Weighted projective spaces and minimal nilpotent orbits

Author:
Carlo A. Rossi

Journal:
Represent. Theory **12** (2008), 208-224

MSC (2000):
Primary 13N10

Published electronically:
April 17, 2008

MathSciNet review:
2403559

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Abstract: We investigate (twisted) rings of differential operators on the resolution of singularities of an irreducible component of (where is the (Zariski) closure of the minimal nilpotent orbit of and is the Borel subalgebra of ) using toric geometry, and show that they are homomorphic images of a certain family of associative subalgebras of , which contains the maximal parabolic subalgebra determining . Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same family of subalgebras. Finally, we investigate this family of subalgebras from the representation-theoretical point of view and, among other things, rediscover in a different framework irreducible highest weight modules for the UEA of .

**[1]**Walter Borho and Jean-Luc Brylinski,*Differential operators on homogeneous spaces. I. Irreducibility of the associated variety for annihilators of induced modules*, Invent. Math.**69**(1982), no. 3, 437–476. MR**679767**, 10.1007/BF01389364**[2]**Nicolas Bourbaki,*Lie groups and Lie algebras. Chapters 4–6*, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002. Translated from the 1968 French original by Andrew Pressley. MR**1890629****[3]**Giovanni Felder and Carlo A. Rossi,*Differential operators on toric varieties and Fourier transform*(2007), available at`http://arxiv.org/abs/math/0705.1709v3`.**[4]**William Fulton,*Introduction to toric varieties*, Annals of Mathematics Studies, vol. 131, Princeton University Press, Princeton, NJ, 1993. The William H. Roever Lectures in Geometry. MR**1234037****[5]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****[6]**T. Levasseur, S. P. Smith, and J. T. Stafford,*The minimal nilpotent orbit, the Joseph ideal, and differential operators*, J. Algebra**116**(1988), no. 2, 480–501. MR**953165**, 10.1016/0021-8693(88)90231-1**[7]**Ian M. Musson,*Actions of tori on Weyl algebras*, Comm. Algebra**16**(1988), no. 1, 139–148. MR**921946**, 10.1080/00927878808823565**[8]**Ian M. Musson,*Differential operators on toric varieties*, J. Pure Appl. Algebra**95**(1994), no. 3, 303–315. MR**1295963**, 10.1016/0022-4049(94)90064-7**[9]**Ian M. Musson and Sonia L. Rueda,*Finite dimensional representations of invariant differential operators*, Trans. Amer. Math. Soc.**357**(2005), no. 7, 2739–2752 (electronic). MR**2139525**, 10.1090/S0002-9947-04-03573-1**[10]**Michel Van den Bergh,*Differential operators on semi-invariants for tori and weighted projective spaces*, Topics in invariant theory (Paris, 1989/1990) Lecture Notes in Math., vol. 1478, Springer, Berlin, 1991, pp. 255–272. MR**1180993**, 10.1007/BFb0083507

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Additional Information

**Carlo A. Rossi**

Affiliation:
Department of mathematics, ETH Zürich, 8092 Zürich, Switzerland

Email:
crossi@math.ethz.ch

DOI:
https://doi.org/10.1090/S1088-4165-08-00328-2

Received by editor(s):
August 17, 2007

Received by editor(s) in revised form:
November 8, 2007

Published electronically:
April 17, 2008

Article copyright:
© Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.