By R. Keith Dennis

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Suppose that |un (x)−1| < n for all x in X. Then 2 πn (x)ξn − ξn = πn (x)ξn − ξn , πn (x)ξn − ξn = 2 − 2 Re πn (x)ξn , ξn , and so πn (x)ξn − ξn ≤ (2 )1/2 . It follows, by induction, that πn (x1 . . xm )ξn − ξn ≤ m(2 n )1/2 ∀x1 , . . , xm ∈ X; indeed πn (x1 . . xm )ξn − ξn ≤ πn (x1 . . xm−1 )(πn (xm )ξn − ξn ) + πn (x1 . . xm−1 )ξn − ξn . Thus if un (x) → 1 as n → ∞ for all x in X, un → 1 as n → ∞ locally uniformly on G. Property T may be expressed in the following form: if none of the unitary representations πn has a trivial subrepresentation, then the corresponding matrix coeﬃcients un cannot tend to 1 locally uniformly.

17. M. Burger and P. Sarnak, Ramanujan duals. II, Invent. Math. 106 (1991), 1–11. 18. W. Casselman and D. Miliˇci´c, Asymptotic behavior of matrix coeﬃcients of admissible representations, Duke Math. J. 49 (1982), 869–930. 19. A. P. Veselov, Integrability and Huygens’ principle on symmetric spaces, Comm. Math. Phys. 178 (1996), 311–338. 20. J. Cheeger, M. E. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J.

It is worth pointing out that the signiﬁcant diﬀerence in the behaviour of the solutions of the heat and Laplace equations is due to the fact that the function exp(−tQθ (·)) extends to an entire function in a∗C while the function exp(−tQθ (·)1/2 ) extends into a tube Tδ , where δ = (1 − θ)1/2 , but into no larger tube. This shows clearly that harmonic analysis on a noncompact symmetric space involves phenomena with no Euclidean analogue. 6 Approaches to the Wave and Schr¨ odinger Equations Dealing with the wave equation is tricky.