DOI: 10.1017/s1474748026101881 ISSN: 1474-7480
SYMPLECTIC DETERMINANT LAWS AND INVARIANT THEORY
Mohamed Moakher, Julian Quast 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
over an arbitrary commutative ring
A
, generalising results of Zubkov.