After sexual misconduct allegations were leveled against his husband late last year, Senate President Stan Rosenberg reassured his colleagues about a key concern: His husband, Bryon Hefner, had no influence on Senate business. But according to interviews with those who dealt with Hefner, and communications reviewed by the Globe, Hefner was deeply involved in matters that Rosenberg had vowed to bar him from when he ascended to the presidency in 2015. They reveal that Hefner had full access to Rosenberg’s e-mails, attempted to affect the state budget, and involved himself in the workings of his husband’s office, as well as...