## T. Kobayashi and G. Mano, *The Schrödinger model for the minimal
representation of the indefinite orthogonal group **O*(*p*, *q*), Mem. Amer. Math. Soc. **212**, no. 1000, 2011, vi+132 pp., (accepted for publication on 31 July, 2008; published online first, on 4 February 2011).
DOI: 10.1090/S0065-9266-2011-00592-7.
arXiv:0712.1769 [math.RT]..

We introduce the 'Fourier transform' *F*_{C} on the isotropic cone *C*
associated to an indefinite quadratic form of signature (*n*_{1},*n*_{2}) on **R**^{n} (*n*=*n*_{1}+*n*_{2}: even).
This transform is in some sense the unique and natural unitary operator on *L*^{2}(*C*),
as is the case with the Euclidean Fourier transform *F*_{R^n} on *L*^{2}(**R**^{n}).
Inspired by recent developments of algebraic representation theory of reductive groups,
we shed new light on classical analysis on the one hand,
and give the global formulas for the *L*^{2}-model of the minimal representation of the
simple Lie group *G*=*O*(*n*_{1}+1,*n*_{2}+1) on the other hand.
The transform *F*_{C} expands functions on *C* into joint
eigenfunctions of *fundamental differential operators*
which are mutually commuting, self-adjoint, and of second order.
We decompose *F*_{C} into the singular Radon transform and the Mellin-Barnes integral,
find its distribution kernel, and
establish the inversion and the Plancherel formula.
The transform *F*_{C} reduces to the Hankel transform if
*G* is *O*(*n*,2) or *O*(3,3) ≈ *SL*(4,**R**).

The unitary operator *F*_{C} together with multiplications and translations coming from
the conformal transformation group *CO*(*n*_{1},*n*_{2})*R*^{n_1+n_2}
generates the minimal representation of the indefinite orthogonal group *G*.
Various different models of the same representation have been constructed by
Kazhdan, Kostant, Binegar-Zierau, Gross-Wallach, Zhu-Huang, Torasso,
Brylinski, and Kobayashi-Ørsted, and others.
Among them, our model gives the global formula of
the whole group action on the simple Hilbert space *L*^{2}(*C*),
and generalizes the classic Schrödinger model *L*^{2}(**R**^{n})
of the Weil representation.
Here, *F*_{C} plays a similar role to *F*_{R^n}.

Yet another motif is special functions.
Large group symmetries in the minimal representation
yield functional equations of various special functions.
We find explicit *K*-finite vectors on *L*^{2}(*C*),
and give a new proof of the Plancherel formula for Meijer's *G*-transforms.

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© Toshiyuki Kobayashi