Journal of Lie Theory Volume 6 (1996) 165–177 C ◯1996 Heldermann Verlag

Haar measure on linear groups over local skew fields

Helge Glöckner^{1}

Communicated by K. H. Hofmann

Abstract. General and special linear groups over local fields, as well as their projective counterparts, are prominent examples of totally disconnected, locally compact groups. In this article, an explicit description of Haar measure on these groups is given by computing the measure on special local bases, consisting of open compact subgroups. These data can be used to compute the measure of any open set. In addition to that, I show that all of the groups above are unimodular, and I give a description of Haar measure on the general linear groups in terms of Haar measure on vector spaces over local fields.

1. Notational Conventions

Throughout this article, K denotes a local field, that is, a non–discrete, totally disconnected, locally compact field, commutative or not. As Weil [11] points out, K can be endowed with an ultrametric absolute value which induces the given topology on K, as follows: For any unit x ∈ K^{×}, left multiplication with x yields an automorphism l_{x} of (K, +). We define |x| to be the module of l_{x}, that is, if m is a Haar measure on (K, +) and U is a measurable subset of K such that 0 < m(U) < ∞, we define

|x| := mod_{K} (l_{x}) := \frac{m(l_{x}(U))}{m(U)} = \frac{m(xU)}{m(U)}.

This definition is being completed by setting |0| := 0. We fix the following notation:

R := { x ∈ K : |x| ≤ 1 }, P := { x ∈ K : |x| < 1 }.

Theorem 1.1. R is the unique maximal compact subring of K. Its group of units is R^{×} = { x ∈ K : |x| = 1 }, and P is the unique maximal proper ideal

ISSN 0949–5932 / $2.50 ^{_{C} ◯} Heldermann Verlag of R. The quotient by this ideal, k := R/P, is a finite field. If q denotes its order, then |K^{×}| = ⟨q⟩ ≤ ℝ^{+} . Furthermore, there is an element π ∈ R such that P = πR = Rπ; its absolute value is |π| = q^{−1} . Proof. See Weil loc. cit. Chapter I.–4, Theorem 6.

1 This article is based on several chapters of my Diplomarbeit, Glöckner [7]. I am grateful to Markus Stroppel, my supervisor, for the stimulating discussions which gave rise to this work.

Note that since |.| takes discrete values on K^{×}, the subsets R and R^{×} are open in K. Also, since open subgroups of topological groups are closed, P is closed in R, hence compact. For ℓ ∈ ℕ = {1, 2, … } we define P_{ℓ} := π^{ℓ}R = Rπ^{ℓ} = { x ∈ K : |x| ≤ q^{−ℓ} }. Plainly these sets constitute a local base at 0 of the topology on K; each P_{ℓ} is an open and compact two–sided ideal of R.

Theorem 1.2. Let T be a transversal of R/P such that {0, 1} ⊆ T. Then, for any x ∈ K, there is a unique sequence (a_{i})_{i∈ℤ} in T, with support bounded below, such that

x = ∑_{i∈ℤ} a_{i}π^{i}.

Proof. See Weil loc. cit. Chapter I.–4, Corollary 2 to Theorem 6.

For n ∈ ℕ, M(n, K) = K^{n×n} denotes the ring of n×n–matrices with entries in K. We define

GL(n, K) := M(n, K)^{×}
SL(n, K) := GL(n, K)′
PGL(n, K) := GL(n, K)/Z
PSL(n, K) := SL(n, K)/(Z ∩ SL(n, K)),

where Z = Z(K^{×}) 1 denotes the centre of GL(n, K). The quotient morphisms GL(n, K) → PGL(n, K) and SL(n, K) → PSL(n, K) will be denoted by α and β, respectively. SL(n, K) is the kernel of the Dieudonné determinant det : GL(n, K) → K̄, where K̄ = K^{×}/(K^{×})′. We denote the coset of x ∈ K^{×} in K̄ by x̄. Recall that, for n ≥ 2, SL(n, K) is the subgroup of GL(n, K) generated by the elementary matrices B_{ij}(λ) = 1 + λ E_{ij}, where i ≠ j and λ ∈ K, see Artin [1]. Finally, note that SL(1, K) = (K^{×})′.

2. Special Local Bases at the Identity

With product topology, M(n, K) is a locally compact topological ring. For ℓ ∈ ℕ, the open compact additive subgroups P_{ℓ}^{n×n} = M(n, P_{ℓ}) constitute a local base of the topology at 0. We give GL(n, K) and SL(n, K) the topologies induced by M(n, K), and we give the projective linear groups the respective quotient topologies. For ℓ ∈ ℕ, define

\begin{aligned}U_{ℓ}^{n} & := 1 + P_{ℓ}^{n×n} \\ V_{ℓ}^{n} & := U_{ℓ}^{n} ∩ SL(n, K) \\ Ũ_{ℓ}^{n} & := α(U_{ℓ}^{n}) \\ Ṽ_{ℓ}^{n} & := β(V_{ℓ}^{n}). \end{aligned}

Proposition 2.1. GL(n, K) is a locally compact topological group. For ℓ∈ℕ, the sets U_{ℓ}^{n} constitute a local base for the topology at the identity element, consisting of open compact subgroups. Proof. Any U_{ℓ}^{n} is an open compact subset of M(n, K), which is multiplicatively closed, since P_{ℓ}^{n×n} = (π^{ℓ}1)M(n, R) is an ideal of M(n, R): We have U_{ℓ}^{n}U_{ℓ}^{n} = (1 + P_{ℓ}^{n×n})(1 + P_{ℓ}^{n×n}) ⊆ 1 + P_{ℓ}^{n×n} = U_{ℓ}^{n}. Any matrix A ∈ U_{ℓ}^{n} is invertible, with inverse in U_{ℓ}^{n}, since we presently show that A^{−1} = (1 + B)^{−1} = ∑_{i=0}^{∞}(−B)^{i}, where B := A − 1 ∈ P_{ℓ}^{n×n}. Once we have established the convergence of this so-called von Neumann series, it is clear that its limit is the desired inverse of A. Since K, hence M(n, K), is complete, we only need to check that the series above is a Cauchy series. To this end, note that (−B)^{i} ∈ P_{iℓ}^{n×n} for any i ∈ ℕ. Therefore ∑_{i=ν}^{ν′}(−B)^{i} ∈ P_{νℓ}^{n×n}, for all ν ≤ ν′ ∈ ℕ. This implies the Cauchy property. Fixing ν = 0 and letting ν′ tend to infinity, the previous formula also shows that ∑_{i=0}^{∞}(−B)^{i} ∈ 1 + P_{ℓ}^{n×n} = U_{ℓ}^{n}. Since GL(n, K) contains the open subset U_{1}^{n} of M(n, K) and left multiplication by a unit is a homeomorphism of M(n, K), we conclude that GL(n, K) is an open subset of M(n, K). Now since multiplication is continuous, inversion is continuous if it is continuous at the identity element. But this is guaranteed, since the von Neumann series converges absolutely and uniformly on U_{1}^{n}.

Lemma 2.2. The Dieudonné determinant det: GL(n, K) → K̄ is continuous. Proof. Since det is a homomorphism of groups, it suffices to check continuity at the identity. We claim det (U_{ℓ}^{n})⊆(U_{ℓ}^{1})^{−}. To see this, let A = (a_{ij}) ∈ U_{ℓ}^{n}. Left multiplication with the elementary matrix B_{n1}(−a_{n1}a_{11}^{−1}) ∈ U_{ℓ}^{n} yields a matrix A′ ∈ U_{ℓ}^{n} whose (n, 1)–entry vanishes; here we used that a_{11} ∈ U_{ℓ}^{1} is a unit. Continuing in this way, we can replace a_{n1}, …, a_{21} by zero, and then we apply the same procedure to the other columns. We conclude that there is a lower triangular matrix B ∈ U_{ℓ}^{n} ∩ SL(n, K) and an upper triangular matrix (c_{ij}) = C ∈ U_{ℓ}^{n} such that BA = C. Then det (A) = det (C) = c̄_{11} · . . . · c̄_{nn} ∈ (U_{ℓ}^{1})^{−}.

Lemma 2.3. The commutator subgroup (K^{×})′ is closed in K^{×}. Proof. The centre κ of K is a local field, and K has finite dimension over κ, see Weil [11] Chapter I.–4, Proposition 5. Hence K is a central division algebra over κ. Let F be a splitting field for K, that is, a commutative extension field of κ such that there is an isomorphism of F –algebras g : K ⊗_{κ} F → M(n, F) for some n ∈ ℕ. Such a splitting field always exists, and we may assume that F |κ is a finite Galois extension, see Cohn [4] Chapter 7.2. Then F is a local field. Given x ∈ K, one can show that RN_{K/κ}(x) := det_{F^{n}}(g(x ⊗ 1)) ∈ κ, see Cohn loc. cit. Chapter 7.3. The mapping RN_{K/κ} : K → κ is called the reduced norm on K. We claim that RN_{K/κ} is continuous. Recall that any finite dimensional vector space over a local field L admits precisely one topology which makes it a topological L–vector space, and that every linear map between finite dimensional L–vector spaces is continuous, see Weil [11] Chapter I.–2, Corollary 1 to Theorem 3. We give K ⊗_{κ} F and M(n, F) the unique F –vector space topologies. Then g is continuous. With the topology above, K ⊗_{κ} F is a topological κ–vector space as well. Hence the κ–linear map K → K ⊗_{κ} F, x ↦ x ⊗ 1, is continuous. Finally, the determinant mapping M(n, F) → F is continuous since F is a topological field. Hence RN_{K/κ} is continuous, being a composite of continuous maps. The reduced norm induces a continuous homomorphism K^{×} → κ^{×}, also denoted by RN_{K/κ}; its kernel is closed in K^{×}. But ker RN_{K/κ} = (K^{×})′, for any local field K, a fact usually expressed by saying that the reduced Whitehead group SK_{1}(K) := (ker RN_{K/κ})/(K^{×})′ is trivial, see Draxl [6].

Proposition 2.4.

(1) SL(n, K), PGL(n, K), and PSL(n, K) are locally compact groups.

(2) For any ℓ ∈ ℕ, V_{ℓ}^{n} is an open and compact subgroup of SL(n, K). {V_{ℓ}^{n} : ℓ ∈ ℕ} is a local base of the topology at 1.

(3) A similar statement holds for the projective linear groups; here, the open and compact subgroups are Ũ_{ℓ}^{n} and Ṽ_{ℓ}^{n}, respectively.

Proof. By the previous lemmas, the Dieudonné determinant is a continuous homomorphism in a Hausdorff topological group. Hence SL(n, K) = ker det is closed in GL(n, K). Since Z(K^{×}) is a closed subgroup of K^{×}, the respective quotient topologies on PGL(n, K) and PSL(n, K) are locally compact Hausdorff. The remainder is obvious, since quotient morphisms of topological groups are open and continuous.

Remark 2.5. Note that U_{ℓ}^{n} and V_{ℓ}^{n} are normal subgroups of GL(n, R) and GL(n, R) ∩ SL(n, K), respectively, for any ℓ ∈ ℕ. This is due to the fact that since any A ∈ GL(n, R) is a unit of M(n, R) and P_{ℓ}^{n×n} is an ideal of this ring, AU_{ℓ}^{n} = A(1 + P_{ℓ}^{n×n}) = A + P_{ℓ}^{n×n} = U_{ℓ}^{n}A. Now R is an open and compact subring of K, hence M(n, R) is an open and compact subring of M(n, K). If A ∈ M(n, R)\GL(n, R), then A + P_{1}^{n×n} ∩ GL(n, R) = Ø. In fact, if there was some matrix B in this intersection, then A ∈ B +P_{1}^{n×n} = B (1+P_{1}^{n×n}) would be invertible, a contradiction. Hence GL(n, R) is a closed subset of M(n, R), from which we conclude that GL(n, R) is an open and compact subgroup of GL(n, K).^{2} This in turn implies that GL(n, R) ∩ SL(n, K) is an open and compact subgroup of SL(n, K).

3. Some Quotients and their Orders

Definition 3.1. For ℓ ∈ ℕ, ψ_{ℓ} : R → R/P_{ℓ} : x ↦ x + P_{ℓ} denotes the quotient morphism. We define ψ_{ℓ}^{n}: M(n, R) → M(n, R/P_{ℓ}) by the prescription A = (a_{ij}) ↦ (ψ_{ℓ}(a_{ij})). We also write x_{[ℓ]} := ψ_{ℓ}(x), A_{[ℓ]} := ψ_{ℓ}^{n}(A).

Lemma 3.2. Let ℓ ∈ ℕ.

(1) Assume that a, b ∈ R are given, with expansions a = ∑_{i=0}^{∞} a_{i}π^{i} and b = ∑_{i=0}^{∞} b_{i}π^{i}, respectively. Then a_{[ℓ]} = b_{[ℓ]} iff a_{i} = b_{i} for 0 ≤ i ≤ ℓ−1.

(2) (∀A, B ∈ M(n, R)) A + P_{ℓ}^{n×n} = B + P_{ℓ}^{n×n} ⇔ A_{[ℓ]} = B_{[ℓ]} .

2 Serre [10] shows that, in the commutative case, GL(n,R) is a maximal compact subgroup of GL(n,K), and that all maximal compact subgroups of GL(n,K) are conjugate.

(3) ψ_{ℓ}^{n} is a quotient morphism of topological rings, and ker ψ_{ℓ}^{n} = P_{ℓ}^{n×n}.

(4) (∀A, B ∈ GL(n, R)) AU_{ℓ}^{n} = BU_{ℓ}^{n} ⇔ A_{[ℓ]} = B_{[ℓ]} .
(5) (∀A, B ∈ SL(n, K) ∩ GL(n, R)) AV_{ℓ}^{n} = BV_{ℓ}^{n} ⇔ A_{[ℓ]} = B_{[ℓ]} .

Proof. (1) and (2) are obvious. Ad (3). It is clear that ψ_{ℓ}^{n} is a surjective morphism of rings with ker ψ_{ℓ}^{n} = (ker ψ_{ℓ})^{n×n} = P_{ℓ}^{n×n}. Since this is an open subset of M(n, R), we conclude that ψ_{ℓ}^{n} is continuous. Ad (4). P_{ℓ}^{n×n} is an ideal of M(n, R), and A is a unit. Therefore AU_{ℓ}^{n} = A(1 + P_{ℓ}^{n×n}) = A + AP_{ℓ}^{n×n} = A + P_{ℓ}^{n×n}. We conclude that AU_{ℓ}^{n} = BU_{ℓ}^{n} ⇔ A + P_{ℓ}^{n×n} = B + P_{ℓ}^{n×n}. Now apply (2). Ad (5). Since V_{ℓ}^{n} ≤ U_{ℓ}^{n}, we infer from (3) that AV_{ℓ}^{n} = BV_{ℓ}^{n} ⇒ A_{[ℓ]} = B_{[ℓ]} . Conversely, if A_{[ℓ]} = B_{[ℓ]} , where A, B ∈ SL(n, K) ∩ GL(n, R), then there is some C ∈ U_{ℓ}^{n} such that AC = B, by (4). But this implies C = A^{−1}B ∈ U_{ℓ}^{n} ∩ SL(n, K) = V_{ℓ}^{n}.

Lemma 3.3. Let a_{ij} ∈ δ_{ij} + P for i, j ∈ J, where J := {1, …, n}^{2}\{(1, 1)}, and set A := {a_{11} ∈ U_{1}^{1} : (a_{ij}) ∈ V_{1}^{n}}. Then A is the union of y := [V_{1}^{1} : V_{ℓ}^{1}] equivalence classes modulo U_{ℓ}^{1} , for every ℓ ∈ ℕ. In particular, A is not empty.

Proof. We may assume n ≥ 2, since the case n = 1 is trivial. Let a_{11} ∈ U_{1}^{1} and set A := (a_{ij}). Then there is a lower unitriangular matrix B ∈ V_{1}^{n}, independent of a_{11} , such that AB =: (c_{ij}) is an upper triangular matrix. We can construct B as follows. Multiplication of A on the right by the product of elementary matrices B_{n} := B_{n1}(−a_{nn}^{−1}a_{n1}) · . . . · B_{nn−1}(−a_{nn}^{−1}a_{nn−1}) yields a matrix (a′_{ij}) such that a′_{ni} = 0 for i = 1, …, n−1. We proceed analogously with columns n − 1, …, 2, obtaining lower triangular matrices B_{n−1}, …, B_{2} ∈ V_{1}^{n}, all of whose diagonal entries are 1 and which are independent of a_{11} , such that AB_{n} ⋯ B_{2} is upper triangular. Now set B := B_{n} … B_{2} .

Note that c_{11} = a_{11} + r for some r ∈ P, where r is independent of a_{11} . Also, c_{22}, …, c_{nn} are independent of a_{11} . Set s := c_{22} · . . . · c_{nn} ∈ U_{1}^{1} . Then A ∈ SL(n, K) if and only if (a_{11} + r)s ∈ (K^{×})′, and, indeed, if and only if (a_{11} + r)s ∈ (K^{×})′ ∩ U_{1}^{1} = V_{1}^{1} . Now choose representatives t_{1}, …, t_{y} of the cosets of V_{ℓ}^{1} = (K^{×})′ ∩ U_{ℓ}^{1} in V_{1}^{1} . For k = 1, …, y, set a_{k} := t_{k}s^{−1} − r ∈ U_{1}^{1} . Then (a_{k} + r)s = t_{k}, whence a_{k} ∈ A. Note that for a, b ∈ A, a_{[ℓ]} = b_{[ℓ]} ⇔ ((a + r)s)_{[ℓ]} = ((b + r)s)_{[ℓ]} ; here we use that ψ_{ℓ} is a ring homomorphism, and that s, hence s_{[ℓ]} , is a unit. From this the claim follows.

Lemma 3.4. With notation as in Lemma 3.3, assume A′ = (a′_{ij}) ∈ V_{ℓ}^{n} such that (a′_{ij})_{[ℓ]} = (a_{ij})_{[ℓ]} for (i, j) ∈ J. Then there is a_{11} ∈ A such that (a_{11})_{[ℓ]} = (a′_{11})_{[ℓ]} .

Proof. Let a_{11} ∈ U_{1}^{1} and set A = (a_{ij}). Define B, C, s as above, and let B′, C′, s′ be the corresponding expressions for A′. Then B_{[ℓ]} = B′_{[ℓ]} . Also, s_{[ℓ]} = s′_{[ℓ]} , and (c_{ij})_{[ℓ]} = (c′_{ij})_{[ℓ]} for (i, j) ∈ J. Further, c_{11} = a_{11}+r and c′_{11} = a′_{11}+r′ for some r, r′ ∈ P_{ℓ}, where r_{[ℓ]} = r′_{[ℓ]} . We have t := (a′_{11} + r′)s′ ∈ (K^{×})′ ∩ U_{1}^{1} , since A′ ∈ V_{1}^{n}. Then (a_{11})_{[ℓ]} = (a′_{11})_{[ℓ]} if and only if ((a_{11} + r)s)_{[ℓ]} = t_{[ℓ]} . We may replace a_{11} by ã_{11} := ts^{−1} − r without changing B, C, r, and s. Then ã_{11} ∈ A, and (ã_{11})_{[ℓ]} = (a′_{11})_{[ℓ]} .

Proposition 3.5. For any ℓ ∈ ℕ, (1) [U_{1}^{n} : U_{ℓ}^{n}] = q^{n^{2}(ℓ−1)}

(2) [Ũ_{1}^{n} : Ũ_{ℓ}^{n}] = x_{ℓ}^{−1}q^{n^{2}(ℓ−1)}
(3) [V_{1}^{n} : V_{ℓ}^{n}] = y_{ℓ} q^{(n^{2}−1)(ℓ−1)}
(4) [Ṽ_{1}^{n} : Ṽ_{ℓ}^{n}] = z_{ℓ,n}^{−1} [V_{1}^{n} : V_{ℓ}^{n}].

Here x_{ℓ} := [U_{1}^{1} ∩ κ^{×} : U_{ℓ}^{1} ∩ κ^{×}] divides q^{ℓ−1} , and so do y_{ℓ} := [V_{1}^{1} : V_{ℓ}^{1}] and z_{ℓ,n} := [U_{1}^{1} ∩ W_{n} : U_{ℓ}^{1} ∩ W_{n}], where κ denotes the centre of K, and where W_{n} := {x ∈ κ^{×} : x^{n} ∈ (K^{×})′}.

Remark 3.6. Note that if K is commutative, W_{n} is just the group of n–th roots of unity in K, which is finite. Hence z_{ℓ,n} becomes stationary for ℓ ∈ ℕ sufficiently large. Also note that y_{ℓ} = 1 and x_{ℓ} = q^{ℓ−1} in the commutative case. Proof. Ad (1). By Lemma 3.2 (1) and (4), the set of matrices with entries of the form a_{ij} = δ_{ij} + ∑_{k=1}^{ℓ−1} α_{ijk}π^{k}, where α_{ijk} ∈ T, is a transversal of U_{1}^{n}/U_{ℓ}^{n}. But this set has q^{n^{2}(ℓ−1)} elements.

Ad (2). Set Z := Z(GL(n, K)) = κ^{×} 1. Since U_{ℓ}^{n}Z∩U_{1}^{n} = U_{ℓ}^{n}(U_{1}^{1}∩κ^{×})1 and [U_{ℓ}^{n}(U_{1}^{1} ∩κ^{×})1 : U_{ℓ}^{n}] = [(U_{1}^{1} ∩κ^{×})1 : (U_{1}^{1} ∩κ^{×})1∩U_{ℓ}^{n}] = [U_{1}^{1} ∩κ^{×} : U_{ℓ}^{1} ∩κ^{×}], we have [Ũ_{1}^{n} : Ũ_{ℓ}^{n}] = [U_{1}^{n} : U_{ℓ}^{n}Z ∩ U_{1}^{n}] = [U_{1}^{n} : U_{ℓ}^{n}] · [U_{1}^{1} ∩ κ^{×} : U_{ℓ}^{1} ∩ κ^{×}]^{−1} , from which (2) follows.

Ad (3). For n = 1, we compute [V_{1}^{1} : V_{ℓ}^{1}] = [(K^{×})′ ∩ U_{1}^{1} : (K^{×})′ ∩ U_{ℓ}^{1}] = [((K^{×})′ ∩ U_{1}^{1})U_{ℓ}^{1} : U_{ℓ}^{1}], which divides [U_{1}^{1} : U_{ℓ}^{1}] = q^{ℓ−1} . Now assume n ≥ 2. Set J := {1, …, n}^{2}\{(1, 1)}. We consider the set R of matrices A = (a_{ij}) such that a_{ij} = δ_{ij} + ∑_{k=1}^{ℓ−1} α_{ijk}π^{k} for (i, j) ∈ J, where α_{ijk} ∈ T, and where, for fixed (a_{ij})_{(i,j)∈J}, we let a_{11} run through a set of representatives modulo U_{ℓ}^{1} of the possible (1, 1)–entries, A, as in the proof of Lemma 3.3 (where the representatives were denoted by a_{k}). We claim that R is a transversal of V_{1}^{n}/V_{ℓ}^{n}. To see this, let B = (b_{ij}) ∈ V_{1}^{n}, and, for i, j ∈ {1, …, n}, let b_{ij} = δ_{ij} + ∑_{k=1}^{∞} β_{ijk}π^{k} be the expansion of b_{ij}, where β_{ijk} ∈ T. Then there is precisely one A = (a_{ij}) ∈ R such that AV_{ℓ}^{n} = BV_{ℓ}^{n}, since, by Lemma 3.2, this condition is equivalent to a_{ij} = δ_{ij} + ∑_{k=1}^{ℓ−1} β_{ijk} for (i, j) ∈ J, and (a_{11})_{[ℓ]} = (b_{11})_{[ℓ]} . By Lemma 3.4, we can choose a_{11} as required. To obtain (4), note that V_{ℓ}^{n}Z ∩V_{1}^{n} = V_{ℓ}^{n}(W_{n}∩U_{1}^{1})1. Now copy the proof of (2).

Lemma 3.7. Assume that K is commutative. If det_{ℓ} : M(n, R/P_{ℓ}) → R/P_{ℓ} and det: M(n, R) → R denote the determinant mappings, then

det_{ℓ}(A_{[ℓ]}) = (det A)_{[ℓ]},

for any matrix A ∈ M(n, R) and any ℓ ∈ ℕ. In particular, det U_{ℓ}^{n} ⊆1+P_{ℓ} =U_{ℓ}^{1} . Proof. By an easy computation.

Since ψ_{ℓ}^{n} is a morphism of rings by Lemma 3.2, it maps units to units. If K is commutative, Lemma 3.7 implies that A_{[ℓ]} ∈ SL(n, R/P_{ℓ}), for every A ∈ SL(n, R). Hence, for ℓ ∈ ℕ, we can consider

ϕ_{ℓ}^{n} := ψ_{ℓ}^{n}|_{GL(n,R)}^{GL(n,R/P_{ℓ})}
χ_{ℓ}^{n} := ψ_{ℓ}^{n}|_{SL(n,R)}^{SL(n,R/P_{ℓ})}.

Proposition 3.8. Assume that K is commutative. Then, for all ℓ ∈ ℕ, the mappings ϕ_{ℓ}^{n} and χ_{ℓ}^{n} are quotient morphisms of groups, and their kernels are ker ϕ_{ℓ}^{n} = U_{ℓ}^{n} and ker χ_{ℓ}^{n} = V_{ℓ}^{n}. In particular,

[GL(n, R) : U_{1}^{n}] = |GL(n, R/P_{1})| = (q^{n} − 1) · . . . · (q^{n} − q^{n−1})

and, for n ≥ 2,

[SL(n, R) : V_{1}^{n}] = |SL(n, R/P_{1})| = (q^{n} − 1) · . . . · (q^{n} − q^{n−2}) q^{n−1}.

Proof. ϕ_{ℓ}^{n} is onto: For any A ∈ GL(n, R/P_{ℓ}), there are Ã, B̃ ∈ M(n, R) such that ψ_{ℓ}^{n}(Ã) = A and ψ_{ℓ}^{n}(B̃) = A^{−1} , since ψ_{ℓ}^{n} is onto by Lemma 3.2 (3). Lemma 3.7 shows that (det ÃB̃)_{[ℓ]} = det_{ℓ} 1 = 1_{[ℓ]} , from which we conclude that (det Ã)(det B̃) = det ÃB̃ ∈ 1 + P_{ℓ} ⊆ R^{×} is a unit. This implies Ã ∈ GL(n, R), and we have proved that ϕ_{ℓ}^{n} is onto. Now another application of Lemma 3.2 (3) shows ker ϕ_{ℓ}^{n} = (1 + ker ψ_{ℓ}^{n}) ∩ GL(n, R) = U_{ℓ}^{n}. χ_{ℓ}^{n} is onto: Let A ∈ SL(n, R/P_{ℓ}). Then there is some B ∈ GL(n, R) such that ϕ_{ℓ}^{n}(B) = A, as has just been shown. Now (det B)_{[ℓ]} = det_{ℓ} (A) = 1_{[ℓ]} , whence det B ∈ 1 + P_{ℓ} and r := (det B)^{−1} ∈ 1 + P_{ℓ}. Set C := diag(r, 1, …, 1). Then BC ∈ SL(n, R), and we have χ_{ℓ}^{n}(BC) = ϕ_{ℓ}^{n}(BC) = ϕ_{ℓ}^{n}(B)ϕ_{ℓ}^{n}(C) = A, because C ∈ U_{ℓ}^{n} = ker ϕ_{ℓ}^{n}. Since A was arbitrary, χ_{ℓ}^{n} is onto. Finally, one computes ker χ_{ℓ}^{n} = ker ϕ_{ℓ}^{n} ∩ SL(n, R) = U_{ℓ}^{n} ∩ SL(n, R) = V_{ℓ}^{n}. The remainder of the proposition follows easily now: simply note that R/P_{1} = R/P = k is a finite field with q elements, and that the general and special linear groups over these fields have the orders stated above.

4. Computation of Haar Measure

In this section, we give an explicit description of Haar measure on the linear groups over local fields. To this end, the measure of the open compact subgroups introduced in Section 2 is being computed. Lemma 4.1 and Proposition 4.4 show how the Haar measure of any open compact subset can be determined from these data. In fact, the measure of any open subset can be expressed in terms of the values of Haar measure on the local bases, but in a less explicit way. An alternative description of Haar measure on the general linear groups will be given in Section 6.

We recall that if G is a locally compact topological group, a positive measure μ on the σ –algebra of Borel sets of G is called a Haar measure on G, if it is finite on all compact sets, regular, left–invariant, and if there is an open subset U of G such that 0 < μ(U) < ∞. Then, by inner regularity, μ(V) > 0 for any non–empty open subset V of G, since any compact subset of U is covered by finitely many translates of V . It can be shown that on any locally compact group G, there exists a Haar measure μ, which is unique up to a multiplicative positive constant (see Hewitt and Ross [8], Section 15.8). Note that all of the topological groups discussed in this article satisfy the second countability axiom, since K is metric and has a dense countable subset by Theorem 1.2. Now, in any locally compact, second countable group, any open subset is σ –compact. By Rudin [9], Theorem 2.18, any Borel measure on a locally compact space with this property is regular, provided it is finite on compact sets. Hence, as regards the groups we are interested in, regularity of Haar measure is a consequence of the other axioms.

Lemma 4.1. Let G be a locally compact group and μ a Haar measure on G. If U and V are open compact subgroups of G such that V ≤ U, then the index of V in U is finite, and μ(V) = [U : V]^{−1}μ(U).

Proof. Since V is an open subgroup of the compact group U, finitely many cosets of V cover U, that is, the index of V in U is finite. If F is a transversal of U/V , then U = ⋃_{x∈F} xV , where the union is disjoint. Now, by left invariance of Haar measure, μ(U) = ∑_{x∈F} μ(xV) = ∑_{x∈F} μ(V) = [U : V] μ(V).

Theorem 4.2. Let μ_{1} , μ_{2} , μ_{3} , and μ_{4} denote Haar measure on GL(n, K), PGL(n, K), SL(n, K), and PSL(n, K), respectively. Then, with notation as in Lemma 3.5,

(1) μ_{1}(U_{ℓ}^{n}) = q^{−n^{2}(ℓ−1)} μ_{1}(U_{1}^{n})

(2) μ_{2}(Ũ_{ℓ}^{n}) = x_{ℓ} q^{−n^{2}(ℓ−1)} μ_{2}(Ũ_{1}^{n})
(3) μ_{3}(V_{ℓ}^{n}) = y_{ℓ}^{−1}q^{−(n^{2}−1)(ℓ−1)} μ_{3}(V_{1}^{n})
(4) μ_{4}(Ṽ_{ℓ}^{n}) = z_{ℓ,n} y_{ℓ}^{−1}q^{−(n^{2}−1)(ℓ−1)} μ_{4}(Ṽ_{1}^{n}).

If K is commutative and μ_{1} , μ_{3} are chosen such that μ_{1}(GL(n, R)) = 1 and μ_{3}(SL(n, R)) = 1, respectively, then

(5) μ_{1}(U_{ℓ}^{n}) = γ q^{−n^{2}(ℓ−1)} , where γ = |GL(n, 𝔽_{q})|^{−1} ;

(6) μ_{3}(V_{ℓ}^{n}) = δ q^{−(n^{2}−1)(ℓ−1)} , where δ = |SL(n, 𝔽_{q})|^{−1} .

Proof. The assertions follow immediately from Proposition 3.5, Proposition 3.8 and Lemma 4.1.

Remark 4.3. If K is commutative, the image of the measure μ_{3} under the quotient morphism β: SL(n, K) → PSL(n, K) also yields a Haar measure, λ say, on PSL(n, K), defined by λ(Ω) := μ_{3}(β^{−1}(Ω)) for Borel sets Ω of PSL(n, K). Since ker β = K^{×}1 ∩ SL(n, K) is finite, λ inherits the required properties from μ_{3} .

Proposition 4.4. Let G be a totally disconnected, locally compact group satisfying the first countability axiom. Then there is a descending countable local base W_{1} ⊇ W_{2} ⊇ ⋯ of the topology, where W_{i} is an open and compact subgroup of G, for any i ∈ ℕ. If W is any open and compact subset of G, then there exists r ∈ ℕ such that W is the (disjoint) union of finitely many cosets of W_{r}.

Proof. Since G is locally compact and totally disconnected, the open and compact subgroups constitute a local base of the topology. By first countability, a countable subbase can be selected. Replacing its elements by suitable finite intersections, we obtain a local base with the required properties. Now since W is open, there is an i_{x} ∈ ℕ such that xW_{i_{x}} ⊆ W, for any x ∈ W. Due to compactness of W, there is a finite subset F of W such that W = ⋃_{x∈F} xW_{i_{x}}. Then W is a disjoint union of cosets of W_{r}, where r := max{i_{x} : x ∈ F }.

Arbitrary open sets can be decomposed into open and compact ones as well:

Proposition 4.5. Let G and W = {W_{i} : i ∈ ℕ} be as in Proposition 4.4, and assume that G is σ –compact. Then, for any open subset U of G, there are a countable set J and families (g_{j})_{j∈J} ∈ G^{J}, (V_{j})_{j∈J} ∈ W^{J} such that U = ⋃_{j∈J} g_{j}V_{j}, where the union is disjoint. Hence, if μ is a Haar measure on G, we have μ(U) = ∑_{j∈J} μ(V_{j}).

Proof. For any g ∈ U, there is an i_{g} ∈ ℕ such that gW_{i_{g}} ⊆ U, being minimal with respect to this property. The set J := {gW_{i_{g}} : g ∈ U} is countable, since, for any i ∈ ℕ, the index of the open subgroup W_{i} in the σ –compact group G is countable. We claim that U is the disjoint union of the sets V ∈ J. For let h, g ∈ U be given such that hW_{i_{h}} ∩ gW_{i_{g}} ≠ Ø. We may assume that i_{g} ≤ i_{h}. Then hW_{i_{g}} = gW_{i_{g}} ⊆ U, whence i_{h} ≤ i_{g}, by minimality. We conclude i_{h} = i_{g}, hence hW_{i_{h}} = gW_{i_{g}}.

Lemma 4.6. Let G denote one of the groups K^{×}, GL(n, K), SL(n, K), PGL(n, K), or PSL(n, K), where n ≥ 2, let μ be a Haar measure on G, and choose a non–empty open and compact subset U of G. We know that |R/P | = q is a power of some prime p.

(1) There is γ ∈ ℚ, such that for any open and compact subset V of G, there are unique numbers z ∈ ℕ and i ∈ ℕ_{0} such that gcd (p, z) = 1 and μ(V) μ(U)^{−1} = γzp^{−i}.

(2) For any i_{0} ∈ ℕ_{0} , there exists an open and compact subset V of G, such that i ≥ i_{0} , with notations as in (1).

Proof. The assertions follow immediately from Proposition 2.1, Theorem 4.2, and Proposition 4.4.

Corollary 4.7. With notation as in Lemma 4.6, let Γ be the set of all rationals μ(V)μ(U)^{−1} , where V ranges through the open and compact subsets of G. Let p′ be a prime number, and let ν_{p′} denote the p′ –adic valuation on ℚ. Then p = p′ if and only if ν_{p′}(Γ) is not bounded below.

Remark 4.8. The following application demonstrates the usefulness of the ideas presented in this article. For i ∈ {1, 2}, consider a local field K_{i} with valuation ring R_{i} and valuation ideal P_{i}. Then q_{i} := |R_{i}/P_{i}| is a power of some prime p_{i}. Let G_{i} be a general linear group GL(n_{i}, K_{i}), where n_{i} ∈ ℕ, or one of the groups SL(n_{i}, K_{i}), PGL(n_{i}, K_{i}), or PSL(n_{i}, K_{i}), where n_{i} ≥ 2. Assume that θ : G_{1} → G_{2} is a topological isomorphism. We choose a non–empty open and compact subset U_{1} of G_{1} and set U_{2} := θ(U_{1}). For i ∈ {1, 2}, we define Γ_{i} as in Corollary 4.7. Then Γ_{1} = Γ_{2} , since if μ_{1} is a Haar measure on G_{1} , then the image of μ_{1} under θ is a Haar measure on G_{2} . The conclusion of the corollary shows p_{1} = p_{2} .^{3} An interesting special case is stated in Corollary 4.10.

3 For n_{1},n_{2}≥2, this also follows from general investigations on the isomorphy problem of linear groups, cf. Dieudonné [5], which show that the local fields K_{1} and K_{2} are algebraically isomorphic or antiisomorphic. This actually implies q_{1}=q_{2} in the commutative case, see Glöckner [7].

Lemma 4.9. Let F be a commutative field, where char F ≠ 2, and n ∈ ℕ. Then Δ_{2} := { diag(α_{1}, …, α_{n}) : (α_{i})^{2} = 1 } is a maximal elementary abelian 2–subgroup of GL(n, F), and all of these are conjugate. An analogous statement holds for SL(n, F) and its subgroup Δ_{2} ∩ SL(n, F).

Proof. Let H be an elementary abelian 2–subgroup of GL(n, F), and A ∈ H. Then spec (A) ⊆ {1, −1}, since A^{2} = 1. For any x ∈ F^{n}, we have x = (x + Ax)/2 + (x − Ax)/2, where A(x ± Ax) = ±(x ± Ax). Hence F^{n} is the sum of the eigenspaces of A, that is, A is diagonalizable. Note that since H is abelian, the matrices A ∈ H are simultaneously diagonalizable: there is S ∈ SL(n, F) such that SHS^{−1} ≤ Δ_{2} . If H ≤ SL(n, F), then SHS^{−1} ≤ Δ_{2} ∩ SL(n, F). The assertions follow easily from this.

Corollary 4.10. For any n_{1}, n_{2} ∈ ℕ and prime numbers p_{1} and p_{2} , the topological groups GL(n_{1}, ℚ_{p_{1}}) and GL(n_{2}, ℚ_{p_{2}}) are isomorphic if and only if n_{1} = n_{2} and p_{1} = p_{2} . An analogous statement holds for the special linear groups, provided n_{1}, n_{2} ≥ 2. (Here ℚ_{p_{i}} denotes the field of p_{i} –adic numbers).

Proof. Assume that θ : GL(n_{1}, ℚ_{p_{1}}) → GL(n_{2}, ℚ_{p_{2}}) is an isomorphism. By Lemma 4.9, there exists a maximal elementary abelian 2–subgroup H of GL(n_{1}, ℚ_{p_{1}}), of order 2^{n_{1}} . Then θ(H) is a maximal elementary abelian 2– subgroup of GL(n_{2}, ℚ_{p_{2}}), whose order is 2^{n_{2}} . This implies n_{1} = n_{2} . Now since R_{i} = ℤ_{p_{i}} is the valuation ring of ℚ_{p_{i}}, and P_{i} = p_{i}ℤ_{p_{i}} its valuation ideal, we infer |R_{i}/P_{i}| = |𝔽_{p_{i}}| = p_{i}, for i ∈ {1, 2}. Hence p_{1} = p_{2} , by Remark 4.8.

5. Computation of the modular functions

Let G be a locally compact topological group, μ a Haar measure on G and ϕ an (algebraical and topological) automorphism of G. We obtain another Haar measure, ν, on G by defining ν(Ω) := μ(ϕ(Ω)) for Borel sets Ω of G. By uniqueness, there is a positive real number mod_{G}(ϕ), the module of ϕ, such that ν = mod_{G}(ϕ)μ. For g ∈ G, we set mod(g) := mod_{G}(I_{g}), where I_{g} : G → G denotes the inner automorphism x ↦ g^{−1}xg of G. The mapping mod: G → ℝ^{+} is a morphism of topological groups; it is called the modular function of G. If mod ≡ 1, G is called unimodular (cf. Hewitt and Ross [8], Chapter 15).

Theorem 5.1. All of the groups GL(n,K), SL(n,K), PGL(n,K), PSL(n,K) are unimodular.

Proof. Let G denote one of the groups above. Since mod : G → ℝ^{+} is a homomorphism into an abelian group, we conclude that G′ ≤ ker mod, where G′ denotes the derived group.

Now, for n ≥ 2, we have SL(n, K)′ = SL(n, K), hence PSL(n, K)′ = PSL(n, K), which shows that these groups are unimodular. Note that SL(1, K) = (K^{×})′ is a closed subset of R^{×}, hence compact. Also, PSL(1, K) is compact, and we conclude that these groups are unimodular. As regards GL(n, K), note that GL(n, K)′ = SL(n, K) ≤ ker mod. For x ∈ K^{×}, we define A_{x} := diag(x, 1, …, 1). Since SL(n, K) ∪ {A_{x} : x ∈ K^{×}} generates GL(n, K), it suffices to show that mod (A_{x}) = 1, for all x ∈ K^{×}. Hence let x ∈ K^{×} and A := A_{x}, where we may assume |x| = q^{−ℓ} ≤ 1, that is, ℓ ≥ 0 (otherwise compute mod (A_{x}^{−1}) = mod (A_{x^{−1}})). For any B = (b_{ij}) ∈ GL(n, K), we have

A^{−1}BA = ( \begin{matrix}x^{−1}b_{11}x & x^{−1}b_{12} & ⋯ & x^{−1}b_{1n} \\ b_{21}x & b_{22} & ⋯ & b_{2n} \\ ⋮ & ⋮ & & ⋮ \\ b_{n1}x & b_{n2} & ⋯ & b_{nn} \end{matrix} ) ,

from which we infer that A^{−1}U_{ℓ+1}^{n}A is the set of all matrices B = (b_{ij}) such that b_{11} ∈ 1+P_{ℓ+1} , b_{ij} ∈ δ_{ij} +P_{ℓ+1} , b_{i1} ∈ P_{2ℓ+1} , and b_{1j} ∈ P_{1} , where i, j ∈ {2, …, n}. Hence

U_{2ℓ+1}^{n} ≤ A^{−1}U_{ℓ+1}^{n}A ≤ U_{1}^{n} ≤ GL(n, R).

We wish to determine the index [A^{−1}U_{ℓ+1}^{n}A : U_{2ℓ+1}^{n}]. By Lemma 3.2 (1) and (4), this can be achieved by counting the possible choices for the first 2ℓ + 1 coefficients occuring in the power series expansions of the entries b_{ij} of matrices (b_{ij})∈A^{−1}U_{ℓ+1}^{n}A. The first row yields q^{ℓ}(q^{2ℓ})^{n−1} possible choices, the remaining coefficients of the first column allow for only one possible choice, and the remainder of the matrix yields another (q^{ℓ})^{(n−1)^{2}} choices, whence [A^{−1}U_{ℓ+1}^{n}A : U_{2ℓ+1}^{n}] = q^{ℓ+2ℓ(n−1)+ℓ(n−1)^{2}} = q^{ℓn^{2}} holds. Now, if μ denotes Haar measure on G, we obtain μ(A^{−1}U_{ℓ+1}^{n}A) = q^{ℓn^{2}}μ(U_{2ℓ+1}^{n}) = q^{ℓn^{2}}q^{−2ℓn^{2}}μ(U_{1}^{n}) = μ(U_{ℓ+1}^{n}), which implies mod(A) = 1.

Now let λ denote Haar measure on PGL(n, K). We have PGL(n, K)′ = α(GL(n, K)′) = α(SL(n, K)), and by the preceding argument, it suffices to show mod(α(A)) = 1 for A = A_{x} as above. Proceeding as in the proof of Proposition 3.5 (2), we obtain

[α(A)^{−1}Ũ_{ℓ+1}^{n}α(A) : Ũ_{2ℓ+1}^{n}] = [A^{−1}U_{ℓ+1}^{n}A : (U_{2ℓ+1}^{n}Z) ∩ A^{−1}U_{ℓ+1}^{n}A]
= [A^{−1}U_{ℓ+1}^{n}A : U_{2ℓ+1}^{n}(U_{ℓ+1}^{1} ∩ κ^{×})1]

= [A^{−1}U_{ℓ+1}^{n}A : U_{2ℓ+1}^{n}]·[U_{2ℓ+1}^{n}(U_{ℓ+1}^{1}∩ κ^{×})1 : U_{2ℓ+1}^{n}]^{−1}

= [U_{ℓ+1}^{n} : U_{2ℓ+1}^{n}] · [U_{2ℓ+1}^{n}(U_{ℓ+1}^{1} ∩ κ^{×})1 : U_{2ℓ+1}^{n}]^{−1}
= [Ũ_{ℓ+1}^{n} : Ũ_{2ℓ+1}^{n}],

whence λ(α(A)^{−1}Ũ_{ℓ+1}^{n}α(A)) = λ(Ũ_{ℓ+1}^{n}), that is, mod(α(A)) = 1.

6. An alternative description of Haar measure on GL(n, K)

It is well–known that if K is a commutative local field, a Haar measure μ on GL(n, K) is given by dμ = ρ dλ, where ρ(A) := |det(A)|^{−n} for A ∈ GL(n, K), and where λ denotes the restriction of Haar measure on (K^{n}, +) to the Borel sets of GL(n, K), see Bourbaki [3], Chap. VII–3. We wish to drop the hypothesis that K be commutative.

Lemma 6.1. Let K be a local field, commutative or not, |.| = mod_{K} the absolute value on K described in Section 1, and det : GL(r, K) → K̄ the Dieudonné determinant. Since ℝ^{+} is abelian, the restriction of mod_{K} to K^{×} factors through K̄ = K^{×}/(K^{×})′, via a homomorphism f : K̄ → ℝ^{+} . Claim: mod_{K^{r}}(A) = f(det(A)), for any A ∈ GL(r, K).

Proof. Any elementary matrix A is in the commutator subgroup SL(r, K) = GL(r, K)′, hence in the kernel of mod_{K^{r}} : Aut(K^{r}) → ℝ^{+} . This implies mod_{K^{r}}(A) = 1 = f(det(A)). Now if D = diag(1, …, 1, a) and W is an open subset of K of finite positive measure, then mod_{K^{r}}(D) λ(W^{r}) = λ(DW^{r}) = mod_{K}(a) λ(W^{r}), using that Haar measure λ on K^{r} is the r–fold product of Haar measure on K. We infer mod_{K^{r}}(D) = mod_{K}(a) = f(det(D)). Since the matrices above generate GL(r, K), the claim follows.

Theorem 6.2. With notations as in 6.1, define ρ(A) := (f(det A))^{−n} for A ∈ GL(n, K). Let λ′ denote Haar measure on K^{n×n}. Then dμ = ρ dλ is a Haar measure on GL(n, K), where λ denotes the restriction of λ′ to the Borel sets of GL(n, K).

Proof. Since μ(U_{1}^{n}) = λ(U_{1}^{n}) = λ(1 + P_{1}^{n×n}) = λ′(P_{1}^{n×n}) > 0, we only need to check that μ is left invariant. Let V := K^{n×n}. Then

\begin{matrix}l : GL(n, K) & → GL(n, V), \\ C & ↦ (l_{C}: A ↦ CA) \end{matrix}

defines a morphism of topological groups. Note that l_{C} = C ⊕ ⋯ ⊕ C, since the action of l_{C} on each column is multiplication by C. Hence det_{V} (l_{C}) = (det_{K^{n}}(C))^{n}, and Lemma 6.1 shows that mod_{V} (l_{C})=f(det(l_{C}))=(f(det(C)))^{n}. Now let Ω be a Borel set of GL(n, K) and B ∈ GL(n, K). Then

μ(BΩ) = ∫ 1_{BΩ}(A) · (f(det A))^{−n} dλ(A)
= ∫ (1_{Ω} ∘ l_{B^{−1}})(A) · mod_{V}^{−1}(l_{A}) dλ(A)
= ∫ (1_{Ω} ∘ l_{B^{−1}})(A) · mod_{V}^{−1}(l_{BB^{−1}A}) dλ(A)
= ∫ (1_{Ω} · (mod_{V}^{−1} ∘ l ∘ l_{B})) ∘ l_{B^{−1}} dλ
= ∫ 1_{Ω} · (mod_{V}^{−1} ∘ l ∘ l_{B}) dl_{B^{−1}}λ,

by transformation of integrals, cf. Bauer [2], Satz 19.1. Here l_{B^{−1}}λ denotes the image of λ under the mapping l_{B^{−1}} , defined by (l_{B^{−1}}λ)(ω) := λ((l_{B^{−1}})^{−1}(ω)) for Borel sets ω ⊆ GL(n, K), see Bauer loc. cit. Definition 7.6. Now we have λ((l_{B^{−1}})^{−1}(ω)) = λ(l_{B}(ω)) = mod_{V} (l_{B}) λ(ω), hence l_{B^{−1}}λ = mod_{V} (l_{B}) · λ. Since mod_{V} : Aut(V) → ℝ^{+} is a homomorphism, one computes

(mod_{V} ∘ l ∘ l_{B})(A) = mod_{V} (l_{l_{B}(A)}) = mod_{V} (l_{BA})
= mod_{V} (l_{B}l_{A}) = (mod_{V} (l_{B}) · (mod_{V} ∘ l))(A).

With these replacements, μ(BΩ) = ∫ 1_{Ω}·(mod_{V} (l_{B}))^{−1}·(mod_{V}^{−1}∘l)·mod_{V} (l_{B}) dλ = ∫ 1_{Ω} · (mod_{V}^{−1} ∘ l) dλ = μ(Ω). Since Ω was arbitrary, μ is left invariant, and we have proved that μ is a Haar measure on GL(n, K).

Remark 6.3. In particular, Theorem 6.2 shows that a Haar measure on the open compact subgroup GL(n, R) of GL(n, K) can be obtained by restricting the Haar measure on (K^{n×n}, +) to the Borel sets of GL(n, R). Similar phenomena occur in every standard group, see Serre [10] Part II, Chapter IV, Exercise 5. A Haar measure λ on the additive group K^{n×n} can be described explicitely. We consider the open and compact subgroups P_{ℓ}^{n×n} introduced in Section 2, which constitute a local base of the topology. As in the proof of Proposition 3.5, one computes [P_{1}^{n×n}:P_{ℓ}^{n×n}] = q^{n^{2}(ℓ−1)} . This implies λ(P_{ℓ}^{n×n}) = q^{−n^{2}(ℓ−1)}λ(P_{1}^{n×n}).

References

[1]  Artin, E., “Geometric Algebra,” Interscience Publishers, New York, 1957.

[2] Bauer, H., ,,Maß– und Integrationstheorie“, Walter de Gruyter, Berlin, 1992.

[3]  Bourbaki, N., “Intégration, Chap. 7, 8,” Hermann Paris, 1963.
[4]  Cohn, P. M., “Algebra 3,” John Wiley & Sons, New York, 1991.

[5] Dieudonné, J. A., “La Géométrie des Groupes Classiques,” Springer-Verlag, Berlin etc., 1971.

[6]  Draxl, P. K., “Skew fields,” Cambridge University Press, 1983.

[7] Glöckner, H., ,,Zum Isomorphieproblem linearer Gruppen über lokalen Körpern“, Diplomarbeit, TH Darmstadt, 1995.

[8] Hewitt, E., and K. A. Ross, “Abstract Harmonic Analysis I,” Springer-Verlag, Berlin etc., 1979.

[9] Rudin, W., “Real and Complex Analysis,” MacGraw–Hill, New York etc., 1987.

[10] Serre, J. P., “Lie Algebras and Lie Groups,” Springer-Verlag, Berlin etc., 1992.

[11]  Weil, A., “Basic Number Theory,” Springer-Verlag, Berlin etc., 1967.

Helge Glöckner Fachbereich Mathematik, AG 5 Technische Hochschule Darmstadt Schloßgartenstraße 7 D-64289 Darmstadt gloeckner@mathematik.th-darmstadt.de

Received March 4, 1996 and in final form June 18, 1996