Hermitian K-theory of Lagrangian Grassmannians via reducible Gorenstein models

TL;DR

This paper develops a method to compute Hermitian K-theory of Lagrangian Grassmannians using reducible Gorenstein models called generalized Lagrangian flag schemes, revealing a direct sum decomposition related to shifted Young diagrams.

Contribution

It introduces generalized Lagrangian flag schemes as reducible Gorenstein models that facilitate explicit Hermitian K-theory computations for Lagrangian Grassmannians.

Findings

Hermitian K-theory of Lagrangian Grassmannians splits into a direct sum of base's K-theories.

Construction of reducible Gorenstein models enables explicit Hermitian K-theory calculations.

The approach applies to regular schemes with intermediate reducible models.

Abstract

We construct a family of moduli spaces, called generalized Lagrangian flag schemes, that are reducible Gorenstein (hence singular), and that admit well-behaved pushforward and pullback operations in Hermitian $K$-theory. These schemes arise naturally in our computations. Using them, we prove that the Hermitian $K$-theory of a Lagrangian Grassmannian over a regular base splits as a direct sum of copies of the base's (Hermitian) $K$-theory, indexed by certain shifted Young diagrams. The isomorphism is realized via pullback to each generalized Lagrangian flag scheme followed by pushforward to the Lagrangian Grassmannian. This yields an unusual example in which both the base and the target are regular schemes, while the intermediate reducible Gorenstein models remain sufficient to allow explicit computations in Hermitian K-theory of regular schemes.