The Bloch-Okounkov correlation functions, a classical half-integral case

Bloch and Okounkovs correlation function on the infinite wedge space has connections to Gromov-Witten theory, Hilbert schemes, symmetric groups, and certain character functions of $hgl_infty$-modules of level one. Recent works have calculated these character functions for higher levels for $hgl_infty$ and its Lie subalgebras of classical type. Here we obtain these functions for the subalgebra of type $D$ of half-integral levels and as a byproduct, obtain $q$-dimension formulas for integral modules of type $D$ at half-integral level.

The Bloch-Okounkov correlation functions of negative levels

Bloch and Okounkov introduced an $n$-point correlation function on the fermionic Fock space and found a closed formula in terms of theta functions. This function affords several distinguished interpretations and in particular can be formulated as correlation functions on irreducible $hat{gl}_infty$-modules of level one. These correlation functions have been generalized for irreducible integrable modules of $hat{gl}_infty$ and its classical Lie subalgebras of positive levels by the authors. In this paper we extend further these results and compute the correlation functions as well as the $q$-dimensions for modules of $hat{gl}_infty$ and its classical subalgebras at negative levels.