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  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 (L2 model) of minimal representations [2,8].
3. Geometric quantization of minimal nilpotent orbits .
4. ''New'' special functions arising from minimal representations .
5. Generalized Fourier-Hankel transofrm [2,7].
6. Interpolation of minimal representations and a deformation theory .
7. Broken symmetries of minimal representations [3,8].
 T. Kobayashi, Algebraic analysis on minimal representations, Publ. RIMS 42, Special issue in commemoration of the golden jubilee of algebraic analysis, (2011), 585-611.
 T. Kobayashi and G. Mano, Schrödinger model of minimal representations of O(p,q), Memoirs of Amer. Math. Soc. (2011), 212, no. 1000, vi+132 pp.
 T. Kobayashi, M. Pevzner, and B. Ørsted, Analysis of small unitary reps of GL(n, R), J. Funct. Anal. 260 (2011), 1682-1720,
 J. Hilgert, T.Kobayashi, and J. Möllers, Minimal representations via Bessel operators, 66 pp. arXiv:1106.3621
 S. Ben Saïd, T. Kobayashi, and B. Ørsted, Laguerre semigroup and Dunkl operators, 74 pp. arXiv:0907.3749 to appear in Compositio Math.
 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
 T. Kobayashi and G. Mano, Inversion and holomorphic extension, R. Howe 60th birthday volume (2007), 65 pp ISBN 978-9812770783. (cf. math. RT/0607007)
 T. Kobayashi and B. Ørsted, Analysis on minimal representations of O(p,q), Adv. Math. (2003) I, II, III, 110 pp.
 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