By Audrey Terras

Well, eventually, right here it is-the long-promised "Revenge of the better Rank Symmetric areas and Their basic Domains." while i started paintings on it in 1977, i'd most likely have stopped instantly if somebody had instructed me that ten years may cross ahead of i'd claim it "finished." certain, i'm mentioning it finished-though by no means perfected. there's a great amount of labor happening for the time being because the piles of preprints achieve the ceiling. however, it really is summer season and the sea calls. So i'm really not going to spend one other ten years revising and sharpening. yet, light reader, do ship me your corrections or even your preprints. because of your paintings, there's an Appendix on the finish of this quantity with corrections to quantity I. I acknowledged all of it within the Preface to quantity I. So i'm going to try out to not repeat myself right here. definite, the "recent developments" pointed out in that Preface are nonetheless simply as recent.

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M (3) K-Bessel Functions with a Singular Argument Reduce to Lower Rank K-Bessel Functions. lIn-m' Then = n m(n-m)/2IBI-m/2 PUl (A -1 )fm(0"1)K n- m(0"2IB, D). We need to assume that Re 0"1 is sufficiently large for the convergence of f m• (4) The Argument in ,ql)n of the Matrix k-Bessel Function Can Be Reduced to I. , g is upper triangular with positive diagonal. lim, W > 0, then ~). a) = p_s(V-l) IVI-l/2wm/2km, km,l (s, 0 \ (~ 1 (s, 0IIm+1, w1/2g- 1a). Here SEem. (5) An Inductive Formula for k-Bessel Functions.

Here db is the right or left (they are equal) Haar measure on G as in Exercise 3. 103) in Vol. I is due to the fact that we are thinking that &'n has a right G-action while the Poincare upper half plane H has a left G-action. Lemma 1 (Properties of Convolution Operators). , 9 E C';'(G/K). We will ultimately need to generalize this, however. 24) commutes with the action of c E GL(n, /R) on functions f(a), a E G, defined by fC(a) = f(ac). Thus we say that Cg is a G-invariant integral operator.

It is also useful to view Bessel and Whittaker functions in the light of the theory of the operators intertwining pairs of representations (see Dieudonne [1, Vol. VIJ, Hashizume [1J, Kirillov [2J, Mackey [1, pp. J, and Vilenkin [1, Ch. VIIIJ). However, we will not delve into group representations in this volume. 4X», (n-m>}. 18) for X E IRm x(n-m). , f(Y[UJ) = XA(U)f(Y) for all Y E 811.. 19) (c) f grows at most like a power function at the boundary. 12) in Volume I. (y), as y approaches infinity, one might think that the growth condition (c) is somewhat weak.