Jack-Laurent symmetric functions

Dunkl Operators at Infinity and Calogero–Moser Systems

“…Finally it is interesting to understand the precise relation of our formulae for quantum CMS integrals at infinity in trigonometric type A case with the results of the recent paper [15] by Nazarov and Sklyanin, whose main tool was the quantum Lax operator for the periodic Benjamin-Ono equation, which they have introduced. 2 We believe that their integrals (which do not depend on p 0 ) are simply related to the stable integrals H (r) k from our recent paper [21], which were constructed using the infinite-dimensional version of Polychronakos operator (rather than Dunkl-Heckman operator used in the present paper). Note that the relation between H (r) k and our quantum CMS integrals (34) is non-trivial (see the formulae in section 5 of [21]).…”

“…2 We believe that their integrals (which do not depend on p 0 ) are simply related to the stable integrals H (r) k from our recent paper [21], which were constructed using the infinite-dimensional version of Polychronakos operator (rather than Dunkl-Heckman operator used in the present paper). Note that the relation between H (r) k and our quantum CMS integrals (34) is non-trivial (see the formulae in section 5 of [21]).…”

“…In this section we follow mainly to our paper [21], where a more general Laurent case is considered. Since it is largely parallel to the rational case we will omit most of the proofs.…”

Symmetric Lie superalgebras and deformed quantum Calogero–Moser problems

“…From the results of [33] it follows that L p generate the same algebra D n,m as commuting CMS integrals from [30], which gives another proof of their commutativity.…”

“…The type AI/AII we have considered in that sense is different since the corresponding deformed root system is of type A(n − 1, m − 1). To describe the zonal spherical functions in this case we can use the theory of Jack-Laurent symmetric functions developed in [33,34].…”