The GRE vocabulary is infrequent to meet in academic life. What is the purpose of learning them? My question stems from this paper: An analysis on the validity of the lexicon required by GRE® test takers.

According to the paper, "the great majority of these words are too infrequent to be deemed as useful tools for graduate life". Only 15 out of 302 analyzed words (about 5%) found to be frequent in academic corpus.

So why do they exist in the Graduate Record Examination?

My experience is that those words are especially helpful when you read articles from reputed newspapers and magazines, where the demand of complex language is high. Sometimes I do meet a GRE word that I've learned in a scientific paper, and thanks for my studying in GRE I don't have to look up the dictionary. I do use those words to show off in my TOEFL test, and I have a high score in writing and reading skills. In short, I am wholeheartedly advocate to learn advanced vocabulary, for the sake of getting fluent in the language. However, I don't understand why the GRE requires the test-takers to learn words which are too infrequent for their academic jobs.
I think it helps a lot here to understand the American cultural context, and in particular the liberal arts model of education that is quite common in the US.

The GRE, and the departments and universities that care most about them, thinks of graduate training as pointing towards a career as a professor, not as a researcher.  In the US, the majority of jobs as professors have teaching (at the university level) as a valued component, and in the US, students at university are supposed to take a broad range of courses rather than courses focused on a single area of study.  This means that professors are also teaching an audience that isn't solely focused on their subject but rather students with a wide range of interests (at least in the ideal case - in non-ideal cases, they may be teaching students with no interests).

Even though I'm a mathematician and teach only mathematics courses, I think I am a better teacher (and academic advisor) because I know a fair amount about philosophy, religion, history, literature, economics, sociology, computer science, and so on.  I'm not an expert in any of these things, and I make this clear, but I can make analogies to a broad range of ideas and use them in conversation with students.

For an American, the vocabulary section of the GRE is meant to test, not whether you can memorize the meaning of a large number of words for the GRE, but rather whether you are a widely read person who has already encountered many of these words in the course of their life and studies.

Certainly the verbal GRE scores are generally taken into account much less seriously (or not at all) when making graduate admissions decisions for non-American students, particularly when the student has no plans to remain in the US after their studies.