On the Bruhat-Tits-Building of Unitary Groups: Simultaneous Diagonalization of Real Norms

Winfried FRISCH

Abstract

Let K be a local field and G the group of K-rational points of an unitary group over K. In [4], there is given a concrete interpretation of the Bruhat-Tits-building of G as a space of “maximinorant” norms on the vector-space associated to the natural representation of G. I describe a method to find for any given two points in this building an apartment, that contains both of them.

Introduction In [1] and [2] F. Bruhat and J. Tits construct to any connected reductive algebraic group G defined over a local field K an affine building, called the Bruhat-Tits-building of G, using root-data with valuation. In the case that G is one of the classical groups, they give a concrete realization of this building in [3] and [4] as an suitable space 𝒩 = 𝒩(G(K)) of norms on the K-vector-space X of the natural representation of the group G(K) of K-rational points of G as classical group. They do this defining a G(K)-equivariant bijection from the building (resp. the ”enlarged building” in the case that G is of type A_{n} , cf. [3] 2.4) onto 𝒩 on which G(K) acts by righttranslation. In this way one gets a very natural description of the apartements of the building in terms of norms on X: To the apartements correspond the sets of those norms in 𝒩, which are ”split” by a given ”canonical” basis of X (I will give the correct definitions later). Since in a building any two points are contained in a common apartement, it follows, that, given two norms in 𝒩, there is at least one ”canonical” basis of X splitting them simultaneously. In the case, that G is of type A_{n}, a direct and constructive proof of this fact is easily found in the literature (e.g. [6] prop. 1.3 or [5] prop. 2.3.4). Therefore I will give here an analogous proof for the unitary groups, which is a generalization of a similar proof for maximal lattices in orthogonal spaces by [7], p. 51-54 and a part of a direct proof, that these spaces of norms can be endowed with the structure of a building (as [5] does for the A_{n}-case), and may be usefull for computations.

Notations Let (K, ω) be a (commutative) field with a discrete valuation, such that ω(K^{×}) = ℤ. Let X be a finite-dimensional vector space over K.

A norm α : X → ℝ ∪ {∞} is a map which satisfies:

1. α(x) = ∞ ⇒ x = 0, for x ∈ X.

2. α(λx) = ω(λ) + α(x) for λ ∈ K, x ∈ X.

3. α(x + y) ≥ inf(α(x), α(y)) for x, y ∈ X.

Let α be a norm of X and Y a subspace of X. A further subspace Y′ of X is called splitting complement of Y for α, if X = Y ⊕ Y′ and α(y + y′) = inf(α(y), α(y′)), for all y ∈ Y, y′ ∈ Y′. A norm α of X is called split, if there is a basis x_{1}, …, x_{n} of X, such that

α(∑_{i=1}^{n} λ_{i}x_{i}) = inf_{i}(ω(λ_{i}) + α(x_{i})) for λ_{1}, …, λ_{n} ∈ K.  (1)

If α is a split norm of X, then by [3] 1.5, all subspaces of X have a splitting complement for α.

A norm is called discrete, if their values are discrete in ℝ ∪ {∞}. A split norm is discrete, clearly.

Further, let the pair (f, q) consist of a bilinear map and a quadratic form on X, such that one of the following cases are given^{1}:

1. f is non-degenerate and alternating. Then put q = 0.

2. q is a semiregular quadratic form with associated symmetric bilinearform f, i.e. f(x, y) = q(x+y)−q(x)−q(y) for x, y ∈ X, where semiregular means that either X as quadratic space is regular or char(K) = 2 and X = X′⊥Kx with a nonisotropic element x ∈ X and a regular quadratic subspace X′ of codimension 1.

The Witt-index r of (X, f, q) is by definition the dimension of a maximal isotropic subspace of X, which is well-defined by the Witt-theorem. Given a maximal isotropic subspace X_{+} in X, there is another, but not unique maximal isotropic subspace X_{−} “dual” to X_{+}, such that X_{+} ⊕ X_{−} is hyperbolic and X_{0} := (X_{+} ⊕ X_{−})^{⊥} is anisotropic, in particular there is a basis e_{1}, …, e_{r} of X_{+} and a basis e_{−1}, …, e_{−r} of X_{−} with

1My conventions above are more restrictive than those in [4] section 2, which include hermitian forms. But I do not want to repeat all the notations here. Although the theorem and his proof stay valid without changes in the more general situation, too.

f(e_{i}, e_{−i}) = 1, for i = 1, …, r, and
f(e_{i}, e_{j}) = 0, for i, j ∈ I := {±1, …, ±r}, i ≠ −j. (2)

Let x_{1}, …, x_{k} be a basis of X_{0}. We call a decomposition of the form X = X_{0} ⊥ (X_{+} ⊕ X_{−}) the Witt-decomposition of X (which is by the Witttheorem unique up to isometry) and a basis x_{1}, …, x_{k}, e_{1}, …, e_{r}, e_{−1}, …, e_{−r} of X as above a canonical basis of (X, f, q).

If X_{0} ≠ {0} we assume further, that

1. 1 ∈ q(X_{0}).

2. 2ω(f(x, y)) ≥ ω(q(x)) + ω(q(y)), for x, y ∈ X_{0}.

Condition 2. is satisfied, if K is henselian (see [4] 1.15).

As in [4] 2.1, a norm α is called maximinorant with respect to (f, q), if α is maximal under the conditions:

\begin{matrix}α(x) + α(y) ≤ ω(f(x, y)) & for x, y ∈ X. \\ 2α(x) ≤ ω(q(x)) & for x ∈ X. \end{matrix}  (3)

Now, the appropriate space of norms is the set of all maximinorant norms with respect to (f, q), which will be denoted by 𝒩_{mm}, cf. [4] 2.9 and 2.12, where is proven, that for any α ∈ 𝒩_{mm} there is a Witt-decomposition X = X_{0} ⊥ (X_{+}⊕X_{−}) and an associated canonical basis x_{1}, …, x_{k}, e_{1}, …, e_{r}, e_{−1}, …, e_{−r} as above which splits α. Moreover

α(x) = \frac{1}{2}ω(q(x)) for x ∈ X_{0}  (4)

and

α(e_{i}) = −α(e_{−i}) for i = 1, …, n.  (5)

Theorem 1 If α, β ∈ 𝒩_{mm} are maximinorant norms with respect to (f, q), then there is a canonical basis of X which splits α and β simultaneously.

The proof is by induction on the Witt-index. But first I prove some lemmas:

Lemma 1 Let 𝒜 (resp. ℬ) be a canonical basis of (X, f, q), which splits α (resp. β). If the Witt-index of (X, f, q) is positiv, then there is a pair of isotropic vectors e_{1} ∈ 𝒜, ẽ_{−1} ∈ ℬ and a λ ∈ K^{×}, such that with e_{−1} := λẽ_{−1}

f(e_{1}, e_{−1}) = 1, α(e_{1}) = −α(e_{−1}), β(e_{1}) = −β(e_{−1})

and

α(e_{1}) − β(e_{1}) = sup_{x∈X\{0}}(α(x) − β(x)).

Proof. First I assume, that α ≠ β and

sup_{x∈X\{0}}(α(x) − β(x)) ≥ sup_{x∈X\{0}}(β(x) − α(x)),  (6)

in particular

sup_{x∈X\{0}}(α(x) − β(x)) > 0.  (7)

Denote the vectors of ℬ by u_{1}, …, u_{k}, f_{1}, …, f_{r}, f_{−1}, …, f_{−r} in the obvious manner. As remarked in [3] 1.26, we can choose an u ∈ X \ {0}, such that α(u) − β(u) is maximal. I put u = ∑_{i=1}^{k} λ_{i}u_{i} + ∑_{i∈I} μ_{i}f_{i} (where I = {±1, …, ±r}), u_{0} := ∑_{i=1}^{k} λ_{i}u_{i} and f_{0} := ∑_{i∈I} μ_{i}f_{i}, hence u = u_{0} + f_{0}.

By (7), we have \frac{1}{2}ω(q(u)) ≥ α(u) > β(u). Assume, that β(u_{0}) < β(f_{0}) would hold. Then, by (4) and since β is minorant (f, q),

\frac{1}{2}ω(q(u_{0})) = β(u_{0}) < β(f_{0}) ≤ \frac{1}{2}ω(q(f_{0})),

and therefore

\frac{1}{2}ω(q(u)) = \frac{1}{2}ω(q(u_{0})) = β(u_{0}) = β(u),

contradiction. Thus β(u_{0}) ≥ β(f_{0}) and β(u) = β(f_{0}), in particular f_{0} ≠ 0.

Take i_{0} ∈ I with β(u) = ω(μ_{i_{0}}) + β(f_{i_{0}}) and put e_{−1} := ±μ_{i_{0}}^{−1}f_{−i_{0}} with a suitable sign, such that f(u, e_{−1}) = 1. Also q(e_{−1}) = 0 and

β(e_{−1}) = ω(μ_{i_{0}}^{−1}) + β(f_{−i_{0}}) = −ω(μ_{i_{0}}) − β(f_{i_{0}}) = −β(u)

holds by (5).

Now put e_{1} = u − q(u)e_{−1}. This yields q(e_{1}) = 0, and since ω(q(u)) + β(e_{−1}) > 2β(u) − β(u) = β(u), it follows, that

β(e_{1}) = inf(β(u), ω(q(u)) + β(e_{−1})) = β(u).

Now α(e_{1}) − β(u) = α(e_{1}) − β(e_{1}) ≤ α(u) − β(u) implies

α(e_{1}) ≤ α(u)

and from α(u) − β(u) ≥ β(e_{−1}) − α(e_{−1}) = −β(u) − α(e_{−1}) we get

α(e_{−1}) ≥ −α(u).  (8)

But this yields also

α(e_{1}) ≥ inf(α(u), ω(q(u)) + α(e_{−1})) ≥ inf(α(u), 2α(u) + α(e_{−1})) = α(u).

Thus α(e_{1}) = α(u) and

α(e_{1}) − β(e_{1}) = α(u) − β(u) = sup_{x∈X\{0}}(α(x) − β(x)).

Finally, since α is minorant f, one gets α(e_{−1}) ≤ ω(f(e_{1}, e_{−1})) − α(e_{1}) = −α(e_{1}). On the other hand α(e_{−1}) ≥ −α(u) = −α(e_{1}), by (8).

Now, since α(e_{1}) − β(e_{1}) = β(e_{−1}) − α(e_{−1}), in (6) holds equality. So this assumption is made without loss of generality. Moreover, it follows from the proof above, that the difference β(x) − α(x) takes his supremum on an isotropic vector of an arbitrary canonical basis which splits β. Hence we can choose e_{1} = u ∈ 𝒜, by symmetry, as stated in the lemma. □

Lemma 2 Let α be a split maximinorant norm with respect to (f, q) and X = X_{1} ⊥ X_{2} an orthogonal decomposition, which splits α. Then α|_{X_{1}}, α|_{X_{2}} are also split maximinorant norms with respect to the restricted forms.

Proof. We only have to show, that if α_{1} and α_{2} are minorant norms with respect to (f, q) on X_{1} and X_{2} respectively, then α_{1} ⊥ α_{2}, where

α_{1} ⊥ α_{2}(x_{1} + x_{2}) = inf(α_{1}(x_{1}), α_{2}(x_{2})) for x_{1} ∈ X_{1}, x_{2} ∈ X_{2}

is also minorant with respect to (f, q). But this follows immediatly from

f(x_{1}, x_{2}) = 0 and q(x_{1} + x_{2}) = q(x_{1}) + q(x_{2}).

The restrictions of α are split by [3] 1.5. □

Now we can finish the proof of the theorem 1 by induction on r: Proof. If (X, f, q) is anisotropic, then every basis of X is canonical and we can proceed as [6] in the A_{n}-case. So assume that r > 0 and choose e_{1} and e_{−1} as in lemma 1. Then H := K[e_{1}, e_{−1}] is a hyperbolic plane. Take x = λ_{1}e_{1} + λ_{−1}e_{−1} + z ∈ X with z ∈ H^{⊥}. Then since α is minorant f and f(e_{1}, e_{−1}) = 1, for j = ±1 we get, if λ_{j} ≠ 0:

α(∑_{i=±1} λ_{i}e_{i} + z) ≤ −α(λ_{j}^{−1}e_{−j}) = α(λ_{j}e_{j}),

otherwise α(∑_{i=±1} λ_{i}e_{i} + z) ≤ α(λ_{j}e_{j}) = ∞ holds, clearly. This implies

α(∑_{i=±1} λ_{i}e_{i} + z) ≤ α(z).

So the decomposition Ke_{1} ⊕ Ke_{−1} ⊥ H^{⊥} splits α and the same holds for β, clearly. Hence, using lemma 2, the theorem follows by induction. □

References

[1] Bruhat F., Tits J., Groupes réductifs sur un corps local I: Données radicielles valluées, Publ. Math. I.H.E.S. 41 (1972) 5-521.

[2] Bruhat F., Tits J., Groupes réductifs sur un corps local I: Données radicielles valluées, Publ. Math. I.H.E.S. 60 (1984) 259-301.

[3] Bruhat F., Tits J., Schémas en groupes et immeubles des groupes classiques sur un corps local I: le groupe linéaire générale, Bull. Soc. Math. France 112 (1984) 259-301.

[4] Bruhat F., Tits J., Schémas en groupes et immeubles des groupes classiques sur un corps local II:groupes unitaires, Bull. Soc. Math. France 115 (1987) 141-195.

[5] Gerardin P., Immeubles des groupes linéaires généraux, in: Carmona J. Vergne M., Harmonic Analysis and Lie Groups, Springer Verlag LNM 880,1981,pp. 138-178.

[6] Goldman O., Iwahori N., The space of p-adic norms, Acta Math. 109 (1963) 137-177.

[7] Eichler M., Quadratische Formen und orthogonale Gruppen, Springer-Verlag, 1952.

Winfried Frisch Mathematisches Institut der Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany E-mail: frisch@uni-math.gwdg.de