By F. M. Hall

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479–480], and Muirhead [468, p. 252]. 18 of Vol. I). 3. 4. 5. It will also be necessary when we consider analogues of Hecke’s correspondence between modular forms and Dirichlet series in later sections of this chapter and the next. 45) for n was found in 1928 by the statistician Wishart [670]. A more general result is due to Ingham [320]. 45) in his work on quadratic forms (see [565, Vol. I, pp. 326–405]). Such gamma functions for Pn and more general domains of positivity are considered by Gindikin [218].

YŒk/ for all Y 2 Pn and k 2 K. Pn / is a commutative algebra. Proof. I/. Pn / are identical on radial or K bi-invariant functions. a/: Thus L D M. 18). 2, we will show that L D L D the complex conjugate adjoint operator: Next we need to prove the following fact. Claim. a; b/: Proof of Claim. XŒh/ D I: The solution is thus g D D 1=2 . Since D is a positive diagonal matrix, we can indeed take its square root. Y; X/jY! Y; X/jY! X; Y/: This completes the proof of the claim. Pn /, we can write L1 M1 k D L1 M2 k D M2 L1 k D M1 L1 k; since differential operators acting on different arguments certainly commute.

24). 28. Suppose that g W Pn ! C is infinitely differentiable with compact support. 1. Pn /. 24). Hint. 2 in Volume I. Now consider what happens to the Laplace operator in the various coordinate systems which have been introduced. gV /. 32) is: ijkl @ 2 . 29. 33). 32). 29) to obtain Â Ã Â Ã @ @ @ @ Tr dY D Tr dF C dg C t dh ; @Y @F @g @h and compare the result with Ã Â Ã Â @ @ @ @ C t dx : D Tr dV C dw Tr dY @Y @V @w @x This leads to the following formulas: Ât Ã Â Â t ÃÃ @ @ @t @ @ 1 t D Cx xC x C ; x @V @F @g 2 @h @h @ @ D ; @w @g @ @ @ D 2Vx C V : @x @g @h t@ Dx It follows that, setting Á @x , @ @g D @ ; @w @ @h @ @F D @ @V @ t C x @w x DV 1 @ @x 1 2 ˚ @ 2x @w ; V 1 CV 1 t « : The preceding calculation is a little tricky since @=@F must be symmetric.