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It is shown by Berest-Etingof-Ginzburg that there exist finite-dimensional irreducible representations of rational Cherednik algebra $H_c(S_n)$ of $A_{n-1}$ type if and only if the deformation parameter $c$ takes the rational numbers of the form $c=m/n$.

Since the rational Cherednik algebra is the rational degeneration of double affine Hecke algebra (DAHA), the question is as follows:

Question: Are there the corresponding finite-dimensional representations of DAHA? Or is any finite-dimensional representation of DAHA known?

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I believe that you should consult the book "Double affine Hecke algebras" by Cherednik, more specifically Section 3.7, more specifically Theorem 3.7.2.

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