Explicit description of irreducible \(\mathrm{GL}_2(\mathbb{Q}_p)\)-representations over \(\bar {\mathbf F}_{p}\)
In this paper, a description of irreducible mod \(p\) representations of \(\mathrm{GL}_2(\mathbb{Q}_p)\) using their socle filtration as \(\mathrm{GL}_2(\mathbb{Z}_p)\mathbb{Q}_p^\times\)-representations is given. This is done through the study of certain functions on the Bruhat-Tits tree. This allows one to describe the elements fixed by the congruence subgroups of \(\mathrm{GL}_2(\mathbb{Z}_p)\) which has been pursued in a different paper of the author [Invariant elements under some congruence subgroups for irreducible \(\mathrm{GL}_2(\mathbb{Q}_p)\)-representations over \(\overline{\mathbb{F}_p}\), preprint (2009)]. The techniques of the present paper generalize to unramified extensions of \(\mathbb{Q}_p\) giving the Iwahori structure for the canonical Hecke operators in terms of euclidean structures, see [the author, J. Algebra 339, No. 1, 252--303 (2011; Zbl 1239.22012)].
scientific article; zbMATH DE number 6009306
scientific article