By Allan G. Silberger

Based on a chain of lectures given via Harish-Chandra on the Institute for complicated research in 1971-1973, this publication offers an creation to the speculation of harmonic research on reductive p-adic groups.

Originally released in 1979.

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**Extra resources for Introduction to Harmonic Analysis on Reductive P-adic Groups. (MN-23): Based on lectures by Harish-Chandra at The Institute for Advanced Study, 1971-73 (Mathematical Notes)**

**Sample text**

1 T(6U(h) 2 ~(h) 2 a(h- )'(). First we shall show that = fcy(xh 1,xyh2 )d1 x -1 = f G y(xh1,xyh2 )6G (x)drx = fcy(x,xh~ 1 yh2 )6i(xh; 1 )drx -1 = 6 c(hl)'(O(hl yh2). Next, using Lemma 1. 7. 6 and the first part of this lemma, we have and 55 Our reformulation follows immediately from Theorem 1. 2 and Lemma 1. 9. 3 together with the obvious fact that yt- y 0 is surjective on c:(G : E). Theorem 1. 4. for all h 'J Let = (h1,h2 ) e "H- be the space of all E-distributions There exists an injective linear and y 0 e mapping of "IO(ir1, ir2 ) into TO on G such that J.

K1lK I -.... 1. Thus, in the special case in which, in effect, we induce from H = (1), the theorem is true. To extend the argument to the case of general H and cr, we first use Lemma 1. 7. ). ). if

To prove the surjectivity it suffices to show that, C an arbitrarily small open compact subset of G and there exists a e c 00c (G : V) such that Pa = c- 1CilHCK. /J. /J e 1e. /J (xh)d 1h Pa(x) = 0, x ( CH. This implies the lemma. Lemma 1. 7. 6. and a e C 00 (G : V) c .! -1 P(P(h)a) = 6 2 (h)6G(h) 2 P(a(h )a). For every h e H 1 Proof. P(p(h)a)(x) = JHx(h' )2 a(h' )(p(h)a)(xh' )dth' Theorem 1. 7. 7. 1 = JHx(h' )2 a(h' )a(xh' h)d h' l = JH x(h' )2 a(h' )a(xh'h)6- 1 (h' )d h' 1 r Let G, H and a be as before.