Sₙ-equivariant L-classes of Cartesian powers of Witt spaces and the Hodge index conjecture for symmetric products

For a compact oriented even-dimensional Witt pseudomanifold X, we compute the Sₙ-equivariant Banagl--Zagier L-classes of Xⁿ in terms of the Goresky--MacPherson L-classes of X and the intersection cohomology Euler characteristic of X. This generalizes a theorem of Zagier for manifolds. Our approach is based on analyzing the equivariant structure of the signature operator of Xⁿ using equivariant $KK$-theory and power operations in analytic K-homology. We establish an analytic K-homology analogue of Moonen's algebraic localization theorem by proving that the localized Chern character of the n-th power operation, evaluated at an n-cycle, agrees with the n-th homological Adams operation applied to the Chern character. As an application, we prove that the characteristic class Hodge index conjecture for singular varieties, formulated by Brasselet--Schürmann--Yokura, is preserved under taking symmetric products of complex projective varieties.