Generating functions for the universal Hall-Littlewood $P$- and $Q$-functions

Abstract

Recently, P. Pragacz described the ordinary Hall-Littlewood $P$-polynomials by means of push-forwards (Gysin maps) from flag bundles in the ordinary cohomology theory. Together with L. Darondeau, he also gave push-forward formulas (Gysin formulas) for all flag bundles of types $A$, $B$, $C$ and $D$ in the ordinary cohomology theory. In this paper, we introduce a generalization of the ordinary Hall-Littlewood $P$- and $Q$-polynomials, which we call the {\it universal $($factorial$)$ Hall-Littlewood $P$- and $Q$-functions}, and characterize them in terms of Gysin maps from flag bundles in the complex cobordism theory. We also generalize the (type $A$) push-forward formula due to Darondeau-Pragacz to the complex cobordism theory. As an application of our Gysin formulas in complex cobordism, we give generating functions for the universal Hall-Littlewood $P$- and $Q$-functions and their factorial analogues. Using our generating functions, classical determinantal and Pfaffian formulas for Schur $S$- and $Q$-polynomials, and their $K$-theoretic or factorial analogues can be obtained in a simple and unified manner.Comment: 46 pages, AMSLaTeX; Section 6 added, An error of the generating function for the universal factorial Hall-Littlewood $P$-functions was correcte