Fights over which agencies will be involved in the investigation into the fatal shooting of a woman by a federal immigration officer in Minneapolis are roiling as calls for a thorough investigation and accountability mount. Experts say the conflict Thursday between Minnesota officials and federal authorities exemplifies the complicated question of whether a federal agent performing a federally sanctioned operation can be criminally charged by a state. Federal authorities have blocked the participation of Minnesota officials in the investigation. In the past year as they have deployed a growing number of federal agents into cities across the country, many for immigration crackdowns, federal officials have said agents are immune from state prosecution while performing those duties.