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Society has its insiders and its outsiders, and in my experience the insiders, those at the high end of the power structures, tend to identify with the interests of other members of different power structures. Let me give an example. I was expelled from university and I am suing my professors. The judges are very quick to take the side of the university, even where the facts show that I have an arguable case. I'm not saying the university has paid them off, or even that they go to the same cocktail parties. I'm just saying theirs a subconscious class identification (not in the Marxist sense), a sense of "us vs them", and the judges aren't going to go against the respectable pillars of society in favor of a renegade troublemaker.

I wonder if people can think of words or expressions which succinctly describe this phenomenon.

In your question you say:

the judges aren't going to go against the respectable pillars of society in favor of a renegade troublemaker

Going by this, (small-c) conservatism ("a tendency to dislike change")¹ seems relevant, if not bang on the money, as does snobbery ("the attitudes or behaviour of someone who thinks they are better than other people");¹ perhaps, if you don't find a satisfactory term, you can make do with something like conservative snobbery.

More specific than conservatism or snobbery, classism ("unfair treatment of people because of their social class")¹ is definitely applicable. Classism doesn't presuppose a Marxist definition of class (or Weberian or any other), nor does it necessitate any conscious class identification - on the part of the judges with the professors, in your example.

classism
(noun, mass noun)

Prejudice against people belonging to a particular social class.

They are told to be on watch against the evils of classism.

The leftist types I hang out with can discuss this movie for hours with themes of sexism, racism, classism running throughout.

More importantly, there still exist many non-financial barriers to post-secondary education including institutionalized classism, racism
and sexism.

Moving from the societal viewpoint of classism to an internal, individual one, bias has been suggested previously in the comments. More specific terms can be drawn from the typology of cognitive biases used in psychology and behavioural economics.

In-group favoritism, sometimes known as in-group–out-group bias, in-group bias, or intergroup bias, is a pattern of favoring members of one's in-group over out-group members. This can be expressed in evaluation of others, in allocation of resources, and in many other ways.

Whether this is applicable to your example depends on what extent the judges and the academics count as members of the same group... although the premise of the question is that they are of the same group.

In this kind of context you also see the term affinity bias used, but this doesn't seem to carry any academic weight or to be as common as in-group bias, for instance. Then again, perhaps you'd prefer the flexibility of a term which isn't weighed down with the baggage of a precise definition.

Authority bias
is the tendency to attribute greater accuracy to the opinion of an authority figure (unrelated to its content) and be more influenced by that opinion. The Milgram experiment in 1961 was the classic experiment that established its existence.

One point to note: you need not be part of an authority to labour under an authority bias - both a shoplifter and a shopkeeper could equally have an authority bias in favour of the police, for instance. So, while authority bias is applicable to your example, the term is wider than authority deferring to authority.

I don't think there is an expression for this, so I'm going to propose the "cronyism of the respectable classes". I think it conveys the idea that people in high positions are not going to allow individuals from the rabble to challenge memebers of their own social class.

Nepotism.

Nepotism describes a society or behavior in which the rulers look after their own interests, rather than those of the people that they are meant to serve.

Definition:

Nepotism is the unfair use of power in order to get jobs or other benefits for your family or friends.
[disapproval]
Many will regard his appointment as the kind of nepotism British banking ought to avoid.
Synonyms: favouritism, bias, patronage, preferential treatment More Synonyms of nepotism

https://www.collinsdictionary.com/dictionary/english/nepotism