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Finite dimensional representations of invariant differential operators

Authors: Ian M. Musson and Sonia L. Rueda
Journal: Trans. Amer. Math. Soc. 357 (2005), 2739-2752
MSC (2000): Primary 16S32
Published electronically: July 22, 2004
MathSciNet review: 2139525
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Abstract: Let $k$ be an algebraically closed field of characteristic $0$, $Y=k^{r}\times {(k^{\times})}^{s}$, and let $G$ be an algebraic torus acting diagonally on the ring of algebraic differential operators $\mathcal{D} (Y)$. We give necessary and sufficient conditions for $\mathcal{D}(Y)^G$ to have enough simple finite dimensional representations, in the sense that the intersection of the kernels of all the simple finite dimensional representations is zero. As an application we show that if $K\longrightarrow GL(V)$ is a representation of a reductive group $K$ and if zero is not a weight of a maximal torus of $K$ on $V$, then $\mathcal{D} (V)^K$ has enough finite dimensional representations. We also construct examples of FCR-algebras with any integer GK dimension $\geq 3 $.

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Additional Information

Ian M. Musson
Affiliation: Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201

Sonia L. Rueda
Affiliation: Departamento de Matemáticas, E.T.S. Arquitectura, Universidad Politécnica de Madrid, Avda. Juan Herrera, 4, 28040 Madrid, Spain

Received by editor(s): June 3, 2003
Received by editor(s) in revised form: October 27, 2003
Published electronically: July 22, 2004
Additional Notes: The first author was partially supported by NSF grant DMS-0099923.
Article copyright: © Copyright 2004 American Mathematical Society

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