Symplectic Determinant Laws and Invariant Theory

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 Z [ 1 2 ] Z[1/2] double struck upper Z left bracket one half right bracket -algebras. We prove that this notion satisfies the properties expected from a good theory of pseudorepresentations, and we compare it to V. Lafforgue’s Sp 2 d Sp2d upper S p Subscript 2 d -pseudocharacters. In the process, we compute generators of the invariant algebras A [ M 2 d m ] Sp 2 d A[M2dᵐ]^Sp2d upper A left bracket upper M Subscript 2 d Superscript m Baseline right bracket Superscript upper S p Super Subscript 2 d and A [ G m ] G A[Gᵐ]G upper A left bracket upper G Superscript m Baseline right bracket Superscript upper G when G ∈ { Sp 2 d , O d , GSp 2 d a n d GO d } G ∈ 2d, Od, GSp2d and\ GOd\ upper G element of StartSet upper S p Subscript 2 d Baseline comma normal upper O Subscript d Baseline comma upper G upper S p Subscript 2 d Baseline a n d upper G upper O Subscript d Baseline EndSet </jats:inline-form