Minimal representations are the smallest infinite dimensional unitary representations. The Weil representation for the metaplectic group, which plays a prominent role in number theory, is a classic example.Whereas minimal representations are of importance in algebraic representation theory, we initiated another new line of investigations, namely, ''geometric analysis of minimal representations''. A recent progress on geometric analysis of minimal representations shows that this new area is surprisingly rich and seems promising through the interactions with various areas of mathematics.

I plan to take a few topics from analysis of minimal representations based on the philosophy [1] that minimal representations are ''maximal symmetries'' on representation spaces. Possible topics will be

1. Conformal construction of minimal representations, conservative quantities of ultrahyperbolic equations [8,9].

2. Schrödinger model (

L^{2}model) of minimal representations [2,8].3. Geometric quantization of minimal nilpotent orbits [4].

4. ''New'' special functions arising from minimal representations [6].

5. Generalized Fourier-Hankel transofrm [2,7].

6. Interpolation of minimal representations and a deformation theory [5].

7. Broken symmetries of minimal representations [3,8].

References:

[1] T. Kobayashi,

Algebraic analysis on minimal representations, Publ. RIMS 42, Special issue in commemoration of the golden jubilee of algebraic analysis, (2011), 585-611.[2] T. Kobayashi and G. Mano,

Schrödinger model of minimal representations of, Memoirs of Amer. Math. Soc. (2011), 212, no. 1000, vi+132 pp.O(p,q)[3] T. Kobayashi, M. Pevzner, and B. Ørsted,

Analysis of small unitary reps of, J. Funct. Anal. 260 (2011), 1682-1720,GL(n,)R[4] J. Hilgert, T.Kobayashi, and J. Möllers,

Minimal representations via Bessel operators, 66 pp. arXiv:1106.3621[5] S. Ben Saïd, T. Kobayashi, and B. Ørsted,

Laguerre semigroup and Dunkl operators, 74 pp. arXiv:0907.3749 to appear in Compositio Math.[6] J. Hilgert, T.Kobayashi, and G. Mano, and J. Möllers,

Special functions associated to a fourth order differential equation, Ramanujan J. Math ( 2011);Orthogonal polynomials associated to a certain fourth order differential equation, to appear in Ramanujan Journal. arXiv:0907.2612[7] T. Kobayashi and G. Mano,

Inversion and holomorphic extension, R. Howe 60th birthday volume (2007), 65 pp ISBN 978-9812770783. (cf. math. RT/0607007)[8] T. Kobayashi and B. Ørsted,

Analysis on minimal representations of, Adv. Math. (2003) I, II, III, 110 pp.O(p,q)[9] T. Kobayashi,

Conformal geometry and global solutions to the Yamabe equations on classical pseudo-Riemannian manifolds, Rendiconti del Circolo Matematico di Palermo, Serie II 71 (2003), 15-40, Lecture Notes of the 22th Winter School 2002 on Geometry and Physics, Czech Republic.

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© Toshiyuki Kobayashi