Bispectrality of \(AG_2\) Calogero-Moser-Sutherland system (Q2678232)

The aim of this paper is to consider the generalised Calogero-Moser-Sutherland quantum integrable system associated to the configuration of vectors \(AG_2\), which is a union of the root systems \(A_2\) and \(G_2\). The authors establish the existence of and construct a Baker-Akhiezer function for the system, and they show that it satisfies bispectrality. They also find two corresponding dual difference operators of rational Macdonald-Ruijsenaars type in an explicit form. This paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with Baker-Akhiezer functions. The authors recall the notion of a multidimensional Baker-Akhiezer function associated with a configuration following [\textit{O. A. Chalykh} and \textit{A. P. Veselov}, Commun. Math. Phys. 126, No. 3, 597--611 (1990; Zbl 0746.47025); \textit{A. P. Veselov} et al., Theor. Math. Phys. 94, No. 2, 1 (1993; Zbl 0805.47070); translation from Teor. Mat. Fiz. 94, No. 2, 253--275 (1993)], and they generalise it to a case when the root system has proportional vectors. It is shown that if such a function exists then it is an eigenfunction for the generalised Calogero-Moser-Sutherland operator. Section 3 is devoted to generalised Macdonald-Ruijsenaars operators. Here, the authors give a general Ansatz for the dual generalised Macdonald-Ruijsenaars difference operators with rational coefficients, and they find sufficient conditions for these operators to preserve a space of quasi-invariant analytic functions. Section 4 deals with bispectral dual difference operator for \(AG_2\). Here the authors find a difference operator \(\mathcal{D}_1\) related to the configuration \(AG_2\) which satisfies the conditions from Section 3. In Section 5, they use this operator and employ the method from [\textit{O. A. Chalykh}, J. Math. Phys. 41, No. 8, 5139--5167 (2000; Zbl 0987.81038)] to show that the Baker-Akhiezer function for the configuration \(AG_2\) exists, and they express this function by iterated action of the operator. They also show that the Baker-Akhiezer function is an eigenfunction of the operator \(\mathcal{D}_1\), thus establishing bispectral duality. More precisely, the Baker-Akhiezer function will be an eigenfunction for the difference operator from Section 4, which establishes bispectrality of the Calogero-Moser-Sutherland \(AG_2\) Hamiltonian for integer coupling parameters. In Section 6, the authors present another difference operator (dual operator \(\mathcal{D}_2\)) for the configuration \(AG_2\) which preserves the quasi-invariants. They also give the corresponding second construction of the Baker-Akhiezer function, thus establishing the corresponding statements for this operator analogous to the ones from Sections 4 and 5. Section 7 is devotes to a relation with \(A_2\) and \(A_1\) Macdonald-Ruijsenaars Systems. They consider a version of the operator \(\mathcal{D}_1\) from Section 4 for the root system \(A_1\) and decompose it into a sum of two non-symmetric commuting difference operators. They relate these operators with the standard Macdonald-Ruijsenaars operator for the ``minuscule weight'' of the root system \(A_1\).