The 1999 Britton Lectures at McMaster University on ”p-adic representation theory” by Peter Schneider

Lecture 1: The Lazard ring

We fix a field K with a complete nonarchimedean absolute value | |. The unique extension of | | to an algebraic closure \overline{K} of K again will be denoted by | |. Let G_{K} := Gal(\overline{K}|K) be the absolute Galois group of K.

Disk algebras

For r ∈ |\overline{K}^{×}| let

X_{r}(\overline{K}) := \{a ∈ \overline{K} : |a| ≤ r\}

be the closed disk around 0 of radius r and

X_{r} := space of G_{K}-orbits in X_{r}(\overline{K})

Definition:

O(X_{r}) := all power series f(T) = ∑_{n≥0} a_{n}T^{n} with a_{n} ∈ K which converge
on X_{r}(\overline{K})
= all power series f(T) = ∑_{n≥0} a_{n}T^{n} with lim_{n→∞} |a_{n}|r^{n} = 0

is called the ring of holomorphic functions on X_{r}.

Facts:

1. O(X_{r}) is an integral domain;

2. O(X_{r}) is a K-Banach algebra w.r.t. the (multiplicative) norm

|| ∑_{n≥0} a_{n}T^{n}||_{r} := max_{n≥0} |a_{n}|r^{n}.

An effective divisor on X_{r} is a map D : X_{r} ⟶ ZZ_{≥0} with finite support; the effective divisors form a semigroup Div^{+}(X_{r}) which is partially ordered by

D ≤ D′ iff D(x) ≤ D′(x) for any x ∈ X_{r} .

Fact:

all monic polynomials f = all monic polynomials →^{∼} Div^{+}(X_{r})
∈ K[T] with ||f||_{r} = r^{deg f}  in K[T] with all zeros
in X_{r}(\overline{K})
f ↦ (f)(x) :=

multiplicity of a ∈ x as a zero of f

is a bijection.

Weierstrass division: Fix a 0 ≠ g(T) = ∑_{n≥0} a_{n}T^{n} ∈ O(X_{r}) and let m ≥ 0 be the largest index such that |a_{m}|r^{m} = ||g||_{r}; then for any f ∈ O(X_{r}) there are uniquely determined elements q ∈ O(X_{r}) and R ∈ K[T] of degree < m such that

f = qg + R

moreover: i. ||f||_{r} = max(||qg||_{r}, ||R||_{r}); ii. if f and g are polynomials and g has degree m then q is a polynomial.

Weierstrass preparation: Let g and m as above; then there is a unique monic polynomial g_{0} ∈ K[T] of degree m and a unique e ∈ O(X_{r})^{×} such that

g = e · g_{0};

moreover: i. ||g_{0}||_{r} = r^{m}; ii. if g is a polynomical so, too, is e. (Proof: Apply division to T^{m} = e′g + R and put g_{0} := T^{m} − R.) Reference: U. Güntzer, Modellringe in der nichtarchimedischen Funktionentheorie, Indagationes math. 29, 334-342 (1967)

In particular we may define the divisor of any 0 ≠ g ∈ O(X_{r}) by

(g) := (g_{0}) ∈ Div^{+}(X_{r}).

Consequences:

1. g|g′ if and only if (g) ≤ (g′);

2. O(X_{r}) is a principal ideal domain; 3.

nonzero ideals in O(X_{r}) ↔^{∼} Div^{+}(X_{r})
gO(X_{r}) ↔ (g)

is an order reversing bijection; it restricts to a bijection

maximal ideals in O(X_{r}) ↔^{∼} X_{r} ;

4. for r ≤ r′ in |\overline{K}^{×}| the diagram

O(X_{r′}) \ {0} ⟶^{( )}  Div^{+}(X_{r′})
⊆↓  ↓ restriction
O(X_{r}) \ {0} ⟶^{( )}  Div^{+}(X_{r})

is commutative.

Note: Since || ||_{r} is multiplicative the map O(X_{r}) ⟶^{g·} O(X_{r}), for any 0 ≠ g ∈ O(X_{r}), is a homeomorphism onto its image. It follows that any principal ideal and hence any ideal in O(X_{r}) is closed.

The open unit disk

Let

X(\overline{K}) := \{a ∈ \overline{K} : |a| < 1\}

be the open unit disk and

X := G_{K}-orbits in X(\overline{K}) .

Definition:

O(X) := all power series f(T) = ∑_{n≥0} a_{n}T^{n} with a_{n} ∈ K which converge
on X(\overline{K})

is called the ring of holomorphic functions on X.

Since

O(X) = lim_{⟵_{r<1}} O(X_{r}) = ⋂_{r<1} O(X_{r})

the ring O(X) is an integral domain as well as a K-Fréchet algebra w.r.t. the family of norms {|| ||_{r}}_{r<1}. We note that this family of norms is increasing, i.e., ||f||_{r} ≤ ||f||_{r′} for f ∈ O(X) and r ≤ r′. This leads to the following technical observation which will be used later on.

Remark: An element f ∈ O(X) lies in the closure \overline{A} of a subset A ⊆ O(X) if and only if for any r < 1 in |\overline{K}^{×}| and any ε > 0 we find a g ∈ A such that ||f − g||_{r} < ε.

We define

Div^{+}(X) := lim_{⟵_{r<1}} Div^{+}(X_{r})
= all maps D : X ⟶ ZZ_{≥0} such that, for any r < 1, the

restriction D|X_{r} has finite support

to be the partially ordered semigroup of effective divisors on X and obtain the divisor map

\begin{matrix}O(X) \ \{0\} & ⟶ & Div^{+}(X) \\ f & ⟼ & (f) \end{matrix}

as the projective limit of the divisor maps for the X_{r} with r < 1. It is immediately clear that, for any 0 ≠ f, g ∈ O(X), we have

f|g if and only if (f) ≤ (g) ;

in particular:

(f) = (g) if and only if f = ug for some u ∈ O(X)^{×} .

Theorem: (Lazard)

If K is spherically complete then the divisor map O(X) \ {0} ⟶ Div^{+}(X) is surjective.

Proof: (M. Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Publ. Math. IHES 14, 47-75 (1962); we follow the argument in J. Fresnel/M. van der Put, Géométrie Analytique Rigide et Applications, Progress in Math. vol. 18, Birkhäuser 1981, p. 44.)

We fix a strictly increasing sequence r_{1} < r_{2} < … < 1 in |\overline{K}^{×}| tending towards 1 and put X_{n} := X_{r_{n}}. Given a divisor D ∈ Div^{+}(X) we may choose, for any n ∈ IIN, an element f_{n} ∈ O(X_{n}) \ {0} such that (f_{n}) = D|X_{n}. We then have

f_{n+1}|X_{n} = u_{n} · f_{n} with appropriate units u_{n} ∈ O(X_{n})^{×}.

We claim that

(∗)  there are v_{n} ∈ O(X_{n})^{×} such that v_{n} · (v_{n+1}|X_{n})^{−1} = u_{n} .

If so we modify our original lifts f_{n} by posing f̃_{n} := v_{n} · f_{n} and obtain (f̃_{n}) = D|X_{n} and

f̃_{n+1}|X_{n} = (v_{n+1}|X_{n}) · u_{n} · f_{n} = v_{n} · f_{n} = f̃_{n} .

This shows that there is a (unique) f ∈ O(X) \ {0} such that f|X_{n} = f̃_{n} for any n ∈ IIN. By construction we have (f) = D which proves our assertion.

Before we start establishing the claim (∗) we remark that, by a standard Čech covering argument, it amounts to the vanishing of the Picard group H^{1}(X, O_{X}^{×}) of the rigid analytic variety X.

As part of Weierstrass preparation we know that any unit u_{n} ∈ O(X_{n})^{×} is of the form

u_{n} = c_{n} · h_{n} with c_{n} ∈ K^{×}, h_{n}(0) = 1, and ||h_{n} − 1||_{r_{n}} < 1 .

We would like to define

v_{n} := (c_{1} · . . . · c_{n−1})^{−1} · ∏_{m≥n} h_{m}|X_{n}

but the infinite product may not converge. Our assumption on the field K of being spherically complete will enable us to “correct” this expression in such a way that it becomes well defined.

We first consider, for any n, m ∈ IIN, the finite product

h_{n,m} := ∏_{l=n+1}^{m} h_{l}|X_{n} = ∑_{i≥0} a_{i}^{(n,m)}T^{i}

in O(X_{n})^{×} (in particular h_{n,m} = 1 if n ≥ m). We have

\begin{matrix}|a_{i}^{(n,m)}| · r_{n}^{i} = & |a_{i}^{(n,m)}| · r_{n+1}^{i} · (\frac{r_{n}}{r_{n+1}} )^{i} \\ ≤ & (\frac{r_{n}}{r_{n+1}} )^{i} · ∏_{l=n+1}^{m} ||h_{l}|X_{n+1}||_{r_{n+1}} \\ ≤ & (\frac{r_{n}}{r_{n+1}} )^{i} \end{matrix}

and in particular

|a_{i}^{(n,m)}| ≤ r_{n+1}^{−i} for any n, m ≥ 1 and i ≥ 0 .

The sequence (a_{i}^{(n,1)}, a_{i}^{(n,2)}, a_{i}^{(n,3)}, …) then is, for any fixed n ≥ 1 and i ≥ 0, an element in the K-Banach space ℓ^{∞}(K) of all bounded sequences (b_{1}, b_{2}, …) in K with norm ||(b_{1}, b_{2}, …)|| := sup_{m∈IIN} |b_{m}|. On the subspace of converging

sequences we have the continuous linear form λ(b_{1}, b_{2}, …) := lim_{m→∞} b_{m} of norm 1. Since over a spherically complete K the Hahn-Banach theorem is valid we may extend λ to a continuous linear form λ of norm 1 on all of ℓ^{∞}(K). We now define

a_{i}^{(n)} := λ(a_{i}^{(n,1)}, a_{i}^{(n,2)}, …)

and

H_{n}(T) := a_{0}^{(n)} + a_{1}^{(n)}T + a_{2}^{(n)}T^{2} + …

Clearly a_{0}^{(n)} = λ(1, 1, …) = 1. Moreover

|a_{i}^{(n)}| · r_{n}^{i} ≤ sup_{m∈IIN} |a_{i}^{(n,m)}| · r_{n}^{i} ≤ (\frac{r_{n}}{r_{n+1}} )^{i} < 1 for i ≥ 1

which shows that

H_{n} ∈ O(X_{n})^{×}.

We also see that, for any a ∈ K with |a| ≤ r_{n}, the sequence {(a_{i}^{(n,1)}a^{i}, a_{i}^{(n,2)}a^{i}, …)}_{i≥0} in ℓ^{∞}(K) is a zero sequence. By continuity we therefore have

λ(h_{n,1}(a), h_{n,2}(a), …)

= λ(∑_{i≥0} a_{i}^{(n,1)}a^{i}, ∑_{i≥0} a_{i}^{(n,2)}a^{i}, …) = λ(∑_{i≥0} a^{i} · (a_{i}^{(n,1)}, a_{i}^{(n,2)}, …))

= ∑_{i≥0} a^{i} · λ(a_{i}^{(n,1)}, a_{i}^{(n,2)}, …) = ∑_{i≥0} a_{i}^{(n)}a^{i}

= H_{n}(a)

and hence

\begin{matrix}\frac{H_{n}(a)}{H_{n+1}(a)} = & \frac{λ(1,...,1,h_{n,n+1}(a),h_{n,n+2}(a),...)}{λ(1,...,1,1,h_{n+1,n+2}(a),...)} \\ = & h_{n+1}(a) · \frac{λ(1,...,1,h_{n+1,n+2}(a),...)}{λ(1,...,1,h_{n+1,n+2}(a),...)} \\ = & h_{n+1}(a) . \end{matrix}

Here we have used that, by construction, λ has the same value on any two bounded sequences which differ only in finitely many entries. Since any nonzero power series in O(X_{n}) can have at most countably many zeros it follows that we have the identity

H_{n} · H_{n+1}^{−1} = h_{n+1} in O(X_{n}) .

One easily verifies that the units

v_{n} := (c_{1} · . . . · c_{n−1})^{−1} · h_{n} · H_{n} in O(X_{n})^{×}

satisfy the claim

v_{n} · (v_{n+1}|X_{n})^{−1} = u_{n}

Complement: (Lazard, loc. cit, Prop. 5) If K is arbitrary then there is, for any D ∈ Div^{+}(X), an element 0 ≠ f ∈ O(X) such that (f) ≥ D.

Proposition: i. The map

Div^{+}(X) ⟶^{∼} all nonzero closed ideals in O(X)
D ⟼ I_{D} := {f ∈ O(X) : (f) ≥ D} ∪ {0}

is a well defined bijection; ii. any principal ideal in O(X) is closed; iii. if K is spherically complete then any closed ideal in O(X) is principal. Proof: First of all we note that, for any nonzero ideal I ⊆ O(X), the divisor

D(I)(x) := min_{0≠f∈I} (f)(x) for x ∈ X

is well defined. Given a divisor D ∈ Div^{+}(X) we find, according to the Complement, a 0 ≠ f ∈ O(X) such that (f) ≥ D. It follows that I_{D} is a nonzero ideal. For trivial reasons we have D(I_{D}) ≥ D. To see the opposite inequality we fix a 0 ≠ f ∈ I_{D}. For any r < 1 in |\overline{K}^{×}| let P_{r} ∈ K[T] be the unique monic polynomical with all zeros in X_{r}(\overline{K}) and

(P_{r}) + D|X_{r} = (f|X_{r}) .

Then fP_{r}^{−1} ∈ I_{D} with (fP_{r}^{−1}|X_{r}) = D|X_{r}. This shows that D(I_{D}) ≤ D. The ensuing equality D(I_{D}) = D implies that the map I ⟼ I_{D} is injective. If K is spherically complete then, by the Theorem, D = (f) for some 0 ≠ f ∈ O(X); in this case the ideal I_{D} = fO(X) is principal. In general, being the intersection

I_{D} = ⋂_{r<1} I_{D}O(X_{r})

of closed ideals in the disk algebras O(X_{r}) the ideal I_{D} is closed in O(X).

Given an arbitrary nonzero ideal I in O(X) we claim that I_{D(I)} is the closure of I. We clearly have \overline{I} ⊆ I_{D(I)}. Consider therefore an f ∈ I_{D(I)}. In order to show that f lies in \overline{I} we fix an r < 1 in |\overline{K}^{×}| and an ε > 0 and we have to find a g ∈ I such that ||f − g||_{r} < ε (compare the earlier Remark). Since IO(X_{r}) = I_{D(I)}O(X_{r}) we find g_{1}, …, g_{m} ∈ I and h_{1}, …, h_{m} ∈ O(X_{r}) such that

f = g_{1}h_{1} + … + g_{m}h_{m} .

We now choose polynomials R_{i} ∈ K[T] such that ||g_{i}||_{r} · ||h_{i} − R_{i}||_{r} < ε for 1 ≤ i ≤ m and set g := g_{1}R_{1} + … + g_{m}R_{m} ∈ I. Then ||f − g||_{r} < ε.

Applying this claim to a nonzero closed ideal I we obtain that I = I_{D(I)} lies in the image of our map. If, on the other hand, I = fO(X) is nonzero principal then D(I) = (f) and I_{D(I)} = I_{(f)} = fO(X) = I is closed.

Remark:

In the above as well as the subsequent proofs we repeatedly use the simple observation that for any nonzero ideal I ⊆ O(X) and any r < 1 in |\overline{K}^{×}| we have

IO(X_{r}) = {f ∈ O(X_{r}) : (f) ≥ D(I)|X_{r}} ∪ {0} .

Proof: Fixing r we find finitely many nonzero g_{1}, …, g_{m} ∈ I such that

D(I)(x) = min_{1≤i≤m} (g_{i})(x) for x ∈ X_{r} .

Since O(X_{r}) is a principal ideal domain the g_{i} generate the ideal on the right hand side of the asserted identity.

Lemma:

Given a closed ideal I ⊆ O(X) and elements g_{r} ∈ O(X_{r}) such that

g_{r′} − g_{r} ∈ IO(X_{r}) for any r ≤ r′ < 1 in |\overline{K}^{×}|

there is a g ∈ O(X) such that

g − g_{r} ∈ IO(X_{r}) for any r < 1 in |\overline{K}^{×}| .

Proof: If I = {0} the assertion is a consequence of the fact that the X_{r} form an admissible covering of the rigid analytic variety X. Otherwise we fix a 0 ≠ f ∈ I and put D := (f) − D(I). Then {g_{r}f^{−1}}_{r} is a family of principal parts with respect to D on X. Since X is a rigid analytic Stein space the invertible sheaf O(D) on X has no cohomology which implies that any such family is the family of principal parts with respect to D of a meromorphic function on X, i.e., there is a meromorphic function h on X such that

h − g_{r}f^{−1} ∈ O(D)(X_{r}) for any r < 1 in |\overline{K}^{×}| .

We set g := fh. Then

g − g_{r} = f(h − g_{r}f^{−1}) ∈ f · O(D)(X_{r}) ⊆ I_{D(I)}O(X_{r}) = IO(X_{r})

and g ∈ g_{r} + O(X_{r}) ⊆ O(X_{r}) for any r < 1 which shows that g ∈ O(X).

Proposition:

The sum I + J of any two closed ideals I and J in O(X) is closed.

Proof: We may assume that both I and J are nonzero so that, by the previous Proposition I = I_{D_{0}} and J = I_{D_{1}} for two divisors D_{0}, D_{1} ∈ Div^{+}(X). From the proof of that Proposition we also know that the closure of I + J is the ideal I_{D} where the divisor D is defined by

D(x) := min(D_{0}(x), D_{1}(x)) for x ∈ X .

We therefore consider any g ∈ I_{D}. Since IO(X_{r}) + JO(X_{r}) = I_{D}O(X_{r}) we find g_{0,r} ∈ IO(X_{r}) and g_{1,r} ∈ JO(X_{r}) such that

g = g_{0,r} + g_{1,r} .

For r < r′ we then have

g_{i,r′} − g_{i,r} ∈ IO(X_{r}) ∩ JO(X_{r}) = (I ∩ J)O(X_{r})

so that, by the above Lemma, there are g_{0}, g_{1} ∈ O(X) such that

g_{i} − g_{i,r} ∈ (I ∩ J)O(X_{r})

and, in particular, g_{i} ∈ I_{D_{i}}O(X_{r}) for any r < 1. Hence g_{0} ∈ I and g_{1} ∈ J. Since

g − (g_{0} + g_{1}) = (g_{0,r} − g_{0}) + (g_{1,r} − g_{1}) ∈ (I ∩ J)O(X_{r})

for any r < 1 we have g − (g_{0} + g_{1}) ∈ I ∩ J and consequently g ∈ I + J. Corollary: (Lazard) Any finitely generated ideal in O(X) is closed.

Lecture 2: The Fourier transform

The character group of ZZ_{p}

Let C_{p} be the completion of an algebraic closure \overline{Q}_{p} of Q_{p}. We will determine the character group Ĝ of the compact group G := ZZ_{p}. The character group

Ĝ := Hom_{an}(G, C_{p}^{×})

is defined to be the group of all locally C_{p}-analytic group homomorphisms κ : G ⟶ C_{p}^{×}. Let o_{p} ⊆ C_{p} be the ring of integers and 𝔪_{p} ⊆ o_{p} be the maximal ideal. We begin with a few very simple observations. Since C_{p}^{×}/o_{p}^{×} = Q is discrete and torsionfree whereas ZZ_{p} is compact we must have κ(ZZ_{p}) ⊆ o_{p}^{×}. But o_{p}^{×}/1 + 𝔪_{p} = IF_{p}^{×} is a discrete torsion group without p-torsion; so we actually have κ(ZZ_{p}) ⊆ 1 + 𝔪_{p}. The group ZZ_{p} being topologically cyclic generated by 1 ∈ ZZ_{p} any continuous character κ of ZZ_{p} is determined by its value κ(1). All this together means that

\begin{matrix}Ĝ & ⟶ & 𝔪_{p} \\ κ & ⟼ & κ(1) − 1 \end{matrix}

is a well defined injective map. On the other hand, for any z ∈ 𝔪_{p} and any a ∈ ZZ_{p} the series

κ_{z}(a) := (1 + z)^{a} := ∑_{n≥0} \binom{a}{n}z^{n}

converges. The notation (1 + z)^{a} is justified by the fact that

(1 + z)^{a} = lim_{i→∞} (1 + z)^{m_{i}}

for any sequence of integers m_{i} ∈ ZZ converging to a. The latter formula shows that κ_{z} : ZZ_{p} ⟶ C_{p}^{×} is a homomorphism of groups. Later on we will show that κ_{z} is locally C_{p}-analytic. (For more details see W. Schikhof, Ultrametric calculus, Cambridge Univ. Press 1984, §§32 and 47). Since κ_{z}(1) − 1 = z we obtain that the above map

\begin{matrix}Ĝ & ⟶^{∼} & 𝔪_{p} \\ κ & ⟼ & κ(1) − 1 \end{matrix}

is a bijection. Actually we can be a little bit more precise. Let us fix a complete intermediate field Q_{p} ⊆ K ⊆ C_{p} and let X denote, as in Lecture 1, the rigid analytic open unit disk over K. Its K-points are

X(K) := {z ∈ K : |z| < 1} = 𝔪_{p} ∩ K .

We will see that κ_{z}, for z ∈ X(K), in fact is locally K-analytic. This means we have the bijection

\begin{matrix}Hom_{an}(G, K^{×}) & ⟶^{∼} & X(K) \\ κ & ⟼ & κ(1) − 1 . \end{matrix}

The Fourier transform

This last fact is the starting point for the theory of the Fourier transform from (the dual of) locally K-analytic functions on G to rigid K-analytic functions on X. The locally convex K-vector space

C^{an}(ZZ_{p}, K) := all locally K-analytic functions on ZZ_{p}

is constructed in the following way. For any b ∈ K and any r ∈ |\overline{K}^{×}| let X_{r}(b) denote the closed rigid K-analytic disk of radius r around b and let O_{b,r} := O(X_{r}(b)) be the K-Banach space (with respect to the multiplicative spectral norm || ||_{b,r}) of holomorphic functions on X_{r}(b). Then

C^{an}(ZZ_{p}, K) := lim_{⟶_{j≥1}} ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}}

is the locally convex inductive limit of the Banach spaces ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}} with

respect to the obvious restriction maps. We define the K-Fréchet space

D(ZZ_{p}, K) := continuous dual C^{an}(Z_{p}, K)′ of C^{an}(Z_{p}, K)
= lim_{⟵_{j≥1}} ∏_{b∈ZZ/p^{j}ZZ} O′_{b,p^{−j}}

of K-valued distributions on ZZ_{p} as the projective limit of the dual Banach spaces ∏_{b∈ZZ/p^{j}ZZ} O′_{b,p^{−j}}. Without proof we state the following result.

Proposition:

If K is spherically complete then D(ZZ_{p}, K) = C^{an}(ZZ_{p}, K)′_{b} is the strong dual of C^{an}(ZZ_{p}, K) and is reflexive and nuclear.

Using the embedding

\begin{matrix}X(K) & ↪ & C^{an}(ZZ_{p}, K) \\ z & ↦ & κ_{z} \end{matrix}

any linear form λ ∈ D(ZZ_{p}, K) gives rise to the function

F_{λ}(z) := λ(κ_{z})

which is called the \underline{Fourier transform} of λ. It will turn out that F_{λ} in fact is a holomorphic function on X. The main aim of this lecture is to prove the following result.

Theorem: (Amice)

The Fourier transform

\begin{matrix}D(ZZ_{p}, K) & ⟶^{≅} & O(X) \\ λ & ⟼ & F_{λ} \end{matrix}

is an isomorphism of locally convex K-vector spaces.

The technical heart of the proof consists in the computation of the spectral norms ||P ||_{b,p^{−j}} of certain concrete polynomials P. We fix integers j ≥ 1 and 0 ≤ b < p^{j} and consider the polynomials

P_{n,l}(T) := ∏_{i=0}^{n−1}(T − lp^{j} − i) for n, l ∈ IIN_{0}

over K viewed as elements in O_{b,p^{−j}}. By definition we have

||P_{n,l}||_{b,p^{−j}} = sup{|P_{n,l}(z)| : z ∈ b + p^{j}o_{p}}.

Given a z ∈ b + p^{j}o_{p} and a 0 ≤ i < n we distinguish two cases. If i ∉ b + p^{j}o_{p} then |z − lp^{j} − i| = |b − i|. If i ∈ b + p^{j}o_{p} then |z − lp^{j} − i| ≤ p^{−j}. Moreover, since the residue class field \overline{IF}_{p} of o_{p} is infinite we certainly find a z ∈ b + p^{j}o_{p} such that |z − lp^{j} − i| = p^{−j} whenever i ∈ b + p^{j}o_{p}. This shows that, setting

ν(b, n, k) := #{0 ≤ i < n : i ∈ b + p^{k}o_{p}} for 1 ≤ k ≤ j ,

we have

(0) \begin{matrix}||P_{n,l}||_{b,p^{−j}} = p^{−j·ν(b,n,j)} · ∏_{0≤i<n_{i∉b+p^{j} op}} |b − i| \\ = p^{−j·ν(b,n,j)} · p^{−(j−1)[ν(b,n,j−1)−ν(b,n,j)]} · . . . · p^{−[ν(b,n,1)−ν(b,n,2)]} \\ = p^{− ∑_{k=1}^{j} ν(b,n,k)} . \end{matrix}

Writing

n = m_{k}p^{k} + n_{k} with 0 ≤ n_{k} < p^{k} and
b = c_{k}p^{k} + b_{k} with 0 ≤ b_{k} < p^{k}

we find that the numbers b_{k}, b_{k} +p^{k}, …, b_{k} +(m_{k} −1)p^{k} lie between 0 and n−1. This implies that

(1)  ν(b, n, k) ≥ m_{k} = [\frac{n}{p^{k}}] .

If b = n_{j} then b_{k} = n_{k} and hence b_{k} + m_{k}p^{k} = n. Therefore

(2)  ν(n_{j}, n, k) = [\frac{n}{p^{k}}] for any 1 ≤ k ≤ j .

If b < n_{j} then b_{j} + m_{j}p^{j} < n and consequently

(3)  ν(b, n, j) = [\frac{n}{p^{j}}] + 1 for b < n_{j} .

A standard calculation for the absolute value of any factorial m! (Schikhof, loc. cit., 25.5) shows that

|n!/ [\frac{n}{p^{j}} ]!| = p^{− ∑_{k=1}^{j}[ \frac{n}{p^{k}} ]} .

Inserting these computations into (0) we obtain the following formulas:

(1′)  ||P_{n,l}||_{b,p^{−j}} ≤ |n!/[\frac{n}{p^{j}}]!|

(2′)  ||P_{n,l}||_{n_{j},p^{−j}} = |n!/[\frac{n}{p^{j}}]!| , and

(3′)  ||P_{n,l}||_{b,p^{−j}} < |n!/[\frac{n}{p^{j}}]!| for b < n_{j} .

We in particular have

(4)  ||P_{n,l}||_{(j)} = |n!/[\frac{n}{p^{j}}]!|

where || ||_{(j)} = max_{b∈ZZ/p^{j}ZZ} || ||_{b,p^{−j}} denotes the norm on the Banach space

∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}} .

Proposition:

The polynomials \binom{T}{n} := \frac{T (T −1)·...·(T −n+1)}{n!} for n ≥ 0 form an orthogonal basis of

the K-Banach space ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}} with ||\binom{T}{n}||_{(j)} = |[\frac{n}{p^{j}}]!|^{−1}.

Proof: The assertion about the norm is a special case of (4). Let IF denote the residue class field of K and define O_{b,p^{−j}}^{0}, resp. O_{b,p^{−j}}^{00}, to be the ball of elements of norm ≤ 1, resp. < 1, in O_{b,p^{−j}}. Then

E := ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}}^{0}/O_{b,p^{−j}}^{00}

in a natural way is an IF-algebra. For any f ∈ ∏_{b} O_{b,p^{−j}} with ||f||_{(j)} ≤ 1 let \overline{f}

denote its class in E. Our assertion about the \binom{T}{n} is equivalent to the statement that the classes \overline{Q}_{n} ∈ E of the normalized polynomials

\overline{Q}_{n} := [\frac{n}{p^{j}} ]! · \binom{T}{n}

form a basis of E as an IF-vector space (van Rooij, Non-Archimedean Functional Analysis, M. Dekker 1978, 5.A). But E has the obvious basis consisting of the classes \overline{χ}_{k,c} of the elements

χ_{k,c} := ∏_{b∈ZZ/p^{j}ZZ} χ_{k,c,b} with χ_{k,c,b} := \{ \begin{matrix}p^{−j}(T − c)^{k} if b = c, \\ 0 otherwise \end{matrix}

for k ≥ 0 and c ∈ ZZ/p^{j}ZZ. We let E_{l} ⊆ E, for l ≥ 0, denote the subspace of dimension (l + 1)p^{j} spanned by the \overline{χ}_{k,c} for 0 ≤ k ≤ l and c ∈ ZZ/p^{j}ZZ; put E_{−1} := {0}. One has E_{l} · E_{l′} ⊆ E_{l+l′}. We will show that

(5)  E_{l} = E_{l−1} + ∑_{m=0}^{p^{j}−1} IF\overline{Q}_{lp^{j}+m} for any l ≥ 0 .

For 0 ≤ m < p^{j} one has

Q_{lp^{j}+m} = Q_{m,l} · Q_{lp^{j}} with Q_{m,l} := \frac{(lp^{j})!}{(lp^{j} + m)!}P_{m,l} .

Since |\frac{(lp^{j})!}{(lp^{j}+m)!}| = |m!|^{−1} we may use (1’)-(3’) and obtain, for 0 ≤ b, m < p^{−j}, that

||Q_{m,l}||_{b,p^{−j}} ≤ 1, ||Q_{m,l}||_{m,p^{−j}} = 1, and ||Q_{m,l}||_{b,p^{−j}} < 1 if b < m .

Moreover, if ||Q_{m,l}||_{b,p^{−j}} = 1 then, because of b ≥ m, Q_{m,l} has no zeros in b + p^{−j}o_{p}; in this situation it is an immediate consequence of Weierstrass preparation in the ring O_{b,p^{−j}} that Q_{m,l} ∈ O_{b,p^{−j}}^{0} is a constant modulo O_{b,p^{−j}}^{00}. It follows that

(6)  \overline{Q}_{m,l} = ∑_{m≤c<p^{j}} γ_{m,c} · \overline{χ}_{0,c} with γ_{m,c} ∈ IF and γ_{m,m} ≠ 0 .

In particular, for any l ≥ 0, the \overline{Q}_{m,l} with 0 ≤ m < p^{j} span E_{0}. Our claim (5) follows from (6) by a straightforward induction argument with respect to l. This finishes the proof of the Proposition.

The Proposition says that ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}} is the Banach space of all series

φ = ∑_{n≥0} c_{n}\binom{T}{n} with c_{n} ∈ K and lim_{n→∞} |c_{n}/ [\frac{n}{p^{j}} ]!| = 0

and that

||φ||_{(j)} = max_{n≥0} |c_{n}/[\frac{n}{p^{j}}]!|

(Schikhof, loc. cit., §50). Hence the dual Banach space D_{j} := ∏_{b∈ZZ/p^{j}ZZ} O′_{b,p^{−j}}

can be identified with the K-Banach space of all sequences

\underline{a} = (a_{0}, a_{1}, a_{2}, …) in K such that sup_{n≥0} |a_{n} · [\frac{n}{p^{j}} ]!| < ∞

with the norm

||\underline{a}|| := sup_{n≥0} |a_{n} · [\frac{n}{p^{j}} ]!| < ∞ .

The duality is given by the continuous pairing (φ, \underline{a}) ⟼ ∑_{n≥0} a_{n}c_{n}. We put

r_{j} := |p|^{1/p^{j}(p−1)} ∈ |\overline{K}^{×}| .

Consider any n ≥ 1 and write n = lp^{j} + m with 0 ≤ m < p^{j}. Then (using again 25.5 in Schikhof)

| [\frac{n}{p^{j}} ]!|^{1/n} = |l!|^{1/(lp^{j}+m)} ≥ |l!|^{1/lp^{j}} > |p|^{1/p^{j}(p−1)}

which means that |[\frac{n}{p^{j}}]!|/r_{j}^{n} ⟶ ∞ with n ⟶ ∞. It follows that

\begin{matrix}D_{j} & ⟶ & O(X_{r_{j}}) \\ \underline{a} & ⟼ & ∑_{n≥0} a_{n}T^{n} \end{matrix}

is an injective continuous linear map. Passing to the projective limit with respect to j gives rise to an injective continuous linear map

F : D(ZZ_{p}, K) ⟶ O(X) .

On the other hand one checks that

| [\frac{n}{p^{j}} ]!| < r_{j+1}^{n} for n ≥ p^{j+2} .

Hence

\begin{matrix}O(X_{r_{j+1}}) & ⟶ & D_{j} \\ ∑_{n≥0} a_{n}T^{n} & ⟼ & (a_{0}, a_{1}, …) \end{matrix}

also is an injective continuous linear map. By a cofinality argument F therefore must be bijective. Since both sides are Fréchet spaces the open mapping theorem finally implies that F is a topological isomorphism. Let z ∈ K with |z| < r_{j}. We then have lim_{n→∞} |z^{n}/|[\frac{n}{p^{j}}]!| = 0 so that ∑_{n≥0} z^{n}\binom{T}{n} is a convergent expansion

in the Banach space ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}}. This shows that κ_{z} is locally K-analytic,

as was claimed earlier, and in fact lies in ∏_{b∈ZZ/p^{j}ZZ} O_{b,p^{−j}}. Moreover, for any

λ ∈ D(ZZ_{p}, K) and F(λ) = ∑_{n≥0} a_{n}T^{n} we obtain

λ(κ_{z}) = ∑_{n≥0} a_{n}z^{n} = F(λ)(z).

This establishes the identity F (λ)(z) = F_{λ}(z) for any z ∈ X(K) and completes the proof of Amice’s theorem.

As to be expected the multiplication of functions in O(X) corresponds to a “convolution” of linear forms in D(ZZ_{p}, K).

Lecture 3: The principal series

We fix a spherically complete intermediate field Q_{p} ⊆ K ⊆ C_{p}. In this lecture we will construct (rather in the form of a survey) certain topologically irreducible representations of the compact group G := GL_{2}(ZZ_{p}) in locally convex K-vector spaces.

Distributions

In a completely analogous way as for the group ZZ_{p} one can construct the locally convex K-vector space

C^{an}(G, K) := all locally K-analytic functions on G

as a countable inductive limit of Banach spaces. Its strong dual

D(G, K) := C^{an}(G, K)′_{b}

- the space of K-valued distributions on G - is a reflexive and nuclear K-Fréchet space. In fact, the usual convolution product of distributions makes D(G, K) into a K-Fréchet algebra. This algebra should be considered as some kind of analytic version of the group algebra of G. And, indeed, sending a group element g ∈ G to the Dirac distribution δ_{g} defines an injective ring homomorphism K[G] ↪ D(G, K) whose image is dense in D(G, K).

The Lie algebra 𝔤 := M_{2}(Q_{p}) of G is another source for explicit elements in D(G, K). Any 𝔵 ∈ 𝔤 acts on C^{an}(G, K) via

(𝔵f)(g) := \frac{d}{dt}f(exp(−t𝔵)g)_{|t=0}

where exp : 𝔤 − − − > G denotes the exponential map defined locally around 0. This induces the embedding

\begin{matrix}𝔤 & ⟶ & D(G, K) \\ 𝔵 & ⟼ & [f ↦ (−𝔷(f))(1)] \end{matrix}

which by the universal property of the universal enveloping algebra U(𝔤) of 𝔤 extends to a homomorphism of K-algebras

U(𝔤) ⊗_{Q_{p}} K ⟶ D(G, K) .

It can be shown to be injective. Technically more important is that D(G, K) even contains a certain kind of explicit completion of U(𝔤). To make this precise we fix an ordered basis 𝔵_{1}, …, 𝔵_{4} of 𝔤 as a K-vector space. A power series in 4 variables over K is called entire if it converges on the whole affine space C_{p}^{4} ; this amounts to the requirement that the coefficients b\underline{n} of such a power series satisfy sup_{\underline{n}} |b\underline{n}|r^{−|\underline{n}|} < ∞ for any r > 0.

Lemma:

Any entire power series in the 𝔵_{1}, …, 𝔵_{4} converges in D(G, K).

The group G acts continuously on C^{an}(G, K) via left translation giving rise to the (left) regular representation of G in the present context. If the philosophy about D(G, K) being some kind of group algebra is correct then this G-action should extend to a D(G, K)-module structure. And indeed it does: For μ ∈ D(G, K) and f ∈ C^{an}(G, K) define

(μ ∗ f)(h) := ∫_{G} f(g^{−1}h)dμ(g) .

The principal series

The principal series of G consists of certain subrepresentations of the regular representation which are defined as follows. Let P ⊆ G denote the subgroup of lower triangular matrices and T ⊆ G the subgroup of diagonal matrices. We fix a locally K-analytic character

χ : T → K^{×}

and consider the space

Ind_{P}^{G}(χ) := {f ∈ C^{an}(G, K) : f(gp) = χ(p^{−1})f(g) for any g ∈ G, p ∈ P } .

This clearly is a closed G-invariant subspace of C^{an}(G, K) and hence is a continuous G-representation as well as a D(G, K)-module in its own right. It will turn out to be much preferable to study the strong dual

M_{χ} := Ind_{P}^{G}(χ)′_{b}

which (by functoriality) carries a continuous D(G, K)-module structure. Since all these spaces are reflexive this certainly will not mean any loss of information.

The most important numerical invariant of a principal series representation is the number c(χ) ∈ K defined by the expansion

χ(( \begin{matrix}t^{−1} & 0 \\ 0 & t \end{matrix} )) = exp(c(χ) log(t))

for t sufficiently close to 1 in ZZ_{p}.

Main Theorem:

If c(χ) ∉ −IN_{0}, then M_{χ} is an (algebraically) simple D(G, K)-module. In particular, Ind_{P}^{G}(χ) is a topologically irreducible G-representation.

In a first step one reduces the assertion to a similar assertion for a certain subgroup of G. This is the so called Iwahori subgroup B consisting of all matrices

which are lower triangular modulo p. We set w := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) ∈ G and P^{−} :=

B ∩ wP w. Using the Bruhat decomposition

G = BP ∪^{·} BwP

we obtain by the Mackey formula the B-equivariant decomposition

Ind_{P}^{G}(χ) = Ind_{P}^{B}(χ) ⊕ Ind_{P^{−}}^{B}(wχ) .

Defining

M_{χ}^{+} := Ind_{P}^{B}(χ) and M_{χ}^{−} := Ind_{P^{−}}^{B}(χ)

this dualizes into a decomposition

M_{χ} = M_{χ}^{+} ⊕ M_{wχ}^{−}

as D(B, K)-modules. Since the action of w is easily seen not to respect this decomposition the Main Theorem is a consequence of the following slightly stronger result.

Theorem:

i. If c(χ) ∉ ±IN_{0}, then M_{χ}^{∓} is an (algebraically) simple D(B, K)-module;

ii. M_{χ}^{+} and M_{χ′}^{−} never are isomorphic as D(B, K)-modules.

The second assertion reduces via a Frobenius reciprocity argument to the fact that there are no p-adic valued Haar measures.

The two cases in the first assertion are completely analogous. So let us look at M_{χ}^{−}. The reason why it is easier to deal with the group B (and not G directly) is the fact that the map

\begin{matrix}ι : ZZ_{p} & ⟶ & B \\ b & ⟼ & ( \begin{matrix}1 & 0 \\ b & 1 \end{matrix} ) \end{matrix}

is a section of the projection map B → B/P^{−} = ZZ_{p}. Pushing forward distributions via ι therefore yields a linear isomorphism

ι_{∗} : D(ZZ_{p}, K) ⟶^{≅} M_{χ}^{−} .

Under this isomorphism the D(ZZ_{p}, K)-module structure on M_{χ}^{−} coming from the ring inclusion

ι_{∗} : D(ZZ_{p}, K) → D(B, K)

becomes simply the ring multiplication in D(ZZ_{p}, K). Thus M_{χ}^{−} is a copy of D(ZZ_{p}, K) with the additional structure corresponding to the action of D(B, K). One then calculates this additional structure explicitly. The most important formula one gets is the following. Identifying D(ZZ_{p}, K) with O(X) under the

Fourier transform the action of the Lie algebra element 𝔲^{+} := ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ∈ 𝔤

satisfies

(𝔲^{+})^{m}F ≡ [ ∏_{i=0}^{m−1}(c(χ) − i)] · ((1 + T)\frac{d}{dT} )^{m} F (mod log(1 + T)O(X))

for any m ≥ 0 and F ∈ O(X). Under our assumption on c(χ) this congruence means we have control over the divisor of (𝔲^{+})^{m}F in terms of the divisor of F. Now, each nonzero D(B, K)-submodule of M_{χ}^{−} corresponds to a (not necessarily closed) nonzero ideal in O(X) with extra invariance properties. In particular it is invariant under all the operators (𝔲^{+})^{m} and hence under all operators which are given as an entire series in 𝔲^{+}. This information then is enough to conclude that such an ideal necessarily has to be the full ring.

No two of the simple D(G, K)-modules in the Main Theorem are isomorphic. This follows from the next result. In order to formulate it we introduce the character ϵ defined by ϵ(( \begin{matrix}t_{1} & 0 \\ 0 & t_{2} \end{matrix} )) := t_{2}/t_{1}.

Theorem:

For any two locally K-analytic characters χ ≠ χ′ of T the vector space Hom_{D(G,K)}(M_{χ′}, M_{χ}) of all D(G, K)-module homomorphisms is 1-dimensional if c(χ) ∈ −IN_{0} and χ′ = ϵ^{1−c(χ)}χ and is zero otherwise.

We close with the following remark. Let us consider an algebraic character

χ(( \begin{matrix}t_{1} & 0 \\ 0 & t_{2} \end{matrix} )) = t_{1}^{m_{1}}t_{2}^{m_{2}} with m_{i} ∈ ZZ such that c(χ) = m_{2} − m_{1} ≤ 0. In

this case we have the finite dimensional K-rational representation of GL_{2} of highest weight χ (w.r.t. the Borel subgroup of upper triangular matrices). It can be described as an ”algebraic” induction and as such is a G-invariant closed subspace of Ind_{P}^{G}(χ).

(The details of this Lecture 3 can be found in the forthcoming joint paper with J. Teitelbaum on ”Locally analytic distributions and p-adic representation theory, with applications to GL_{2}”.)