A higher-dimensional Chevalley restriction theorem for orthogonal groups

TL;DR

This paper extends the Chevalley restriction theorem to higher dimensions for orthogonal groups, proving conjectures and deriving new algebraic properties with implications for trace identities and Pfaffian multiplicativity.

Contribution

It establishes a higher-dimensional Chevalley restriction theorem for orthogonal groups, confirming a conjecture and exploring characteristic-dependent variants.

Findings

Proved the higher-dimensional Chevalley restriction theorem for orthogonal groups.

In characteristic zero, showed the quotient of a commuting scheme is integral and normal.

Derived trace identities and multiplicative properties of the Pfaffian.

Abstract

We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which was conjectured by Chen and Ng\^{o} for reductive groups. In characteristic $p>2$, we also prove a weaker statement. In characteristic $0$, the theorem implies that the categorical quotient of a commuting scheme by the diagonal adjoint action of the group is integral and normal. As applications, we deduce some trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.