By Jeffrey Adams (auth.), Prof. William H. Barker, Prof. Paul J. Sally Jr. (eds.)

A convention on Harmonic research on Reductive teams used to be held at Bowdoin university in Brunswick, Maine from July 31 to August eleven, 1989. The acknowledged target of the convention was once to discover fresh advances in harmonic research on either actual and p-adic teams. It used to be the 1st convention because the AMS summer time Sym posium on Harmonic research on Homogeneous areas, held at Williamstown, Massachusetts in 1972, to hide neighborhood harmonic research on reductive teams in such element and to such an volume. whereas the Williamstown convention was once longer (three weeks) and a bit of broader (nilpotent teams, solvable teams, in addition to semisimple and reductive groups), the constitution and timeliness of the 2 conferences used to be remarkably comparable. this system of the Bowdoin convention consisted of 2 components. First, there have been six significant lecture sequence, each one along with numerous talks addressing these themes in harmonic research on genuine and p-adic teams which have been the focal point of extensive study throughout the earlier decade. those lectures all started at an introductory point and complex to the present country of analysis. Sec ond, there has been a sequence of unmarried lectures during which the audio system awarded an outline in their most recent research.

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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).