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We prove a criterion for the reality of irreducible representations of therational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterionfor the real loci of the Calogero-Moser spaces C_n in terms of theEtingof-Ginzburg finite maps \Upsilon \colon C_n \to C^n/S_n \times C^n/S_n,recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequencewe obtain a criterion for the real locus of the Wilson's adelic Grassmannian ofrank one bispectral solutions of the KP hierarchy. Using Wilson's firstparametrisation of the adelic Grassmannian, we give a new proof of a result of[MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture forGrassmannians is equivalent to a special case of our result for Calogero-Moserspaces, namely for the fibres of \Upsilon over C^n/S_n \times 0.

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture