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Download Modular Functions of One Variable II: Proceedings by W. Casselman (auth.), Pierre Deligne, Willem Kuijk (eds.) PDF

By W. Casselman (auth.), Pierre Deligne, Willem Kuijk (eds.)

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Read Online or Download Modular Functions of One Variable II: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972 PDF

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Extra resources for Modular Functions of One Variable II: Proceedings International Summer School University of Antwerp, RUCA July 17–August 3, 1972

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I If ~ A(o) = (n, n), then PS(~) contains exactly two primitive principal series representations of G(o), nCdet) and sp(n), If ~ each exactly once. I A(o) = M = (M , M ) with M . M;1 ~ 1, then 1 1 2 contains exactly one primitive principal series representation, PS(~) ps(M), exactly once. The following two results are the main results of this §, and most of the rest of the § will be concerned with proving them. 5. The representation PS(~) is irreducible and equivalent to PS(~w), (1) unless (2) Then -1 ~1~2 = v or v -1 If ~1~2 PS(~1' ~2) = v -1 -1 1 set ~1 = wv -'2 1 '~2 '2 = wv .

1 to obtain a few representations of G. Let k (See also §1 of [llJ). K its be the residue field olp of q elements, unique quadratic ex- Fix an additive character $ of k as e2nix/p if k = F tension. prime field) and the composite of this with Tr kIF p in general. the map * : x ~ x* to be the conjugation of K/k, and N K/j( x ~ xx·. N_,= {x k Define the additive character ~ of E K I N__ (x) = functions on K. K/k 1}. K as $ . Tr p (the Define as the map K/j( Define Define L(K) to be the space of all C-valued There exists on L(K) a twisted Fourier transform f(y) = 1 L q x E which satisfies f(y) = f (-y) and K If 12 ~(xy·) f(x) = For -47Cas-47 X E R X L[ : 1If] , L[x]f(y) = f(x- l y) and define L[x] on L(R) L[ 1If] f(x) = f(x llf ).

PROOF. Also left as an exercise. 12 follows directly from these lemmas. As a final topic, let me mention here the primitive principal series representations of SL (0). 2 If M is a character of 0 x N of conductor p , let it also be the character of Go(pN) n SL (0) taking (~ ~) to M(a), 2 unless M = 1 when pSSL(l) will be Ind(l I I Go(pN) n SL 2 (0), SL 2 (0», Go(p) n SL 2 (0), SL 2 (0». 15. (1) If M = 1, pSSL(M) splits into two components, the trivial one and one of degree q called sPSL' (2) If M2 = 1 but M ~ 1, pSSL(M) splits into two distinct components of equal dimension.

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