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Weighted projective spaces and minimal nilpotent orbits

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      Abstract

      We investigate (twisted) rings of differential operators on the resolution of singularities of a particular irreducible component of the (Zarisky) closure of the minimal orbit \(\bar O_{\mathrm{min}}\) of \(\mathfrak{sp}_{2n}\), intersected with the Borel subalgebra \(\mathfrak n_+\) of \(\mathfrak{sp}_{2n}\), using toric geometry and show that they are homomorphic images of a subalgebra of the Universal Enveloping Algebra (UEA) of \(\mathfrak{sp}_{2n}\), which contains the maximal parabolic subalgebra \(\mathfrak p\) determining the minimal nilpotent orbit. Further, using Fourier transforms on Weyl algebras, we show that (twisted) rings of well-suited weighted projective spaces are obtained from the same subalgebra. Finally, investigating this subalgebra from the representation-theoretical point of view, we find new primitive ideals and rediscover old ones for the UEA of \(\mathfrak{sp}_{2n}\) coming from the aforementioned resolution of singularities.

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      The minimal nilpotent orbit, the Joseph ideal, and differential operators

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        Actions of tori on weyl algebras

         Ian Musson (1988)
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          Author and article information

          Journal
          2007-08-13
          2007-11-06
          0708.1714
          Custom metadata
          13N10
          15 pages; misprints corrected; some terminology mistakes have been corrected
          math.RA math.AG

          Algebra

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