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


Authors: Toshihisa Kubo and Bent Ørsted
Journal: Represent. Theory 23 (2019), 213-248
MSC (2010): Primary 22E46, 17B10
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.


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Toshihisa Kubo
Affiliation: Faculty of Economics, Ryukoku University, 67 Tsukamoto-cho, Fukakusa, Fushimi-ku, Kyoto 612-8577, Japan
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

DOI: https://doi.org/10.1090/ert/527
Keywords: Intertwining differential operators, generalized Verma modules, duality theorem, $K$-finite solutions, Peter--Weyl-type formulas, small representations, Torasso's representation, hypergeometric polynomials
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.
Article copyright: © Copyright 2019 American Mathematical Society