A sample sentence would be:
We can't ban him if there is no genuine reason.
Is there a single word to replace "[if / where] there is no genuine reason" with? Indiscriminately is not an option for me.
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Unjustifiably comes to mind: a reason is needed to justify an action. Or possibly perversely. Or unreasonably.
unjustifiable : Not able to be shown to be right or reasonable
perverse : Showing a deliberate and obstinate desire to behave in a way that is unreasonable or unacceptable
unreasonable : Not guided by or based on good sense
Oxford Dictionaries Online
However, without providing a reason doesn't mean that there isn't one and the ban cannot be justified.
Your action in banning me was arbitrary and capricious, you'll be hearing from my solicitors shortly.
Consider gratuitously and unwarrantedly for "without any reason," and "unexplainedly" for without providing a reason.
gratuitous: being without apparent reason, cause, or justification.
unwarranted: without a basis for reason or fact; unjustified.
unexplainedly: without explanation.
Sources: Random House Dictionary, The Free Dictionary, Wiktionary
I think "arbitrarily" fits.
I think that FumbleFingers asks a very cogent question in his comment beneath the original question:
How can it possibly make sense to ask Why...? about an action which is specifically identified as not having a reason?
The answer, I think, is that the author isn't saying to the person or entity whom he is addressing, "Please explain why you did this inexplicable thing" (which would, as FumbleFingers observes, be highly illogical), but rather "Please explain why you did this unexplained thing" (which simply asks for information not heretofore provided).
Unfortunately, most of the answers given here (including the one that the OP accepted and the one that received the most votes) suggest modifiers that mean "without any good, fair, or rational reason," not "without providing a reason." It seems to me that the suggestion that most accurately expresses the idea of "without providing a reason" is Elian's unexplainedly.
Unfortunately, as Mari-Lou A says in a comment to that answer, unexplainedly is not an especially widely used word in everyday English, though it has appeared in print since at least 1811, when Laetitia Hawkins used it in The Countess and Gertrude, volume 2:
We could point out many of these insular situations, where every one knows, guesses, or fabricates, what is done in their neighbors' homes, where nothing can occur unexpectedly and unexplainedly, without, like a comet, carrying an inflammable train.
Overall, a Google Books search of the years from 1914 through 2008 finds more than 450 matches for unexplainedly, including at least one match from every publication year since 1945. For example, from Earl Rovit and Gerry Brenner, "Of Tyros and Tutors" in Harold Bloom, ed., Ernest Hemingway (2005):
He [Nick Adams] also undergoes a puppy-love affair with a “nice” girl, which he is tremulously strong enough to break off. Unexplainedly “on the road,” he comes into contact with sentimental whores, sinister homosexuals, and a vaudeville team of professional assassins.
Nevertheless, though unexplainedly is a useful, precise, and relevant word, many people are not familiar with it. Another option is peremptorily, used in much the same sense that the adjective peremptory is in the phrase "peremptory challenge" (which refers to rejecting a potential juror from a trial jury "without assigning a cause" as Merriam-Webster puts it. But peremptorily is another rather flashy word, and that distracting quality may not suit the OP's situation (it also has connotations of imperiousness that may be unwelcome).
Ultimately, if you want to know why you were banned from some activity or location, the crucial question isn't "Why did the message notifying me of my banning not include an explanation?" but "Why was I banned?" If you want to ask both questions, I recommend putting the one about the lack of explanation second, and dispensing with any adverbial synonym for "without giving a reason." For example:
Why was I banned, and why didn't you tell me the reason when you imposed the ban?
But keep in mind that the more questions you ask, the less likely you are to get a satisfactory answer to any of them.
Try the word inexplicable.
Cambridge Dictionary defines it as follows:
unable to be explained or understood:
For some inexplicable reason, he's decided to cancel the project.