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Let  A  be a ring with1  ≠  0    ${1\neq 0}$   , not necessarily finite, endowed with an involution*   ${*}$   , that is, an anti-automorphism of order≤  2    ${\leq 2}$   . LetH  n   ⁢   (  A  )${H_{n}(A)}$    be the additive group of alln  ×  n    ${n\times n}$    hermitian matrices over  A  relative to*   ${*}$   . Let  n   ⁢   (  A  )${{\mathcal{U}}_{n}(A)}$    be the subgroup ofGL  n   ⁢   (  A  )${{\mathrm{GL}}_{n}(A)}$    of all upper triangular matrices with ones along the main diagonal. LetP  =H  n   ⁢   (  A  )    ⋊     n    ⁢   (  A  )${P=H_{n}(A)\rtimes{\mathcal{U}}_{n}(A)}$   , where  n   ⁢   (  A  )${{\mathcal{U}}_{n}(A)}$    acts onH  n   ⁢   (  A  )${H_{n}(A)}$    by*   ${*}$   -congruence transformations. We may view  P  as a unipotent subgroup of either a symplectic groupSp   2  ⁢  n    ⁢   (  A  )${{\mathrm{Sp}}_{2n}(A)}$   , if*  =   1  A${*=1_{A}}$    (in which case  A  is commutative), or a unitary groupU   2  ⁢  n    ⁢   (  A  )${{\mathrm{U}}_{2n}(A)}$    if*  ≠   1  A${*\neq 1_{A}}$   . In this paper we construct and classify a family of irreducible representations of  P  over a field  F  that is essentially arbitrary. In particular, when  A  is finite andF  =  ℂ    ${F={\mathbb{C}}}$   , we obtain irreducible representations of  P  of the highest possible degree.

Irreducible representations of unipotent subgroups of symplectic and unitary groups defined over rings