Imperial Mandate

The Imperial system of government operates on two parallel chains of command. Both chains originate with the Emperor and pass down through many layers of authority to the billions of ordinary sophonts that do the everyday work of running the government. One chain is the Imperial bureaucracy, which is run through the various civilian and military ministries. The other chain of command runs through the high nobility.
These paired chains of command allow the Imperial government to be robust and flexible. The bureaucracy works according to established procedure and precedent and keeps the government working during normal times. When there is an emergency that the bureaucracy can't manage according to their policies, the high nobles have the authority bring Imperial resources to bear and resolve the crisis.
Briefly, the bureaucracy and military manage the government, but it is the nobility that controls it in the name of the Emperor. The legal authority to exert this control is the Imperial Mandate.
Every high noble wields the Imperial Mandate over specific Imperial territories or activities. This realm where they can wield the Mandate is called their demesne.
Demesnes are carefully arranged so that nobles of the same rank do not have demenes that overlap. If there is an overlap and a dispute arises, a higher noble must intervene to resolve the dispute.
At the same time, every noble has a demesne that falls under the demesne of a high noble of higher rank, and is under the higher noble's authority. This chain leads all the way up to the Emperor.
A baron is the lowest ranking noble, and their demesne in a single star system, within which they wield the Mandate and have a fief. In most cases, there are several barons in a system, or on a single world. Their demesnes are carefully proscribed to avoid overlap and conflict.
A marquis will have a demesne over the whole star system, and any barons in the system answer to the marquis' authority. This will be repeated with viscounts and counts and so on up to the Emperor, who can wield the Imperial Mandate to override the any policies or decisions of any other noble.
The offices and ministries of the Imperial government are typically organized at the subsector level instead of by individual star systems. Subsector capitals will haev offices of various Imperial ministries as well as headquarters facilities for the Imperial military. Generally this means that they are within the demesne of a subsector duke. Higher level headquarters and ministerial offices are organized at the sector and domain level, being commanded by sector dukes and archduke. The top levels of the Imperial ministries and military are on Capital and are within the demesne of the Emperor.
Every Imperial world has some members of the civil service assigned to them. Officials of the Starport Authority will run the Imperial Starport, and employees of the Ministry of the Exchequer will collect Imperial taxes and fees due the Imperium. Important worlds will also have Imperial courts, along with the judges, baliffs, administrators and members of the Imperial Bar. Depending on the history of the world entering the Imperium, there will also be other offices or facilities that are under direct Imperial control, like a terraforming project or power generation and distribution networks.
All of these local facilities and personnel are overseen by at least one noble wielding the Imperial Mandate who can give them legal orders alogside their superiors in the bureaucracy.
High nobles are also responsible for ensuring that Imperial law is enforced within their demesne, able to investigate possible violations, arrest suspects, and bring them to the courts for prosecution and judgement. A high noble can use Imperial resources to help him in this work, as well as using his personal huscarles.
Most of this law enforcement has to do interstellar trade and commerce, and most citizens will not interact with a noble in this work. The exceptions are the Imperial laws that affect every citizen, primarily those against slavery, the prohibitions on psionic powers, and the Imperial Rules of War. High nobles are not only able but required to enforce these laws.
When a high noble is acting legally in the use of his authority no one of lesser rank can interfere; doing so is a serious violation of Imperial law. Assaulting a high noble while they are "on duty" is considered an act of treason against the Imperium.
However, a high noble's authority has its limits. The Mandate must be wielded within Imperial law, and any noble above them in the noble hierarchy can countermand orders and change policies. This negation of a subordinate's use of the Imperial Mandate is not usually exercised since the whole point of the lower noble having the Mandate is to decisions made as low in the chain of command as possible. But it is still done, and any citizen who disagrees with the actions of a high noble can always appeal to a superior noble.
Only the Emperor, archdukes and dukes are allowed to issue an Imperial Warrant, which extends to someone the authority to wield the noble's Imperial Mandate on their behalf.