Minimal representations via Bessel operators
Hilgert J, Kobayashi T, Möllers J (2014)
Publication Type: Journal article
Publication year: 2014
Journal
Book Volume: 66
Pages Range: 349-414
Journal Issue: 2
URI: http://projecteuclid.org/euclid.jmsj/1398258176
DOI: 10.2969/jmsj/06620349
Abstract
We construct an L2-model of “very small” irreducible unitary representations of simple Lie groups G which, up to finite covering, occur as conformal groups Co(V) of simple Jordan algebras V. If V is split and G is not of type An, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where G does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of SO(n,1)0. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand-Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schrödinger models in L2-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.
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APA:
Hilgert, J., Kobayashi, T., & Möllers, J. (2014). Minimal representations via Bessel operators. Journal of the Mathematical Society of Japan, 66(2), 349-414. https://doi.org/10.2969/jmsj/06620349
MLA:
Hilgert, Joachim, Toshiyuki Kobayashi, and Jan Möllers. "Minimal representations via Bessel operators." Journal of the Mathematical Society of Japan 66.2 (2014): 349-414.
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