Another reference that goes along the lines of what Professor Humphreys mentioned in his answer is a paper by Bill Casselman "the restriction of a representation of $GL_2(k)$ to $GL_2(O)$"Math. Ann. 206, 311- 318 (1973). Also with respect Kutzko's results, mentioned in previous answers, even though they were not published one can find them in his Ph.D thesis at Madison. On convenient feature about Kutzko's thesis is that it contains at the end very explicit character tables for the groups in question. If you are interested I have a scan copy of such tables.

Another reference that goes along the lines of what Professor Humphreys mentioned in his answer is a paper by Bill Casselman "the restriction of a representation of $\operatorname{GL}_2(k)$ to $\operatorname{GL}_2(O)$"Math. Ann. 206, 311- 318 (1973). Also with respect Kutzko's results, mentioned in previous answers, even though they were not published one can find them in his Ph.D thesis at Madison. On convenient feature about Kutzko's thesis is that it contains at the end very explicit character tables for the groups in question. If you are interested I have a scan copy of such tables.