A triangulation of 𝐺𝐿(𝑛,𝐹)

Abstract

Let F F be a non-Archimedian field. We prove that each open and compact subset of G L n ( F ) \mathrm {GL}_n(F) can be decomposed into finitely many open, compact, and self-conjugate subsets. As a corollary, we obtain a short, elementary proof of a well-known theorem of I.M. Gelfand and D.A. Kazhdan.