Published online by Cambridge University Press: 20 November 2018
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL\left( 2,\,k \right)$, for
$k$ a
$p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of
$SL\left( 2,\,k \right)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.