Geometric interplay between function subspaces and their rings of differential operators

Bögvad, Rikard; Källström, Rolf

doi:10.1090/S0002-9947-06-03949-3

Geometric interplay between function subspaces and their rings of differential operators
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by Rikard Bögvad and Rolf Källström PDF

Trans. Amer. Math. Soc. 359 (2007), 2075-2108 Request permission

Abstract:

We study, in the setting of algebraic varieties, finite-dimensional spaces of functions $V$ that are invariant under a ring $\mathcal {D}^V$ of differential operators, and give conditions under which $\mathcal {D}^V$ acts irreducibly. We show how this problem, originally formulated in physics, is related to the study of principal parts bundles and Weierstrass points, including a detailed study of Taylor expansions. Under some conditions it is possible to obtain $V$ and $\mathcal {D}^V$ as global sections of a line bundle and its ring of differential operators. We show that several of the published examples of $\mathcal {D}^V$ are of this type, and that there are many more—in particular, arising from toric varieties.

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