Compactly Supported Distributions on \(p\)-Adic Lie Groups

Let \(K\) be a finite extension of \(\mathbb{Q}_p\) and let \(G\) be a \(p\) -adic Lie group. In this paper, we define the Iwasawa algebra \(K[[G]]\) and prove that it is isomorphic to the convolution algebra of compactly supported distributions on \(G\) . This has important applications in the theory of admissible representations of \(G\) on \(p\) -adic Banach spaces. In particular, we prove the Frobenius reciprocity for continuous principal series representations.