The Hilbert series of the irreducible quotient of the polynomial representation of the rational Cherednik algebra of type An−1 in characteristic p for p|n−1.

In this paper, we study the irreducible quotient ℒ t , c of the polynomial representation of the rational Cherednik algebra ℋ t , c (S n ,) of type A n − 1 over an algebraically closed field of positive characteristic p where p | n − 1. In the t = 0 case, for all c ≠ 0 we give a complete description of the polynomials in the maximal proper graded submodule ker ℬ , the kernel of the contravariant form ℬ , and subsequently find the Hilbert series of the irreducible quotient ℒ 0 , c . In the t = 1 case, we give a complete description of the polynomials in ker ℬ when the characteristic p = 2 and c is transcendental over 2 , and compute the Hilbert series of the irreducible quotient ℒ 1 , c . In doing so, we prove a conjecture due to Etingof and Rains completely for p = 2 , and also for any t = 0 and n ≡ 1 (mod p). Furthermore, for t = 1 , we prove a simple criterion to determine whether a given polynomial f lies in ker ℬ for all n = k p + r with r and p fixed. [ABSTRACT FROM AUTHOR]