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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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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.
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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