House Speaker Robert A. DeLeo has launched an investigation into reports that a freshman state representative had inappropriate contact with a female staff member in the House chamber. DeLeo is looking into reports that a court officer found the lawmaker and the staff member near the rostrum, after a late-night debate over the state budget in late April. The court officer was reporting for work in the morning. Both the lawmaker and the staff member were clothed. “The House will conduct an internal review of this matter,’’ said Seth Gitell, DeLeo’s spokesman. DeLeo is also looking into reports that the lawmaker gave the name of another state representative when confronted by the court officer.