Superanalogs of the Calogero Operators and Jack Polynomials

DOI

10.2991/jnmp.2001.8.1.7How to use a DOI?

Abstract

A depending on a complex parameter k superanalog SL of Calogero operator is costructed; 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 indterminates and parameter k the operator constructed by Veselov, Chalykh and Fegin [2, 3]. For k = 1, 1 2 the operator SL is the radial part of the 2nd order Laplace oerator 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 speranalogs 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 coresponding to the above mentioned symmetric superspaces. We also study the inner product induced by Berezin's integral on these superspaces.

Copyright

© 2006, the Authors. Published by Atlantis Press.

Open Access

This is an open access article distributed under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

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TY  - JOUR
AU  - A. Sergeev
PY  - 2001
DA  - 2001/02/01
TI  - Superanalogs of the Calogero Operators and Jack Polynomials
JO  - Journal of Nonlinear Mathematical Physics
SP  - 59
EP  - 64
VL  - 8
IS  - 1
SN  - 1776-0852
UR  - https://doi.org/10.2991/jnmp.2001.8.1.7
DO  - 10.2991/jnmp.2001.8.1.7
ID  - Sergeev2001
ER  -