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SYMPLECTIC DETERMINANT LAWS AND INVARIANT THEORY
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 25 / Issue 5 / September 2026
- Published online by Cambridge University Press:
- 31 July 2026, pp. 2853-2918
- Print publication:
- September 2026
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- Article
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We introduce the notion of symplectic determinant laws in analogy to Chenevier’s definition of determinant laws. Symplectic determinant laws are a way to define pseudorepresentations for symplectic representations of algebras with involution over arbitrary $\mathbb Z[\tfrac {1}{2}]$
-algebras. We prove that this notion satisfies the properties expected from a good theory of pseudorepresentations, and we compare it to V. Lafforgue’s $\operatorname {Sp}_{2d}$
-pseudocharacters. In the process, we compute generators of the invariant algebras $A[M_{2d}^m]^{\operatorname {Sp}_{2d}}$
and $A[G^m]^G$
when $G \in \{\operatorname {Sp}_{2d}, \mathrm O_d, \operatorname {GSp}_{2d} and\ \operatorname {GO}_d\}$
over an arbitrary commutative ring A, generalising results of Zubkov.
