Superanalogs of the Calogero Operators and Jack Polynomials

Abstract A depending on a complex parameter k superanalog SL of Calogero operator is constructed; it is related with the root system of the Lie superalgebra gl(n|m). For m = 0 we obtain the usual Calogero operator; for m = 1 we obtain, up to a change of indeterminates and parameter k the operator constructed by Veselov, Chalykh and Feigin [2, 3]. For k = 1, 1 2 the operator SL is the radial part of the 2nd order Laplace operator for the symmetric superspaces corresponding to pairs (GL(V )×GL(V ), GL(V )) and (GL(V ), OSp(V )), respectively. We will show that for the genericm and n the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of SL; for k = 1, 1 2 they coinside with the spherical functions corresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin’s integral on these superspaces.