 
Summary: THE REAL LOCI OF CALOGEROMOSER SPACES,
REPRESENTATIONS OF RATIONAL CHEREDNIK ALGEBRAS
AND THE SHAPIRO CONJECTURE
IAIN GORDON, EMIL HOROZOV, AND MILEN YAKIMOV
Abstract. We prove a criterion for the reality of irreducible representations
of the rational Cherednik algebras H0,1(Sn ). This is shown to imply a criterion
for the real loci of the CalogeroMoser spaces Cn in terms of the Etingof
Ginzburg finite maps # : Cn # C n /Sn ×C n /Sn , recovering a result of Mikhin,
Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for
the real locus of the Wilson's adelic Grassmannian of rank one bispectral
solutions of the KP hierarchy. Using Wilson's first parametrisation 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 for Grassmannians is
equivalent to a special case of our result for CalogeroMoser spaces, namely
for the fibres of # over C n /Sn × 0.
1. Introduction
The nth CalogeroMoser space C n is the geometric quotient of
C n = {(X, Z) # gl n (C) ×2  rank([X, Z] + I n ) = 1}
by the action of GL n (C) by simultaneous conjugation. It is a smooth, irreducible,
complex, a#ne variety, [W2]. The space C n is the phase space of the (complex)
