By Jonathan David Rogawski
The function of this e-book is to increase the strong hint formulation for unitary teams in 3 variables. The reliable hint formulation is then utilized to procure a category of automorphic representations. This paintings represents the 1st case within which the strong hint formulation has been labored out past the case of SL (2) and similar teams. Many phenomena so that it will look within the common case current themselves already for those unitary groups.
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The aim of this publication is to increase the reliable hint formulation for unitary teams in 3 variables. The sturdy hint formulation is then utilized to procure a class of automorphic representations. This paintings represents the 1st case during which the sturdy hint formulation has been labored out past the case of SL (2) and comparable teams.
Extra info for Automorphic Representation of Unitary Groups in Three Variables
3). a; H (! H, /) is called a transfer factor. It depends only on the stable conjugacy class of /H in H and the conjugacy class of I in G. 2) A general definition of the transfer factor has been proposed by Langlands and Shelstad ([LS]). The definition specifies D.. 0 ; H up to a non-zero constant that depends on certain choices. a;H(/H,::Y) is independent of the choice of compatible collection. Furthermore, D.. 3. >st(IH )) and let f = Ilvfv E C(G,w). 3). >st( I/ A). The sum is finite by the above remark.
Suppose that E /Fis local. Let / E M be a singular, non-central element. , = H. 2, there exists a Cartan subgroup T of type (1) contained in H such that O(T, G) = Z /2. Suppose that 1' E T has the same eigenvalues as/· Then 1' is stably conjugate to I in G, but since OF(T, G) = S 3 , we can choose 1' so that it is not conjugate to / in G. In particular, / 1 is regular as an element of H. Let bH be the nontrivial element in 'DH(T/F) and set 1" = (-y') 0n. 1. This proves the next proposition. 2: Assume that E/F is local.
2. Some results of Kottwitz. In this section, we assume that c: is trivial. Let / E G be a semisimple element and let I= G'Y. By a theorem of Steinberg ([St], pg. 57), the assumption that Gder is simply connected implies that I is connected. 1): If G is quasisplit and fuer is simply connected, then every conjugacy class in G(F) which is defined over F contains an element of G. THEOREM Suppose that G' is an inner form of G and let '¢ : G --+ G' be an inner twisting. /J(r(g))x-; 1 for all g E G. ) is therefore defined over F.