Projective modules in the category \(\mathcal O\) for the Cherednik algebra. (Q1399143)

For any finite complex reflection group acting on a finite dimensional complex vector space one can define the rational Cherednik algebra \({\mathbf H}_c\), which can be viewed as a deformation of the double affine Hecke algebra introduced by I. Cherednik. In this paper the author considers the category \({\mathcal O}_c\) of \({\mathbf H}_c\)-modules. Using a criterion given by \textit{A. Beilinson}, \textit{V. Ginzburg} and \textit{W. Soergel} [J. Am. Math. Soc. 9, No. 2, 473--527 (1996; Zbl 0864.17006)], he proves that this category has enough projective modules. The second main result of the paper is that \({\mathcal O}_c\) is a highest weight category and hence it satisfies a BGG-type reciprocity formula. An upper bound on the cohomological dimension of \({\mathcal O}_c\) is also given. Reviewer's remark: On p.~215, line \(-11\) of the paper, ``Theorem 4'' actually means Theorem 3.