On Some Problems Suggested Bv the Trace Formula

where c is t h e i d e n t i t y component of C Then C i s a f i n i t e 4' 4' 4' group which is known t o be abe l ian . ( [ l l ( c ) I . See a l s o [ 5 ] . ) I t can t h e r e f o r e be c a n o n i c a l l y i d e n t i f i e d w i t h an a b s t r a c t Froup which de-pends o n l y on t h e c l a s s of 4. For each on n4' x C$, such t h a t t h e r a n i s an i n j e c t i o n from. i n t o t h e uroun 8 of c h a r a c t e r s of . 0) 6 Unfor tuna te ly , t h e p a i r i n g cannot be d e f i n e d c a n o n i c a l l v . HoweverShelstad shows that there is a function c from cG/zG to {?I}, which is invariant on conjuqacy classes, such that is independent of the pairin?. Here, s is the projection of s onto C This latter function can be used to map functions on G ( p ) to 4) ' functions on endoscopic Froups. Given a parameter ' ' 'temp (G/IR) and a senisinple element s â CG/ZGr one can check that there is a uniaue endoscopic Froun H = H such that s $I then defines a parameter C Otep-p (~133 ) . For a viven H, ever17 parameter in atemp(H/IR) arises in this wav. For anv Function f â c(G(IR) , Shelstad defines a function f â C;(H(TR)) , uniaue H up to stable distributions on H(I!?) . To do so, it is enouyh to specify the value for every such .I$-,. This is done bv settino Actuall~, Shelstad defines fH bv transferrino orbital intevralsr and then proves the formula (1.2.3) as a theorem. Fowever, we shall take the formula as a definition. Shelstad shows that the riaoriinrf -r f H is canonically defined up to a sicrn. (It also depends on the enbedrlincr H c G which we have fixed.) Pie shall fix the sir-ns in anv wav, asking only that in the case H = GI f be consistent with the notation above. That is, c(l) = 1. 1.3. It is important for the trace formula to understand how the notions above relate to nontempere6 parameters (o. Shelstaci fiefined the pairings <>> only for tempered , but it is easv enouoh to extend the definition to arbitrary parameters. For one can show that there is a natural way to decompose anv narameter (o bv ( w ) = (oo (w) (o+ (w) I ^ O ' ̂tei-o (Wp ) , <))+ c ~ ( C / S R I ) , so that the images of (oo and @ + commute, and so that (o itself is tempered whenever '(. (Wn) = { 11. The centralizer in L~ of the imase L of + will be the Levi component ' M of a parabolic subqroup of G , and n\ will consist of a positive quasi-character v+ of M(B). The image of @ must lie n 'Â¥M so that (oo defines an element in Ètem (M/B) . There will be a bijection between ~ > n d a , the 0 e n n I] being the Lanflands quotients obtained froi" the tempered representations in y o and the positive ~uasi-character v+ of M(E) . On the other hand C" equals C: , so we can define the $0 p a r on c x to be the one obtained fro^ the pairin? on (o -IT: ô 0 ' However, simply defining the pairing for nontempered 4 is not satisfactory. For it could well happen that the distribution is not stable if the parameter (t is not terioered. A related $i^ficulty is that (1.2.3) no lonoer ~akes sense if 41. is not a tempered parameter for H. We shall define a subset o-F a('";/:[") -For which these difficulties are likely to have nice solutions. The subset will contain Ã̂ temp(G/ll?) and oucrht also to account for the representations of G(B) which are of interest in crlobal annlications. Let Y(G/IR) be the set of "GO-conjuoacv classes o-F maps such t h a t t h e r e s t r i c t i o n o f I/J t o raTm beloncrs t o @tern ( R / P 1 . For any ip ? Y ( G / I R ) d e f i n e a parameter 4 ' i n @ ( G / l R ) bv Here it i s h e l p f u l t o r e c a l l t h a t i s t h e map f r o n 1' 7 t o IR SL(2,C) = L ( ~ r ^ ( 2 ) ) 0 which a s s i g n s t h e t r i v i a l r e p r e s e n t a t i o n t o P R L ( 2 , P ) . R e c a l l a l s o t h a t t h e u n i p o t e n t conjucracv c l a s s e s i n anv cor'nlex crroun a r e b i j e c t i v e w i t h t h e conjuoacy c l a s s e s o f m a p s o f SL(2,C) i n t o t h e oroun. "he u n i p o t e n t conjuoacv c l a s s e s f o r coriolex crroups have been c l a s s i f i e c 1 hv weighted Dynkin diacrrans. (See [I31 .) Mow anv ip c Y ( R / B ) can be i d e n t i f i e d wi th a p a i r (4>,p), i n which 4> (. QePn ( W P ) and p is a map from SL(2,C) i n t o C criven up t o coniur'acv bv C prom 4)' 4' ' t h e c l a s s i f i c a t i o n of n i l p o t e n t s it fo l lows t h a t p i s c'eterninec' bv i t s r e s t r i c t i o n t o t h e diacronal suborouo of SL(2.C) . We o b t a i n p r o p o s i t i o n 1.3.1: The nap Thus, V ( G / I R ) can be regarded a s a s u b s e t o f @(P/l? ) . I t c o n t a i n s @temp ( G / R ) a s t h e set of $ = ($I, p ) w i t h p t r i v i a l . Conjecture 1.3.2: For any $ ? f ( G / I P ) , the representations in are all unitarv. suppose that $ = ( 4 ,p) is an arbitrary parapeter in CJ ( P / P ) . Copying a previous definition we set