Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces
Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces
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某些幂零齐次空间上不变微分算子的谱分解
DOI：
10.1016/0022-1236(92)90030-m
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发表时间：
1992
影响因子：
1.7
通讯作者：
F. Greenleaf
中科院分区：
文献类型：
--
作者：
L. Corwin;F. Greenleaf
如果K是幂零李群G的连通子群，则L2(KG)上的作用的不可约分解具有纯无限重数或有界有限重数。在有限情形下，作者最近证明了KG上G不变微分算子的代数D(KG)是交换的，即使作用不是重数自由的，并证明了猜想D(KG)同构于零化子上的所有Ad∗(K)不变多项式的代数，其中K是K的李代数。这里对一大类数据(K，G)证明了这一猜想。对于这样的对，可以找到同构的显式构造；它是一种具有一些不寻常的非线性方面的傅里叶变换。此外，运营商Ind(KG)已经缓和了基本解。
IfKis a connected subgroup of a nilpotent Lie groupG, the irreducible decompositionof the action onL2(KG) has either pure infinite or boundedly finite multiplicities. In the finite case the authors recently proved that the algebraD(KG) ofG-invariant differential operators onKGis commutative, even if the action is not multiplicity free, and produced evidence for the conjecture thatD(KG) is isomorphic to the algebra of allAd∗(K)-invariant polynomials on the annihilator , where is the Lie algebra ofK. Here the conjecture is proved for a large class of data (K,G). For such pairs an explicit construction of the isomorphism can be found; it is a type of Fourier transform with some unusual nonlinear aspects. Furthermore the operators inD(KG) have tempered fundamental solutions.