The classical Harish-Chandra theorem describes the center of the universal enveloping algebra via Weyl-group-invariant polynomials on the Cartan subalgebra. This theorem admits two natural generalizations: one to invariant differential operators on symmetric spaces and one to Lie superalgebras. In the super-setting, we can also define the anti-center which contains certain square roots of central elements. Its Harish-Chandra image was computed by Gorelik and the construction was generalized to symmetric superspaces by Sherman, linking them to certain invariant "ghost" distributions.

In this talk, we will combine these generalizations and describe the Harish-Chandra theorem for symmetric superspaces as well as for ghost distributions.

Joint work with Siddhartha Sahi, Vera Serganova and Alexander Sherman.