Discriminants of symplectic graded involutions on graded simple algebras

In this article, we define and study the discriminant of symplectic graded involutions on non-inertially split graded simple algebras with simple 0-component. In particular, we show that if F is a graded field of characteristic different from 2, D is a graded central division algebra over F with exp(D)=2 and |ker(θD)|>4 (see the preliminaries below), A=Mn(D), and σ is a graded involution of symplectic type on A, then there is only a finite number of values for the discriminants Δσ(τ), where τ describes all graded involutions of symplectic type on A (see Proposition 2.11). Consequently, for any graded central simple algebra C over F with C0 simple non-split, exp(C)=2, |ker(θC)|>4 and deg(C)ind(C) even, we have Δσ(τ)=0 for any graded involutions of symplectic type σ and τ on C (see Corollary 2.12). We prove also that if E is a Henselian valued field with residue characteristic different from 2, D is a central division algebra of exponent 2 over E with |ker(θD)|>4, and B=Mn(D) with n even, then for any symplectic involutions σ, τ on B, preserving a tame gauge defined on B, we have Δσ(τ)=0 (see Corollary 3.5).