Exercise 4, § 3

Integration, VII, p. 91

Let K be a nondiscrete commutative locally compact field. Let A be an algebra of finite rank over K.

a) Let $T(n, A)$ be the algebra of matrices $(x_{ij}) \in \mathbf{M}_n(A)$ such that $x_{ij} = 0$ for $i > j$ . Show that if $M = (m_{ij}) \in T(n, A)$ , then

N_{T(n, A)/K}(M) = \prod_{i=1}^n N_{A/K}(m_{ii}^{i-n+1}).

(Use Lemma 6.)

b) Let $T(n, A)^*$ be the group of invertible elements of $T(n, A)$ . Let $M = (m_{ij}) \in T(n, A)$ . Show that $M \in T(n, A)^*$ if and only if $m_{ii}$ is invertible in A for all $i$ .

c) Show that a left Haar measure on $T(n, A)^*$ is given by

\left( \prod_{i=1}^n \mathrm{mod}\, N_{A/K}(m_{ii})^{i-n-1} \right) \cdot \bigotimes_{i \leq j} d\alpha(m_{ij}),

where $\alpha$ denotes a Haar measure of the additive group of A.

d) Show that if $N_{A/K} = N_{A^0/K}$ , then

\Delta_{T(n,A)^*}((m_{ij})) = \prod_{i=1}^n \operatorname{mod} N_{A/K}(m_{ii})^{2i-n-1}.

Solution

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