# Geometric analysis on small unitary representations of GLN,R.

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1 Graduate School of Mathematics 2 Mathematical Institute 3 LMR - Laboratoire de MathÃ©matiques de Reims Reims

Abstract : The most degenerate unitary principal series representations $\pi {i\lambda,\delta}$ $\lambda\in \R,\,\delta\in\mathbb Z-2\mathbb Z$ of $G = GLN,\R$ 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 $\pi {i\lambda,\delta}| H$ \textit{branching law} with respect to all symmetric pairs $G,H$. For $N=2n$ with $n \ge 2$, the restriction $\pi {i\lambda,\delta}| H$ remains irreducible for $H=Spn,\R$ if $\lambda e0$ and splits into two irreducible representations if $\lambda=0$. The branching law of the restriction $\pi {i\lambda,\delta}| H$ is purely discrete for $H = GLn,\C$, consists only of continuous spectrum for $H = GLp,\R \times GLq,\R$ $p+q=N$, and contains both discrete and continuous spectra for $H=Op,q$ $p>q\ge1$. Our emphasis is laid on geometric analysis, which arises from the restriction of `small representations- to various subgroups.

Keywords : Small representation Branching law Symmetric pair Reductive group Phase space representation Symplectic group Degenerate principal series representations

Author: ** Toshiyuki Kobayashi - Bent Orsted - Michael Pevzner - **

Source: https://hal.archives-ouvertes.fr/