## T. Kobayashi, B. Ørsted, and M. Pevzner, *Geometric analysis on
small unitary representations of **GL*(*n*,*R*), J. Funct. Anal.
**260** (2011), no. 6, 1682-1720, (published online first, on 28
December 2010). DOI:
10.1016/j.jfa.2010.12.008.
arXiv:1002.3006 [math.RT]..

The most degenerate unitary principal series representations π_{iλ,δ} (λ∈**R**, δ∈**Z**/2**Z**)
of *G* = *GL*(*N*,**R**) attain the minimum of the Gelfand-Kirillov
dimension among all irreducible unitary representations of *G*.
This article gives an explicit formula of the irreducible
decomposition of the restriction π_{iλ,δ}|_{H} (*branching law*)
with respect to all symmetric pairs (*G*,*H*).
For *N*=2*n* with *n* ≥ 2, the restriction
π_{iλ,δ}|_{H} remains irreducible for *H*=*Sp*(*n*,**R**)
if λ≠0
and splits into two irreducible representations if
λ=0.
The branching law of the restriction π_{iλ,δ}|_{H} is purely
discrete for *H* = *GL*(*n*,**C**),
consists only of continuous spectrum for
*H* = *GL*(*p*,**R**) × *GL*(*q*,**R**) (*p*+*q*=*N*),
and contains
both
discrete and continuous spectra for *H*=*O*(*p*,*q*) (*p*>*q*≥1). Our emphasis is laid on geometric analysis, which arises from the restriction of 'small representations' to various subgroups.

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© Toshiyuki Kobayashi