SYMPLECTIC DETERMINANT LAWS AND INVARIANT THEORY

Abstract 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 ] $\mathbb Z[\tfrac {1}{2}]$ 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 $\operatorname {Sp}_{2d}$ 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 $A[M_{2d}^m]^{\operatorname {Sp}_{2d}}$ 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 $A[G^m]^G$ 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 } $G \in \{\operatorname {Sp}_{2d}, \mathrm O_d, \operatorname {GSp}_{2d} and\ \operatorname {GO}_d\}$ 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