dorsal/arxiv
View SchemaExotic Differential Operators on Complex Minimal Nilpotent Orbits
| Authors | A. Astashkevich, R. Brylinski |
|---|---|
| Categories | |
| ArXiv ID | q-alg/9711023 |
| URL | https://arxiv.org/abs/q-alg/9711023 |
Abstract
Let O be the minimal nilpotent adjoint orbit in a classical complex semisimple Lie algebra g. O is a smooth quasi-affine variety stable under the Euler dilation action $C^*$ on g. The algebra of differential operators on O is D(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \cite{jos} and \cite{bkHam} for some results on the geometry and quantization of O. We construct an explicit subspace $A_{-1}\subset D(O)$ of commuting differential operators which are Euler homogeneous of degree -1. The space $A_{-1}$ is finite-dimensional, g-stable and carries the adjoint representation. $A_{-1}$ consists of (for $g \neq sp(2n,C)$) non-obvious order 4 differential operators obtained by quantizing symbols we obtained previously. These operators are "exotic" in that there is (apparently) no geometric or algebraic theory which explains them. The algebra generated by $A_{-1}$ is a maximal commutative subalgebra A of D(X). We find a G-equivariant algebra isomorphism R(O) to A, $f\mapsto D_f$, such that the formula $(f|g)=({constant term of}D_{\bar{g}} f)$ defines a positive-definite Hermitian inner product on R(O). We will use these operators $D_f$ to quantize O in a subsequent paper.
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"abstract": "Let O be the minimal nilpotent adjoint orbit in a classical complex\nsemisimple Lie algebra g. O is a smooth quasi-affine variety stable under the\nEuler dilation action $C^*$ on g. The algebra of differential operators on O is\nD(O)=D(Cl(O)) where the closure Cl(O) is a singular cone in g. See \\cite{jos}\nand \\cite{bkHam} for some results on the geometry and quantization of O.\n We construct an explicit subspace $A_{-1}\\subset D(O)$ of commuting\ndifferential operators which are Euler homogeneous of degree -1. The space\n$A_{-1}$ is finite-dimensional, g-stable and carries the adjoint\nrepresentation. $A_{-1}$ consists of (for $g \\neq sp(2n,C)$) non-obvious order\n4 differential operators obtained by quantizing symbols we obtained previously.\nThese operators are \"exotic\" in that there is (apparently) no geometric or\nalgebraic theory which explains them. The algebra generated by $A_{-1}$ is a\nmaximal commutative subalgebra A of D(X). We find a G-equivariant algebra\nisomorphism R(O) to A, $f\\mapsto D_f$, such that the formula $(f|g)=({constant\nterm of}D_{\\bar{g}} f)$ defines a positive-definite Hermitian inner product on\nR(O).\n We will use these operators $D_f$ to quantize O in a subsequent paper.",
"arxiv_id": "q-alg/9711023",
"authors": [
"A. Astashkevich",
"R. Brylinski"
],
"categories": [
"q-alg",
"math.QA"
],
"title": "Exotic Differential Operators on Complex Minimal Nilpotent Orbits",
"url": "https://arxiv.org/abs/q-alg/9711023"
},
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