Symplectic determinant laws and invariant theory

Abstract

We introduce the notion of $\textit{symplectic determinant laws}$ by analogy with Chenevier's definition of determinant laws. Symplectic determinant laws are a way to define pseudorepresentations for symplectic representations of algebras with involution over arbitrary $\mathbb{Z}[\frac{1}{2}]$-algebras. We prove that this notion satisfies the properties expected from a good theory of pseudorepresentations, and we compare it to Lafforgue's $\text{Sp}_{2d}$-pseudocharacters. In the process, we compute generators of the invariant algebras $A[M_d^m]^{G}$ and $A[G^m]^G$ over an arbitrary commutative ring $A$ when $G \in \{\text{Sp}_d, \mathrm O_d, \text{GSp}_d, \text{GO}_d\}$, generalizing results of Zubkov.