On some subspaces in representations of degree two matrix groups with coefficients from a locally compact disconnected field

The commutativity of certain Hecke algebras of subgroups of $\operatorname{PL}(2,F)$ containing the commutator subgroup is established (here $F$ is a discrete normed field with finite field of residues). Application of these results to representations of $p$-adic groups by automorphic forms on the upper complex halfplane allows one to distinguish subrepresentations belonging to the discrete series.
Bibliography: 8 titles.