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Transactions of the American Mathematical Society

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Symplectic Stiefel harmonics and holomorphic representations of symplectic groups


Author: Tuong Ton-That
Journal: Trans. Amer. Math. Soc. 232 (1977), 265-277
MSC: Primary 22E45
DOI: https://doi.org/10.1090/S0002-9947-1977-0476926-1
MathSciNet review: 0476926
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Abstract: Let $ {I_k}$ denote the identity matrix of order k and set

$\displaystyle {s_k} = \left[ {\begin{array}{*{20}{c}} 0 & { - {I_k}} \\ {{I_k}} & 0 \\ \end{array} } \right].$

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.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1977-0476926-1
Keywords: Symmetric algebras of polynomial functions, irreducible holomorphic representations of symplectic groups, symplectic Stiefel harmonics
Article copyright: © Copyright 1977 American Mathematical Society

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