By Michael Barr

ISBN-10: 0387095632

ISBN-13: 9780387095639

ISBN-10: 3540095632

ISBN-13: 9783540095637

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**Example text**

The homogeneous dimension doublecounts the dimension of the “missing directions”. The Cayley transform is the map σ : N → S, given by σ(x, t) = 1+ |x|2 4 2 −1 + |z|2 1+ |x|4 |x|4 − z x, −1 + + |z|2 + 2z 4 16 for all (x, t) in N . It is a (nontrivial) exercise in calculus to show that σ is Carnot–Carath´eodory conformal when N is given the left-invariant Carnot– Carath´eodory structure which at the group identity (0, 0) is Fn (that is, {(x, 0) : x ∈ Fn }) with its standard inner product. The Jacobian of the transformation is given by the Qth power of the dilation factor, that is, Jσ (x, t) = 1+ |x|2 4 2 −Q/2 + |z|2 .

The group GL(n, Qp ) is not semisimple, and this is not quite the analogue of a Riemannian symmetric space. One example of a discrete symmetric space may be obtained by restricting attention to the space L1 of lattices whose volume is equal to that of the standard lattice. It follows from the invariant factor theorem that L1 may be identiﬁed with the coset space SL(n, Qp )/SL(n, Op ). The more standard example of a discrete symmetric space is a quotient space of L. Deﬁne an equivalence relation ∼ on L by the stipulation that L1 ∼ L2 if L1 = λL2 for some λ in Qp ; the equivalence class of L is written [L].

Bekka, On uniqueness of invariant means, Proc. Amer. Math. Soc. 126 (1998), 507–514. 10. B. Bekka and M. Cowling, Some unitary representations of G(K) for a simple algebraic group over a ﬁeld K, Math. Z. 241 (2002), 731–741. 11. B. Bekka and M. Cowling, Addendum to “Some unitary representations of G(K) for a simple algebraic group over a ﬁeld K”, in preparation. 12. A. Borel and H. Garland, Laplacian and the discrete spectrum of an arithmetic group, Amer. J. Math. 105 (1983), 309–335. ´ 13. T. Branson, G.

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