Representation of a finite group in a vector space is a homomophism from the finite group into the vector space.
Clearly, it is a square matrice. Weil representation is a kind of special representation. By decomposing Weil representation of Sp2 and Sp4, we can find the theta correspondence of (GL1(F),GL1(F)) and (GL1(F),GL2(F)) over a finite field F. In the process, one can know the correspondence between GL1(F) and GL1(F) (resp. GL1(F) and GL2(F)) in Sp2 (resp. Sp4).

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