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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Invariant differential operators on a real semisimple Lie algebra and their radial components

Author: Mohsen Pazirandeh
Journal: Trans. Amer. Math. Soc. 182 (1973), 119-131
MSC: Primary 22E45; Secondary 17B20
MathSciNet review: 0335689
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Abstract: Let $ S({\mathfrak{g}_C})$ be the symmetric algebra over the complexification $ {\mathfrak{g}_C}$ of the real semisimple Lie algebra $ \mathfrak{g}$. For $ u\;\epsilon \;S({\mathfrak{g}_C}),\partial (u)$ is the corresponding differential operator on $ \mathfrak{g}$. $ \mathcal{D}(\mathfrak{g})$ denotes the algebra generated by $ \partial (S({\mathfrak{g}_C}))$ and multiplication by polynomials on $ {\mathfrak{g}_C}$. For any open set $ U \subset \mathfrak{g},{\text{Diff}}(U)$ is the algebra of differential operators with $ {C^\infty }$-coefficients on U. Let $ \mathfrak{h}$ be a Cartan subalgebra of $ \mathfrak{g},\mathfrak{h}'$ the set of its regular points and $ \pi = {\Pi _{\alpha \epsilon P}}\alpha $, P some positive system of roots. Let $ W = {(\mathfrak{h}')^G}$, G the connected adjoint group of $ \mathfrak{g}$.

Harish-Chandra showed that, for each $ D\;\epsilon \;{\text{Diff}}(W)$, there is a unique differential operator $ \delta {'_\mathfrak{h}}(D)$ on $ \mathfrak{h}'$ such that $ (Df){\left\vert {_\mathfrak{h}' = \delta {'_\mathfrak{h}}(D)(f} \right\vert _\mathfrak{h}})$ for all G-invariant $ f\epsilon \;{C^\infty }(W)$, and that if $ D\;\epsilon \mathcal{D}(\mathfrak{h})$, then $ \delta {'_\mathfrak{h}}(D) = {\pi ^{ - 1}} \circ \bar D \circ \pi $ for some $ \bar D\epsilon \mathcal{D}(\mathfrak{g})$. In particular $ \overline {\partial (u)} = \partial (u{\vert _\mathfrak{h}}),u\;\epsilon \;S({\mathfrak{g}_C})$ and invariant.

We prove these results by different, yet simpler methods. We reduce evaluation of $ \delta {'_\mathfrak{h}}(\partial (u))\;(u\;\epsilon \;S({\mathfrak{g}_C})$, invariant) via Weyl's unitarian trick, to the case of compact G. This case is proved using an evaluation of a family of G-invariant eigenfunctions on:

$\displaystyle \pi (H)\pi (H')\int_G {\exp B({H^x},H')dx = c\sum\limits_{S\epsil... ...hfrak{h}_C})} {\epsilon (s)\exp B(sH,H'),H,H'\epsilon \;\mathfrak{g},c > 0.} } $

For G-invariant $ D\;\epsilon \;\mathcal{D}(\mathfrak{g})$, we prove $ {\pi ^{ - 1}} \circ \delta '(D) \circ \pi \;\epsilon \;\mathcal{D}(\mathfrak{h})$ using properties of derivations $ E \to \left[ {\partial (u),E} \right]$ of $ \mathcal{D}(\mathfrak{g})$ induced by $ \partial (u)\;(u\;\epsilon \;S({\mathfrak{g}_C}))$ and of the algebra of polynomials on $ {\mathfrak{h}_C}$ invariant under the Weyl group.

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Keywords: Radial components, invariant differential operators
Article copyright: © Copyright 1973 American Mathematical Society