By Jonathan David Rogawski

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**Extra resources for Automorphic Representation of Unitary Groups in Three Variables**

**Example text**

Let I be a regular element in T. 1, there exists 1' E H which is stably conjugate but not conjugate to I in G. (T, G) = Z/2. 8. Singular semisimple elements. Let G = U(3). For ~ E F*, let He be the unitary group in two variables defined by the Hermitian form: He The isomorphism class of depends only on ~ modulo NE* and we obtain a bijection between F* /NE* and the set of isomorphism classes of unitary groups in two variables over F with respect to E. The group H 1 is quasisplit and is isomorphic to U(2).

Let IP(T) be the set of unramified characters of T. (T, G)-orbit and every element of E"( G) is of the form 7r x for some x E Il"(T). Fix an element Wp E Wp whose projection to r(Fun /F) is the Frobenius element. (T, G) (and hence E"(G)) and the set of semisimple G- conjugacy classes in LG of the form {g x wp }. The conjugacy class {g(7r)} in LG associated to a representation 7r E E" ( G) is called the Langlands class of 7r. We can choose a representative g x wp with g E f. Let (H, s, T/) be an endoscopic datum for G.

Regular class in G. Suppose that 1' E TH. Then 1' is called ( G, H)-regular if a('ef;(I')) f. 1 for each root a of T which is not the image of a root of TH in H. Let 1' be a ( G, H)-regular element of TH. Suppose that 'lj; is defined over F (this entails no loss of generality since the choice of Tis arbitrary) and let I= 'ef;(1'). H in H -Y, and that of T in G-y. It follows that 'lj; extends to an isomorphism of H-y' with G-Y which is an inner twisting over F. In particular, we can identify Z(H-y') with Z(G-y) and, if Fis local, we can choose compatible measures on H-y' and G-y.