Symplectic Stiefel harmonics and holomorphic representations of symplectic groups

Abstract:

Let ${I_k}$ denote the identity matrix of order k and set \[ {s_k} = \left [ {\begin {array}{*{20}{c}} 0 & { - {I_k}} \\ {{I_k}} & 0 \\ \end {array} } \right ].\] Let ${\text {Sp}}(k,{\mathbf {C}})$ denote the group of all complex $2k \times k$ matrices which satisfy the equation $g{s_k}{g^t} = {s_k}$. Let E be the linear space of all $n \times 2k$ complex matrices with $k \geqslant n$, and let $S({E^\ast })$ denote the symmetric algebra of all complex-valued polynomial functions on E. The study of the action of ${\text {Sp}}(k,{\mathbf {C}})$, which is obtained by right translation on $S({E^\ast })$, leads to a concrete and simple realization of all irreducible holomorphic representations of ${\text {Sp}}(k,{\mathbf {C}})$. In connection with this realization, a theory of symplectic Stiefel harmonics is also established. This notion may be thought of as a generalization of the spherical harmonics for the symplectic Stiefel manifold.