We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, 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 $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:

representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types

MSC Classifications:

20G25, 22E35, 20H25 show english descriptions Linear algebraic groups over local fields and their integers
Analysis on $p$-adic Lie groups
Other matrix groups over rings 20G25 - Linear algebraic groups over local fields and their integers
22E35 - Analysis on $p$-adic Lie groups
20H25 - Other matrix groups over rings

top of page | contact us | social media | privacy | site map