1.2 Example: \(\mathrm{GL}_n(K)\) and Its Closed Subgroups

Let \(K\) be a non-archimedean local field. We use \(M_n(K)\) to denote the ring of square matrices of size \(n\) with entries in \(K\), and \(\mathrm{GL}_n(K)\) to denote the general linear group.

Lemma 15

✓

The local field \(K\) is T2, locally compact, and totally disconnected.

Proof ▶

Lemma 16 Topological Properties of Matrices

✓

For any \(n \in \mathbb {N}\), the space \(M_n(K)\) is totally disconnected, locally compact, and Hausdorff.

Proof ▶

This follows from the finite product topology of \(K\), which possesses these properties.

Lemma 17 Topological Properties of \(\mathrm{GL}_n(K)\)

✓

For any \(n \in \mathbb {N}\), the group \(\mathrm{GL}_n(K)\) is a totally disconnected, locally compact, Hausdorff topological group.

Proof ▶

The group \(\mathrm{GL}_n(K)\) is an open subset of \(M_n(K)\) (as the preimage of \(K^\times \) under the continuous determinant map), so it inherits local compactness and total disconnectedness. It is Hausdorff as a subspace of a Hausdorff space.

Theorem 18

✓

The group \(\mathrm{GL}_n(K)\) has a compact open subgroup basis.

Proof ▶

Since \(\mathrm{GL}_n(K)\) is a td group (Lemma 17), the result follows directly from the forward direction of Van Dantzig’s theorem (Theorem 10).

Lemma 19

Let \(G\) be a td group (totally disconnected, locally compact, Hausdorff). Any closed subgroup of \(G\) is also a td group.

Proof ▶

Closed subgroups of locally compact groups are locally compact. Subspaces of totally disconnected Hausdorff spaces are totally disconnected and Hausdorff.