# K-theory formulas for orthogonal and symplectic orbit closures

## TL;DR

This paper develops K-theory formulas for orbit closures under orthogonal and symplectic groups, introducing polynomials that generalize Grothendieck polynomials and connect to Schur Q-functions.

## Contribution

It introduces new polynomials representing K-theory classes of orbit closures, characterizes them uniquely, and provides explicit formulas including Pfaffian and degeneracy locus expressions.

## Key findings

- Polynomials uniquely characterize orbit closure classes.
- Explicit Pfaffian formulas derived for special cases.
- Limit recovers K-theoretic Schur Q-functions.

## Abstract

The complex orthogonal and symplectic groups both act on the complete flag variety with finitely many orbits. We study two families of polynomials introduced by Wyser and Yong representing the $K$-theory classes of the closures of these orbits. Our polynomials are analogous to the Grothendieck polynomials representing $K$-classes of Schubert varieties, and we show that like Grothendieck polynomials, they are uniquely characterized among all polynomials representing the relevant classes by a certain stability property. We show that the same polynomials represent the equivariant $K$-classes of symmetric and skew-symmetric analogues of Knutson and Miller's matrix Schubert varieties. We derive explicit expressions for these polynomials in special cases, including a Pfaffian formula relying on a more general degeneracy locus formula of Anderson. Finally, we show that taking an appropriate limit of our representatives recovers the $K$-theoretic Schur $Q$-functions of Ikeda and Naruse.