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 Author Xu, Bin Title Endoscopic Classification of Representations of GSp(2n) and GSO(2n) URL http://hdl.handle.net/1807/68169 Publication Date 2014 Date Available 2015-04-22 00:00:00 Degree PhD Degree Level doctoral University/Publisher University of Toronto Abstract In 1989 Arthur conjectured a very precise description about the structure of automorphic representations of reductive groups using Arthur packets and endoscopy theory. In his recent monograph [Art13], he proved this conjecture for symplectic groups and orthogonal groups G. The results are conditional on the stabilization of twisted trace formula, which is a project in progress under Moeglin and Waldspurger (see [MW14], [Wal14a]-[Wal14e]). Our goal is to extend Arthur's result to general symplectic groups and general even orthogonal groups eG, by studying the restriction of representations from $\tilde{G}$ to G. This idea goes back to Labesse and Langlands [LL79], where they considered G to be SL(2) and $\tilde{G}$ to be GL(2).To extend Arthur's result, there are two main problems that we have solved in this thesis. One is local and the other is global. Locally, we have determined the tempered Arthur packets for $\tilde{G}$, and shown they satisfy certain character relations under the endoscopic transfer. Globally, we have proven the functoriality of endoscopic transfer for a large family of tempered automorphic representations of $\tilde{G}$. Subjects/Keywords Arthur packet; automorphic representation; endoscopy; multiplicity; similitude group; trace formula; 0405 Contributors James, Arthur; Mathematics Country of Publication ca Record ID handle:1807/68169 Repository toronto-diss Date Retrieved 2020-03-09 Date Indexed 2020-03-09

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…there exists a discrete automorphic representation π of G, then < ·, π̃ >=< ι(·), π >= ε|Sψ̃ = εψ̃ and hence π̃ is also a discrete automorphic representation under Arthur’s global conjectures. Motivated by this we make the following conjecture…

…e π̃ is a discrete automorphic Conjecture 1.1.4. Suppose π̃ ∈ Πψ̃ for some ψ̃ ∈ Ψ(G), e if and only if there exists a discrete automorphic representation π of representation of G G in the restriction of π̃ to G. In fact Conjecture 1.1.4 is…

…representations of G as H̄(G)-module if and only if < ·, π >= 1. Moreover, if π is a discrete automorphic representation of G, its multiplicity is m(π) = 1 or 2, and m(π) = 2 only when G is even orthogonal and φ is not O(2n, C…

…we let Y be the group of idele class characters over F , identified with the characters of the global Weil group WF under the global class field theory. e and there Theorem 1.2.3. Suppose that π̃ is a discrete automorphic representation of G exists φ…

…x28;G) b Γ is finite. Let Ψ2 (G) be the set of parameters in Ψ(G) whose centralizer in G The global conjectures of Arthur assert that all discrete automorphic representations π of G are contained in a certain global Arthur…

…trivial on the SL(2, C) factor. The existence of the packet structure for automorphic forms is motivated by the trace formula and its stabilization. Let ZG be the centre of G, and χ be a character of ZG (F )\ZG (AF )…

…on G(Fv ) defined by the trace of πv (f ) := f (g)πv (g)dg on the space of representation πv . And it is clear from this conjecture G(F ) that stability plays a key role in characterizing the packet…

…vice versa. us to parametrize the automorphic representations of G by that of G, So in the absence of the global Langlands group, one can still try to deduce Arthur’s e through conjectures (in a certain modified form) from one to the other…