ADMISSIBLE SEMI-LINEAR REPRESENTATIONS

M.ROVINSKY

Let k be an algebraically closed field^{1} of characteristic zero, F be a universal domain over k, i.e., an algebraically closed extension of k of countable transcendence degree, and G_{F/k} be the field automorphism group of F over k with the base of open subgroups {G_{F/k(x)} | x ∈ F }.

An F -semi-linear representation of G_{F/k}, i.e., an F -vector space V endowed with an additive semi-linear (g(fv) = gf ·gv for any f ∈ F, g ∈ G and v ∈ V) G-action G×V → V, is called admissible if it is smooth (with open stabilizers) and dim_{F^{U}} V^{U} < ∞ for any open subgroup U ⊆ G.

In this note we study the category A of admissible F -semi-linear representations of G_{F/k}. Clearly, A is an additive category and it is shown in [R] that it is a tensor category.

Theorem 3.5 asserts that the category A is abelian.

This is deduced from the acyclicity of the objects of A: H_{smooth}^{1}(G_{F/k}, V) = 0 for any object V of A, cf. Proposition 3.4, meaning that F is a projective object of A.

Other results of this note are motivated by the following conjecture.

Conjecture. Let C be the category of smooth F -semi-linear representations of G_{F/k}.

(1) The functors Hom_{C}(⨂_{F}^{q} Ω_{F}^{1} , −) are exact on A for any q ≥ 0.

(2) Irreducible objects of A are direct summands of the tensor algebra ⨂_{F}^{•} Ω_{F/k}^{1}.

It follows from this conjecture that if V, V′ are irreducible objects of A and Ext_{A}^{j}(V, V′) ≠ 0 then there is a surjection V′ ⊗_{F} ⨂_{F}^{j} Ω_{F/k}^{1} → V and Ext_{A}^{j}(Ω_{F/k}^{j+q}, Ω_{F/k}^{q}) = Hom_{k}(Sym_{k}^{j}Ω_{k}^{1}, k). If k = \overline{ℚ} is the field of algebraic numbers then A is semi-simple.

Let n ≥ 1 be an integer, K = k(ℙ) be the function field of an n-dimensional projective k-space ℙ and G = Aut(ℙ/k) be its automorphism group. Fix a k-field embedding K ↪ F.

We show, in particular, that if V is an admissible F -semi-linear representation of G_{F/k} with no sub-objects isomorphic to F then any irreducible subquotient of the K-semi-linear representation V^{G_{F/K}} of G is a direct summand of ⨂_{K}^{≥1} Ω_{K/k}^{1} (this is Theorem 2.5 and Proposition 3.4 below). This is an evidence for the part (2) of the conjecture. Lemma 3.9 gives an evidence for the above mentioned corollary of the conjecture.

0.1. Notation. Let k, F and G_{F/k} be as above. We fix a transcendence basis x_{1}, x_{2}, x_{3}, … of F over k. For each n ≥ 1 set Y_{n} = Speck[x_{1}^{±1}, …, x_{n}^{±1}] ⊂ 𝔸_{k}^{n} = Speck[x_{1}, …, x_{n}] ⊂ ℙ_{k}^{n} = ℙ(Q_{n}) = Projk[X_{0}, …, X_{n}] with x_{j} = X_{j}/X_{0}, K_{n} = k(x_{1}, …, x_{n}), G_{n} = Aut(ℙ_{k}^{n}/k). Let Aff_{n} = G_{n} ∩ Aut(𝔸_{k}^{n}/k) be the affine subgroup. Let H_{n} = Aff_{n} ∩ Aut(𝔸_{k}^{n}/𝔸_{k}^{n−1}) be the subgroup fixing the coordinates x_{1}, …, x_{n−1} on 𝔸_{k}^{n}. Let T_{n} ⊂ G_{n} be the maximal torus acting freely on Y_{n}. Let L_{n} ⊂ 𝔸_{k}^{n} be the line of common zeroes of x_{1}, …, x_{n−1}.

For a field extension L/K we denote by Der(L/K) the Lie K-algebra of derivations of L over K. The subgroup of ℓ-th roots of unity in k is denoted by μ_{ℓ}.

1In fact, we shall only use the existence in k of all ℓ-primary roots of unity for a prime ℓ.

1. Equivariantness of irreducible P GL-sheaves

Let 𝔖𝔏_{n}^{u} be the category of finite-dimensional K_{n}-semi-linear G_{n}-representations whose restrictions to the maximal torus T_{n} in G_{n} are induced by unipotent representations.

In [R], for n > 1, a fully faithful functor 𝔖𝔏_{n}^{u} →^{S} {coherent G_{n}-sheaves on ℙ_{k}^{n}} is constructed. (One says that V is a G_{n}-sheaf if it is endowed with a collection of isomorphisms α_{g} : V ⟶^{∼} g^{∗}V for each g ∈ G_{n} satisfying the chain rule: α_{hg} = g^{∗}α_{h} ∘α_{g} for any g, h ∈ G_{n}.) The composition of S with the generic fibre functor is the identical full embedding of 𝔖𝔏_{n}^{u} to the category of finite-dimensional K_{n}-semi-linear G_{n}-representations.

In this section we show that the category 𝔖𝔏_{n}^{u} is abelian and its irreducible objects are generic fibres of irreducible coherent G_{n}-equivariant sheaves on ℙ_{k}^{n}, i.e., direct summands of Hom_{K_{n}}((Ω_{K_{n}/k}^{n})^{⊗M}, ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}) for appropriate M.

Lemma 1.1. The category 𝔖𝔏_{n}^{u} is closed under taking K_{n}-semi-linear subquotients.

Proof. Let V ∈ 𝔖𝔏_{n}^{u} and 0 → V_{1} → V →^{π} V_{2} → 0 be a short exact sequence of K_{n}- semi-linear representations of G_{n}. Then the sequences 0 → V_{1}^{T_{n}^{tors}} → V^{T_{n}^{tors}} → V_{2}^{T_{n}^{tors}} and 0 → V_{1}^{T_{n}^{tors}} ⊗_{k} K_{n} → V = V^{T_{n}^{tors}} ⊗_{k} K_{n} →^{π′} V_{2}^{T_{n}^{tors}} ⊗_{k} K_{n} are also exact. As the surjection π factors through π′, we see that V_{2} = V_{2}^{T_{n}^{tors}} ⊗_{k}K_{n}, and therefore, V_{1} = V_{1}^{T_{n}^{tors}} ⊗_{k}K_{n}, i.e., that restriction of V_{1} (resp., of V_{2}) to T_{n} are induced by a sub- (resp., quotient) representation of V^{T_{n}^{tors}}. □

Lemma 1.2. Let E be the total space of a vector bundle on ℙ_{k}^{n}, Aut_{lin}(E) be the group of automorphisms of E over k inducing linear transforms between the fibres, and τ : G_{n} → Aut_{lin}(E) be an irreducible G_{n}-structure on E, i.e., a homomorphism splitting the projection Aut_{lin}(E) → G_{n}. Then the Zariski closure \overline{τ(G_{n})} is reductive.

Proof. Let Aut_{τ} be the kernel of the projection \overline{τ(G_{n})} →^{π} G_{n}.

For each point p ∈ ℙ_{k}^{n} let ρ_{p} : R_{p} := π^{−1}(Stab_{p}) → GL(E_{p}) be the natural representation. As we suppose that E is an irreducible G_{n}-bundle, ρ_{p} is irreducible, since otherwise

B := \overline{τ(G_{n})}B_{p} ⊂ E is a G_{n}-subbundle for any proper R_{p}-invariant k-subspace B_{p} ⊂ E_{p}.

In particular, ρ_{p} is trivial on the unipotent radical of R_{p}. The unipotent radical of any algebraic group contains the unipotent radical of its arbitrary normal subgroup, so ρ_{p} is trivial on the unipotent radical of Aut_{τ}. As the action of Aut_{τ} on E is faithful, ⋂_{p} ker ρ_{p}|_{Aut_{τ}} = \{1\}, i.e., Aut_{τ} is reductive. As G_{n} is also reductive, so is \overline{τ(G_{n})}. □

We need the following particular case of Théorème 8.16 of [BT]. For a commutative finite k-algebra A denote by R_{A/k} the Weil functor of restriction of scalars on A-schemes, cf. [DG], I, §1, 6.6.

Theorem 1.3. Let G be a simply connected absolutely almost simple k-group, and G′ be a reductive k-group. Let τ : G(k) → G′(k) be a homomorphism with Zariski dense image. Let G′_{1}, …, G′_{m} be the almost simple normal subgroups of G′.

Then there exist finite field extensions k_{i}/k, field embeddings φ_{i} : k → k_{i}, a special isogeny β : ∏_{i=1}^{m} R_{k_{i}/k}^{φ_{i}}G → G′ (here ^{φ_{i}}G := G ×_{k,φ_{i}} k_{i}) and a homomorphism μ : G(k) → Z_{G′}(k) such that β(R_{k_{i}/k}^{φ_{i}}G) = G′_{i} and τ(h) = μ(h) · β(∏_{i=1}^{m} φ_{i}^{∘}(h)) for any h ∈ G(k) (here φ_{i}^{∘} : G(k) → R_{k_{i}/k}^{φ_{i}}G(k) is the canonical homomorphism). □

Corollary 1.4. Under assumptions of Theorem 1.3, for any torus T ⊂ G the Zariski closure of τ(T (k)) is a torus in G′. □

Proposition 1.5. Let V be an irreducible subquotient of an object of 𝔖𝔏_{n}^{u} for n ≥ 2. Then V is a direct summand of Hom_{K_{n}}((Ω_{K_{n}/k}^{n})^{⊗M}, ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}) for an appropriate M.

Proof. The functor S, mentioned in the beginning of this §, associates to V a coherent G_{n}-sheaf V on ℙ_{k}^{n} with generic fibre V.

Let, as before, T_{n} be a maximal torus in G_{n} and Y_{n} ⊂ ℙ_{k}^{n} be the n-dimensional T_{n}-orbit. As V^{T_{n}^{tors}} = Γ(Y_{n}, V)^{T_{n}^{tors}}, cf. [R], is a unipotent representation of T_{n}, Lemma 1.2 and Corollary 1.4 imply that Γ(Y_{n}, V)^{T_{n}^{tors}} is a trivial representation of T_{n}.

As G_{n} is generated by a finite number of conjugates of T_{n}, we see that V is equivariant, i.e., the action G_{n} × tot(V) → tot(V) is a morphism of k-varieties. The generic fibres of irreducible G_{n}-equivariant sheaves on ℙ_{k}^{n} are exactly of the desired type. □

Remark. There can exist, a priori, non-equivariant irreducible coherent G_{n}-sheaves on ℙ_{k}^{n}, e.g. the one induced by a non-rational representation of G_{n} is seemingly of this type.

2. “Positivity”

In this section we show that for any admissible F -semi-linear representation V of G_{F/k} any irreducible subquotient of the K_{n}-semi-linear representation V^{G_{F/Kn}} of G is a direct summand of ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}.

It is shown in [R] that any finite-dimensional K_{n}-semi-linear G_{n}-representation extendable to End(K_{n}/k), e.g. V^{G_{F/Kn}}, is an object of 𝔖𝔏_{n}^{u}. By Proposition 1.5, we only need to eliminate the negative twists by Ω_{K_{n}/k}^{n} in irreducible subquotients of V^{G_{F/Kn}}.

To do that we show first that the generic fibres of irreducible coherent G_{n}-equivariant sheaves are determined by their restrictions to a subgroup Aff_{n} = G_{n} ∩ Aut(𝔸_{k}^{n}/k).

Lemma 2.1 ([R], Lemma 2.2). Let P be a subgroup of Aut(K/k) such that the H-orbit of any element of K − k spans a k-subspace in K of dimension > r^{2}. Then the map H^{1}(P, GL_{r}k) → H^{1}(P, GL_{r}K) is injective. □

Lemma 2.2. Let Aff_{n} be the group of affine transformations of an affine space 𝔸_{k}^{n} with the function field K_{n}. Then the natural morphism

(1) {rational k-linear Aff_{n}-representations} ⟶^{⊗_{k}K_{n}} {K_{n}-semi-linear Aff_{n}-representations},

injective by Lemma 2.1, transforms isomorphism classes of irreducible k-representations of Aff_{n} to isomorphism classes of irreducible K_{n}-semi-linear representations of Aff_{n}.

Proof. Let W be an irreducible k-representations of Aff_{n}, and U ⊂ W ⊗_{k} K_{n} a non-zero K_{n}-semi-linear subrepresentation of Aff_{n}. Let α = ∑_{j=1}^{N} w_{j}α_{j} ∈ U be a non-zero element with minimal possible N. Multiplying α by an element of K_{n} we may assume that all α_{j} are polynomials: α = ∑_{I} w′_{I}z^{I}. The elements of the unipotent radical of Aff_{n}, i.e., σ such that σx − x ∈ k for any linear function x on 𝔸_{k}^{n}, act trivially on W.

Applying an appropriate composition of difference operators σ − τ for some σ, τ in the unipotent radical of Aff_{n} to α, we get a non-zero element of W. As any non-zero element of W generates W ⊗_{k} K_{n}, this means that U = W ⊗_{k} K_{n}. □

Corollary 2.3. Let Aff_{n}, 𝔸_{k}^{n} and K_{n} be as in Lemma 2.2. Then there is a natural injection

(2) \{ \begin{matrix}isomorphism classes of \\ irreducible G_{n}-subrepresentations in \\ ⨁_{M} Hom_{K}((Ω_{K_{n}/k}^{n})^{⊗M}, ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}) \end{matrix} \} ⟶ \{ \begin{matrix}isomorphism classes of \\ irreducible rational k-linear \\ Aff_{n}-representations \end{matrix} \}

such that its composition with the morphism (1) is the restriction map. □

Let W be an (n + 1)-dimensional k-vector space, L ⊂ W be a one-dimensional subspace, H_{lin} = ker[GL(W, L) → GL(W/L)] ≅ k^{×} ⋉ Hom(W/L, L). Lemma 2.4. For any Young diagram λ with no columns of height n + 1 one has

(S^{λ}W^{∨} ⊗_{k} (det W)^{⊗s})^{H_{lin}} = \{ \begin{matrix}S^{λ}(W/L)^{∨} & if s = 0, \\ 0 & otherwise \end{matrix}

Proof. Denote by X = AF (W) ≅ GL(W)/R_{u}(B), the variety of complete affine flags in W. An affine flag is a filtration W_{•} = (0 = W_{0} ⊂ W_{1} ⊂ W_{2} ⊂ ⋯ ⊂ W_{n+1} = W) with dim_{k} W_{j} = j and a collection of l_{j} ∈ W_{j}/W_{j−1} − {0}.

Let Y ≅ GL(W)/B be the variety of complete linear flags in W. Then the natural projection X →^{π} Y is a principal (𝔾_{m})^{n+1}-bundle, and there is a decomposition π_{∗}O_{X} = ⨁_{μ} M(μ) into a direct sum of invertible sheaves on Y, where μ runs over the group ℤ^{n+1} of characters of (𝔾_{m})^{n+1}, so O(X) = ⨁_{μ} Γ(Y, M(μ)).

Set X^{∘} = {(V_{•}, l_{•}) ∈ X | V_{n} ∩ L = 0 ⟺ “l_{n+1} ∈ L”}. Then reduction modulo L defines a principal L^{n} × 𝔾_{m}-bundle X^{∘} → AF (W/L), (l_{1}, …, l_{n+1}) ↦ (l_{1}, …, l_{n}).

Let X^{∘} ⟶^{“l_{n+1}”} L − \{0\} be the natural H- (or 𝔾_{m}-) equivariant map, and \overline{l}_{n+1} be the composition of “l_{n+1}” with an isomorphism L − {0} ≅ 𝔾_{m}.

Set SH := H_{lin} ∩ SL(W) = Hom(W/L, L). Then O(X)^{SH} = O(X) ∩ O(X^{∘})^{SH} = O(X) ∩ O(AF (W/L))[\overline{l}_{n+1}, \overline{l}_{n+1}^{−1}], so O(X)(μ)^{SH} = O(X) ∩ O(AF (W/L))(μ′)\overline{l}_{n+1}^{μ_{n+1}}, where μ′ ∈ ℤ^{n} is the restriction of μ to the first n multiples of (𝔾_{m})^{n+1}.

For any μ this is an irreducible representation of GL(W/L), and thus, O(X)(μ)^{SH} = O(AF (W/L))(μ′)\overline{l}_{n+1}^{μ_{n+1}} if O(X)(μ) ≠ 0.

As any irreducible representation of SL(W) coincides with O(X)(μ) for some μ, this implies that (S^{λ}W^{∨})^{SH} = S^{λ}(W/L)^{∨}. □ Theorem 2.5. For any F -semi-linear G_{F/k}-representation V ∈ A any irreducible subquotient of the K_{n}-semi-linear G_{n}-representation V^{G_{F/Kn}} is a direct summand of ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}.

Proof. Let K′ = k(W) be purely transcendental over K_{n} of degree 1. By Proposition 1.5 and Corollary 2.3, the restrictions to Aff(W) (or to its reductive quotient GL(W)) of irreducible subquotients of the K′-semi-linear PGL(W ⊕ k)-representation V^{G_{F/K′}} are of type (S^{λ}W^{∨} ⊗ (det W)^{⊗s}) ⊗_{k} K′ for a Young diagram λ with no columns of height n + 1 and some integer s.

Let H ⊂ Aff(W) be the subgroup fixing the functionals x_{1}, …, x_{n} in W^{∨} vanishing on L. Let Aff_{u} be the unipotent radical of Aff(W), i.e. the group of translations of W.

Then the restrictions to Aff_{n} of the irreducible subquotients of the K_{n}-semi-linear G_{n}- representation V^{G_{F/Kn}} are contained in ((S^{λ}W^{∨} ⊗ (det W)^{⊗s}) ⊗_{k} K′)^{H}. As H ∩ Aff_{u} = ⟨1⟩_{k} ≅ k, we get ((S^{λ}W^{∨} ⊗ (det W)^{⊗s}) ⊗_{k} K′)^{H∩Aff_{u}} = (S^{λ}W^{∨} ⊗ (det W)^{⊗s}) ⊗_{k} K_{n}, so ((S^{λ}W^{∨} ⊗ (det W)^{⊗s}) ⊗_{k} K′)^{H} = (S^{λ}W^{∨} ⊗ (det W)^{⊗s})^{H} ⊗_{k} K_{n}.

By Lemma 2.4, (S^{λ}W^{∨} ⊗ (det W)^{⊗s})^{H} coincides with S^{λ}(W/L)^{∨} if s = 0, and vanishes otherwise. This means that any representation of Aff_{n} obtained this way is a direct summand of the tensor algebra of the representation (W/L)^{∨} = (Ω_{K_{n}/k}^{1})^{{translations}} of GL_{n}k. As any irreducible subquotient U of the K_{n}-semi-linear G_{n}-representation V^{G_{F/Kn}} is determined by its restriction U|_{Aff_{n}} to Aff_{n} and U|_{Aff_{n}} is a direct summand of ⨂_{K_{n}}^{•} Ω_{K_{n}/k}^{1}, the same holds for U. □

3. Extensions in 𝔖𝔏_{n}^{u} and in A

Denote by Aff_{n}^{(ℓ)}ℚ the subgroup of Aff_{n} consisting of the ℚ(μ_{ℓ})-affine substitutions of x_{1}, …, x_{n} with Jacobian in μ_{ℓ}, and by SAff_{n}^{(ℓ)}ℚ the evident subgroup of index ℓ.

Lemma 3.1. Let n ≥ 2 and ℓ ≥ 2. Then restriction of any object V ∈ 𝔖𝔏_{n}^{u} to SAff_{n}^{(ℓ)}ℚ is induced by a rational representation of its reductive quotient SL_{n}ℚ(μ_{ℓ}) over k. Irreducible rational representations of SL_{n}ℚ(μ_{ℓ}) over k induce irreducible semi-linear representations of SAff_{n}^{(ℓ)}ℚ over K_{n}. In particular, any extension in 𝔖𝔏_{n}^{u} splits as an extension of K_{n}- semi-linear representations of SAff_{n}^{(ℓ)}ℚ.

Proof. It is shown in [R] that H^{0}(U_{0}, −) is the fibre functor on 𝔖𝔏_{n}^{u}, where U_{0} is the unipotent radical of SAff_{n}^{(ℓ)}ℚ, so V = V^{U_{0}} ⊗_{k} K_{n} and the restrivction of V to SAff_{n}^{(ℓ)}ℚ is induced by a k-linear representation V^{U_{0}} of SL_{n}ℚ(μ_{ℓ}) for any V ∈ 𝔖𝔏_{n}^{u}.

By Proposition 1.5, the irreducible subquotients of V restricted to SAff_{n}^{(ℓ)}ℚ are induced by rational irreducible representations of SL_{n}ℚ(μ_{ℓ}), so the irreducible subquotients of V^{U_{0}} are rational and induce irreducible semi-linear representations of SAff_{n}^{(ℓ)}ℚ over K_{n}.

It follows from Theorems 1.3 and 3.8 that the k-linear representation V^{U_{0}} of SL_{n}ℚ(μ_{ℓ}) is semi-simple. □

Remarks. 1. Let V = Ω_{K_{n}}^{1}/Λ ⊗_{k} K_{n}, where Λ ⊂ Ω_{k}^{1} is a proper k-subspace. Let the extension 0 → V → U → K_{n} → 0 be given by the cocycle (ω_{σ} = d log \frac{σω}{ω}) ∈ Z^{1}(G_{n}, V), where ω = dx_{1} ∧ ⋯ ∧ dx_{n} ∈ Ω_{K_{n}/k}^{n}. Then the restriction of (ω_{σ}) to GL_{n}k is non-trivial.

2. The convolution with the Euler vector field ∑_{j=1}^{n} x_{j}∂/∂x_{j} defines a GL_{n}k-equivariant morphism Ω_{K_{n}/k}^{1} → K_{n} given by dx_{j} ↦ x_{j}. It is non-split for n ≥ 3, since (Ω_{K_{n}/k}^{1})^{SL_{n}ℚ} = 0.

Lemma 3.2. Let n, ℓ ≥ 2 and s be some integers such that ℓ does not divide s + 1. Let V = S_{K_{n}}^{λ}Ω_{K_{n}/k}^{1} ⊗_{K_{n}} (Ω_{K_{n}/k}^{n})^{⊗s} for a Young diagram λ with columns of height < n.

Then (V^{H_{n}})^{Aff_{n−1}^{(ℓ)}ℚ} = V^{Aff_{n}^{(ℓ)}ℚ}, where H_{n} := G_{K_{n}/K_{n−1}} ∩ Aff_{n}^{(ℓ)}ℚ.

Proof. Let W be the k-span of dx_{1}, …, dx_{n} in Ω_{K_{n}/k}^{1}. Then S_{K_{n}}^{λ}Ω_{K_{n}/k}^{1} = S_{k}^{λ}W ⊗_{k}K_{n} and

Ω_{K_{n}/k}^{n} = det_{k} W ⊗_{k} K_{n}. Set SH_{n} = H_{n} ∩ SAff_{n}^{(ℓ)}ℚ. Then H_{n} ≅ μ_{ℓ} ⋉ SH_{n}, and therefore, V^{H_{n}} = (V^{SH_{n}})^{μ_{ℓ}}.

One has V^{SH_{n}} = (S_{k}^{λ}W ⊗_{k} K_{n})^{SH_{n}} ⊗_{k} (det_{k} W)^{⊗s}. As the unipotent radical of SH_{n} acts trivially on S_{k}^{λ}W and fixes exactly K_{n−1} in K_{n}, if n ≥ 1, we get V^{SH_{n}} = (S_{k}^{λ}W)^{SH_{n}} ⊗_{k} (det_{k} W)^{⊗s} ⊗_{k} K_{n−1} = S_{k}^{λ}(W^{SH_{n}}) ⊗_{k} (det_{k} W)^{⊗s} ⊗_{k} K_{n−1}. Then V^{H_{n}} = S_{k}^{λ}(W^{SH_{n}}) ⊗_{k}

((det_{k} W)^{⊗s})^{μ_{ℓ}} ⊗_{k} K_{n−1} = \{ \begin{matrix}S_{K_{n−1}}^{λ}Ω_{K_{n−1}/k}^{1} ⊗_{K_{n−1}} (Ω_{K_{n−1}/k}^{n−1})^{⊗s} & if ℓ|s \\ 0 & otherwise. \end{matrix}

On the other hand, V^{Aff_{n}^{(ℓ)}ℚ} = (S_{k}^{λ}W ⊗_{k} (det_{k} W)^{⊗s}⊗_{k} K_{n})^{Aff_{n}^{(ℓ)}ℚ} coincides with (S_{k}^{λ}W ⊗_{k} (det_{k} W)^{⊗s})^{Aff_{n}^{(ℓ)}ℚ}, since the unipotent radical of Aff_{n}^{(ℓ)}ℚ acts trivially on S_{k}^{λ}W ⊗_{k}(det_{k} W)^{⊗s}

and fixes exactly k in K_{n}. Thus, for n ≥ 1, we get V^{Aff_{n}^{(ℓ)}ℚ} = \{ \begin{matrix}k & if λ = 0 and ℓ|s, \\ 0 & otherwise. \end{matrix} This

implies that (V^{H_{n}})^{Aff_{n−1}^{(ℓ)}ℚ} = \{ \begin{matrix}k & if either λ = 0 and ℓ|s, or S^{λ} = ∧^{n−1} and ℓ|(s + 1), \\ 0 & otherwise \end{matrix} for

n ≥ 2. □

Lemma 3.3 ([R], Lemma 7.1). Let n > m ≥ 0 be integers and H be a subgroup of G_{F/k} preserving K_{n} and projecting onto a subgroup of G_{K_{n}/k} containing the permutation group of the set {x_{1}, …, x_{n}}. Then the subgroup in G_{F/k} generated by G_{F/K_{m}} and H is dense. □

For any U ∈ A and m ≥ 0 set U_{m} = U^{G_{F/Km}}.

Proposition 3.4. If U ∈ A and there is a subquotient of U_{n} ∈ 𝔖𝔏_{n}^{u} isomorphic to K_{n} then there is an embedding F ↪ V in A. One has H_{smooth}^{1}(G_{F/k}, V) = 0 for any V ∈ A.

Proof. By Lemma 3.3, U^{G_{F/k}} = U_{n+1}^{Aff_{n+1}^{(ℓ)}ℚ} ∩ U_{n} for any even ℓ ≥ 2. By Theorem 2.5,

Lemma 3.1 and Lemma 3.2, (U_{n+1}^{H_{n+1}})^{Aff_{n}^{(ℓ)}ℚ} = U_{n+1}^{Aff_{n+1}^{(ℓ)}ℚ} for any sufficiently big ℓ, so, as U_{n} ⊆ U_{n+1}^{H_{n+1}}, one has U^{G_{F/k}} = U_{n}^{Aff_{n}^{(ℓ)}ℚ}, and thus, U^{G_{F/k}} ≠ 0 if there is a subquotient of U_{n} ∈ 𝔖𝔏_{n}^{u} isomorphic to K_{n}.

Clearly, H^{0}(G, −) = Hom_{G}(ℚ, −) = Hom_{C}(F, −) and H_{smooth}^{1}(G, −) = Ext_{Sm_{G}}^{1}(ℚ, −) = Ext_{C}^{1}(F, −) on C, so we have to show that any smooth F -semi-linear extension 0 → V → U → F → 0 splits. Fix n > 1 such that U_{n} surjects onto K_{n}. By Theorem 2.5 and Lemma 3.1 the semi-linear representation U_{n} of Aff_{n}^{(ℓ)}ℚ over K_{n} splits as K_{n} ⊕ V_{n}, and thus, U^{G_{F/k}} projects onto k. □

Theorem 3.5. The category A is abelian. The functor H^{0}(G_{F/L}, −) is exact on A for any subfield L in F containing k.

Proof. Let V ∈ A and V →^{π} V′ be a surjection of F -semi-linear representations of G_{F/k}. By Proposition 3.4, for any K ⊂ F of finite type over k and any v ∈ (V′)^{G_{F/\overline{K}}} − \{0\}, the extension 0 → ker π → π^{−1}(F · v) → F → 0 of F -semi-linear representations of G_{F/\overline{K}} splits. This implies that the natural projection V^{G_{F/K}} ⟶^{π_{K}} (V′)^{G_{F/K}} is surjective, and thus, V′ is also an admissible semi-linear representation.

The functor H^{0}(G_{F/L}, −) on A is the composition of the forgetful functor Φ : A_{k} → C_{L}, the functor H^{0}(G_{F/\overline{L}}, −) on C_{L} and the exact functor H^{0}(G_{\overline{L}/L}, −) on Sm_{G_{\overline{L}/L}}. If L is of finite transcendence degree over k then the forgetful functor Φ factors through A\overline{L}, so the composition H^{0}(G_{F/\overline{L}}, −) ∘ Φ is exact. If L is of infinite transcendence degree over k then H^{0}(G_{F/\overline{L}}, −) induces an equivalence of categories Sm_{G_{F/k}} ⟶^{∼} Sm_{G_{\overline{L}/k}}, so H^{0}(G_{F/\overline{L}}, −) is also exact. □

Corollary 3.6. H_{smooth}^{1}(G, Ω_{F/k,closed}^{•}) = H_{smooth}^{1}(G, Ω_{F/k,exact}^{•}) = 0.

Proof. This follows from Proposition 3.4 and the following pair of short exact sequences 0 → Ω_{F/k,closed}^{q} → Ω_{F/k}^{q} → Ω_{F/k,exact}^{q+1} → 0 and 0 → Ω_{F/k,exact}^{q} → Ω_{F/k,closed}^{q} → H_{dR/k}^{q}(F) → 0. □

3.1. Extensions in 𝔖𝔏_{n}^{u}. Now we need the following particular case of Bott’s theorem.

Theorem 3.7 (Bott). If V is an irreducible G_{n}-equivariant coherent sheaf on ℙ_{k}^{n} then there exists at most one j ≥ 0 such that H^{j}(ℙ_{k}^{n}, V) ≠ 0. If H^{j}(ℙ_{k}^{n}, V)^{G_{n}} ≠ 0 then V ≅ Ω_{ℙ_{k}^{n}/k}^{j}. □

As well as the following elaboration of Remark 8.19 of [BT] and [T], §5.1.

Theorem 3.8 ([LR], Theorem 3). Let G be a simple simply connected Chevalley k-group, and G be a connected algebraic k-group. Let τ : G(k) → G(k) be a homomorphism with Zariski dense image. Assume that the unipotent radical G_{u} of G is commutative and the composition G(k) →^{τ} G(k) → G′(k), where G′ = G/G_{u}, is induced by a rational k-morphism λ : G → G′. Let A = k[ε_{1}, …, ε_{r}]/(ε_{1}^{2}, …, ε_{r}^{2}) and H = R_{A/k}(G ×_{k} A) ≅ G ⋉ 𝔤^{⊕r}, where 𝔤 = Lie(G) is the adjoint representation of G, and r = dim G_{u}/ dim G′.

Then G_{u} splits into a direct sum of r copies of the adjoint representation of G′; there exist derivations δ_{1}, …, δ_{r} of k and a k-isogeny μ : H → G such that τ = μ ∘ η_{δ},

where η_{δ} : G(k) → H(k) is induced by the ring embedding id + ∑_{j=1}^{r} δ_{j}ε_{j} : k → A. □

Lemma 3.9. Let n ≥ 2. Suppose that Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, V_{∘}) ≠ 0 for some irreducible object V_{∘} of 𝔖𝔏_{n}^{u}. Then either V_{∘} ≅ Ω_{K_{n}/k}^{1}, or V_{∘} ≅ Der(K_{n}/k). One has Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, Ω_{K_{n}/k}^{1}) = k and Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, Der(K_{n}/k)) = Der(k).

Proof. Let V = S(V_{∘}) be the irreducible coherent G_{n}-equivariant sheaf on ℙ(Q) = ℙ_{k}^{n} with the generic fibre V_{∘}, and V ∈ Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, V_{∘}).

Suppose that the short exact sequence 0 → V → S(V) → O → 0 of coherent sheaves on ℙ(Q) splits. Let E be the total space of S(V) ≅ O ⊕ V. Then, as Aut(S(V), V) ≅ (𝔾_{m} × 𝔾_{m}) ⋉ Γ(ℙ_{k}^{n}, V), the G_{n}-structure on V corresponds to a splitting of the sequence

(3)  1 ⟶ (𝔾_{m} × 𝔾_{m}) ⋉ Γ(ℙ(Q), V) ⟶ Aut_{lin}(E, tot(V)) ⟶ G_{n} ⟶ 1.

As H^{1}(G_{n}, 𝔾_{m} × 𝔾_{m}) = 1, Theorem 3.8 (with G = SL_{n+1}k, G′ = G_{n} and G_{u} ⊆ Γ(ℙ_{k}^{n}, V)) implies that a non-standard splitting of (3) can exist only if Γ(ℙ_{k}^{n}, V) is isomorphic to the adjoint representation of G_{n}, i.e., if V ≅ T_{ℙ_{k}^{n}/k}. The identity Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, Der(K_{n}/k)) = Der(k) follows also from Theorem 3.8.

If V ≅ Ω_{ℙ_{k}^{n}/k}^{1} then the target of the homomorphism Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, V_{∘}) ⟶^{α} Ext_{O}^{1}(O_{ℙ_{k}^{n}}, V) = k induced by the functor S is generated by the class of the Euler extension 0 → Ω_{ℙ(Q)/k}^{1} → Q^{∨} ⊗_{k} O_{ℙ(Q)}(−1) → O_{ℙ(Q)} → 0. Let E be the total space of the vector bundle with the sheaf of sections Q^{∨} ⊗_{k} O(−1). Any G_{n}-structure on the middle term of this extension corresponding to an element of Ext_{𝔖𝔏_{n}^{u}}^{1}(K_{n}, Ω_{K_{n}/k}^{1}) is a splitting of the short exact sequence

(4) 1 ⟶ Aut(Q^{∨} ⊗_{k} O(−1), Ω_{ℙ(Q)/k}^{1}) ⟶ Aut_{lin}(E, tot(Ω_{ℙ(Q)/k}^{1})) ⟶ G_{n} ⟶ 1.

As any point of ℙ(Q) determines a hyperplane in Q^{∨}, the group Aut(Q^{∨} ⊗_{k} O(−1), Ω_{ℙ(Q)}^{1}) coincides with the subgroup of GL(Q) stabilizing all hyperplanes in Q^{∨}, i.e., with the centre 𝔾_{m} of GL(Q). Then Aut_{lin}(E, tot(Ω_{ℙ(Q)/k}^{1})) is a central 𝔾_{m}-extension of G_{n}, so the splitting of (4) is unique and corresponds to the usual G_{n}-equivariant structure. □

Acknowledgement. I am grateful to the Max-Planck-Institut für Mathematik in Bonn for its hospitality and the financial support in September–November 2004 when this work was done.

References

[BMS] H.Bass, J.Milnor, J.-P.Serre, Solution of the congruence subgroup problem for SL_{n} (n ≥ 3) and Sp_{2n} (n ≥ 2), I.H.É.S. Publ. Math. No. 33, 1967 59–137.

[BT] A.Borel, J.Tits, Homomorphismes ”abstraits” de groupes algébriques simples, Ann. of Math. (2) 97 (1973), 499–571.

[DG] M.Demazure, P.Gabriel, Groupes algébriques. Tome I. Masson; North-Holland, 1970.

[LR] L.Lifschitz, A.Rapinchuk, On abstract homomorphisms of Chevalley groups with nonreductive image. I, J. Algebra 242 (2001), no. 1, 374–399.

[R] M.Rovinsky, Semi-linear representations of P GL, math.RT/0306333.

[T] J.Tits, Homorphismes ”abstraits” de groupes de Lie, Symposia Mathematica, Vol. XIII (Convegno di Gruppi e loro Rappresentazioni, INDAM, Rome, 1972), pp. 479–499. Academic Press, London, 1974.

Independent University of Moscow 121002 Moscow B.Vlasievsky Per. 11

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and Institute for Information Transmission Problems of Russian Academy of Sciences