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By P. Deligne (auth.), Roger Howe (eds.)

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Extra resources for Discrete Groups in Geometry and Analysis: Papers in Honor of G.D. Mostow on His Sixtieth Birthday

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If o 0 G, as its square. and the normalized Haar measure on o A If we put we obviously have vo' G is a eonneeted simply eonneeted simple CL l ; and We may assume that ~ is the produet of the o Tamagawa measure on §l. [gx,gy] = vo(g) [x,y] such that vo [x,y] ). in 40 w = ws' it has poles of order Z(w s ) other words 13, 14. and We observe that if (x - A) 's; cf. Z(ws ) at 1 at has poles of order 0, 2, 9, 11, 17, 19, 26, 28. In 1 1 1 1 1 at 0, 1, 42 , ~, SZ, 82, b(s) = rr(s + A), then they are the Kimura [22], p.

Borel [6]; we put JG /G A k The integrand is a continuous function on but in general it is 1 not in L (GA/Gk ). In fact all triplets such that similar integrals relative to algebraic extensions of k remain convergent are so special that they have been classified; and by using the classification and the theory of algebras the following theorem has been proved: "If a triplet (G,X,p) satisfies the above condition of convergence, the ring of invariants of p is generated by algebraically independent homogeneous elements, say ••• , f r , algebra of f:X ~; if we define a k-morphism f(x) = (fl(x), ••• , fr(x», such that at every ~ in X' are transversal and such that U(i) codimension of f-l(i) r Aff .

However, we have not been able to prove this. Section 7 is a technical section proving the existence of nicely intersecting totally geodesic submanifolds in the standard arithmetic examples. 2) that if p + q # n - 1 there exist totally geodesic submanifolds of codimension respectively intersecting in a single component. p and q The reader may find this section difficult - he is advised to refer to O'Meara [18] for background information on the Strong Approximation Theorem and the spinor norm. Section 8 is concerned with the interaction of Riemann geometry.

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