Complex Analysis by Eells J. (ed.)

By Eells J. (ed.)

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We write (2-26) 23(x,y) for the block corresponding to the admissible pair (x, y). (2-27) Example: SL(2). The blocks of Example (2-22) are obtained as follows: (1) G = SL(2): (a) ~(o, Vo) is the block {PSe, 11"+, 11"_} of the trivial representation of SU(I, 1), (b) ~(too,vo) = {"';;~via,} for SU(2,O), (c) ~(-too,Vo) = {1I"~;~vial} for SU(O,2), (d) ~(o,to Vo) = {pso} for SU(I, 1). (2) G = PGL(2): (a) ~(o, V o) is the block {1I"d,PStrivial,PS8gn} containing the trivial representation of PU(I, 1), (b) ~(o,to Vo) = {ps+(2p)} for PU(I, 1), (c) ~(o, -to vo) = {ps-(2p)} for PU(I, 1), (d) 23(t oo, Vo) = {1I";;~vial} for PU(2,O).

5]) we may in fact choose y E vD, and furthermore require that int(y)lvH be a principal involution. Conjugating by vG we obtain the following Lemma (ef. 8]) which we use to compute weak endoscopic data. (4-3) Lemma. Fix v6 0 E vD, let Vo = int(V6 o), and letVK = (VG(8. (1) Given an elliptic element 8 with 11"(8') E vK, let vH r be the group generated by the centralizer vH of s = 11"(8) and v6 ' Then (8', VH r ) 0 is a set of weak endoscopic data, and every set of weak endoscopic data is equivalent to a set of this form.

We obtain a duality mapping from (g,kz ) modules to (Vg,Vky ) modules, and similarly dual blocks. Then Theorem (2-36) holds as stated. Note that we allow the E-groups Gf' and vGf' to vary independently. For example, let Gf' be an L-group, but let vGf' be a more general E-group. Then we see a block 23 of (g, /(z) modules is dual to a block of (Vg,Vk y ) modules for various covering groups of vG, including the trivial one. This flexibility will be important in §6. (2-46) Example: 8L(2). The block containing the trivial representation of SU(1, 1) is dual to the block of the trivial representation of PU(I, 1).

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