Abstract:
Pairs of $n\times n$ matrices whose commutator differ from the identity by a matrix of rank $r$ are used to construct bispectral differential operators with $r\times r$ matrix coefficients satisfying the Lax equations of the Matrix KP hierarchy. Moreover, the bispectral involution on these operators has dynamical significance for the spin Calogero particles system whose phase space such pairs represent. In the case $r=1$, this reproduces well-known results of Wilson and others from the 1990's relating (spinless) Calogero-Moser systems to the bispectrality of (scalar) differential operators. This new class of pairs $(L, \Lambda)$ of bispectral matrix differential operators is different than those previously studied in that $L$ acts from the left, but $\Lambda$ from the right on a common $r\times r$ eigenmatrix.

Abstract:
Explicit formulae are given for the Airy and Bessel bispectral involutions, in terms of Calogero-Moser pairs. Hamiltonian structure of the motion of the poles of the operators is discussed.

Abstract:
The space of solutions of the rational Calogero-Moser hierarchy, and the space of solutions of the KP hierarchy whose tau functions are monic polynomials in $t_1$ with coefficients depending on $t_n$, $n > 1$, are identified, generalizing earlier results of Airault-McKean-Moser and Krichever.

Abstract:
Trigonometric non-isospectral flows are defined for KP hierarchy. It is demonstrated that symmetry constraints of KP hierarchy associated with these flows give rise to trigonometric Calogero-Moser system.

Abstract:
The complete solutions of the spin generalization of the elliptic Calogero Moser systems are constructed. They are expressed in terms of Riemann theta-functions. The analoguous constructions for the trigonometric and rational cases are also presented.

Abstract:
We introduce an integrable time-discretized version of the classical Calogero-Moser model, which goes to the original model in a continuum limit. This discrete model is obtained from pole solutions of a semi-discretized version of the Kadomtsev-Petviashvili equation, leading to a finite-dimensional symplectic mapping. Lax pair, symplectic structure and sufficient set of invariants of the discrete Calogero-Moser model are constructed for both the rational and elliptic cases. The classical $r$-matrix is the same as for the continuum model. An exact solution of the initial value problem is given for the rational discrete-time Calogero-Moser model. The pole-expansion and elliptic solutions of the fully discretized Kadomtsev-Petviashvili equation are also discussed.

Abstract:
Analytic-bilinear approach is used to study continuous and discrete non-isospectral symmetries of the generalized KP hierarchy. It is shown that M\"obius symmetry transformation for the singular manifold equation leads to continuous or discrete non-isospectral symmetry of the basic (scalar or multicomponent KP) hierarchy connected with binary B\"acklund transformation. A more general class of multicomponent M\"obius-type symmetries is studied. It is demonstrated that symmetry constraints of KP hierarchy defined using multicomponent M\"obius-type symmetries give rise to Calogero-Moser system.

Abstract:
We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165].

Abstract:
We show that the trigonometric solitons of the KP hierarchy enjoy a differential-difference bispectral property, which becomes transparent when translated on two suitable spaces of pairs of matrices satisfying certain rank one conditions. The result can be seen as a non-self-dual illustration of Wilson's fundamental idea [Invent. Math. 133 (1998), 1-41] for understanding the (self-dual) bispectral property of the rational solutions of the KP hierarchy. It also gives a bispectral interpretation of a (dynamical) duality between the hyperbolic Calogero-Moser system and the rational Ruijsenaars-Schneider system, which was first observed by Ruijsenaars [Comm. Math. Phys. 115 (1988), 127-165].

Abstract:
We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-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 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 Calogero-Moser spaces, namely for the fibres of \Upsilon over C^n/S_n \times 0.