Example 1

Tag 09H4, permanent. Cite it as [Bourbaki, Tag 09H4].

Integration, VII, p. 60, § 3. Applications and examples

General linear group.

Let $A$ be the algebra $M_n(K)$ . The group $A^*$ of invertible elements of $A$ is none other than the general linear group $\mathbf{GL}(n, K)$ . For every $X \in A$ , the reduced norm $\operatorname{Nrd}_{A/K}(X)$ is $\det X$ ; consequently $N_{A/K}(X) = (\det X)^n$ (Alg., Ch. VIII, §12, No. 3, Prop. 8; cf. A, III, §9, No. 3, Example 3). Since $X \mapsto {}^t X$ is an isomorphism of $A$ onto the opposite algebra,

N_{A^0/K}(X) = N_{A/K}({}^t X) = \det ({}^t X)^n = (\det X)^n.

Then, Prop. 16 of §1, No. 11 proves that the measure

\operatorname{mod}(\det X)^{-n} \cdot \bigotimes_{i,j} dx_{ij} \quad (X = (x_{ij}))

is a left and right Haar measure on $\mathbf{GL}(n, K)$ .

To determine the relatively invariant measures on $\mathbf{GL}(n, K)$ , we shall rely on the following lemma:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

Prop. 16 of §1, No. 11