TY - JOUR

T1 - Spectral decomposition of invariant differential operators on certain nilpotent homogeneous spaces

AU - Corwin, Lawrence

AU - Greenleaf, Frederick P.

N1 - Funding Information:
* Research supported in part by National Science Foundation Grants DMS 86-03169, DMS 89-02993 and DMS 87-03334, 89-07776, respectively.

PY - 1992/9

Y1 - 1992/9

N2 - If K is a connected subgroup of a nilpotent Lie group G, the irreducible decompositionof the action on L2(KG) has either pure infinite or boundedly finite multiplicities. In the finite case the authors recently proved that the algebra D(KG) of G-invariant differential operators on KG is commutative, even if the action is not multiplicity free, and produced evidence for the conjecture that D(KG) is isomorphic to the algebra of all Ad*(K)-invariant polynomials on the annihilator {A figure is presented}, where {A figure is presented} is the Lie algebra of K. 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 in D(KG) have tempered fundamental solutions.

AB - If K is a connected subgroup of a nilpotent Lie group G, the irreducible decompositionof the action on L2(KG) has either pure infinite or boundedly finite multiplicities. In the finite case the authors recently proved that the algebra D(KG) of G-invariant differential operators on KG is commutative, even if the action is not multiplicity free, and produced evidence for the conjecture that D(KG) is isomorphic to the algebra of all Ad*(K)-invariant polynomials on the annihilator {A figure is presented}, where {A figure is presented} is the Lie algebra of K. 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 in D(KG) have tempered fundamental solutions.

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U2 - 10.1016/0022-1236(92)90030-M

DO - 10.1016/0022-1236(92)90030-M

M3 - Article

AN - SCOPUS:0040198755

SN - 0022-1236

VL - 108

SP - 374

EP - 426

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -