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In this article we construct Weil representations of quasi-split unitarygroups $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associated to quadraticextensions of finite fields. We define these representations by using anadequate presentation Bruhat like of those groups. More precisely, we defineWeil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ associating toeach generator a linear map of a suitable $\mathbb{C}$-vector space satisfyingthe relations of the aforementioned presentation. In addition, we also addressthe natural question on the compatibility of our representation of$U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$ with the classical one constructed byG\'erardin.

Weil representations of $U(n,n)(\mathbb{F}_{q^2}/\mathbb{F}_q)$, $q>3$ odd via presentation and compatibility of methods