ A becomes an LF-space in this case.We have the following basic Lemma 1.9.Let (π, V ) be a smooth G(A)-representation, and let U ⊆ V be a subrepresentation.Then, U is a smooth G(A)-representation.Moreover, if V /U is a G(A)-representation (which holds, e.g., if each quotient V Kn /U Kn is complete and barrelled, cf.Rem.1.7), then V /U is a smooth G(A)-representation.
Proof.Since U is a smooth G(A f )-representation by Lem.1.4 and is obviously a smooth G ∞ -representation, it is a smooth G(A)-representation. Next, suppose that V /U is a representation of G(A).To prove that it is a smooth G ∞ -representation, we note that by restricting the natural exact sequence of G ∞ -representations 0 → U ֒→ V ։ V /U → 0 to the subspaces of G ∞ -smooth vectors, we obtain a sequence of homomorphisms of remains surjective, whence this sequence is exact, too.Thus, the identity map defines a continuous linear bijection We also recall the notion of a (g ∞ , K ∞ , G(A f ))-module: That is a (g ∞ , K ∞ )-module V 0 , which carries a smooth linear action of G(A f ) (i.e., each vector v ∈ V 0 has a smooth orbit map c v : G(A f ) → V 0 ), which commutes with the actions of g ∞ and K ∞ , see [Bor-Wal00], XII, §2.2.A (g ∞ , K ∞ , G(A f ))-module is called admissible, if for all irreducible (and hence finite-dimensional) representations (ρ, W ) of K A the space The following lemma will be useful later: Lemma 1.10.Let (π, V ) be a smooth G(A)-representation.Let V 0 be an admissible (g ∞ , K ∞ , G(A f ))submodule of V (K∞) that is dense in V .Then, V (K∞) = V 0 and so (π, V ) is admissible.
Proof.Let us prove that for all irreducible (hence finite-dimensional) representations (ρ ∞ , W ∞ ) of K ∞ and n ∈ N, we have the equality of ρ ∞ -isotypic components To this end recall the operator E Kn ρ∞ : V → V , (1.12) where d(ρ ∞ ) and ξ ρ∞ are, respectively, the degree and character of ρ ∞ .Then E Kn ρ∞ is a continuous projection onto V Kn (ρ ∞ ) and restricts to a projection of V 0 onto V Kn 0 (ρ ∞ ), cf.[Mui-Žun20], Lem.19.Thus, we have , where the inclusion holds by the continuity of E Kn ρ∞ , and the last equality is true because V Kn 0 (ρ) is finitedimensional.Since the reverse inclusion is obvious, this proves (1.11).Therefore, which shows the first claim.As V is smooth, by our above observation, it is admissible, if and only if its underlying However, by what we have just proved, V (K∞) = V 0 , and so admissibility of V follows from the assumption that V 0 is admissible.1.5.4.Casselman-Wallach representations.Following [Cas89I,WalII92], we say that a G ∞ -representation (π, V ) on a Fréchet space V is of moderate growth, if for every continuous seminorm p on V , there exist a real number d = d p and a continuous seminorm q = q p on V such that Here, g ∞ is the group norm from §1.3.2 as applied to g ∞ = (g ∞ , id) ∈ G(A).We refer to [WalII92], §11.5 for the basic properties of smooth representations of moderate growth.We will call a smooth admissible G ∞ -representation (π, V ) of moderate growth a Casselman-Wallach representation of G ∞ , if its underlying (g ∞ , K ∞ )-module V (K∞) is finitely generated (cf.[WalII92] §11.6.8) or equivalently Z(g)-finite (cf.[Vog81], Cor.5.4.16).
A smooth representation (π, V ) of G(A) such that for every n ∈ N, V Kn is a Casselman-Wallach representation of G ∞ , will be called a Casselman-Wallach representation of G(A).By the above said, every Casselman-Wallach representation (π, V ) of G(A) is admissible and V is an LF-space.Moreover, the following well-known, classical results of Wallach translate into the "global" setup: and W (K∞) are isomorphic. Proof.
(1) The analogous assertion about Casselman-Wallach representations of G ∞ follows from [WalII92], Lem.11.5.2 and implies the claims about U and V /U using Lem.1.9 together with the following elementary fact: For every n ∈ N, the assignment v By a celebrated result due to Casselman and Wallach, cf.see [WalII92], Thm.11.6.7(2),for every n ∈ N we can extend the By the uniqueness of this extension, T n+1 extends T n .Hence, going over to the direct limit over n ∈ N, we obtain a well-defined isomorphism of G ∞ -representations T : V → W , which extends T 0 (and is uniquely determined by this property).Since 2. The LF-space of smooth-automorphic forms 2.1.Spaces of functions of uniform moderate growth.Let C ∞ (G(F )\G(A)) be the space of smooth functions G(A) → C, which are invariant under multiplication by G(F ) from the left.For n, d ∈ N we define the space ) that satisfy the following uniform moderate growth condition: For every X ∈ U(g), We note that the space Kn with the locally convex topology, defined by the seminorms p d,X , X ∈ U(g).Let us point out that even with this natural topology, (2.2) does not necessarily define an embedding of closed subspaces.
For fixed n and d, it will be useful to relate the space C ∞ umg,d (G(F )\G(A)) Kn to the spaces of smooth functions of uniform moderate growth on G ∞ , which is done in the following standard way: We recall that there exists a finite set Again, we equip C ∞ umg,d (Γ c,Kn \G ∞ ) with the locally convex topology defined by the seminorms p ∞,d,X , X ∈ U(g).With this setup in place, we obtain which restricts to an isomorphism of their closed subspaces of A R G -invariant functions.Proof.It follows from [Wal94], Lemma 2.7, that the spaces C ∞ umg,d (Γ c,Kn \G ∞ ) (and hence their finite direct sum) are Fréchet.The map f → (f ( • c)) c∈C is obviously a continuous bijection, whose inverse is given by the following construction: For a In order to show that this inverse is continuous (only from which it will follow that Kn for the space of smooth, left-G(F )-invariant functions, which satisfy (2.1) for a given d ∈ N, and let a subscript "(Z(g))" denote the subspace of Z(g)-finite vectors.We take the opportunity to point out that -although each of the corresponding claims can be found in the sparse literature on smooth-automorphic forms -actually none of the following three spaces is Fréchet, when equipped with the seminorms p d,X , X ∈ U(g): . Indeed, one may easily verify that none of the above spaces is complete.
In our eyes, the resulting topological intricacies (as well as the resulting issues in providing completely accurate references) make it necessary to lay down some details on the topological structure of the (in this paper yet to be defined) space of smooth-automorphic forms and the canonical action of G(A) by right translations, which we will do in the remainder of this section.The reader, who is familiar with these functional-analytic subtleties, may skip them and head on to §3.

Smooth-automorphic forms and absolute bounds of exponents of growth.
2.2.1.Spaces of smooth-automorphic forms.Recall our arbitrary, but fixed ideal J ⊳ Z(g) of finite codimension.For each n ∈ N, we let Kn of functions, which are annihilated by the ideal J n .We obtain Proposition 2.5.Acted upon by right translation R ∞ , the spaces Proof.The arguments of [Wal94], §2.5 and Lem.2.7 imply that ] and is obviously Z(g)-finite.This shows the claim.
Definition 2.6.A smooth-automorphic form is an element of one of the spaces where J runs over the ideals of finite codimension in Z(g).
Remark 2.7.Our definition is obviously compatible with the usual one: A smooth function f : , annihilated by an ideal J ⊳ Z(g) of finite codimension and of uniform moderate growth.See [Wal94], §6.1, where to our knowledge the notion of smooth-automorphic forms has first been introduced (in the context of real reductive groups and their arithmetic subgroups) or [Cog04], §2.3 (for G = GL n ).
We want to consider A ∞ J (G) as a representation of G(A) under right translation, whence we have to specify a locally convex topology on A ∞ J (G), making it into a complete LCTVS, cf.§1.5.In a very first try, it is tempting to put an ordering on the set of tuples (d, n) ∈ N × N, with defining condition that ,J n+1 and then equip A ∞ J (G) with the inductive limit topology given by the natural, continuous inclusions.However, estimating the respective seminorms in question, it is a priori not clear at all that the locally convex topology on A ∞ d+1 (G) Kn+1,J n+1 (defined by the seminorms p d+1,X , X ∈ U(g)), induces the original topology on A ∞ d (G) Kn,J n (defined by the seminorms p d,X , X ∈ U(g)), Whence, it is a priori even unclear whether the latter inductive limit topology yields a Hausdorff space at all, and even less, if the resulting space is a complete LCTVS.
In order to overcome this problem, we shall make use of the following general result: Proposition 2.8.Let J ⊳ Z(g) be an arbitrary, but fixed ideal of finite codimension.Then there exists In other words, having fixed J (or, equivalently, the string of ideals {J n } n∈N ), then there exists an exponent d, such that all smooth-automorphic forms in A ∞ J (G) satisfy (2.1) for the same such d.In order to prove Prop.2.8 we shall need the following preparatory considerations: 2.2.2.Growth conditions for constant terms.Let P = LN ∈ P. Let us introduce coordinates a P ∼ −→ R nP , λ → λ, and the following multi-index notation: For a multi-index α as above, we write |α| := nP i=1 α i .
For every subset Λ of ǎP,C and N ∈ Z ≥0 , let P A R P (Λ, N ) be the space of functions φ : where Λ ′ runs through the finite subsets of Λ, and c λ,α ∈ C ("polynomial exponential functions").For a continuous, left-G(F )-invariant function f : G(A) → C, we write as usual for the constant term along P , cf. [Moe-Wal95], I.2.6.
Lemma 2.9.For every P ∈ P, let Λ P be a finite subset of ǎP,C .Then there exists an r ∈ R >0 such that Proof.This follows from the proof of [Ber-Lap20], Lem.6.12, noting that the constant r whose existence is proved in said lemma depends, in the notation of loc.cit., only on dim a 0 and on the constant max P max (λ1,...,λn P )∈ΛP max nP i=1 λ i , where • is a fixed norm on X P , and does not depend on the numbers n P .
Proof.Lacking a good reference, we sketch the proof.Arguing inductively, let n = 2.Then, by the lemma of Artin-Rees [Bos12, §2.3, Lem.1], there exists an l ∈ Z ≥0 such that I k+l The induction-step is obvious.Consequently, We are now ready to give the Proof of Prop.2.8.By Lem.2.9, it suffices to prove the following claim: For every P ∈ P there exists a finite set Λ P ⊆ ǎP,C with the following property: For every ϕ ∈ A ∞ J (G) there exists N ∈ Z ≥0 such that (2.11) First, we recall that, denoting by L the left action of U(g) on C ∞ (G(A)), we have where (•) # is the unique anti-automorphism of U(g) such that 1 # = 1 and X # = −X for all X ∈ g ∞ .Now, let P = LN ∈ P.Then, denoting the Harish-Chandra homomorphism by ν : Z(g) → Z(l), we get Y ∈ ν(Y ) + U(g)n C for all Y ∈ Z(g), and hence (2.12) Let ϕ ∈ A ∞ J (G), and let k ∈ N be such that J k ϕ = 0.For every g ∈ G(A), the function ϕ( • g) : G(A) → C is also annihilated by J k , hence by (2.12) the function ϕ Since the algebra Z(a P ) embeds canonically into Z(l), it follows that the function ϕ P ( • g) : A R P → C is annihilated by the following ideals in Z(a P ): J k l ∩ Z(a P ) ⊇ (J l ∩ Z(a P )) k ⊇ Z(a P ; Λ P , M ) for some M ∈ N, where Λ P is chosen such that J l ∩ Z(a P ) ⊇ Z(a P ; Λ P , N ) for some N ∈ N, see §2.
).We will henceforth fix such an exponent d.Recall that for each n ∈ N, A ∞ d (G) Kn,J n has been given the structure of a Fréchet space.Having fixed d, we obtain canonical continuous inclusions ι n : with closed images and we will hence equip the space of smooth-automorphic forms A ∞ J (G) with its natural LF-space topology given by (2.13) As remarked in §1.4,this makes A ∞ J (G) into a complete, barrelled, bornological LCTVS (which, as we recall, for us includes the property of being Hausdorff).It is easy to see that if (ϕ i ) i∈I is a convergent net in A ∞ J (G) with limit ϕ, then the net of complex numbers ((Xϕ i )(g)) i∈I converges in C to (Xϕ)(g) for every X ∈ U(g) and g ∈ G(A).In particular, every convergent net of smooth-automorphic forms is pointwise convergent everywhere.
Remark 2.14.Our particular choice of an exponent of growth d is only made for the convenience of normalization and can be replaced by any other integer d ′ ≥ d, without change of topology, as the following argument shows: Let us for now denote G) Kn,J n as sets for every n ∈ N. Estimating the seminorms in question, using only the fact that the adelic group norm g is bounded away from 0, cf.(1.1), one shows that the bijection given by the identity map A ∞ d (G) Kn,J n → A ∞ d+1 (G) Kn,J n is continuous for every n ∈ N and hence -domain and target space being Fréchet -a topological isomorphism by the Open Mapping Theorem.Here we note that the observation that A ∞ d (G) Kn,J n = A ∞ d+1 (G) Kn,J n as sets, which is all based on Prop.2.8, is crucial for the latter conclusion.Having said this, going over to the direct limits, G) d+1 also topologically.The reader may also find an analogue of this discussion (for real reductive groups and arithmetic sugroups) in [Li-Sun19], §3.1, there, however, without proofs.

The right regular action of G(A) on smooth-automorphic forms. The following result on A ∞
J (G) is well-known and central for all our considerations later on.However, lacking a good reference and given certain topological issues and incompatibilities in the literature, cf.Remark-Warning 2.4, we prefer to give a full proof of it.
Proposition 2.15.Acted upon by right translation R, the LF-space A ∞ J (G) becomes a smooth representation of G(A).
Proof.We will first verify that the right regular action R of G(A) defines a continuous map and hence a representation of G(A): As A ∞ J (G) is barrelled, it suffices to show that the latter map is separately continuous, cf.§1.5.To this end, let g = (g ∞ , g f ) ∈ G(A) arbitrary, but fixed, and consider the linear operator R(g) : . By construction of the LF-space topology on A ∞ J (G) it suffices to show the continuity of the restrictions R(g) : for each n, in order to obtain the desired continuity of R(g), §1.4.So, let n ∈ N be arbitrary.Prop.2.5 implies that R((g is a topological automorphism of the Fréchet space A ∞ d (G) Kn,J n .On the other hand, R((id, g f )) obviously defines a linear operator R((id, g f )) : .
is continuous for all g ∈ G(A).In order to prove continuity in the other variable, let ϕ ∈ A ∞ J (G) be an arbitrary, but fixed smoothautomorphic form and consider its orbit map c ϕ .Choose any n ∈ N such that ϕ ∈ A ∞ d (G) Kn,J n .Then, by Prop.2.5, c ϕ restricts to a smooth, and hence in particular continuous map, c ϕ : On the other hand, ϕ being right-invariant under the open compact subgroup K n , we also obtain a continuous restriction c ϕ : is continuous as it is a composition of continuous maps.Hence, ) is separately continuous and hence, by barrelledness of A ∞ J (G), also jointly continuous.
We now prove smoothness.Combining Lem.1.8 and Prop.2.5, right translation on the LF-space A ∞ J (G) is a smooth representation of G ∞ .To prove that it is also a smooth representation of G(A f ), i.e., that topologically A ∞ J (G) = lim n→∞ A ∞ J (G) Kn , we need to argue that that the identity map is bicontinuous: By definition of the strict inductive limit topology, this is equivalent to that the restrictions to the limit-steps are continuous for all n ∈ N. The second inclusion is continuous by construction, as is the map However, as the image of the latter map lands inside Km is continuous by construction, also the first map in (2.16) is continuous.
Remark 2.17.We recall that any smooth function f : G(A) → C is automatically continuous.In fact, for smooth-automorphic forms this also follows from the the fact, shown in the proof of Prop.2.15, that the orbit maps c ϕ : G(A) → A ∞ J (G) are all continuous, as ϕ = ev id • c ϕ .2.4.(Relation to) K ∞ -finite automorphic forms.Obviously, the space of (usual) automorphic forms A J (G), cf.[Bor-Jac79], §4.2, which are annihilated by a power of the fixed ideal J , identifies as the dense module (but it also follows directly from Prop.2.15 above, as ).Although dense in A ∞ J (G), and hence, topologically "almost all" of A ∞ J (G), the space A J (G) is much smaller than A ∞ J (G) from the perspective of vector spaces: Indeed, A J (G) is of countable dimension by a theorem of Harish-Chandra, cf.[HCh68], Thm. 1 (see also [Bor-Jac79], Thm.1.7 and §4.3.(i)therein), whereas A ∞ J (G) is of uncountable dimension (since it contains infinite-dimensional Fréchet spaces A ∞ d (G) Kn,J n ). 3. Smooth-automorphic representations 3.1.Smooth-automorphic representations.In this paper we propagate the idea to give preference to A ∞ J (G), rather than to A J (G), as the former allows a representation of G(A) and not only the structure of a (g ∞ , K ∞ , G(A f ))-module (which breaks the symmetry between the role of the archimedean and the non-archimedean factors of G(A)).Moreover, to round this up by a subtlety (which may suit the taste of purists among the readers), the space of functions A J (G) changes with the very choice of K ∞ , whereas A ∞ J (G) is independent of any particular additional choices: Once J is given, A ∞ J (G) depends on nothing else than the group-scheme G/F itself (which we believe is a much more satisfactory setup).
In order to substantiate this approach, we will show here that the representation theory evolving out of A ∞ J (G) is rich enough in order to recover the representation-theoretical phenomena in automorphic forms.
Indeed, as a first step and as some sort of ground-work, in this section we shall prove an "automorphic analogue" of a famous result of Harish-Chandra (on admissible G ∞ -representations and their underlying (g ∞ , K ∞ )-modules); moreover, we will establish a natural 1-to-1 correspondence between the irreducible smooth-automorphic representations of G(A) (i.e., irreducible G(A)-subquotients of A ∞ J (G)) and the usual irreducible automorphic representations (i.e., irreducible (g ∞ , K ∞ , G(A f ))-module subquotients of A J (G)); finally, we will also verify the fundamental and all-important local-global property of irreducible smoothautomorphic representations, provided by a topologized version of the restricted tensor product theorem.
In analogy with the classical definition of automorphic representations, we introduce the following Definition 3.1.A G(A)-representation (π, V ) is a smooth-automorphic representation if it is equivalent to a quotient U/W , where W ⊆ U are G(A)-subrepresentations of A ∞ J (G) (for some ideal J of Z(g) of finite codimension).Moreover, if W = 0, we say that (π, V ) is a smooth-automorphic subrepresentation.
We shortly observe that by Lem.1.9 and Prop.2.15 the word "smooth" in our terminus "smoothautomorphic representation" does not amount to an abuse of terminology as every smooth-automorphic representation is indeed a smooth G(A)-representation in the sense of §1.5.3.Remark 3.2.By definition, the notion of a smooth-automorphic representation entails the assumption that the quotient U/W is complete, as it is supposed to be a representation of G(A).We warn the reader that in general, the quotient of a complete LCTVS by a closed subspace does not need to be complete, nor is it automatic that the quotient of two G(A)-representations is again a representation, cf.Rem.1.7.It is our conjecture, however, that for all G(A)-subrepresentations W ⊆ U ⊆ A ∞ J (G), the quotient U/W is complete and defines a smooth-automorphic representation.Let us point out that according to Lem. 1.9, this is certainly the case, if each quotient U Kn /W Kn is complete and barrelled.It follows that U/W is in particular complete, if there exists an m ∈ N such that J m U = 0, as then U Kn is a closed subspace of the Fréchet space A ∞ d (G) Kn,J n for all n ≥ m and hence Fréchet itself.Since the actions of Z(g) and G(A) commute, the latter applies in particular, if U is finitely generated as a G(A)-representation.
The following lemma gives a first glimpse into the relationship between the classical automorphic representations and smooth-automorphic representations.
Proof.We need to prove that V is G(A)-invariant.Let U be the smallest closed G(A)-invariant subspace of Then, by the Hahn-Banach theorem there exists a non-zero continuous linear functional b : U → C such that b V = 0.For every φ ∈ V 0 , let us look at the function It follows that In particular, for n ∈ N such that J n φ = 0, we have J n ϕ φ = 0. Note also that ϕ φ is K A -finite on the right.Denoting by G • ∞ the identity component of G ∞ , it follows that for every finite on the right and Z(g)-finite, hence it is real analytic (see, e.g., [Bor07], 3.15).Moreover, we have This means that b vanishes on where the second equality holds because K ∞ meets every connected component of G ∞ .Thus, b is identically zero, which is a contradiction.Hence, V = U and so V is G(A)-stable.

3.2.
The general dictionary I: Admissibility and an automorphic variant of a theorem of Harish-Chandra.The following result, see Thm. 3.7 below, provides a fundamental dictionary between smoothautomorphic subrepresentations of A ∞ J (G) and automorphic subrepresentations of A J (G) (where the expression "automorphic subrepresentation" is used in the usual way, i.e., denoting a (g ∞ , K ∞ , G(A f ))-submodule): We show that the irreducible (and, more generally, even all the admissible) G(A)-subrepresentations of A ∞ J (G) are exactly the topological closures of the irreducible (resp., admissible) automorphic subrepresentations of A J (G) within A ∞ J (G).We invite the reader to view this result of ours as a global, or automorphic analogue of Harish-Chandra's result, providing a 1:1-correspondence between the G ∞ -subrepresentations of a given admissible G ∞ -representation V and the (g ∞ , K ∞ )-submodules of its underlying (g ∞ , K ∞ )-module, cf.[HCh53] or [Var77], Thm.II.7.14.
Theorem 3.7.The admissible smooth-automorphic subrepresentations stand in one-one correspondence with the admissible . Moreover, the above correspondence respects irreducibility.Hence, every admissible (resp., irreducible) automorphic subrepresentation of A J (G) lifts to an admissible (resp., irreducible) smooth-automorphic subrepresentation, recovering the original automorphic representation as its space of K ∞ -finite vectors.
Proof.Firstly, we recall that by [Bor-Jac79], Prop.4.5.(4)every irreducible automorphic subrepresentation V 0 is automatically admissible.Hence, if we manage to show that the underlying (g ∞ , K ∞ , G(A f ))-module of an irreducible smooth-automorphic subrepresentation V of A ∞ J (G) remains irreducible, then -V being a smooth G(A)-representation by Lem.1.9 and Prop.2.15, as already observed above -V is admissible, too.With this observation in mind, it therefore suffices to prove the following four claims: (1) Let V 0 be an admissible ) is an admissible G(A)-representation, and we have V (K∞) = V 0 .
(2) Let V be an admissible smooth-automorphic subrepresentation.Then, V 0 := V (K∞) is an admissible (g ∞ , K ∞ , G(A f ))-module, and we have (4) Let V be an irreducible smooth-automorphic subrepresentation.Then, the (g , and hence, smooth by Lem.1.9 and Prop.2.15.It is obvious from the construction that V 0 is a (g ∞ , K ∞ , G(A f ))-submodule of V (K∞) and dense in V .As V 0 is furthermore assumed to be admissible, Lem.1.10 implies that V (K∞) = V 0 and that V is admissible as a G(A)-representation.
3.3.The general dictionary II: Extension to all irreducibles and smooth-automorphic Casselman-Wallach representations.Thm.3.7 provides (in particular) a dictionary between the irreducible smoothautomorphic subrepresentations of A ∞ J (G) and the irreducible automorphic subrepresentations of A J (G).For a completely general understanding of the internal representation theory of the space of smoothautomorphic forms it is essential, however, to extend this comparison from irreducible subrepresentations to all irreducible subquotients, as only those will capture the representation-theoretical phenomena of A ∞ J (G) in sufficient generality.This section is devoted to such a general comparison of irreducibles.
We begin our analysis by studying smooth-automorphic Casselman-Wallach representations of G(A) (see §1.5.4), examples of which we describe in Prop.3.8 and its fundamental corollary, Cor.3.10 (representations, spanned by one smooth-automorphic form ϕ), below: Proposition 3.8.Let (π, V ) be a smooth-automorphic subrepresentation that is annihilated by a power of J (e.g., an irreducible smooth-automorphic subrepresentation).Then, V is a Casselman-Wallach representation of G(A).In particular, V is an LF-space.
To prove that every irreducible smooth-automorphic subrepresentation (π, V ) satisfies the assumption of the proposition, recall that by the irreducibility of V and the continuity of the right regular action R we have for any non-zero ϕ ∈ V .Hence, by the continuity of the action of U(g), J k V = 0 for every k ∈ N such that J k ϕ = 0.
We shall need the following variant of (1.12):For a G(A)-representation (π, V ) and an irreducible When the ambient representation V is clear from the context, we will write E ρ∞ = E ρ∞,V .The following corollary of Prop.3.8 is fundamental: of V (K∞) equals V (K∞) .To this end, note that by Lem.3.3, is a closed G(A)-invariant subspace of U .As it obviously contains W as a proper subspace, by the irreducibility of V it equals U .In particular, U 0,φ := φ (g∞,K∞,G( by [Bor-Jac79], Prop.4.5.(4), and as V is a smooth G(A)-representation by Lem.1.9 and and Prop.2.15, Lem.1.10 finally implies that V 0,φ = V (K∞) , as desired.Hence, V (K∞) is irreducible.At the same time, the other implication of Lem.1.10 shows that V is admissible as claimed.
We will now prove the second assertion: To prove the existence of V , let Let us fix a φ 0 ∈ U 0 , which is not in W 0 and denote By the irreducibility of U 0 /W 0 , the canonical homomorphism U 1 → U 0 /W 0 is surjective, hence, denoting its kernel by W 1 , we have the following isomorphisms of (g ∞ , K ∞ , G(A f ))modules: (3.13) Let us fix k ∈ N such that J k φ 0 = 0. Since by [Bor-Jac79], Prop.4.5.(4) the Since moreover the spaces W 1 ⊆ U 1 are obviously annihilated by J k , by Prop.3.8 they are Casselman-Wallach representations of G(A), and hence so is their quotient V := U 1 /W 1 by Lem.1.13.(1).Moreover, we have the following isomorphisms of (g ∞ , K ∞ , G(A f ))-modules: where the inverse of the next-to-last isomorphism is just the canonical map that is obviously surjective and is injective because = W 1 .Finally, the irreducibility of V (K∞) ∼ = V 0 implies the irreducibility of G(A)-representation V as in the proof of claim (3) in the proof of Thm.3.7.

3.4.
The general dictionary III: The local-global principle and the restricted tensor product theorem.As it is implicit in the proof of Thm.3.12, the underlying (g ∞ , K ∞ , G(A f ))-module V (K∞) of every irreducible smooth-automorphic representation V allows a completion as a Casselman-Wallach representation of G(A).By Lem.1.13.(2) this completion is in fact unique up to isomorphism of G(A)-representations.
Our next result provides the necessary local-global principle for all such irreducible smooth-automorphic Casselman-Wallach representations.Its underlying algebraic assertion seems well-known to experts (see [Cog04], Thm.3.4, for G = GL n ), albeit the decisive question, which is the correct choice of a locally convex topology on restricted tensor products, seems to remain open in the available literature, whence, lacking a precise reference, we decided to include the result in our paper and fill this gap.In order to state the result, we let ⊗ pr , resp.⊗ in , denote the completed projective tensor product, resp.completed inductive tensor product, of LCTVSs.We refer to [AGro66], I, §1, n • 's 1-3 and [War72], App.2.2, for their basic properties.
Theorem 3.15 (Tensor product theorem).Let (π, V ) be an irreducible smooth-automorphic Casselman-Wallach representation of G(A).Then, for each v ∈ S, there is an irreducible smooth admissible representation where the restricted tensor product ′ v / ∈S∞ π v is endowed with the finest locally convex topology.Among all representations with the aforementioned properties, the π v are unique up to isomorphy.

Main results: Parabolic and cuspidal support
4.1.In this section we will prove our main results on the decomposition of the space of smooth-automorphic forms along the parabolic and the more refined cuspidal support.
As far as the parabolic support of a smooth-automorphic form is concerned, our result provides a topological version of Langlands's algebraic direct sum decomposition of the space of smooth functions of uniform moderate growth, recorded and proved in two different ways in [BLS96], Thm.2.4, and a little earlier in [Cas89II], Thm.1.16 and Cor.4.7, respectively.We refer to [Lan72] for the most original source.
On the other hand, our result on the cuspidal support formally mirrors the main result of [Fra-Sch98], cf.Thm.1.4.(2), on the level of smooth-automorphic forms.As a part of our analysis, we will prove a fine structural, topological decomposition of the space of cuspidal smooth-automorphic forms, which is a "smooth" analogue of the famous theorem of Gelfand-Graev-Piatetski-Shapiro [GGPS69], §7.2, on the decomposition of the space of cuspidal L 2 -functions into a direct Hilbert sum.We refer to Thm. 4.30 below for this characterization of the irreducible cuspidal smooth-automorphic representations and to Thm. 4.33 for our main result on the cuspidal support of a general smooth-automorphic form.

A R
G -invariant smooth-automorphic forms.In order to obtain a meaningful notion of square-integrable smooth-automorphic forms, as usual, we shall pass over from G(F )\G(A) to the smaller quotient G(F )A R G \G(A) ∼ = G(F )\G(A) 1 , which is of finite volume, i.e., we shall work with left-G(F )A R G -invariant functions rather than with left-G(F )-invariant functions from now on.So, for every n ∈ N, let A ∞ d ([G]) Kn,J n be the closed (and hence Fréchet) subspace of A R G -invariant elements in A ∞ d (G) Kn,J n .We define the LF-space (4.1) Remark 4.2.In the notation of Franke, cf.[Fra98], the underlying vector space of the LF-space Franke, however, did not specify any topology on the vector space Observe that it is a priori by no means clear that the above LF-topology on A ∞ J ([G]) agrees with the subspace-topology inherited from the inclusion ) is in fact a smoothautomorphic subrepresentation of A ∞ J (G).However, we shall see now that our ad hoc chosen LF-topology on A ∞ J ([G]) behaves well with respect to the LF-topology on A ∞ J (G).To this end, recall that ), there is hence no loss of generality in assuming that J ⊇ a G .We shall therefore suppose from now on that J ⊇ a G .Doing so, we obtain ), together with the right regular action, is a smooth-automorphic subrepresentation.Otherwise put, the LF-space topology on A ∞ J ([G]), as defined by (4.1), agrees with the subspace topology inherited from A ∞ J (G).
Proof.As indicated by the last sentence, we need to prove that To finish the proof, we show that there exists a continuous retraction ), the continuity here being the subtle point: In order to construct such a retraction, let us fix a Haar measure We claim that the linear operator ), given by is a well-defined, continuous retraction as desired.Indeed, using (1.1) and [Beu20, Prop.A.1.1(i)],one directly checks that for every n ∈ N and where, denoting . Since the latter decomposition restricts to the decomposition , and J ⊇ a G , we have that J = a G Z(g) ⊕ J ∩ Z(g 1 ∞ ) , hence (4.4) Finally, by [Moe-Wal95, I.2.2(viii)] and (1.1) there exist M ∈ R >0 and t ∈ N such that hence we obtain the estimate This proves that for every n ∈ N, P A R G ,d0 restricts to a well-defined, continuous operator ) Kn,J n .Going over to the direct limit, it follows that P A R G ,d0 is a well-defined, continuous retraction ), where the last equality holds by our Prop.2.8 and Rem.2.14 (which applies equally well to A ∞ J ([G])).4.3.LF-compatible smooth-automorphic representations and their direct sums.It will be convenient to introduce the following notion: We will call a smooth-automorphic G(A)-representation (π, V ) LF-compatible, if V = lim n→∞ V Kn,J n topologically.Here we used the suggestive notation V Kn,J n to indicate the closed G ∞ -invariant subspace of K n -invariant elements of V that are annihilated by J n .As we have just observed, ) is an LF-compatible smooth-automorphic representation, as well as every smooth-automorphic G(A)-representation that is annihilated by a power of J , hence in particular every irreducible smooth-automorphic representation.Obviously, if (π, V ) is an LF-compatible smooth-automorphic subrepresentation, then V = lim n→∞ V Kn,J n is an LF-space.
We record the following lemma, which is based on two results of Harish-Chandra and the fact that A ∞ d (G) Kn,J n is a Casselman-Wallach representation: Lemma 4.6.Let n ∈ N, and let {V 0,i } i∈I be a family of (g ∞ , K ∞ )-submodules of A J (G) Kn,J n whose sum is direct.Then, the sum of G ∞ -invariant subspaces V 0,i := Cl A ∞ d (G) Kn,J n (V 0,i ) of A ∞ d (G) Kn,J n is also direct.Proof.For every i ∈ I, by Prop.2.5 and [Var77], Thm.II.7.14, V 0,i is a G ∞ -invariant subspace of A ∞ d (G) Kn,J n , and V 0,i (K∞) = V 0,i .Now, suppose that i∈I φ i = 0 for some φ i ∈ V 0,i , where φ i = 0 for all but finitely many i.Then, for every irreducible representation ρ ∞ of K ∞ , i∈I E ρ∞ (φ i ) = 0. Since E ρ∞ (φ i ) ∈ V 0,i (K∞) = V 0,i , and the sum of V 0,i 's is direct, it follows that E ρ∞ (φ i ) = 0 for all ρ ∞ and i ∈ I. Thus, since [HCh66,Lem. 5] we have that φ i = ρ∞ E ρ∞ (φ i ) = 0 for every i ∈ I.
The next result on LF-compatible smooth-automorphic subrepresentations will be crucial.
Proposition 4.7.Let V be an LF-compatible smooth-automorphic subrepresentation, and denote V 0 := V (K∞) .Suppose that Then, we have the following decomposition into a locally convex direct sum of LF-compatible smooth-automorphic subrepresentations: In particular, for every i ∈ I, we have (4.10) , where V Kn,J n 0,i . For every fixed n ∈ N, (4.12) V Kn,J n i = V Kn,J n 0,i = 0 for all but finitely many i ∈ I.
Proof.Let us first show that for every fixed n ∈ N, (4.13) V Kn,J n 0,i = 0 for all but finitely many i ∈ I.
Since by (4.8) , it suffices to prove that the (g ∞ , K ∞ )-module V Kn,J n 0 is finitely generated, which holds by [Vog81], Cor.5.4.16,since V Kn,J n 0 ⊆ A J (G) Kn,J n is obviously Z(g)-finite and is admissible by [Bor-Jac79], §4.3(i).Next, by Lem.3.3, V i is a smooth-automorphic subrepresentation for every i ∈ I. To prove the proposition, it remains to prove that (4.15) Indeed, (4.15) implies that V i = lim n→∞ V Kn,J n 0,i for every i ∈ I and that (4.9) holds; (4.9) and (4.8) imply (4.10); Thm.3.7 -or, alternatively, a combination of (4.10) and [HCh66], Lem. 4 -implies that V i is irreducible whenever V 0,i is; moreover, (4.10) implies that for every n ∈ N, V Kn,J n i (K∞) , which finishes the proof of (4.11) and, by (4.13), (4.12).Therefore, we are left to prove (4.15).The locally convex direct sum on the right-hand side of (4.15) is well-defined since by Lem.4.6, for every n ∈ N the sum of subspaces V Kn,J n 0,i , which is obvious as an equality of vector spaces, and the topology of both sides is easily seen to be the finest locally convex topology with respect to which the inclusion maps of the subspaces V Kn,J n 0,i are continuous (a detail, which we leave to the reader).To prove (4.15), by (4.16) and the LF-compatibility of V it suffices to prove that for every n ∈ N, (4.17) V Kn,J n = i∈I V Kn,J n 0,i .

The parabolic support of a smooth-automorphic form. Let
) that are cuspidal, i.e., satisfy ϕ P = 0 for every proper parabolic F -subgroup P of G.We say a function Remark 4.19.Usually negligibility is defined as an orthogonality-relation with respect to all cuspidal automorphic forms, i.e., one supposes that (4.18)only holds for all φ ∈ A cusp ([L]).See [BLS96], §2.2, [Bor06], §6.7, [Fra-Sch98], §1.1 and (most explicitly) [Osb-War81], p. 82 (which refers to Langlands's original work [Lan76]).In course of proving Thm.4.20 below, we will show that our definition is in fact equivalent to the common one, but has the advantage to be intrinsic to the notion of smooth-automorphic forms.
For every P ∈ P, let {P } denote the associate class of P , i.e., the set of parabolic ) to be the space of smooth-automorphic forms ϕ ∈ A ∞ J ([G]), which are negligible along all Q / ∈ {P }.The following theorem is our first main result: Theorem 4.20.We have the following, G(A)-equivariant decomposition into a locally convex direct sum of LF-compatible smooth-automorphic subrepresentations: Proof.We proceed in several steps.
Step 1: where l [L] := inf γ∈L(F ) γl and define S([L]) KL,n to be the space, which consists of all functions f ∈ C ∞ (L(F )A R Q \L(A)) KL,n such that q d,X (f ) < ∞ for all d ∈ Z and X ∈ U(l).We equip S([L]) KL,n with the Fréchet topology generated by the seminorms q d,X and define the global Schwartz space we have ) and for all ϕ ∈ S([L]).Hence, we have shown that for each fixed g ∈ G(A), the function λ Q,g,ϕ (f ) is separately continuous in the the arguments f ∈ A ∞ J ([G]) and ϕ ∈ S([L]).
Step 2: As announced in Rem.4.19, we will now show that our definition of negligibility coincides with the usual (weaker) one, i.e., we will prove that a f ∈ C ∞ umg (G(F )\G(A)) is negligible along a parabolic F -subgroup Q = LN of G if and only if λ Q,g,φ (f ) = 0 for all g ∈ G(A) and all φ ∈ A cusp ([L]).Necessity being obvious, we show sufficiency: Suppose λ Q,g,φ (f ) = 0 for all g ∈ G(A) and all φ ∈ A cusp ( f ) for every g ∈ G(A).Thus, it suffices to prove that for every g ∈ G(A), To this end, note that for every ideal J in Z(l) and n ∈ N, we have an equality of Fréchet spaces ).Indeed, it is a simple consequence of [Moe-Wal95], Cor.I.2.11 and I.2.2.(vi), ibidem, that these two spaces coincide as sets.Moreover, for every parabolic F -subgroup KL,n , respectively.Thus, the intersection S cusp ([L]) KL,n,J n of their kernels is a closed subspace of S([L]) KL,n and hence Fréchet.As the identity map S cusp ([L]) KL,n,J n → A ∞ cusp,J ([L]) KL,n,J n is obviously continuous, the claimed equality (4.22) of Fréchet spaces finally follows from the open mapping theorem.Thus, writing A cusp,J ( , we have This proves (4.21) and hence that a function Step 3: Step 2 now allows us to use Langlands's algebraic direct sum decomposition as a G(A)-equivariant decomposition of vector spaces and (4.25) . See also [Fra-Sch98], 1.1.It hence follows from our Prop.4.7 that we have the following decomposition into a locally convex direct sum of LF-compatible smooth-automorphic subrepresentations: Therefore, it remains to prove that for every {P }, . By (4.26) and (4.24) it suffices to prove that Cl A ).We recall from Step 1 above that the linear functional λ Q,g,ϕ : A ).This shows the claim.4.5.Cuspidal smooth-automorphic forms.We will now consider the subspace ).Thus, Thm.4.20 implies the following Corollary 4.27.The space of cuspidal smooth-automorphic forms A ∞ cusp,J ([G]) is an LF-compatible smoothautomorphic subrepresentation.
It is the goal of this subsection to give a refined description of the smooth-automorphic G(A)-representation A ∞ cusp,J ([G]) as a countable locally convex direct sum of irreducible subrepresentations.
First, to settle terminology, we will call a smooth-automorphic representation cuspidal, )) (which may grow unboundedly, though, as H i varies through i ∈ I, cf.[GGJ02], (1.1)).It is finally a consequence of [Bor72], Prop.5.26, that hence there exists a (unique) subset I(J ) ⊆ I such that we have the following decomposition Here we identify each element of the right-hand side, which is by definition an equivalence class of almost everywhere equal measurable functions on G(F )A R G \G(A), with its unique continuous representative.
On the level of cuspidal smooth-automorphic forms, it easily follows from [Bor06], Eq. 6.8.4 in combination with our Prop.2.3, that we have an identification of vector spaces ), of the space of smooth, Z(g)-finite vectors4 in L 2 cusp,J ([G]) := i∈I(J ) H i and the space of cuspidal smoothautomorphic forms, given by assigning each class in L 2 cusp,J ([G]) ∞ A (Z(g)) its unique continuous representative.
It is by no means clear, however, that this identification is compatible with the direct sum decomposition of L 2 cusp,J ([G]), i.e., it is not clear that the LF-spaces of globally smooth vectors in the irreducible subrepresentations H i , i ∈ I(J ), of the Hilbert space representation L 2 cusp ([G]) identify with irreducible subrepresentations of the LF-space A ∞ cusp,J ([G]).It is the goal of this section to establish the following Theorem 4.30.The isomorphism (4.29) extends to a G(A)-equivariant decomposition into a countable locally convex direct sum of LF-compatible smooth-automorphic subrepresentations: Consequently, for each i ∈ I(J ), Kn with its natural LF-space topology embeds as a G(A)-subrepresentation into A ∞ cusp,J ([G]).This characterises the irreducible cuspidal smooth-automorphic subrepresentations of A ∞ J (G) as the subrepresentations isomorphic to the LF-spaces H ∞ A i of smooth vectors in the unitary Hilbert space representations H i , i ∈ I(J ).
We now prove that for every i ∈ I(J ) the map θ i extends to an isomorphism Kn is a Casselman-Wallach representation of G(A).Obviously, θ i is an isomorphism between the (g ∞ , K ∞ , G(A f ))modules H ∞ A i,(K∞) and H ∞ A i,(K∞) .Thus, as argued in the proof of Lem.1.13.(2)θ i must extend to an equivalence of G(A)-representations (and hence, in particular, a bi-continuous map) ) ).In order to complete the proof, it remains to show that θi ([f ]) = θ i ([f ]) for every Kn for some n ∈ N, hence by [HCh66, Lem.4] there exists a sequence ) and hence, replacing the original sequence ([f m ]) m∈N by a suitable subsequence and choosing (any) representatives for our classes in sight, the sequence (f m ) m∈N converges almost everywhere on G(A) to f .On the other hand, by the continuity of θi , ), hence also pointwise everywhere on G(A), cf.§2.2.4.It follows that θi ([f ]) = f almost everywhere on G(A), which, together with the continuity of θi ([f ]), cf.Rem.2.17, implies that θi ([f ]) = θ i ([f ]), which proves the claim.
Remark 4.31.Thm.4.30 has the following consequence: Every cuspidal smooth-automorphic form ϕ is a finite sum of smooth cuspidal functions ϕ i ∈ H ∞ A i .For K ∞ -finite cuspidal automorphic forms this is well-known and an immediate consequence of (4.29), whereas it is in general wrong for elements of the space L 2 cusp,J ([G]).In view of the above mentioned, natural inclusions L 2 cusp,J ([G]) A ∞ cusp,J ([G]) A cusp,J ([G]) this finiteness-statement hence amounts to the dictum that cuspidal smooth-automorphic forms ϕ are more similar to K ∞ -finite cuspidal automorphic forms than to square-integrable cuspidal functions.Remarkably, this is so, though the quotient A ∞ J ([G])/A J ([G]) has uncountable dimension.4.6.The cuspidal support of a smooth-automorphic form.We will now give a definition of the cuspidal support of a smooth-automorphic form.Our notion of cuspidal support -which will be intrinsic to the smooth-automorphic setting -will extend the usual definition for classical automorphic forms to the framework of smooth-automorphic forms.Thm.4.30 will be a crucial ingredient in what follows.
Let {P } denote an associate class of parabolic F -subgroups of G, represented by P = L P N P ∈ P. Given an ideal J of Z(g) of finite codimension, an associate class of cuspidal smooth-automorphic representations is represented by a pair ([π], Λ), where (1) [π] is an equivalence class of an irreducible cuspidal smooth-automorphic subrepresentation π of L P (A) and (2) Λ : A R P → C * is a Lie group character, which is trivial on A R G , such that the following compatibility-hypothesis is satisfied: Let λ 0 := dΛ ∈ ǎG P,C be the derivative of Λ and consider the irreducible smooth-automorphic subrepresentation π := e λ0,HP (•) • π of L P (A).We suppose that the Weyl group orbit of the infinitesimal character of π ∞ := Given J , we denote by Φ J ,{P } the set of all associate classes ϕ([π]), represented by a pair ([π], Λ) as above.
We point out that our notion of associate classes of cuspidal smooth-automorphic subrepresentations extends the usual notion of associate classes (cf.[Fra-Sch98], §1.2) into the context of smooth-automorphic forms, i.e., the collections ϕ([π (K∞) ]) of finite sets of equivalence classes of (g ∞ , K ∞ , G(A f ))-modules of K ∞ -finite vectors in π and its W (L P )-conjugates coincide with the associate classes of cuspidal automorphic representations as defined in [Fra-Sch98], §1.2 (for ideals J stemming from coefficients in automorphic cohomology; however, see also [Fra-Sch98], Rem.3.4): The verification of this claim relies crucially on the characterization of the irreducible cuspidal smooth-automorphic subrepresentations as the spaces of globally smooth vectors H ∞ A i , i ∈ I, provided by Thm.4.30 as applied to the Levi-subgroups L Q (A), and the following lemma: ), was taken.We hope to report on this question in the forthcoming "part 2" of the present paper, which we will devote to the analytic continuation of smooth-automorphic Eisenstein series, i.e., to a smooth-automorphic version of the results in [Moe-Wal95], IV.
2.2.3.Ideals of finite codimension.Recall the symmetric algebra S(a P,C ) ∼ −→ Z(a P ) of a P,C , cf. §1.3.1, which we identify with the algebra of polynomials on ǎP,C .Every ideal I ⊳ Z(a P ) of finite codimension contains an ideal of the form Z(a P ; Λ, N ) := {Y ∈ Z(a P ) | Y vanishes of order ≥ N in each λ ∈ Λ} , where Λ is a finite subset of ǎP,C and N ∈ N, see [Moe-Wal95], I.3.1.Next note that for such I, Λ and N , and for every k ∈ N there exists M ∈ N (depending on I, Λ, N , and k), such that λ∈Λ Z (a P ; λ, N ) M ⊆ λ∈Λ Z (a P ; λ, N ) k .Indeed, the existence of such an exponent M ∈ N is a corollary of the following general Lemma 2.10.Let R be a commutative Noetherian ring.Let n, k ∈ N, and let I 1 , . . ., I n be ideals in R.Then, there exists M ∈ N such that n i=1 2.3 above.But by [Moe-Wal95], I.3.1,

S
([L]) := lim n→∞ S([L]) KL,n .As for any compact set C ⊆ N (A) such that N (A) = N (F )C and d = d 0 ∈ N as in §2.2.4