Invariant differential operators on the tangent space of some symmetric spaces

Thierry Levasseur; J. Toby Stafford

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 6, page 1711-1741
  • ISSN: 0373-0956

Abstract

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Let 𝔤 be a complex, semisimple Lie algebra, with an involutive automorphism ϑ and set 𝔨 = Ker ( ϑ - I ) , 𝔭 = Ker ( ϑ + I ) . We consider the differential operators, 𝒟 ( 𝔭 ) K , on 𝔭 that are invariant under the action of the adjoint group K of 𝔨 . Write τ : 𝔨 → Der 𝒪 ( 𝔭 ) for the differential of this action. Then we prove, for the class of symmetric pairs ( 𝔤 , 𝔨 ) considered by Sekiguchi, that d ∈ 𝒟 ( 𝔭 ) : d 𝒪 ( 𝔭 ) K = 0 = 𝒟 ( 𝔭 ) τ ( 𝔨 ) . An immediate consequence of this equality is the following result of Sekiguchi: Let ( 𝔤 0 , 𝔨 0 ) be a real form of one of these symmetric pairs ( 𝔤 , 𝔨 ) , and suppose that T is a K 0 -invariant eigendistribution on 𝔭 0 that is supported on the singular set. Then, T = 0 . In the diagonal case ( 𝔤 , 𝔨 ) = ( 𝔤 ' ⊕ 𝔤 ' , 𝔤 ' ) this is a well-known result due to Harish-Chandra.

How to cite

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Levasseur, Thierry, and Stafford, J. Toby. "Invariant differential operators on the tangent space of some symmetric spaces." Annales de l'institut Fourier 49.6 (1999): 1711-1741. <http://eudml.org/doc/75400>.

@article{Levasseur1999,
abstract = {Let $\{\frak g\}$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta $ and set $\{\frak k\} = \{\rm Ker\}(\vartheta -I)$, $\{\frak p\} = \{\rm Ker\}(\vartheta +I)$. We consider the differential operators, $\{\cal D\}(\{\frak p\})^K$, on $\{\frak p\}$ that are invariant under the action of the adjoint group $K$ of $\{\frak k\}$. Write $\tau : \{\frak k\}\rightarrow \{\rm Der\}\,\{\cal O\}(\{\frak p\})$ for the differential of this action. Then we prove, for the class of symmetric pairs $(\{\frak g\},\{\frak k\})$ considered by Sekiguchi, that $\left\rbrace d \in \{\cal D\}(\{\frak p\}) : d\bigl (\{\cal O\}(\{\frak p\})^K\bigr )=0\right\lbrace = \{\cal D\}(\{\frak p\})\tau (\{\frak k\})$. An immediate consequence of this equality is the following result of Sekiguchi: Let $(\{\frak g\}_0,\{\frak k\}_0)$ be a real form of one of these symmetric pairs $(\{\frak g\},\{\frak k\})$, and suppose that $T$ is a $K_0$-invariant eigendistribution on $\{\frak p\}_0$ that is supported on the singular set. Then, $T=0$. In the diagonal case $(\{\frak g\},\{\frak k\})=(\{\frak g\}^\{\prime \}\oplus \{\frak g\}^\{\prime \}, \{\frak g\}^\{\prime \})$ this is a well-known result due to Harish-Chandra.},
author = {Levasseur, Thierry, Stafford, J. Toby},
journal = {Annales de l'institut Fourier},
keywords = {symmetric space; reductive Lie algebra; invariant differential operator; invariant eigendistribution},
language = {eng},
number = {6},
pages = {1711-1741},
publisher = {Association des Annales de l'Institut Fourier},
title = {Invariant differential operators on the tangent space of some symmetric spaces},
url = {http://eudml.org/doc/75400},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Levasseur, Thierry
AU - Stafford, J. Toby
TI - Invariant differential operators on the tangent space of some symmetric spaces
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 6
SP - 1711
EP - 1741
AB - Let ${\frak g}$ be a complex, semisimple Lie algebra, with an involutive automorphism $\vartheta $ and set ${\frak k} = {\rm Ker}(\vartheta -I)$, ${\frak p} = {\rm Ker}(\vartheta +I)$. We consider the differential operators, ${\cal D}({\frak p})^K$, on ${\frak p}$ that are invariant under the action of the adjoint group $K$ of ${\frak k}$. Write $\tau : {\frak k}\rightarrow {\rm Der}\,{\cal O}({\frak p})$ for the differential of this action. Then we prove, for the class of symmetric pairs $({\frak g},{\frak k})$ considered by Sekiguchi, that $\left\rbrace d \in {\cal D}({\frak p}) : d\bigl ({\cal O}({\frak p})^K\bigr )=0\right\lbrace = {\cal D}({\frak p})\tau ({\frak k})$. An immediate consequence of this equality is the following result of Sekiguchi: Let $({\frak g}_0,{\frak k}_0)$ be a real form of one of these symmetric pairs $({\frak g},{\frak k})$, and suppose that $T$ is a $K_0$-invariant eigendistribution on ${\frak p}_0$ that is supported on the singular set. Then, $T=0$. In the diagonal case $({\frak g},{\frak k})=({\frak g}^{\prime }\oplus {\frak g}^{\prime }, {\frak g}^{\prime })$ this is a well-known result due to Harish-Chandra.
LA - eng
KW - symmetric space; reductive Lie algebra; invariant differential operator; invariant eigendistribution
UR - http://eudml.org/doc/75400
ER -

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