Applying the embedding of $A_{n-1}$ in $B_n$, $C_n$ and $D_n$ we construct a new supercharacter theory for the Sylow subgroups in orthogonal and symplectic groups over a finite field. The constructed supercharacter appears to be a little bit more precise than the previously known one.
Supercharacter theory for the Borel contraction of the group GL(n,F_q)
Regular characters of $GL_n(O)$ and Weil representations over finite fields
Weil representations over finite fields and Shintani lift
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