Superanalogs of the Calogero Operators and Jack Polynomials
Tóm tắt
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 inde-terminates and parameter k the operator constructed by Veselov, Chalykh and Fei-gin [2, 3]. For
$$k = 1,{1 \over 2}$$
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 generic m and n the superanalogs of the Jack polynomials constructed by Kerov, Okunkov and Olshanskii [5] are eigenfunctions of Sℒ for
$$k = 1,{1 \over 2}$$
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.
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