By Zannier, Umberto (ed.)

Contributions via a number of authors reminiscent of Michael G. Cowling.- Joseph A. Wolf.- Gisbert Wustholz and David Mumford

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8] P. E YMARD, L’alg`ebre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236. [9] P. E YMARD, Alg`ebres A p et convoluteurs de L p , pages 55–72 (expos´e 367) In: “S´eminaire Bourbaki”, Vol. 1969/1970, Expos´es 364– 381, Springer-Verlag, Berlin, Heidelberg, New York, 1971. [10] A. F IG A` -TALAMANCA, Translation invariant operators in L p , Duke Math. J. 32 (1965), 495–501. [11] R. G URALNICK, Invertibility preservers and algebraic groups, Linear Algebra Appl.

The (left regular) representation of G on L 2 (G/K ) is multiplicity free. 29 Classical analysis and nilpotent Lie groups If G is a connected Lie group one can add 6. The algebra D(G, K ) of G–invariant differential operators on G/K is commutative. Commutative spaces G/K are important for a number of reasons. First, they are manageable because their basic harmonic analysis is very similar to that of locally compact abelian groups. We will describe that in a moment. Second, in the Lie group cases, most of the G/K carry invariant weakly symmetric Riemannian metrics, and have properties very similar to those of Riemannian symmetric spaces.

E. f (n)dμ N (n) = N N /Z f (nz)dμ Z (z) dμ N/Z (n Z ). Z Now we have Lebesgue measures ν Z , ν N/Z and ν N on z, n/z and n speciﬁed by the condition that the exponential map have Jacobian 1 at 0, and they satisfy dν N = dν N/Z dν Z . Normalize Lebesgue measures on the dual spaces by the condition that Fourier transform is an isometry; that gives ∗ ∗ and ν N∗ such that dν N∗ = dν N/Z dν Z∗ . 11. Let N have square integrable representations. 2m where 2m is the maximum dimension of the Ad∗ (N )–orbits in n∗ .