Kazhdan constants associated with Laplacian on connected Lie groups

Abstract: Let $G$ be a finite dimensional connected Lie group. Fix a basis $\{ X_i \}_{i=1,\cdots,n}$ of the Lie algebra $\ggot$ and form the associated Laplace operator $\Delta = - \sum_{1\leq i\leq n}\, X_i^2$ in the enveloping algebra $U(\ggot)$. Let $\pi$ be a strongly continuous unitary representation of $G$; let $\overline{d\pi(\Delta)}$ be the closure of the essentially self-adjoint operator $d\pi(\Delta)$. We show that $\pi$ almost has invariant vectors if and only if $0$ belongs to the spectrum of $\overline{d\pi(\Delta)}$. From this, we deduce that $G$ has Kazhdan's property $(T)$ if and only if there exists $\epsilon >0$ such that, for any unitary representation without non zero fixed vectors, one has $\epsilon < \min\{ \Sp(\overline{d\pi(\Delta)})\}$. This answers positively a question of Y. Colin de Verdière. It also allows us to define new Kazhdan constants, that we compare to the classical ones.

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