Remarks on Emerton’s functor by P. Schneider and J. Teitelbaum

These notes grew out of a seminar talk the first author gave in 2003 on the functor constructed by Emerton in [Eme] and called by him the “Jacquet functor”. Their only purpose is to explain the general construction of this functor and Emerton’s computation of it for smooth representations in a way which is, as we like to think, somewhat less technical.

We fix a finite extension L/Q_{p} as well as a spherically complete extension field K of L. We let G be the group of L-rational points of a connected reductive group over L.

Let V be a locally analytic G-representation on a K-vector space of compact type.

We fix a parabolic subgroup P = MN of G with Levi factor M and unipotent radical N. The modulus character δ of P is given by

δ(mn) = vol_{N}(m^{−1}N_{0}m)/vol_{N}(N_{0})

for m ∈ M and n ∈ N where N_{0} ⊆ N is any compact open subgroup. If U is any P -representation we write U(δ) for the twist of U by δ.

Let V (N) denote the vector subspace of V generated by all nv − v for n ∈ N and v ∈ V. Then

V_{N} := V/\overline{V (N)}

is a locally analytic M-representation on a vector space of compact type. Let 𝔫 denote the Lie algebra of N. Then V^{𝔫} := V^{𝔫=0} is closed in V and hence carries a locally analytic P -representation on a vector space of compact type. Note that the N-action on V^{𝔫} is smooth. In an obvious way we may form V^{𝔫}(N) and

(V^{𝔫})_{N} := V^{𝔫}/\overline{V^{𝔫}(N)} .

The latter is a locally analytic M-representation on a vector space of compact type. The inclusion V^{𝔫} ⊆ V passes to an M-equivariant continuous linear map

(V^{𝔫})_{N} ⟶ V_{N} .

First we look at the left hand term. We fix a strictly positive element z in the center Z_{M} of M. This means that there is a compact open subgroup N_{0} ⊆ N such that the compact open subgroups N_{i} := z^{i}N_{0}z^{−i} in N satisfy

… ⊃ N_{−1} ⊃ N_{0} ⊃ N_{1} ⊃ N_{2} ⊃ …, ⋃_{i∈Z Z} N_{i} = N , and ⋂_{i∈Z Z} N_{i} = {1} .

The subspace V^{N_{0}} of N_{0}-fixed vectors is closed in V^{𝔫}, and we may consider the continuous operator

ψ_{z} : V^{N_{0}} ⟶^{z} V^{𝔫} ⟶^{ϵ_{N_{0}}} V^{N_{0}} .

Here we define, quite generally for any compact open subgroup N_{c} ⊆ N, the operator

\begin{matrix}ϵ_{N_{c}} : V^{𝔫} & ⟶ & V^{N_{c}} \\ v & ⟼ & \frac{1}{[N_{c}:N_{v}]} ∑_{n∈N_{c}/N_{v}} nv \end{matrix}

where N_{v} ⊆ N_{c} is a compact open subgroup which fixes v. Below we will repeatedly use the formula

z^{−1} ∘ ϵ_{N_{i}} ∘ z = ϵ_{N_{i−1}}

for any i ∈ ZZ. We set

V_{z−tor}^{N_{0}} := {v ∈ V^{N_{0}} : ψ_{z}^{ℓ}(v) = 0 for some ℓ ∈ IN}

Lemma 1: V_{z−tor}^{N_{0}} = ker(V^{N_{0}} →^{pr} V^{𝔫}/V^{𝔫}(N))

Proof: We first recall (cf. [BZ] 2.33) that

V^{𝔫}(N) = ⋃ V^{𝔫}(N_{c})

where N_{c} runs over all compact open subgroups of N and where

V^{𝔫}(N_{c}) := {v ∈ V^{𝔫} : ϵ_{N_{c}}(v) = 0} .

We further observe that due to our assumption that z^{−1}N_{0}z ⊇ N_{0} we have

(1)  ψ_{z}^{ℓ}(v) = z^{ℓ}(ϵ_{z^{−ℓ}N_{0}z^{ℓ}}(v)) .

If ψ_{z}^{ℓ}(v) = 0 we therefore obtain v ∈ V^{𝔫}(N_{−ℓ}) ⊆ V^{𝔫}(N). Vice versa, if v ∈ V^{N_{0}} ∩V^{𝔫}(N) then v ∈ V^{𝔫}(N_{c}), i.e., ϵ_{N_{c}}(v) = 0, for some compact open subgroup N_{0} ⊆ N_{c} ⊆ N. Since z is strictly positive we find an ℓ ∈ IN such that N_{c} ⊆ N_{−ℓ}. Then ϵ_{z^{−ℓ}N_{0}z^{ℓ}}(v) = 0 and consequently ψ_{z}^{ℓ}(v) = 0.

Since the N-action on V^{𝔫} is smooth the projection map V^{N_{0}} ⟶ V^{𝔫}/V^{𝔫}(N) is surjective. From the lemma we therefore obtain the continuous bijection

V^{N_{0}}/V_{z−tor}^{N_{0}} ⟶ V^{𝔫}/V^{𝔫}(N) .

In the opposite direction we have the map V^{𝔫} ⟶^{ϵ_{N_{0}}} V^{N_{0}}. Because of ϵ_{N_{0}}(V^{𝔫}(N)) ⊆ V^{N_{0}} ∩ V^{𝔫}(N) = V_{z−tor}^{N_{0}} it passes to a map V^{𝔫}/V^{𝔫}(N) ⟶ V^{N_{0}}/V_{z−tor}^{N_{0}} which obviously is inverse to the above bijection. We also note that the operator induced by ψ_{z} on the left hand side of this bijection corresponds to the action of z on the right hand side.

Proposition 2: If V is admissible then the maps

V^{N_{0}}/V_{z−tor}^{N_{0}} ⟶^{≅} V^{𝔫}/V^{𝔫}(N)

and

V^{N_{0}}/\overline{V_{z−tor}^{N_{0}}} ⟶^{≅}(V^{𝔫})_{N}

are topological isomorphisms; in particular, the operator induced by ψ_{z} on the left hand terms is a topological automorphism.

Proof: It suffices to consider the first map. By the preceding discussion it furthermore suffices to show that the map V^{𝔫} ⟶^{ϵ_{N_{0}}} V^{N_{0}} is continuous. Fix a compact open subgroup G_{0} ⊆ G such that N_{0} ⊆ G_{0}. According to [ST1] Prop. 6.5 we can write V, viewed as an admissible G_{0}-representation, as a compact inductive limit of a sequence of locally analytic G_{0}-representations on Banach spaces V_{j} (with injective transition maps). Hence V^{𝔫}, resp. V^{N_{0}}, is the compact inductive limit of the V_{j}^{𝔫}, resp. V_{j}^{N_{0}} (compare [GKPS] 3.1.16). It therefore suffices

to show that each map V_{j}^{𝔫} ⟶^{ϵ_{N_{0}}} V_{j}^{N_{0}} is continuous. But we do find a defining norm on V_{j}^{𝔫} with respect to which the compact group G_{0} acts by isometries. A straightforward computation shows that in this norm the map ϵ_{N_{0}} is norm decreasing.

To avoid the choice of N_{0} and to proceed in a more canonical way we observe that, since the N-action on V^{𝔫} is smooth, the usual Hecke algebra H(N) of K-valued locally constant functions with compact support on N acts on V^{𝔫}. We define

E′_{P} (V) := Hom_{H(N)}(H(N), V^{𝔫})

equipped with the topology of pointwise convergence. Next we observe that the group P acts on E′_{P} (V) via

pA(ϕ) := p(A(ϕ(p . p^{−1})))

through continuous automorphisms. It is well defined since

pA(ϕ_{1} ∗ ϕ_{2}) = p(A((ϕ_{1} ∗ ϕ_{2})(p . p^{−1})))
= δ^{−1}(p) · p(A(ϕ_{1}(p . p^{−1}) ∗ ϕ_{2}(p . p^{−1})))
= δ^{−1}(p) · p(ϕ_{1}(p . p^{−1})(A(ϕ_{2}(p . p^{−1}))))
= ϕ_{1}(p(A(ϕ_{2}(p . p^{−1}))))
= ϕ_{1}(^{p}A(ϕ_{2})) .

Moreover, the restriction to N of this action simply is given by

nA(ϕ) = A(ϕ^{n})

where ϕ^{n}(n′) := ϕ(n′n^{−1}). To see the connection with the previous discussion one checks that the map

\begin{matrix}Hom_{H(N)}(H(N), V^{𝔫}) & ⟶^{≅} & lim_{⟵_{i}} V^{N_{i}} \\ A & ⟼ & (A(ϵ_{N_{i}}))_{i} \end{matrix}

is a topological isomorphism where the projective limit on the right hand side is formed with respect to the ϵ_{N_{i}} as transition maps and is equipped with the projective limit topology. The continuous operators

ψ_{z,i} : V^{N_{i}} ⟶^{z} V^{N_{i+1}} ⟶^{ϵ_{N_{i}}} V^{N_{i}}

form a projective system. Because of

zA(ϵ_{N_{i}}) = δ(z) · z(A(ϵ_{z^{−1}N_{i}z})) = δ(z) · z(A(ϵ_{N_{i−1}}))
= δ(z) · z(ϵ_{N_{i−1}}(A(ϵ_{N_{i}}))) = δ(z) · ϵ_{N_{i}}(z(A(ϵ_{N_{i}})))
= δ(z) · ψ_{z,i}(A(ϵ_{N_{i}}))

its limit lim_{⟵_{i}} ψ_{z,i} corresponds to the action of z twisted by δ^{−1}(z) on the left hand side and in particular is invertible. We have the continuous linear map

\begin{matrix}Hom_{H(N)}(H(N), V^{𝔫})(δ^{−1}) & ⟶ & V^{𝔫}/V^{𝔫}(N) \\ A & ⟼ & A(ϵ_{N_{0}}) + V^{𝔫}(N) . \end{matrix}

Using the two formulas

A(ϵ_{N_{0}}) + V^{𝔫}(N) = A(ϵ_{N_{c}}) + V^{𝔫}(N)

for any compact open subgroup N_{c} ⊆ N and

ϵ_{N_{0}}(p.p^{−1}) = δ(p) · ϵ_{pN_{0}p^{−1}}

for any p ∈ P one easily sees that this map is P -equivariant. It also is surjective since it extends, via the injective linear map

\begin{matrix}V^{𝔫} & ⟶ & Hom_{H(N)}(H(N), V^{𝔫})(δ^{−1}) \\ v & ⟼ & [ϕ ↦ ϕ(v)] , \end{matrix}

the surjective projection map V^{𝔫} ⟶ V^{𝔫}/V^{𝔫}(N). Its kernel, by Lemma 1, is equal to the subspace of all A in Hom_{H(N)}(H(N), V^{𝔫}) for which there is, for any i ∈ ZZ, an ℓ_{i} ∈ IN such that ψ_{z,i}^{ℓ_{i}}(A(ϵ_{N_{i}})) = 0. The latter, by formula (1), is equivalent to ϵ_{N_{i−ℓ_{i}}} (A(ϵ_{N_{i}})) = A(ϵ_{N_{i−ℓ_{i}}}) = 0. This proves the following result for which we introduce the P -invariant subspace

E′_{P} (V)_{0} := {A ∈ Hom_{H(N)}(H(N), V^{𝔫}) : A(ϵ_{N_{c}}) = 0 for some

compact open subgroup N_{c} ⊆ N}

of E′_{P} (V).

Lemma 3: The natural map

E′_{P} (V)/E′_{P} (V)_{0} ⟶^{≅}(V^{𝔫}/V^{𝔫}(N))(δ)

is a continuous linear M-equivariant isomorphism.

(Is the map in Lemma 3 a topological isomorphism for admissible V?)

At this point we first look in more detail at the smooth case. As a piece of general notation, whenever U is a linear representation of some group H a vector u ∈ U is called U-finite if it is contained in a finite dimensional H-invariant subspace of U. Then U_{H−fin} := {u ∈ U : u is H-finite} is an H-invariant subspace.

We now suppose that V is admissible smooth. For any i ∈ ZZ, the commutative Z_{M}-equivariant diagram

\xymatrix{Hom_{H(N)}(H(N), V)(δ^{−1}) \ar@{->>}[rr]^{A↦A(ϵ_{N_{i}})} &  & V/V (N_{i}) \ar@{->>}[r]^{pr} & V_{N} \\  & V^{N_{i}} \ar[ul] \ar[ur]^{≅}_{pr}}

restricts to the commutative diagram

\xymatrix{E′_{P} (V)_{Z_{M} −fin}(δ^{−1}) \ar@{->>}[rr] &  & (V/V (N_{i}))_{Z_{M} −fin} \\  & (V^{N_{i}})_{Z_{M} −fin} . \ar[ul] \ar@{-->}[ur]^{≅}}

In order to establish the surjectivity of the natural map

E′_{P} (V)_{Z_{M} −fin}(δ^{−1}) ⟶ V_{N}

it therefore suffices to show that any element of V_{N} can, for some i ∈ ZZ, be lifted to (V/V (N_{i}))_{Z_{M} −fin}. The fact that V_{N} is admissible smooth as an Mrepresentation implies

(V_{N})_{Z_{M} −fin} = V_{N} .

Consider now an arbitrary vector v ∈ V. It is fixed by some compact open subgroup Z_{c} ⊆ Z_{M}. Since the image of v in V_{N} is Z_{M}-finite the kernel of the map

\begin{matrix}K[Z_{M}/Z_{c}] & ⟶ & V_{N} \\ Q & ⟼ & Q(v) + V (N) \end{matrix}

is an ideal I of finite codimension in the group ring K[Z_{M}/Z_{c}]. But Z_{M}/Z_{c} being a finitely generated abelian group this group ring is noetherian. Hence the ideal I has finitely many generators Q_{1}, …, Q_{r}. We now choose an i ∈ ZZ such that Q_{1}(v), …, Q_{r}(v) ∈ V (N_{i}). Then the above map lifts to a map K[Z_{M}/Z_{c}]/I ⟶ V/V (N_{i}) which shows that the image of v in V/V (N_{i}) is Z_{M}-finite. It follows that the map E′_{P} (V)_{Z_{M} −fin} ⟶ V_{N} is surjective.

On the other hand we claim that

E′_{P} (V)_{Z_{M} −fin} ∩ E′_{P} (V)_{0} = 0 .

Let A be an element in the intersection on the left hand side. By the definition of E′_{P} (V)_{0} we then find a j ∈ ZZ such that A(ϵ_{N_{i}}) = 0 for any i ≤ j. Hence (^{z^{ℓ}}A)(ϵ_{N_{j}}) = 0 for any ℓ ≥ 0. Since A is Z_{M}-finite this forces (^{z^{−1}}A)(ϵ_{N_{j}}) = 0 or, equivalently, A(ϵ_{N_{j+1}}) = 0. Inductively we obtain in this way that A(ϵ_{N_{i}}) = 0 for any i ∈ ZZ which means that A = 0.

Together with Lemma 3 this establishes the following result in the smooth case.

Proposition 4: If V is admissible smooth then the natural map

E′_{P} (V)_{Z_{M} −fin} ⟶^{≅} V_{N}(δ)

is an M-equivariant isomorphism.

Going back to a general V one might ask what properties the M-action on E′_{P} (V) has. We will give an answer for the center Z_{M}.

Lemma 5: The action of Z_{M} on E′_{P} (V) is continuous and extends (uniquely) to a separately continuous action of the distribution algebra D(Z_{M}, K).

Proof: Obviously this needs to be checked only for the action of a sufficiently small compact open subgroup Z_{c} ⊆ Z_{M}. We choose Z_{c} in such a way that it normalizes N_{0} and then also each N_{i} so that the subspaces V^{N_{i}} are Z_{c}-invariant. One easily checks that our above topological isomorphism Hom_{H(N)}(H(N), V^{𝔫}) ≅ lim_{⟵_{i}} V^{N_{i}} is Z_{c}-equivariant. Since the G-action on V is locally analytic so,

too, is the Z_{c}-action on each closed subspace V^{N_{i}}. From this we obtain a unique separately continuous D(Z_{c}, K)-action on each V^{N_{i}} and hence one on their projective limit.

Let Z_{M}^{0} denote the maximal compact subgroup of Z_{M}. We may write Z_{M} = Z_{M}^{0} × Z with a finitely generated free abelian group Z. Then

D(Z_{M}, K) = D(Z_{M}^{0}, K)[Z] = D(Z_{M}^{0}, K) ⊗_{K} K[Z] .

We now view K[Z] as the ring of rational functions on the split K-torus T with character group Z and we introduce the ring O of holomorphic functions on T . Since T is a Stein space the ring O is a commutative nuclear Fréchet-Stein algebra. We introduce the completed tensor product

D^{hol}(Z_{M}, K) := D(Z_{M}^{0}, K) ˆ⊗_{K} O .

One checks that the Fréchet algebra D^{hol}(Z_{M}, K) is (up to natural isomorphism) independent of the choice of Z. It contains D(Z_{M}, K) as a dense subalgebra.

We have seen that E′_{P} (V) is a D(Z_{M}, K)-module. The following definition singles out a submodule on which the D(Z_{M}, K)-action extends (uniquely) to a separately continuous D^{hol}(Z_{M}, K)-action.

Definition: E_{P} (V) := Hom_{D(Z_{M} ,K)}^{cont}(D^{hol}(Z_{M}, K), E′_{P} (V)(δ^{−1})) equipped with the strong topology.

Remarks: 1. The M-action on E′_{P} (V) induces by functoriality an action of M by topological automorphisms on E_{P} (V). It commutes with the obvious D^{hol}(Z_{M}, K)-action.

2. Since D(Z_{M}, K) is dense in D^{hol}(Z_{M}, K) evaluation at 1 ∈ D^{hol}(Z_{M}, K) is a continuous M-equivariant injective linear map

E_{P} (V) ⟶ E′_{P} (V)(δ^{−1}) = Hom_{H(N)}(H(N), V^{𝔫})(δ^{−1}) .

3. The idea to consider the M-analytic vectors in E′_{P} (V)(δ^{−1}) is pointless since, if V is smooth, they contain all of V.

We note that

E_{P} (V) = Hom_{K[Z]}^{cont}(O, E′_{P} (V)(δ^{−1})) .

To explore the projective limit structure of E′_{P} (V) we introduce the submonoid Z^{+} of all positive elements in Z, i.e., all elements which satisfy z^{−1}N_{0}z ⊇ N_{0}. For any z ∈ Z^{+} the continuous operators

ψ_{z,i} : V^{N_{i}} ⟶^{z} V^{zN_{i}z^{−1}} ⟶^{ϵ_{N_{i}}} V^{N_{i}}

still form a projective system and, because of

zA(ϵ_{N_{i}}) = δ(z) · z(A(ϵ_{z^{−1}N_{i}z})) = δ(z) · z(A(ϵ_{z^{−1}N_{i}z} ∗ ϵ_{N_{i}}))
= δ(z) · z(ϵ_{z^{−1}N_{i}z}(A(ϵ_{N_{i}}))) = δ(z) · ϵ_{N_{i}}(z(A(ϵ_{N_{i}})))
= δ(z) · ψ_{z,i}(A(ϵ_{N_{i}})) ,

its limit lim_{⟵_{i}} ψ_{z,i} still corresponds to the action of z twisted by δ^{−1}(z) on the left hand side of the isomorphism

E′_{P} (V) = Hom_{H(N)}(H(N), V^{𝔫}) ≅ lim_{⟵_{i}} V^{N_{i}} .

Observing that, for any two z_{1}, z_{2} ∈ Z^{+}, we have

ψ_{z_{1},i} ∘ ψ_{z_{2},i} = ϵ_{N_{i}} ∘ z_{1} ∘ ϵ_{N_{i}} ∘ z_{2} = ϵ_{N_{i}} ∘ z_{1} ∘ ϵ_{N_{i}} ∘ z_{1}^{−1} ∘ (z_{1}z_{2})
= ϵ_{N_{i}} ∘ ϵ_{z_{1}N_{i}z_{1}^{−1}} ∘ (z_{1}z_{2}) = ϵ_{N_{i}} ∘ (z_{1}z_{2})
= ψ_{z_{1}z_{2},i}

we may also introduce the O-modules

Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i}}) := {α ∈ Hom_{K}^{cont}(O, V^{N_{i}}) : α ∘ z = ψ_{z,i} ∘ α for any z ∈ Z^{+}}

equipped with the strong topology.

Lemma 6: The natural map E_{P} (V) ⟶^{≅} Hom_{K[Z^{+}]}^{cont}(O, V^{N_{0}}) is an O-linear topological isomorphism.

Proof: Using the universal property of the projective limit we have

E_{P} (V) = Hom_{K[Z]}^{cont}(O, E′_{P} (V)(δ^{−1}))

= Hom_{K[Z]}^{cont}(O, lim_{⟵_{i}} V^{N_{i}})

= Hom_{K[Z^{+}]}^{cont}(O, lim_{⟵_{i}} V^{N_{i}})

= lim_{⟵_{i}} Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i}}) .

Here the third equality is due to the fact that any element in Z is a quotient of two elements in Z^{+}. The transition map in the latter projective system is

τ_{i} := Hom(O, ϵ_{N_{i−1}}) : Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i}}) ⟶ Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i−1}}) .

On the other hand, choosing our original strictly positive element z in Z^{+}, we have the continuous linear maps

ζ_{i} := Hom(O, z) : Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i−1}}) ⟶ Hom_{K[Z^{+}]}^{cont}(O, V^{N_{i}}) .

For the two composites we obtain

τ_{i} ∘ ζ_{i} = Hom(O, ψ_{z,i−1}) = Hom(z, V^{N_{i−1}})

and

ζ_{i} ∘ τ_{i} = Hom(O, z ∘ ϵ_{N_{i−1}}) = Hom(O, ψ_{z,i}) = Hom(z, V^{N_{i}})

which both are topological isomorphisms. It follows that the transition maps τ_{i} are topological isomorphisms.

By the way, the same reasoning as in the above proof shows that

E′_{P} (V)(δ^{−1}) ⟶^{≅} Hom_{K[Z^{+}]}(K[Z], V^{N_{0}}) .

Proposition 7: The locally convex vector space E_{P} (V) is of compact type and the M-action on it is locally analytic; moreover, the O-action on E_{P} (V) is separately continuous.

Proof: We fix a compact open subgroup M_{0} ⊆ M which normalizes N_{0} so that V^{N_{0}} is M_{0}-invariant. Since M acts on E_{P} (V) by topological automorphisms the local analyticity only needs to be checked for the M_{0}-action on E_{P} (V). But, by Lemma 6, E_{P} (V) is a closed M_{0}-invariant subspace of L_{b}(O, V^{N_{0}}). It therefore suffices to check that the latter is of compact type and that the M_{0}-action on it is locally analytic. According to [ST0] Cor. 3.3 this is equivalent to checking that the strong dual L_{b}(O, V^{N_{0}})′_{b} is a nuclear Fréchet space and that the M_{0}-action on it extends to a separately continuous D(M_{0}, K)-action. Certainly the strong dual (V^{N_{0}})′_{b} has both these properties. It follows from [NFA] p.134 that

L_{b}(O, V^{N_{0}}) = (O ˆ⊗_{K,π} (V^{N_{0}})′_{b})′_{b} .

On the other hand [NFA] 19.11, 20.4, and 20.14 imply that with O and (V^{N_{0}})′_{b} also O ˆ⊗_{K,π} (V^{N_{0}})′_{b} is a nuclear Fréchet space and, in particular, is reflexive so that

L_{b}(O, V^{N_{0}})′_{b} = O ˆ⊗_{K,π} (V^{N_{0}})′_{b} .

The projective tensor product O ⊗_{K,π} (V^{N_{0}})′_{b} carries obvious separately continuous O- and D(M_{0}, K)-actions through the first and second factor, respectively. By the universal property of the completion and the Banach-Steinhaus theorem these actions extend to separately continuous action on the completed tensor product O ˆ⊗_{K,π} (V^{N_{0}})′_{b}.

Proposition 8: If V is admissible smooth then the image of the injective map E_{P} (V) ↪ E′_{P} (V)(δ^{−1}) given by evaluation in 1 ∈ O is equal to E′_{P} (V)_{Z_{M} −fin}; in particular, we have a natural M-equivariant isomorphism

E_{P} (V) ⟶^{≅} V_{N} .

Proof: Suppose first that A ∈ E′_{P} (V) is Z_{M}-finite. Then the map

\begin{matrix}K[Z] & ⟶ & E′_{P} (V) \\ z′ & ⟼ & ^{z′}A \end{matrix}

has finite dimensional image. Its kernel I therefore is an ideal of finite codimension. Since I is finitely generated the ideal IO it generates in the Fréchet-Stein algebra O is closed. It follows that the dense inclusion K[Z] ⊆ O induces the surjection K[Z]/I ⟶ O/IO of finite dimensional Hausdorff vector spaces. Since O is faithfully flat over K[Z] the latter map also is injective. This shows that A extends continuously to O which means that A = α(1) for some α ∈ E_{P} (V).

Because of Prop. 4 it remains to show that the composed map

E_{P} (V) ⟶ E′_{P} (V) ⟶ V_{N}

is injective. But we have the commutative diagram

\xymatrix{E_{P} (V) \ar[d]_{≅} \ar[r] & E′_{P} (V) \ar[r] & V_{N} \\ Hom_{K[Z^{+}]}^{cont}(O, V^{N_{0}}) \ar[rr]^{β↦β(1)} &  & V^{N_{0}}/V_{z−tor}^{N_{0}} \ar[u]_{≅}}

where the perpendicular arrows are isomorphisms by Lemma 6 and Prop. 2, respectively. So we are reduced to showing that the lower horizontal arrow is injective. Let β ∈ Hom_{K[Z^{+}]}^{cont}(O, V^{N_{0}}). Then β is a continuous map from the Fréchet space O into the vector space V^{N_{0}} with its finest locally convex topology. The latter, in particular, is a countable locally convex inductive limit of finite dimensional Hausdorff vector spaces. Therefore the map β has a finite dimensional image ([NFA] 8.9. Since multiplication by z is invertible on O the map ψ_{z} must restrict to a surjective and hence bijective endomorphism of im(β). It follows that for nonzero β the value β(1) cannot lie in V_{z−tor}^{N_{0}}.

References

[BZ] Bernstein J., Zelevinskii A.V.: Representations of the group GL(n, F) where F is a non-archimedean local field. Russ. Math. Surv. 31 (3), 1-68 (1976)

[GKPS] De Grande-De Kimpe N., Kakol J., Perez-Garcia C., Schikhof W: padic locally convex inductive limits. In p-adic functional analysis, Proc. Int. Conf. Nijmegen 1996 (Eds. Schikhof, Perez-Garcia, Kakol), Lect. Notes Pure Appl. Math., vol. 192, pp. 159-222. New York: M. Dekker 1997

[Eme] Emerton M.: Jacquet modules of locally analytic representations of padic reductive groups I. Construction and first properties. To appear in Ann. Sci. ENS

[Sch] Schneider P.: Nonarchimedean Functional Analysis. Berlin - Heidelberg - New York: Springer 2002

[ST0] Schneider P., Teitelbaum J.: Locally analytic distributions and p-adic representation theory, with applications to GL_{2}. J. AMS 15, 443-468 (2002)

[ST1] Schneider, P., Teitelbaum, J.: Algebras of p-adic distributions and admissible representations. Invent. math. 153, 145-196 (2003)