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# Quantization of the $4$-dimensional nilpotent orbit of $\SL(3,\R)$

Published:1997-10-01
Printed: Oct 1997
• Ranee Brylinski
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## Abstract

We give a new geometric model for the quantization of the 4-dimensional conical (nilpotent) adjoint orbit $\OR$ of $\SL(3,\R)$. The space of quantization is the space of holomorphic functions on ${\C}^2-\{0\})$ which are square integrable with respect to a signed measure defined by a Meijer $G$-function. We construct the quantization out a non-flat Kaehler structure on ${\C}^2-\{0\})$ (the universal cover of $\OR$) with Kaehler potential $\rho=|z|^4$.
 MSC Classifications: 81S10 - Geometry and quantization, symplectic methods [See also 53D50] 32C17 - unknown classification 32C1722E70 - Applications of Lie groups to physics; explicit representations [See also 81R05, 81R10]