Journal of the Mathematical Society of Japan

Minimal representations via Bessel operators

Joachim HILGERT, Toshiyuki KOBAYASHI, and Jan MÖLLERS

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We construct an $L^2$-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 $A_n$, 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 $L^2$-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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J. Math. Soc. Japan, Volume 66, Number 2 (2014), 349-414.

First available in Project Euclid: 23 April 2014

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Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}
Secondary: 17C30: Associated groups, automorphisms 33E30: Other functions coming from differential, difference and integral equations

minimal representation conformal groups Jordan algebras Bessel operators Schrödinger model complementary series representations special functions


HILGERT, Joachim; KOBAYASHI, Toshiyuki; MÖLLERS, Jan. Minimal representations via Bessel operators. J. Math. Soc. Japan 66 (2014), no. 2, 349--414. doi:10.2969/jmsj/06620349.

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