Invariant differential operators on a real semisimple Lie algebra and their radial components
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- by Mohsen Pazirandeh PDF
- Trans. Amer. Math. Soc. 182 (1973), 119-131 Request permission
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 $(Df){\left | {_\mathfrak {h}’ = \delta {’_\mathfrak {h}}(D)(f} \right |_\mathfrak {h}})$ 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 $\overline {\partial (u)} = \partial (u{|_\mathfrak {h}}),u\;\epsilon \;S({\mathfrak {g}_C})$ 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: \[ \pi (H)\pi (H’)\int _G {\exp B({H^x},H’)dx = c\sum \limits _{S\epsilon W({\mathfrak {g}_C},{\mathfrak {h}_C})} {\epsilon (s)\exp B(sH,H’),H,H’\epsilon \;\mathfrak {g},c > 0.} } \] 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.References
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Additional Information
- © Copyright 1973 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 182 (1973), 119-131
- MSC: Primary 22E45; Secondary 17B20
- DOI: https://doi.org/10.1090/S0002-9947-1973-0335689-7
- MathSciNet review: 0335689