Download PDF by Yuval Z Flicker: Automorphic Representations of Low Rank Groups

By Yuval Z Flicker

ISBN-10: 9812568034

ISBN-13: 9789812568038

The realm of automorphic representations is a usual continuation of reviews in quantity conception and modular types. A tenet is a reciprocity legislation pertaining to the endless dimensional automorphic representations with finite dimensional Galois representations. easy family members at the Galois facet mirror deep family members at the automorphic facet, referred to as "liftings". This ebook concentrates on preliminary examples: the symmetric sq. lifting from SL(2) to PGL(3), reflecting the three-d illustration of PGL(2) in SL(3); and basechange from the unitary workforce U(3, E/F) to GL(3, E), [E : F] = 2. The booklet develops the means of comparability of twisted and stabilized hint formulae and considers the "Fundamental Lemma" on orbital integrals of round capabilities. comparability of hint formulae is simplified utilizing "regular" capabilities and the "lifting" is said and proved via personality kinfolk. this allows an intrinsic definition of partition of the automorphic representations of SL(2) into packets, and a definition of packets for U(3), an explanation of multiplicity one theorem and stress theorem for SL(2) and for U(3), a decision of the self-contragredient representations of PGL(3) and people on GL(3, E) fastened via transpose-inverse-bar. particularly, the multiplicity one theorem is new and up to date. There are purposes to building of Galois representations by means of particular decomposition of the cohomology of Shimura forms of U(3) utilizing Deligne's (proven) conjecture at the mounted aspect formulation.

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8). 8)). Now if 8cr(8) is semisimple, with eigenvalues A, 1, A - 1 , then N8 is the stable class in H with eigenvalues A, A - 1 . If A ^ —1 then N18 is the class in Hi with eigenvalues A, 1, A - 1 . 47 II. Orbital integrals 48 Denote by ZG(5a) the group of x in G with 5a = Int(x)(5a), by ZH(J) the centralizer of 7 in H, and by Z^iji) the centralizer of 71 in H\. For / G C™(G), /o G C™(H), /1 G C ~ ( # i ) , define the or&itaZ integrate $(5a,fdg) = f(lnt(x)(5a))^, at JG/ZaiSrr) r (-ji, fidhi) = fi(lnt(x)(ji)) dx — , (i = 0,1), where we put f(gcr) = f(g)- These depend on choices of Haar measures, denoted dg or dx or dhi depending on the context.

Define F(5-f K |det(l-Ad((g)dg (n()(h) = [ f(g)ct>(a(h)g)dg = f JG = f f JG 1 f{a{h- )nak){S T])(a)(p(k)S-1(a)dndadk.

DEFINITION. This sa is called the absolutely semisimple part of ka and u is the topologically unipotent part of ka. PROOF. , then u\ = u% for a = 0ia2- Since (a, q) = 1, there are integers a^, (3N with a^a + (3NqN — 1. Then u2u^ = u«»a+e»oNu-a»°-0»"N = u02NqNu^NqN -+1 as N —> oo. For the existence, recall that the prime-to-p part of the number n of elements in GL(n,F q ) is c = f] {q1 — 1). Let {{ka)q *} be a convergent i=i subsequence in the sequence {{ka) qm ; qm = 1 (mod c«)} in (K, a). Denote the limit by sa,s £ K.

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Automorphic Representations of Low Rank Groups by Yuval Z Flicker

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