Presentation of an Iwasawa Algebra:

The Case of Γ_{1}SL(2, ℤ_{p})

Laurent Clozel ^{1}

Received: July 1, 2010 Revised: February 17, 2011

Communicated by Takeshi Saito

Abstract. We give an explicit presentation of the p-adic Iwasawa algebra of the subgroup of level one of SL(2, ℤ_{p}) for p ≠ 2.

2010 Mathematics Subject Classification: 11F85, 11S31,22E50 Keywords and Phrases: Non-commutative Iwasawa agebras, p-adic groups

Assume G is a semi–simple Chevalley group, so G(ℤ_{p}) ⊂ G(ℚ_{p}) is a maximal compact subgroup. Both the p–adic representation theory of G(ℚ_{p}) and non– commutative Iwasawa theory involve the Iwasawa algebra of G(ℤ_{p}) or suitable congruence subgroups. It seems to have been assumed that explicit descriptions, by generators and relations, of these algebras were inaccessible. However, it is a general principle that natural objects coming from semi–simple (split) groups have explicit presentations. Famous examples are Serre’s presentation of the semi–simple algebras and Steinberg’s presentation of the Chevalley groups [7, 8]. In this paper we will give a presentation for the Iwasawa algebra of the subgroup of level 1 in SL(2, ℤ_{p}) (p ≠ 2). I thank M. Duflo and, once more, J.–P. Serre for useful discussions. Thanks are also due to P. Schneider for reading a first draft and suggesting several corrections, and to the referee for his careful work which eliminated an embarrassing mistake.

1Membre de l’Institut Universitaire de France

Documenta Mathematica 16 (2011) 545–559

1

Let \underline{G} = SL(2) and let G be the subgroup of level 1 in \underline{G}(ℤ_{p}) :

G = {g ∈ SL(2, ℤ_{p}) : g ≡ 1[p]} .

We assume p > 2, so G has no p–torsion. It has a triangular decomposition

G = N^{−} T N^{+}

where N^{−} = (\begin{matrix}1 & 0 \\ ∗ & 1 \end{matrix}), N^{+} = (\begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix}) (entries ∗ in pℤ_{p}) and T = (\begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix})

(entries in 1 + pℤ_{p}). We identify N^{−}, N^{+} with ℤ_{p} by ∗ = px (x ∈ ℤ_{p}). Similarly T ≅ ℤ_{p} by

x ⟼ (\begin{matrix}(1 + p)^{x} \\ & (1 + p)^{−x} \end{matrix}) (x ∈ ℤ_{p}) .

We consider the Iwasawa algebra Λ_{G} of ℤ_{p}–valued measures (or distributions, in the sense of [9]), on G, which we will denote by D(G, ℤ_{p}). The triangular decomposition of G, as an analytic manifold, yields a decomposition of D(G, ℤ_{p}) as a topological ℤ_{p}–module :

(1.1)  D(G, ℤ_{p}) = D(N^{−}, ℤ_{p})ˆ⊗D(T, ℤ_{p})ˆ⊗D(N^{+}, ℤ_{p}) ,

the factors of (1.1) being the spaces of distributions on the factors of G. If f is a function on G and U, V, W distributions on N^{−}, T, N^{+},

(1.2)  < U ⊗ V ⊗ W, f >:=< U ⊗ V ⊗ W, f(uhn) >

where u ∈ N^{−}, h ∈ T, n ∈ N^{+} and f is therefore seen as a function on N^{−} × T × N^{+}. The natural definition of the completed tensor product is equivalent to the explicit description of D(G, ℤ_{p}) reviewed below. The algebra Λ_{ℤ_{p}} = D(ℤ_{p}, ℤ_{p}) is identified with the ring of power series ℤ_{p}[[T]] by Iwasawa’s theorem. For μ ∈ Λ_{ℤ_{p}}, the associated series is given by the Fourier–Amice transform

μ̂(t) = ∫_{ℤ_{p}}(1 + t)^{x}dμ(x) (t ∈ ℤ_{p}, |t| < 1) .

In particular, δ(x) being the Dirac measure at x :

δ̂(1) = 1 + T ,
so T = δ̂(1) − δ̂(0) .

In each factor of the decomposition (1.1), we therefore have the Dirac measures :

μ_{−} = δ(\binom{1}{p 1}) ,  μ̂_{−} = 1 + Y ∈ D(N^{−}, ℤ_{p}) ≅
≅ ℤ_{p}[[Y]]
μ_{+} = δ(\binom{1 p}{1}) ,  μ̂_{+} = 1 + X ∈ D(N^{+}, ℤ_{p}) ≅
≅ ℤ_{p}[[X]]
μ_{0} = δ((\begin{matrix}(1 + p) \\ & (1 + p)^{−1} \end{matrix})) , μ̂_{0} = 1 + H ∈ D(T, ℤ_{p}) ≅
≅ ℤ_{p}[[H]]

For each factor, U = N^{−}, T or N^{+} of G, D(U, ℤ_{p}) is naturally sent to D(G, ℤ_{p}), by integrating a function f ∈ C(G, ℤ_{p}) against μ ∈ D(U, ℤ_{p}) on the U–factor. This map is compatible with the convolution product. We therefore write, unambiguously, Y^{n}, X^{n}, H^{n} (n ≥ 0) in D(G, ℤ_{p}). A distribution λ in this space can then be written uniquely

(1.3)  λ = ∑_{n} λ_{n} Y^{n_{1}} H^{n_{2}} X^{n_{3}}  (n ∈ ℕ^{3})

with λ_{n} ∈ ℤ_{p}. This is the meaning of the completed tensor product (1.1). The expansion is convergent in D(G, ℤ_{p}). Of course the product Y^{n_{1}}H^{n_{2}}X^{n_{3}} := Y^{n_{1}} ⊗H^{n_{2}} ⊗X^{n_{3}} is defined as above. This easily follows from Mahler’s theorem in several variables (cf. Lazard [4, Théorème 1.2.4]). It immediately follows from formula (1.2) that the distributions Y, H, X ∈ D(G, ℤ_{p}) multiply in the obvious fashion when the variables are taken in the “natural order”, i.e.

\begin{aligned}Y ⊗ H & = & Y ∗ H \\ Y ⊗ X & = & Y ∗ X \\ H ⊗ X & = & H ∗ X , \end{aligned}

the convolution product being taken on G. We will simply write, consistent with previous notation :

(1.4)  Y H = Y ∗ H , Y X = Y ∗ X , HX = H ∗ X .

To determine the product structure in D(G, ℤ_{p}) is to understand first the product of monomials in a different order. Consider first the product HY. It suffices to compute, in G, the product μ_{0}μ_{−} = δ(h_{0})δ(u_{0}), say. We compute h_{0}u_{0}h_{0}^{−1}. Since

(\begin{matrix}t \\ & t^{−1} \end{matrix}) \binom{1}{x 1} (\begin{matrix}t^{−1} \\ & t \end{matrix}) = \binom{1}{t^{−2}x 1} ,

we have h_{0}u_{0}h_{0}^{−1} = u_{0}^{(1+p)^{−2}} if we write the group N^{−} multiplicatively. The equation

μ_{0}μ_{−} = δ(h_{0}u_{0}h_{0}^{−1})δ(h_{0}) ,

and the fact that D(N^{−}, ℤ_{p}) ≅ ℤ_{p}[[Y]] is a homomorphism, show that

(1.5)  (1 + H)(1 + Y) = (1 + Y)^{q}(1 + H)

where we have set

(1.6)  q = (1 + p)^{−2} ≡ 1  [p] .

Similarly consider XH. Let n_{0} be the generator of N^{+}. Now δ(n_{0})δ(h_{0}) reduces to h_{0}^{−1}n_{0}h_{0}. Again

(\begin{matrix}t^{−1} \\ & t \end{matrix}) \binom{1 x}{1} (\begin{matrix}t \\ & t^{−1} \end{matrix}) = \binom{1 t^{−2}x}{1} ,

so h_{0}^{−1}n_{0}h_{0} = n_{0}^{(1+p)^{−2}} = n_{0}^{q}, whence

(1.7)  (1 + X)(1 + H) = (1 + H)(1 + X)^{q} .

Finally, to express XY we have to decompose

n_{0}u_{0} = \binom{1 p}{1} \binom{1}{p 1} = (\begin{matrix}1 + p^{2} & p \\ p & 1 \end{matrix}) .

Since

\binom{1}{a 1} (\begin{matrix}t \\ & t^{−1} \end{matrix}) \binom{1 b}{1} = (\begin{matrix}t & tb \\ ta & abt + t^{−1} \end{matrix}) ,

we see that

p = ta = tb

with

1 + p^{2} = t = (1 + p)^{P} , P ∈ ℤ_{p} .

This yields, since t_{0} = 1 + p is the parameter of h_{0} :

(1.8)  (1 + X)(1 + Y) = (1 + Y)^{Q}(1 + H)^{P} (1 + X)^{Q}

with

(1.9)  Q = (1 + p^{2})^{−1} ≡ 1[p^{2}] , P = \frac{log(1 + p^{2})}{log(1 + p)} .

For p > 2, we have

\begin{matrix}log(1 + p) = p − \frac{p^{2}}{2} + \frac{p^{3}}{3} ⋯ = p(1 + O(p)) \\ log(1 + p^{2}) = p^{2}(1 + O(p^{2})) \end{matrix}

whence

(1.10)  P = p(1 + O(p)) .

Note that we have simply written HY for H ∗ Y, etc… This will cause no confusion if we remember that a product such as HY, for variables not in the natural order, is not given by the ostensible product of monomials in the expression (1.3). To summarize, we have :

Proposition 1.1. Set Q = (1 + p^{2})^{−1} , q = (1 + p)^{−2}, P = \frac{log(1+p^{2})}{log(1+p)}. Then the elements X, Y, H of D(G, ℤ_{p}) verify the relations

(a) (1 + H)(1 + Y) = (1 + Y)^{q}(1 + H)

(b) (1 + X)(1 + H) = (1 + H)(1 + X)^{q}

(c) (1 + X)(1 + Y) = (1 + Y)^{Q}(1 + H)^{P} (1 + X)^{Q}.

Consider now the universal, non–commutative p–adic algebra in the variables Y, H, X : thus

A = ℤ_{p}{{Y, H, X}}

is composed of all the non–commutative series

(1.11)  f = ∑_{n≥0} ∑_{i} a_{i}x^{i}

where the coefficients a_{i} ∈ ℤ_{p} and, for all n ≥ 0, i runs over all maps {1, 2, … n} ⟶ {1, 2, 3} ; we set x_{1} = Y, x_{2} = H, x_{3} = X and x^{i} = x_{i(1)} ⋯ x_{i(n)}. The topology on A is the product topology on ∏_{n} ℤ_{p}^{I(n)} where

I(n) is the set of maps (≡ of non–commutative monomials of degree n). The algebra A has a maximal ideal M_{A} generated by (p, x_{1}, x_{2}, x_{3}) and a prime ideal P_{A} generated by (x_{1}, x_{2}, x_{3}). Its topology is given by the powers of M_{A}. The non–commutative polynomial algebra

A = ℤ_{p}{Y, H, X}

is a dense subalgebra of A. Let R be the closed two–sided ideal generated in A by the relations (a, b, c). Our main result is

Theorem 1.2. The Iwasawa algebra Λ_{G} is naturally isomorphic to A/R.

The proof will in fact rely on the corresponding result with coefficients in 𝔽_{p}. So let Ω_{G} = Λ_{G} ⨂_{ℤ_{p}} 𝔽_{p} be the Iwasawa algebra with finite coefficients,

\overline{A} = 𝔽_{p}\{\{Y, H, X\}\} the algebra of non–commutative series with coefficients in 𝔽_{p}, with its natural linearly compact topology, given by its maximal ideal M\overline{A}. Let \overline{R} be the image of R in \overline{A}.

Lemma 1.3. \overline{R} is the closed two–sided ideal generated in \overline{A} by the image of the relations (a, b, c).

Proof.– Denote by I ⊂ A the ideal generated by the relations ; let J ⊂ \overline{A} be the similar ideal. Then J is obviously the image of I in \overline{A} ; we denote it by \overline{I}. Let \overline{R} be the reduction of R, and consider the closure cl(\overline{I}) of \overline{I} in \overline{A}. If f ∈ R, we have f = lim_{n} f_{n} (f_{n} ∈ I) for the topology given by (M_{A}^{N}). This

implies that \overline{f} = lim \overline{f}_{n} for the topology given by MN ; \overline{A} on \overline{A}, thus \overline{f} ∈ cl(\overline{I}). Conversely assume \overline{f} ∈ \overline{A} can be written \overline{f} = lim \overline{f}_{n} with \overline{f}_{n} ∈ \overline{I}. Then \overline{f}_{n} is the reduction of a series f_{n} ∈ I ⊂ R. Since R is closed and A compact, we may assume that f_{n} converges to g ∈ R. Then, by definition of the topologies, \overline{f} = lim \overline{f}_{n} = \overline{g}. Thus cl(\overline{I}) = \overline{R}, which finishes the proof.

Theorem 1.4. The Iwasawa algebra mod p, Ω_{G}, is naturally isomorphic to \overline{A}/\overline{R}.

The proof of these results will occupy § 2, 3.

2

We consider the natural map

A ⟶ Λ_{G}

given by the universal property of A. Note that the topology of Λ_{G}, as a distribution algebra, coincides with its topology when it is seen as the algebra of distributions on the commutative group ℤ_{p}^{3}. In particular a basis of neighbourhoods of 0 is given by the family of ℤ_{p}–modules M_{Λ}^{N}(Λ = Λ_{G}), where

(2.1)  M_{Λ}^{N} = {λ ∈ Λ_{G} , λ = ∑_{n} λ_{n}Y^{n_{1}}H^{n_{2}}X^{n_{3}} , v(λ_{n}) + |n| ≥ N}

with the usual notation |n| = n_{1} + n_{2} + n_{3}. For a linear monomial x = Y, H or X, we have w(x) = 1, w being the function on Λ given by

(2.2)  w(λ) = inf_{n}(v(λ_{n}) + |n|) .

We will use the following deep result of Lazard :

Proposition 2.1 (Lazard). The valuation w is additive : w(λ ∗ μ) = w(λ) + w(μ) (λ, μ ∈ Λ_{G}).

Cf. [4, III 2.3.3]. Lazard proves, in fact, that the associated graded ring is an enveloping algebra, thus an integral domain, and this implies the additivity. I am indebted to the paper of Schneider and Teitelbaum [6] for a lucid exposition of Lazard’s results. In fact, it follows from Lazard’s results that M_{Λ}^{N} is indeed the N-th power of the maximal ideal M_{Λ} of Λ_{G}. Indeed, let J_{N} be defined by w(λ) ≥ N. It is easy to check that J_{1} = M_{Λ}. The additivity implies that M_{Λ}^{N} is contained in J_{N}. Since every linear monomial belongs to the maximal ideal, the expression (2.1) implies the converse inclusion since M_{Λ}^{N} is closed. Consider now the filtration of A by the powers of its maximal ideal. It is defined by a valuation w_{A} given by a formula similar to (2.2) : if

\begin{matrix}f = ∑_{i} a_{i} x^{i} , \\ w_{A}(f) = inf_{i}(v(a_{i}) + |i|) \end{matrix}

where |i| = n is the degree of i (cf. after (1.11)). We now have the following (“ideal” means two–sided ideal unless otherwise indicated).

Proposition 2.2. The natural map φ : A → Λ_{G} extends continuously to a surjective homomorphism A → Λ_{G}. In fact,

φ(M_{A}^{N}) ⊂ M_{Λ}^{N} (N ≥ 0) .

Proof : The continuity is implied by the stronger property

(2.3)  w(φ(x^{i})) = n = |i|

where n, as after (1.11), is the degree of the monomial. By induction on n, this follows from Proposition 2.1. If f ∈ M_{A}^{N}, we have w_{A}(f) ≥ N and the continuity follows from (2.3) by ℤ_{p}–linearity. The surjectivity follows from the fact that φ is already surjective if A is replaced by the set of linear combinations of well-ordered monomials (i increasing).

Corollary 2.3. There is a natural, continuous surjection

B = A/R ⟶ Λ_{G} .

Corollary 2.4. There is a continuous surjection

\overline{φ} : \overline{B} = \overline{A}/\overline{R} ⟶ Ω_{G} .

This follows from Lemma 1.3. It follows from Abelian distribution theory that Ω_{G} is, as a space, isomorphic to

𝔽_{p}[[Y, H, X]]

with the compact topology. An obvious computation shows that

M_{Ω}^{N} = {λ ∈ Ω_{G} : v_{Ω}(λ) ≥ N} ,

v_{Ω} being the usual valuation on power series, is the image of M_{Λ}^{N}. In particular it is an ideal ; for N = 1, M_{Ω} is the maximal (two–sided) ideal, and (M_{Ω})^{N} ⊂ M_{Ω}^{N}. (Reduce mod p the corresponding property for Λ.)

Similarly in A, we find that the reduction mod p (image in \overline{A}) of M_{A}^{N} is the ideal of series

\overline{f} = ∑_{i} α_{i} x^{i} (α_{i} ∈ 𝔽_{p})

such that |i| ≥ N. For N = 1 we obtain the maximal ideal in \overline{A}. Furthermore in this case too (M\overline{A})^{N} = MN ; \overline{A}.

3

In this paragraph we will directly study the quotient algebra \overline{B} = \overline{A}/\overline{R}, using the properties of the relations (a, b, c). Consider the natural filtration of \overline{A} by the powers of M\overline{A}, which we denote by F^{n}\overline{A}. We have F^{n}\overline{A}/F^{n+1}\overline{A} = gr^{n}\overline{A} ≅ 𝔽_{p}^{I(n)} where I(n) is the set of maps {1, …, n} → {1, 2, 3} (§1). The filtration F^{n} induces a filtration on \overline{B} = \overline{A}/\overline{R} :

F^{n}\overline{B} = F^{n}\overline{A} + \overline{R}

whence a graduation

\begin{aligned}gr^{n}\overline{B} & = & F^{n}\overline{A} + \overline{R}/F^{n+1}\overline{A} + \overline{R} \\ & = & F^{n}\overline{A}/F^{n+1}\overline{A} + (F^{n}\overline{A} ∩ \overline{R}) . \end{aligned}

Let S_{n} = S_{n}(X, Y, Z) be the space of commutative polynomials over 𝔽_{p} of degree n ; thus dim S_{n} = \frac{(n+1)(n+2)}{2}. Let Σ_{n} be the space of homogeneous non–commutative polynomials of degree n ; thus Σ_{n} → F^{n}\overline{A}/F^{n+1}\overline{A}, and therefore Σ_{n} → gr^{n}\overline{B}, is surjective.

Proposition 3.1. dim gr^{n}\overline{B} ≤ dim S_{n}.

In order to prove this we consider the relations defining R (or rather \overline{R}). Consider first the relation (a) :

(1 + H)(1 + Y) = (1 + Y)^{q}(1 + H)

with q ≡ 1 [p]. Expanding the power series gives

1 + H + Y + HY = (1 + qY + (\begin{matrix}q \\ 2 \end{matrix}) Y^{2} + ⋯)(1 + H) .

We note that (\begin{matrix}q \\ 2 \end{matrix}) = \frac{q(q − 1)}{2} ≡ 0 [p]. Thus in \overline{A}/\overline{R} :

1 + H + Y + HY = (1 + qY)(1 + H) + R(Y)(1 + H) ,

the term R(Y) being of degree ≥ 3, so

\begin{aligned}HY & = & (q − 1)Y + qY H + R_{1}(Y, H) \\ & = & Y H + R_{1}(Y, H) \end{aligned}

since q ≡ 1, R_{1}(Y, H) of degree ≥ 3. This shows that in \overline{B} = \overline{A}/\overline{R} :

(3.1)  HY = Y H mod F^{3}\overline{B} i.e.
HY = Y H in gr^{2}\overline{B} .

The computation for relation (b) is obviously similar, yielding in \overline{B}

(3.2)  XH = HX  mod F^{3}\overline{B} .

Consider now the identity (c) :

(1 + X)(1 + Y) = (1 + Y)^{Q}(1 + H)^{P} (1 + X)^{Q} .

We have Q ≡ 1 [p^{2}], P ≡ p [p^{2}]. Again the coefficients \frac{Q(Q − 1)}{2} of Y^{2}, X^{2}

in the power series vanish mod p. Modulo M3 ; \overline{A}, whose image is in F^{3}\overline{B}, we then have

(1 + X)(1 + Y) ≡ (1 + QY)(1 + H)^{P} (1 + QX) .

Since P ≡ p [p^{2}] and since 2 is invertible, (1 + H)^{P} ≡ 1 mod (p, H^{3}). Thus

1 + X + Y + XY ≡ 1 + QX + QY + Q^{2}Y X ( mod F^{3}\overline{B}) ,

and since Q ≡ 1 :

(3.3)  XY ≡ Y X  ( mod F^{3}\overline{B}) .

Since gr^{2}\overline{B} is generated by these three monomials and the squares Y^{2}, H^{2}, X^{2} , the identities (3.1)–(3.3) show that dim gr^{2}\overline{B} ≤ 6, whence the result for n = 2. The proposition for general n is deduced from this case. Consider an arbitrary monomial of degree n,

x^{i} = x_{i_{1}}...x_{i_{n}} .

The following lemma is obvious :

Lemma 3.2. We can change x^{i} into a well–ordered monomial x^{i′} (i′ increasing) by a sequence of transpositions x_{i_{α}}x_{i_{α+1}} ↦ x_{i_{α+1}}x_{i_{α}}.

(Consider the set of inversions {α < β : i_{α} > i_{β}}. Assume i_{γ} > i_{γ+1}, and replace in x^{i} the term x_{i_{γ}} x_{i_{γ+1}} by x_{i_{γ+1}}x_{i_{γ}}. It is easy to check that the set of inversions decreases by one element.)

We now write x^{i} = x^{j}x_{i_{α}}x_{i_{α+1}}x^{ℓ}. We will prove by induction

Lemma 3.3. In \overline{B}, x^{i} ≡ x^{i′} mod F^{n+1}\overline{B}, where i′ is well–ordered.

But this is now equally obvious. Let r, s be the degrees of x^{j}, x^{ℓ}, so n = r+s+2. Then x^{j} ≡ x^{j′}[F^{r+1}\overline{B}], x^{ℓ} ≡ x^{ℓ′}[F^{s+1}\overline{B}] and x_{i_{α}}x_{i_{α+1}} ≡ x_{i_{α+1}}x_{i_{α}}[F^{3}\overline{B}] ; we are of course assuming i_{α} > i_{α+1}. Factoring the congruences gives x^{i} ≡ x^{j′}x_{i_{α+1}}x_{i_{α}}x^{ℓ′}[F^{n+1}\overline{B}] since the filtration F^{n}, image of F^{n} on \overline{A}, verifies F^{n}F^{m} ⊂ F^{n+m}. Using induction if necessary, we obtain the Lemma, whence Proposition 3.1.

Proof of Theorem 1.4.– The natural map φ : A → Λ_{G} sends M_{A}^{n} to M_{Λ}^{n}. Since F^{•} is on \overline{B} the filtration inherited from the natural filtration on \overline{A}, we see that \overline{φ} sends F^{n}\overline{B} to M_{Ω}^{n}. We then have a natural map gr\overline{φ} : gr^{•}\overline{B} → gr^{•}Ω_{G}, surjective since \overline{φ} is so. It is an isomorphism since dim gr^{n}\overline{B} ≤ dim S_{n} = dim gr^{n}Ω_{G}. (The last equality follows from the considerations after Cor.2.4 ; cf also [3, Theorem 7.24]). Therefore \overline{φ} is isomorphic since the filtration on \overline{B} is complete. The last point follows from the fact that \overline{B} = \overline{A}/\overline{R} where \overline{R} is closed and therefore complete for the filtration induced from that of \overline{A} : see e.g. [5, Thm 4 (5) p. 31].

Proof of Theorem 1.2.– The reduction of φ : A/R → Λ_{G} is \overline{φ}. Recall that \overline{R} is the image of R in \overline{A}. Assume f ∈ A satisfies φ(f) = 0. We then have \overline{f} ∈ \overline{R} by Theorem 1.3, so f = r_{1} +pf_{1}, r_{1} ∈ R, f_{1} ∈ A. Then φ(f_{1}) = 0. Inductively, we obtain an expression f = r_{n} + p^{n}f_{n} of the same type. Since p^{n}f_{n} → 0 in A and R is closed, we see that f ∈ R, QED.

4

In this section, we show that the description of Λ_{G} given in § 1 allows one to give different proofs of some results of Ardakov and to understand them in terms of the growth of coefficients in the Iwasawa expansion. Ardakov’s main result in [1] is that the centre of the Iwasawa algebra reduces to the Iwasawa algebra of the centre of G, trivial in our case. We will see that the fact of being central is incompatible with the boundedness of the Iwasawa coefficients. It will be instructive to compare this behaviour with what happens for the centre of the enveloping algebra. Recall that instead of the Iwasawa distributions, or measures, we can consider the analytic distributions (or hyperfunctions), dual to the locally analytic functions on G (cf. Schneider–Teitelbaum [6]). They admit an expansion (1.3), but with now

(4.1)  |λ_{n}|r^{|n|} ⟶ 0  ∀r < 1 , |n| = n_{1} + n_{2} + n_{3} .

Among these we have the Casimir operator (seen as a distribution with support at 1)

ω = h^{2} + 2(xy + yx) = h^{2} − 2h + 4xy

(cf. e.g. Borel [2, p. 19]) where h, x, y are the infinitesimal generators of the groups T, N^{+}, N^{−}. It suffices to compute ω on a function f given by

f(utn) = (1 + Y)^{x_{1}}(1 + H)^{x_{2}}(1 + X)^{x_{3}} where u, t, n have parameters x_{1}, x_{2}, x_{3} ∈ ℤ_{p} and Y, H, X belong to the disc |w| < 1 in ℂ_{p} or even ℚ_{p} (such functions are dense). Now

(xyf)(1) = \frac{d}{dt}|_{0}yf(e^{tx}) = \frac{d}{dt}|_{0} \frac{d}{ds}|_{0}f(e^{sy}e^{tx})

= \frac{d}{dt}|_{0} \frac{d}{ds}|_{0}f( \binom{1}{s 1} \binom{1 t}{1} )
= \frac{∂^{2}}{∂s∂t} |_{0}(1 + Y)^{s/p}(1 + X)^{t/p}
= \frac{1}{p^{2}} log(1 + Y) log(1 + X) ,

hf(1) = \frac{d}{dt}|_{0}f (\begin{matrix}e^{t} \\ & e^{−t} \end{matrix})
= \frac{1}{log(1 + p)} \frac{d}{dt}|_{0}f (\begin{matrix}(1 + p)^{t} \\ & (1 + p)^{−t} \end{matrix})
= \frac{1}{log(1 + p)} log(1 + H) ,
h^{2}f(1) = \frac{1}{log^{2}(1 + p)} \frac{d^{2}}{dp^{2}} |_{0}f (\begin{matrix}(1 + p)^{t} \\ & (1 + p)^{−t} \end{matrix})
= \frac{1}{log^{2}(1 + p)}[log(1 + H)] ,

Thus the Amice transform of ω is

F (Y, H, X) =

= \frac{1}{log^{2}(1 + p)} log^{2}(1 + H) − \frac{2}{log(1 + p)} log(1 + H) + \frac{4}{p^{2}} log(1 + Y) log(1 + X) .

This obviously has an expansion (4.1) – and is an element of the ring of convergent series on D(1)^{3}, D(1) ⊂ \overline{ℚ}_{p} being the open unit disc – but it is not an element of Λ_{G}. We will see that the invariance under T suffices to impose such a logarithmic behaviour. This leads to :

Theorem 4.1. The space of elements on Λ_{G} invariant by conjugation under T is equal to the Iwasawa algebra Λ_{T} ⊂ Λ_{G}.

Assume indeed λ ∈ Λ_{G} is T –invariant, with Amice transform

F (Y, H, X) .

We have Y = u_{0} −1, with h_{0}u_{0}h_{0}^{−1} = u_{0}^{(1+p)^{−2}} ; thus the automorphism Ad(h_{0}) of G sends 1 + Y to (1 + Y)^{(1+p)^{−2}}. Similarly, h_{0}n_{0}h_{0}^{−1} = n_{0}^{(1+p)^{2}}, so 1 + X is sent to (1 + X)^{(1+p)^{2}}. Of course H is left invariant. If λ is T –invariant we therefore have

(4.2)  F (Y, H, X) = F (Y′, H, X′)

where 1 + Y′ = (1 + Y)^{(1+p)^{−2}}, 1 + X′ = (1 + X)^{(1+p)^{2}}. Since p ≠ 2, (1 + p)^{2} is a topological generator of 1 + ℤ_{p}. Therefore (4.2) remains true if

(4.3)  1 + Y′ = (1 + Y)^{u} , 1 + X′ = (1 + X)^{u^{−1}} , u ∈ 1 + pℤ_{p} .

In the following computations consider F as an element of the Lazard ring in three variables. If we fix a value of H in ℂ_{p} such that |H| < 1, say H_{0}, F (Y, H_{0}, X) := F_{1}(Y, X) becomes an Iwasawa series in the two variables, still invariant under (4.3). Now set

(4.4)  U = log(1 + Y) , V = log(1 + X) ,

two series convergent in D(1). We have

F_{1}(Y, X) = G_{1}(U, V)

where G_{1} converges absolutely in the domain of convergence of the exponential, i.e. for |U|, |V | < r_{0} = p^{− \frac{1}{p−1}} . Moreover G_{1} is invariant by U ↦ uU, V ↦ u^{−1}V, |u − 1| < p^{−1}. This implies that

G_{1}(U, V) = G_{2}(UV)

with G_{2}(z) convergent for |z| < r_{0}^{2}. Let

G_{2}(z)  = ∑_{0}^{∞} b_{q}z^{q} ,
F_{1}(Y, X) = ∑_{m,n} a_{mn}Y^{m}X^{n} (|a_{mn}| ≤ 1) .
Then F_{1}(Y, X) = G_{2}(log(1 + Y) log(1 + X)) ,
log(1 + Y) = Y ∑_{0}^{∞} \frac{(−1)^{k}}{k + 1} Y^{k} := Y L_{1}(Y)

log(1 + X) = X ∑_{0}^{∞} \frac{(−1)^{ℓ}}{ℓ + 1} Y^{ℓ} := XL_{1}(X)

Thus (log(1+Y) log(1+X))^{q} contains only terms the degree of which in Y and X is at least q. We have of course b_{0} = a_{0}, and the previous remark implies that

∑_{n≥0} a_{1n} Y X^{n} + ∑_{m≥0} a_{m_{1}}Y^{m}X

is identical with the sum of terms of these degrees in

b_{1} Y X L_{1}(Y)L_{1}(X) ,

i.e. with

b_{1} Y X (L_{1}(Y) + L_{1}(X) − 1) .

Since the a_{mn} are integral, this implies that b_{1} = 0 as the denominators in the log–series are not bounded. By induction assume that b_{1} = ⋯ b_{N−1} = 0, so

G_{2} = ∑_{N}^{∞} b_{q}z^{q} .

We then find that

(4.4)  F_{1}(Y, X) = b_{N}Y^{N}X^{N}L_{1}(Y)^{N}L_{1}(X)^{N}

+ terms of degree > N in X and Y.

\begin{aligned}Now & L_{1}(Y) & = & 1 + Y M_{1}(Y) , & say , \\ & L_{1}(X) & = & 1 + XM_{1}(X) \end{aligned}

so (4.4) implies that

F_{1}(Y, X) = b_{N}Y^{N}X^{N}(1 + NY M_{1}(Y) + NXM_{1}(X))

+ terms of degree > N in X and Y .

Since M_{1} does not have bounded denominators, we deduce that b_{N} = 0. Finally we have proved that F_{1} = b_{0}, i.e. F (Y, H, X) ≡ b_{0}(H) for any H ∈ ℂ_{p}, |H| < 1, This implies that F (Y, H, X) = F (H) has no terms involving X or Y, whence the result.

Corollary 4.2. The centre of Λ_{G} is composed of the multiples of the Dirac measure at 1.

For assume that λ ∈ Λ_{G} is central, so invariant by all conjugates of T in G. By Thm. 4.1 its support is contained in the intersection of the tori gT g^{−1} (g ∈ G) . This intersection is reduced to {1}. We note that Theorem 4.1 itself follows from Ardakov’s results [1, Proposition 2.2]: a simple computation shows that the only finite orbits of T in G are the elements of T (use the triangular decomposition).

5

This section is devoted to conjectural remarks on a formal extension of the main result. Consider the formulas of Proposition 1, for example (a) (1 + H)(1 + Y) = (1 + Y)^{(1+p)^{−2}}(1 + H)

(c) (1 + X)(1 + Y) = (1 + Y)^{(1+p^{2})^{−1}}(1 + H)^{\frac{log(1+p^{2})}{log(1+p)}} (1 + X)^{(1+p^{2})^{−1}}.

In the p–adic computation the series for, say, (1 + X)^{x} (x ∈ ℤ_{p}) converges as

an Iwasawa expansion because of the integrality of the binomial function (\begin{matrix}x \\ n \end{matrix}).

However, replace now Λ_{G} by k[[Y, H, X]] where k is a field of characteristic zero. Set p = ε, another formal variable, which should however be considered as a small parameter. The binomial coefficients, namely

(5.1)  (\begin{matrix}(1 + ε)^{−2} \\ n \end{matrix}) = \frac{(1 + ε)^{−2}((1 + ε)^{−2} − 1) ⋯ ((1 + ε)^{−2} − n + 1)}{n!}

and similarly

(\begin{matrix}log(1 + ε^{2})/ log(1 + ε) \\ n \end{matrix})

are well–defined series in k[[ε]]. Formulas (a, b, c) therefore define the products HY, XH and XY in k[[ε]][[Y, H, X]]. The p–adic results do not seem to imply that this extends to an associative product in this ring of power series. Note that if it were so, equations (a, b, c) at ε = 0 would simply yield HY = Y H, XH = HX and XY = Y X. Such an extension would therefore define, quite naturally, a formal deformation of the algebra of power series k[[Y, H, X]] associated to the group SL(2). It would be interesting to understand this deformation in group–theoretic terms (or in terms of the Lie algebra) –assuming, of course, it exists. In this respect one should note that formulas (a, b) allow one to define inductively the products H^{n}Y^{m} and X^{n}H^{m}. However I do not see how to define X^{n}Y^{m}, even granting (c).

References

[1] K. Ardakov, The centre of completed algebras of pro–p groups, Documenta Math. 9 (2004), 599–606.

[2] A. Borel, Automorphic forms on SL(2, ℝ), Cambridge Univ. Press, 1997.

[3] J.D. Dixon, M.P.F. Du Sautoy, A. Mann, D. Segal, Analytic pro-p groups, 2nd edition, Cambridge University Press (1999).

[4] M. Lazard, Groupes analytiques p–adiques, Publ. IHES 26 (1965), 389– 603.

[5] Li Huishi, F. van Oystaeyen, Zariskian filtrations, Kluwer, 1997.

[6] P. Schneider, J. Teitelbaum, Algebras of p–adic distributions and admissible representations, Inv. Math. 153 (2003), 145–196.

[7] J.–P. Serre, Algèbres de Lie semi–simples complexes, Benjamin, 1966.

[8] R. Steinberg, Lectures on Chevalley groups, mimeographed notes, Yale University, 1967.

[9] L. Washington, Introduction to cyclotomic fields, Springer, 1982.

Laurent Clozel Mathématique Université Paris-Sud 11 Bât. 425 91405 Orsay CEDEX France 560

Documenta Mathematica 16 (2011)