## On the space of $K$-finite solutions to intertwining differential operators

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- by Toshihisa Kubo and Bent Ørsted
- Represent. Theory
**23**(2019), 213-248 - DOI: https://doi.org/10.1090/ert/527
- Published electronically: September 10, 2019
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## Abstract:

In this paper we give Peter–Weyl-type decomposition theorems for the space of $K$-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of $\widetilde {\mathrm {SL}}(3,\mathbb {R})$ attached to the minimal nilpotent orbit.## References

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## Bibliographic Information

**Toshihisa Kubo**- Affiliation: Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto 612-8577, Japan
- MR Author ID: 965976
- Email: toskubo@econ.ryukoku.ac.jp
**Bent Ørsted**- Affiliation: Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
- Email: orsted@imf.au.dk
- Received by editor(s): August 29, 2018
- Received by editor(s) in revised form: June 5, 2019
- Published electronically: September 10, 2019
- Additional Notes: The first author was partially supported by JSPS Grant-in-Aid for Young Scientists (B) (JP26800052)

Part of this research was conducted during a visit of the first author to the Department of Mathematics of Aarhus University and a visit of the second author to the Graduate School of Mathematical Sciences of the University of Tokyo. - © Copyright 2019 American Mathematical Society
- Journal: Represent. Theory
**23**(2019), 213-248 - MSC (2010): Primary 22E46, 17B10
- DOI: https://doi.org/10.1090/ert/527
- MathSciNet review: 4001530