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We construct via generators and relations a generalized Weil representationfor the split orthogonal group $\ot_q(2n,2n)$ over a finite field of $q$elements. Besides, we give an initial decomposition of the representationfound. We also show that the constructed representation is equal to therestriction of the Weil representation to $\ot_q(2n,2n)$ for the reductive dualpair $(\Sp_2(\mathbb{F}_q),\ot_q(2n,2n))$ and that the initial decomposition isthe same as the decomposition with respect to the action of$\Sp_2(\mathbb{F}_q)$.

A Generalized Weil Representation for the finite split orthogonal group $O_q(2n,2n)$, $q$ odd greater than $3$