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On semilinear representations of the infinite symmetric group

In this note the smooth (i.e. with open stabilizers) linear and {\sl semilinear} representations of certain permutation groups (such as infinite symmetric group or automorphism group of an infinite-dimensional vector space over a finite field) are studied. Many results here are well-known to the experts, at least in the case of {\sl linear representations} of symmetric group. The presented results suggest, in particular, that an analogue of Hilbert's Theorem 90 should hold: in the case of faithful action of the group on the base field the irreducible smooth semilinear representations are one-dimensional (and trivial in appropriate sense).

Let G be a permutation group, cf. §0.1. In the most general setting, we are interested in the continuous G-actions on the discrete sets (i.e. with open stabilizers; they are called smooth in what follows). In practice, the considered G-sets are endowed with extra structures, e.g., of a vector k-space for a field k. In that case our primary goal is a description of the smooth representations of G in the k-vector spaces, especially of the irreducible ones. The smooth representations form a klinear Grothendieck category with s≫0 W s as one of many possible generators, where W s = W ⊗ s k and W = k [Ψ]. However, the structure of k[G]-modules W s can be quite complicated.
For any group G and any field k there is a field extension K|k endowed with a faithful G-action. Namely, as K one can take the fraction field of the symmetric algebra of a faithful representation W of G over k. Then there is a natural surjection W ⊗ k K → K of K-semilinear representations of G. (The semilinear representations are defined in §6.) Suppose first that G is pre-compact, i.e., any open subgroup of G is of finite index. It is wellknown that any cyclic smooth k[G]-module can be embedded into an arbitrary field extension K of k endowed with a smooth faithful G-action trivial on k. Moreover, Hilbert's Theorem 90 asserts the triviality of smooth semilinear representations of G over K: the functor H 0 (G, −) : V = V G ⊗ K G K ∼ = I K → V G is an equivalence between the category of smooth semilinear representations of G over K and the category of vector spaces over the fixed field K G . (Here I is a basis of the K G -vector space V G .) In particular, the symmetric algebra of any smooth faithful k-representation of G is a generator of the category of smooth k-representations of G.
The above remarks lead to the problem of describing smooth semilinear representations over K for a field K endowed with a smooth faithful G-action.
There is a (rather wild) description of all smooth semilinear G-actions on a given K-vector space in the case when a dense subgroup of G is exhausted by pre-compact subgroups, cf. Appendix A. This description is quite explicit in the case when G is exhausted by open pre-compact subgroups.
Our principal examples of G will be permutation groups of S -type defined on p.4. Typical groups of S -type are the group SΨ of all permutations of a plain set Ψ, or the automorphism group of an infinite-dimensional projective space Ψ over a finite field F q , or the automorphism group GL Fq (Ψ) of an infinite-dimensional vector space Ψ over a finite field F q , or the automorphism group GL Fq (Ψ) of an infinite-dimensional F q -vector space Ψ.
Such groups admit a dense subgroup exhausted by finite subgroups. Let G ⊆ SΨ be a group of S -type for a countable set Ψ. It is well-known (cf., e.g., [8, §6 and references therein]) that (i) any smooth cyclic representation of G is admissible 1 and of finite length 2 (cf., e.g., Lemma 3.4), (ii) the isomorphism classes of irreducible smooth representations of SΨ are in one-to-one correspondence with (finite) Young diagrams. However, the approach of loc.cite is different and I was unable to find in the literature a description of all injective smooth representation of G. This description follows from Lemma 4.1.
One of the purposes of this note is to examine the similar questions in the context of semilinear representations.
The results are as follows (where (4) is my principal motivation): (1) a description of injective smooth representations of a group G of S -type (Corollary 4.3); (2) any smooth finitely generated linear representation of a pseudoprofinite group (this class of permutation groups is introduced in §1.1) is of finite length (Lemma 3.4); (3) the category of smooth K-semilinear representation of the group SΨ is locally noetherian, i.e., any smooth finite K SΨ -module is noetherian (notations are in §6), cf. Corollary 8.6; (4) [an analogue of Hilbert's Theorem 90] any smooth K-semilinear finite-length representation of SΨ is isomorphic to a direct sum of copies of K, if K|k is a purely transcendental field extension with a transcendence base identified with Ψ, cf. Theorem 12.2 and Lemma 9.1; (5) a generalization of the well-known cyclicity of finite-dimensional semilinear representations of infinite semigroups to the cyclicity of certain smooth finite K G -modules (Lemma 11.1); (6) the triple (K SΨ , K[Ψ], K) is an example of a triple (A, M, P ) consisting of an associative unital ring A, a noetherian A-module M and a simple A-module P such that (i) any quotient of M by any non-zero submodule is isomorphic to a finite direct sum of copies of P , (ii) for any integer N ≥ 0 there is a quotient of M isomorphic to a direct sum of N copies of P (Lemma 10.2).
With K as in (4), there are some reasons to expect an explicit description of the indecomposable injectives, cf. Conjecture 11.5. This is compatible with Theorem 12.2.
We endow C B with the maximal topology, i.e. we assume that any sieve is covering. Then the sheaves of sets, groups, etc. on C B are identified with the smooth G-sets, groups, etc.: F → lim − → U ∈B F(U ) (this is a smooth G-set, since any element in it belongs to the image of some F(U ) and the U -action on it is trivial by definition) and W → [U → W U ].

Finiteness conditions on permutation groups
Length of an object of a concrete category is the maximal length of the chains of its subobjects. E.g., length is cardinality for the category of sets; length is dimension for the categories of projective or vector spaces. (1) the category of smooth finitely generated representations of G over a fixed field is tensor, (2) for any commutative ring C endowed with a smooth G-action, any associative C-algebra A endowed with a smooth G-action, any smooth finitely generated A G -module V and any smooth finitely generated C G -module W the A G -module V ⊗ C W is finitely generated, (3) restriction of any smooth finitely generated representation of G to any open subgroup is again finitely generated, (4) G is pseudoprofinite, (5) any open subgroup of G is pseudoprofinite.
Proof. The implications (5)⇒(4) and (2) (1)⇒(4). For any pair of subgroups U, V of G one has the following decomposition of the representations of G: (4)⇒(2). Any smooth finitely generated A G -module is a quotient of a finite sum of A[G/U i ] for some open subgroups U i of G, so tensor product of a pair of finitely generated A G -and C Gmodules is a quotient of a finite sum of

Proof. For any open subgroup
). This set is finite, since (G/U ) V embeds into the finite set V \G/U .
If U is a proper subgroup of gU g −1 for some [g] ∈ (G/U ) U then we get a strictly increasing sequence of subgroups in G: U gU g −1 g 2 U g −2 g 3 U g −3 . . . , so the elements [g], [g 2 ], [g 3 ], . . . of (G/U ) U are pairwise distinct, contradicting the finiteness of (G/U ) U .
This means that the natural inclusion N G (U )/U ֒→ (G/U ) U is bijective.
. . eventually stabilizes, since so does the descending sequence of the orders of these finite sets. In particular, the subset U n \U n+1 /U n ⊆ U n \G/U n maps bijectively onto its image U n+1 \U n+1 /U n+1 ⊆ U n+1 \G/U n+1 for n ≫ 0, i.e., the set U n \U n+1 /U n consists of a single element, which means that U n+1 = U n . (ii) follows from (i).

Groups of
Examples.
The following examples of groups G of S -type are constructed as the groups of all permutations of an infinite set Ψ respecting an extra structure on Ψ. (So G is a closed subgroup of the group SΨ of all permutations of the plain set Ψ.) (1) Ψ is a plain set, i.e., G = SΨ . As a base B of open subgroups we take the subgroups G T for some exhausting collection of finite subsets T ⊂ Ψ.
consisting of a subset T 0 ⊆ T and an embedding T 0 ֒→ T . (2) Ψ = A S is a free A-module of infinite rank with a finite collection of marked elements for a finite commutative ring A. As a base B of open subgroups we take the subgroups G T for sufficiently big finite subsets T of the set S of generators of the A-module Ψ. Then (i) G/G T is a subset in the set of all embeddings T ֒→ Ψ, and (ii) (G/G T ) G T ′ consists of the embeddings T ֒→ Ψ G T ′ = t∈T ′ At inducing splittable embeddings t∈T At ֒→ Ψ, i.e., whose images admit complementary free summands. Clearly, The automorphism group of an infinite-dimensional projective space Ψ = P(F S q ) over a finite field F q . As a base B of open subgroups we take the subgroups G T for the finite subsets T ⊂ S. Then G/G T consists of the embeddings T ֒→ Ψ with (#T − 1)-dimensional projective envelope of the image and (G/G T ) G T ′ consists of the embeddings T ֒→ Ψ G T ′ .  6 Indeed, suppose that the image β ∈ E of an element

Injectivity of trivial representations and admissibility
The support of the element α, i.e., a finite subset in G/U , is pointwise fixed by an open subgroup On the other hand, as the support of α is contained in a (finite) (N G (V )/V )-orbit, α ′ = 0, and thus, β = 0, i.e., the image of R[G/U ] • in E has no non-zero vectors fixed by G. Therefore, E is a quotient of a sum of trivial 3 The condition (i) is the transitivity condition from [3, p.5, §3.1]; (ii) is the bijectivity condition of [3, p.5, §3.2]. 4 By Lemma 1.2, this is automatic if G is pseudoprofinite. 5 The groups of S-type are examples of such groups G. 6 For any ring A, an A-module P and a set S we denote by P  Proof. This is Lemma 2.2, since any open subgroup V of G satisfies assumptions of Lemma 2.1.

Filtered representations and local length-finiteness
, so assume that a > 1. If the annihilators of n i in B are not the same, say Ann(n i ) is not contained in Ann(n j ), then (1 ⊗ ξ)α is a shorter non-zero element for any ξ ∈ Ann(n i ) Ann(n j ), contradicting to the minimality of a. There remains the case of coincident annihilators of n 1 , . . . , n a . In that case B/Ann(n 1 ) ·n i −→ N are isomorphisms. As End B (N ) = k, all these isomorphisms differ by a non-zero multiple, and therefore, the images of the element 1 ∈ B/Ann(n 1 ) under ·n i differ by a non-zero multiple as well, i.e., the elements n 1 , . . . , n a are proportional, so finally, a = 1.
(ii) Set D := End A (M ). This is a division k-algebra. Then D ⊗ k F is a left and right artinian Falgebra, since it is finite-dimensional over D. Therefore, its quotient M ⊗ k F is of finite length. Fix some open subgroup U ⊆ G and set Ψ : for any finite field extension F |Q. The natural projection ( On the other hand, by Lemma 3.

Coinduction and injectives
Let A ⊂ R be a pair of associative unital topological rings with a base of neighbourhood of 0 given by a collection of left ideals. The restriction functor R-mod → A-mod admits a left adjoint (induction): . Let R-mod sm be the category of left R-modules such that any element is annihilated by an open left ideal. Then the restriction functor induces a functor R-mod sm → A-mod sm admitting a right adjoint (coinduction), sending a smooth A-module E to the smooth part of the R-module Hom A (R, E): In particular, if restriction to A of a smooth R-module W is injective the adjunction morphism W → Hom A (R, W ) sm gives an embedding into an injective smooth R-module.
Our only examples of such topological rings will be skew group rings A G , cf.
where to a collection (a τ ) τ :Λ֒→I (consisting of elements of E parametrized by substructures Λ ⊆ J and their embeddings τ : Λ ֒→ I) corresponding to I we associate the function G ∋ g → a τ I (g) ∈ E, where τ I (g) : g(I) ∩ J ֒→ I is the restriction of g −1 . The G-action respects this decomposition: g : T ⊗ k for some T . By Lemma 2.1, the trivial representation k of G J is injective for any J. By adjunction, the coinduced representation of G is injective as well. By Lemma 4.1, this implies that the representation M Ψ T ⊗ k is injective, so finally, any injective indecomposable representation is a direct summand of an appropriate M Ψ T ⊗ k.

Smooth representations of infinite symmetric groups
Let G ⊆ SΨ be a permutation group. For a subset S ⊂ Ψ set S := Ψ G S . We say that (i) a subset S ⊂ Ψ is G-closed if S := S, (ii) elements x 1 , . . . , x n ∈ Ψ are G-independent if {x 1 , . . . , x n } = {y 1 , . . . , y n−1 } for any y 1 , . . . , y n−1 ∈ Ψ. For each integer n ≥ 0 set  Proof. The module W can be embedded into M Ψ T ⊗ k for some minimal T . Then W is embedded into V Ψ T ⊗ k. Let us show that V Ψ T ⊗ k is semisimple. Otherwise, it contains an indecomposable W ′ ⊃ W with simple W ′ /W . There is an embedding ι : W ′ /W ֒→ M n ⊗ k. As M n ⊗ k is injective, ι extends to M Ψ T ⊗ k/W −→ M n ⊗ k, so gives rise to a non-zero morphism M Ψ T ⊗ k ϕ −→ M n ⊗ k, so n ≤ length(T ). If n < length(T ) then ϕ| V Ψ T ⊗k = 0, so ϕ| W ′ = 0, which is a contradiction. Thus, n = length(T ), which means that ker ϕ splits as a direct summand: M Ψ T ⊗ k = ker ϕ ⊕ M ′ , and therefore, assuming that ϕ| W ′ = 0, one has W ′ ⊆ W ⊕ M ′ . Then W ′ = W ⊕ pr M ′ (W ′ ), contradicting to the indecomposability.
It is clear from the identity End k[G] (V Ψ T ⊗ k) = k[Aut(T )] that V Ψ T ⊗ k is multiplicity-free. 7 In the case q = 1 of the plain sets the (Specht) Q[S T ]-modules Specht T λ correspond to the partitions of #T and are absolutely irreducible.
Lemma 5.2. Let k be a field of characteristic zero. Then the representation V Ψ T ⊗ k of G admits no non-zero morphisms to subquotients of M Ψ n−1 ⊗ k. Proof. Assuming the contrary, for any finite J with #J > #T there is a non-zero morphism of The following results should be well-known to the experts.
Theorem 5.3. In notation of Theorem 5.1, More explicitly, the elements of A G are the finite formal sums N i=1 a i [g i ] for all integer N ≥ 0, a i ∈ A, g i ∈ G. Addition is defined obviously; multiplication is a unique distributive one such that (a[g])(b[h]) = ab ρ(g) [gh], where we write a h for the result of applying of h ∈ End ring (A) to a ∈ A.
An additive action of G on an A-module V is called semilinear if g(a · v) = a g · gv for any g ∈ G, v ∈ V and a ∈ A. Then an A-module endowed with an additive semilinear G-action is the same as an A G -module.
The principal example of A will be a field. For a field K endowed with a G-action, a K-semilinear representation of G is a left K G -module. We say that a K-semilinear representation of G is non-degenerate if the action of each element of G is injective. We omit the G-action on K from notation and denote by k := K G the fixed field.
The non-degenerate K-semilinear representations of G form an abelian tensor k-linear category.
Lemma 6.1. Let A be a division ring endowed with a G-action, V be a A G -module and χ : G → End A (V ) be a non-degenerate G-action on the A-module V (e.g., χ ≡ id V ).
Proof. This is well-known: Suppose that elements w 1 , . . . , w m ∈ V χ are A G -linearly independent, i.e., w i ∈ j =i Aw j ⊂ V for all i. Suppose that w = j λ j w j ∈ V χ for some λ j ∈ A. Then σw − χ(σ)w = j (λ σ j − λ j )χ(σ)w j = 0, and therefore, λ σ j = λ j for any j, i.e., λ j ∈ A G . A K-semilinear representation V of G is called trivial, if the natural injective map V G ⊗ k K → V of Lemma 6.1 is bijective, i.e., if V is isomorphic to a direct sum of several copies of K with G-action via ρ.
There is a bijection between k-lattices U in a K-vector space V and the trivial K-semilinear G-actions on V : {structures on V of K-semilinear representation isomorphic to a direct sum of copies of K}.
The set of isomorphism classes of non-degenerate K-semilinear G-actions on a K-vector space V := U ⊗ k K is canonically identified with the set H 1 (G, GL K (V )). Namely, there is a unique K-semilinear G-action ξ on V identical on the k-lattice U . This gives rise to the K-semilinear G-action on End K (V ) by f τ := ξ(τ )f ξ(τ ) −1 ∈ End K (V ) if f ∈ End K (V ), so the matrix of f τ in the basis τ (b) is the result of applying τ to the matrix of f in a basis b of U . For each element σ ∈ G there exists a unique element f σ ∈ GL K (V ) such that σ| U = f σ | U , so σv = f σ ξ(σ)v. This implies Now let the semigroup G be totally disconnected and the homomorphism ρ be continuous. (We endow any set H ⊆ Maps(Ψ 1 , Ψ 2 ) of mappings between sets Ψ 1 and Ψ 2 with topology, where a base {U α } α of open subsets of H is indexed by the mappings α of finite subsets S α ⊂ Ψ 1 to Ψ 2 and U α is the set of all elements of H with restriction α to S α .) Denote by C K (G) the category of smooth K-semilinear representations of G.
The category C K (G) is k-linear abelian. Let F |k be a field extension such that k is algebraically closed in F . Let K ′ be the field of fractions of K ⊗ k F . Then G acts naturally on K ′ and the functor   (N ). The lower bound of d M (N ) is given by the number of subobjects in Ψ N with trivial intersection with a fixed finite subobject of Ψ of length m. Indeed, any non-zero element α = I a I (x i 1 , . . . , x in ) ∈ M is congruent to σ∈Aut(Ψn) b σ τ σ for some collections b σ ∈ A not all zero and a monomorphism τ : Ψ n ֒→ Ψ modulo monomorphisms whose images have a nontrivial intersection with a fixed finite subobject J ⊂ Ψ (of length m).

Proof. As
Let q be either 1 or a primary integer. For an integer s ≥ 0, we denote by Ψ s q the set of subobjects of Ψ length s (i.e., of subsets of order s if q = 1 and of s-dimensional subspaces if q > 1).

Corollary 7.2. Let G be a group of S -type (p.4). Let a ring A endowed with a G-action be as in
(1) follows from the lower growth estimate of Lemma 7.1.
(3) Suppose that there exists a nonzero submodule M ⊆ A ⊗ M Ψ n such that M ∩ V Ξ = 0. Then restriction of some ξ ∈ Ξ to M is nonzero. If ξ| M is not injective, replacing M with ker ξ ∩ M , we can assume that ξ| M = 0. In other words, we can assume that restriction to M of any ξ ∈ Ξ is either injective or zero. In particular, restriction to M of some ξ ∈ Ξ is injective, i.e. ξ embeds M into A ⊗ M Ψ m , contradicting to (2).

Noetherian properties of smooth semilinear representations of SΨ
Examples. 1. Let Ψ be an infinite set, possibly endowed with a structure of a projective space. Let G be the group of automorphisms of Ψ, respecting the structure, if any. Let J be a 'finite substructure' in Ψ, i.e., a finite subset or a finite-dimensional subspace. Let U be the stabilizer of J in G. Then G/U is identified with the set of all substructures in Ψ of the same length as J.

By Lemma 8.1, K[G/U ] is indecomposable in the following examples:
(1) G is the group of projective automorphisms of an infinite projective space Ψ (i.e., either Ψ is infinite-dimensional, or Ψ is defined over an infinite field), U is the setwise stabilizer in G of a finite-dimensional subspace J ⊆ Ψ. Then G/U is identified with the Grassmannian of all subspaces in Ψ of the same dimension as J. Example. Let G be a group acting on a field K; U ⊂ G be a maximal proper subgroup. Assume that K U = K G . Then we are under assumptions of Lemma 8.2.
More particularly, if G = SΨ , U = SΨ,I (so G/U ∼ = Ψ #I ) and K SΨ,I = k := K SΨ for a finite subset I ⊂ Ψ then there are no irreducible K-semilinear subrepresentations in K[ Ψ #I ]. Proof. By Artin's independence of characters theorem (applied to the one-dimensional characters g : is injective. Then, for any non-zero element α ∈ K[G/U ], there exists an element Q ∈ K U such that the morphism K[G/U ] → K, given by g b g [g] → g b g Q g , does not vanish on α. Then α generates a subrepresentation V surjecting onto K. If V is irreducible then it is isomorphic to K, so V G = 0. In particular, K[G/U ] G = 0, which can happen only if index of U in G is finite. This is a normal subgroup of finite index. Then  On the other hand, g(M 0 ) ⊆ g( N j=1 Aα j ) + γ∈{1,...,s}×J A[Ψ s γ ] for g ∈ U J , and therefore, the A U J -module M/ N j=1 A U α j becomes a subquotient of the noetherian, by the induction assumption, A U J -module γ∈{1,...,s}×J A[Ψ s γ ], so the A U J -module M/ N j=1 A U α j is finite, and thus, M is finite as well. The next result generalizes a description of representations k[ Ψ s ] of SΨ from [2]. Lemma 8.7. In notation preceeding Corollary 7.2, let q ≥ 1 be either 1 or a primary number, S be a finite or infinite set (if q = 1) or an F q -vector space, G be the automorphism group of S. Let the morphism of G-modules ∂ S s : Assume first that S is finite. Comparing the cases t = s and t = s − 1 and arguing by induction, we see that (i) Q[ S s q ] is a direct sum of s + 1 pairwise non-isomorphic absolutely irreducible subrepresentation, (ii) Q[ S s q ] embeds into Q[ S s+1 q ]. Assume that length of S is > 2s. The morphisms ∂ S s ⊗ Q, ∂ S s−1 ⊗ Q, . . . , ∂ S 1 ⊗ Q are non-injective, but they cannot drop length of modules by more than 1, since their composition ∂ S s ∂ S s−1 · · · ∂ S 1 is non-zero. This means that their kernels are irreducible, and thus, they are surjective. Now assume that S is infinite. Clearly, the surjectivity of ∂ S s ⊗ Q and the irreducibility of its kernel follows from the case of sufficiently large finite S. By Lemma 3. For any N ≥ M ≥ 0, fix some subobjects J 0 ⊆ I 0 ⊂ Ψ of lengthes M and N , respectively, and set S Let from now on K be a purely transcendental field extension of k with a transcendence base of K|k identified with Ψ. Then Q(J ⊆ I) is a rational function in elements of I symmetric in elements of J and in elements of I J.
Lemma 8.10. There is a canonical isomorphism of k-vector spaces Under this isomorphism, the composition

Triviality of finite-dimensional semilinear representation of SΨ
The following result is analogous to [5,Proposition 5.4], but there one can take any algebraic group. Proof. Set G := SΨ and let b ⊂ V be a K-basis, pointwise fixed by an open subgroup of G, so b ⊂ V I := V G I for a finite subset I ⊂ Ψ. By Lemma 6.1 (with χ ≡ 1), the multiplication maps The composition is an isomorphism, so V I ⊗ K I K J → V J is an isomorphism as well. In particular, f σ = id V if σ ∈ G I , where (f σ ∈ GL K (V )) σ is the 1-cocycle of the G-action in the basis b. Clearly, (i) f σ depends only on the class σ| I of σ in G/G I = {emdeddings of I into Ψ}, (ii) f σ ∈ GL K I∪σ(I) (V I∪σ(I) ).
Assume that I, σ(I), τ σ(I) are disjoint, X, Y, Z are the standard collections of the elementary symmetric functions in I, τ (I), τ σ(I), respectively. Then the cocycle condition is an isomorphism, i.e., any morphism between K SΨ -submodule M of K[Ψ] is induced by an endomorphism of K[Ψ] (identified with an element of k(T )). (3) Any Q ∈ k(T ) × such that t q t Q(t) = 0 determines a non-zero morphism K[Ψ] → K trivial on the submodule generated by α := t q t [t], so α is not a generator. If α generate a proper submodule M ⊂ K[Ψ] then the quotient K[Ψ]/M is finite-dimensional, so by Lemma 9.1 it admits a quotient isomorphic to K. Finally, any morphism K[Ψ] → K is given by some Q ∈ k(T ) × .  a)) m i is a skew symmetric function, i.e., i Φ i (X, Y ) m i is a skew symmetric function.

Cyclicity of finitely generated smooth modules
The following result extends the existence of a cyclic vector in any finite-degree non-degenerate semilinear representation of an endomorphism of infinite order, cf., e.g., [6, Lemma 2.1].
Lemma 11.1. Let G be a permutation group, K be a field endowed with a smooth G-action such that any open subgroup of G contains an element inducing on K an automorphism of infinite order. Then any smooth finite K G -module W admits a cyclic vector.
Proof. A finite system S of generators of the K G -module W is fixed by an open subgroup U ⊆ G. By [6, Lemma 2.1], the K U -module spanned by S admits a cyclic vector v. Then v is a cyclic vector of the K G -module W .
Corollary 11.2. Let K be a field endowed with a smooth faithful SΨ -action. Then any smooth simple left K SΨ -module is isomorphic to K[ Ψ s ]/K SΨ α for some α ∈ K[ Ψ s ]. Proof. By Lemma 6.2, any smooth simple left K SΨ -module is a quotient of K[ Ψ s ] for an appopriate s ≥ 0 by a left K SΨ -submodule V . By Proposition 8.5, the K SΨ -module V is finite, and thus, by Lemma 11.1, it is cyclic, i.e., it is generated by some α ∈ K[ Ψ s ].
If G is not locally compact then the smooth semilinear G-actions on a given K-vector space V are described as compatible systems (Λ U ) U ∈B of K U -lattices in V such that for any vector v ∈ V the intersection over all U ∈ B of the subfields generated over K U by the coordinates of v with respect to the lattice Λ U is of finite type over K G .
Example. If G is countable at infinity then as B one can choose a totally ordered collection U 1 ⊆ U 2 ⊆ U 3 ⊆ . . . of compact subgroups in G such that G = m≥1 U m . Any compatible systems of K U i -lattices Λ U i can be presented (a priori, not uniquely) as a composition, infinite to the left,