Analysis on Minimal Representations. Geometry, Symmetry and Physics. Yale University, USA, 26 March 2009.

Minimal representations are the ''smallest'' infinite dimensional unitary representations. The Weil representation, which plays a prominent role in number theory, is a classic example. Most of these are isolated among the set of all unitary representations, and cannot be built up by induction.

Highlighting indefinite orthogonal groups, I plan to discuss two models of minimal representations, namely, the one is the conformal geometric construction by using the Yamabe operator, and the other is an analog of the Schrödinger model. The latter model leads us to a natural generalization of the ''Fourier transform'' on the isotropic cone, and an extension of some earlier work by R. Howe on the oscillator semigroup for the Weil representation.

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