Selected Title s i n Thi s Serie s

16 Harish-Chandra , Admissibl e invarian t distribution s o n reductiv e p-adi c group s (wit h notes b y Stephe n DeBacke r an d Pau l J . Sally , Jr.) , 199 9

15 Andre w Mathas , Iwahori-Heck e algebra s an d Schu r algebra s o f the symmetri c group , 199 9

14 Lar s Kadison , Ne w example s o f Probeniu s extensions , 199 9

13 Yako v M . Eliashber g an d Wil l ia m P . Thurston , Confoliations , 199 8

12 I . G . Macdonald , Symmetri c function s an d orthogona l polynomials , 199 8

11 Lar s Garding , Som e point s o f analysi s an d thei r history , 199 7

10 Victo r Kac , Verte x algebra s fo r beginners , Secon d Edition , 199 8

9 Stephe n Gelbart , Lecture s o n th e Arthur-Selber g trac e formula , 199 6

8 Bern d Sturmfels , Grobne r base s an d conve x polytopes , 199 6

7 A n d y R . Magid , Lecture s o n differentia l Galoi s theory , 199 4

6 Dus a McDuf f an d Dietma r Salamon , J-holomorphi c curve s an d quantu m cohomology ,

1994

5 V . I . Arnold , Topologica l invariant s o f plan e curve s an d caustics , 199 4

4 Davi d M . Goldschmidt , Grou p characters , symmetri c functions , an d th e Heck e algebra ,

1993

3 A . N . Varchenk o an d P . I . Etingof , Wh y th e boundar y o f a roun d dro p become s a

curve o f orde r four , 199 2

2 Frit z John , Nonlinea r wav e equations , formatio n o f singularities , 199 0

1 Michae l H . Freedma n an d Fen g Luo , Selecte d application s o f geometr y t o

low-dimensional topology , 198 9

http://dx.doi.org/10.1090/ulect/016

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Admissible Invarian t Distributions o n

Reductive p-adi c Group s

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University

LECTURE Series

Volume 1 6

Admissible Invarian t Distributions o n

Reductive p-adi c Group s Harish-Chandra

Notes b y Stephen DeBacke r an d Pau l J . Sally , Jr .

American Mathematical Societ y Providence, Rhode Island

EDITORIAL COMMITTE E

Jerry L . Bon a Nicolai Reshetikhi n

Leonard L . Scot t (Chair )

1991 Mathematics Subject Classification. P r imar y 22E50 , 22E35 .

ABSTRACT. Thi s boo k contain s a faithfu l renderin g o f Harish-Chandra' s lecture s o n admissibl e invariant distribution s (originall y delivere d a t th e Institut e fo r Advance d Stud y i n the 1970s) . Fo r a p-adi c field Q , le t G denot e th e se t o f f2-rationa l point s o f a connecte d reductiv e Q-group . Le t g denote th e Li e algebr a o f G. Th e mai n purpos e o f thes e note s i s t o sho w tha t th e characte r o f an irreducibl e admissibl e representatio n o f G i s represented b y a locall y summabl e functio n o n G. The proo f o f thi s resul t begin s wit h a stud y o f harmoni c analysi s o n g. Th e ke y resul t her e i s tha t the Fourie r transfor m o f a G-invarian t distributio n (wit h compactl y generate d support ) o n g i s itself represente d b y a locall y summabl e functio n o n g. Moreover , th e Fourie r transfor m o f suc h a distributio n ha s a n asymptoti c expansio n abou t an y semisimpl e poin t o f g. Thi s result , an d most o f the wor k o n g, depend s heavil y o n Howe' s Theore m (fo r g) fo r whic h a proo f i s provided . Finally, thes e result s ar e transferre d t o G vi a Howe' s "Kirillo v theory. " Thes e note s ar e intende d for advance d graduat e student s an d mathematician s workin g i n thi s area .

Library o f Congres s Cataloging-in-Publicatio n Dat a

Harish-Chandra. Admissible invarian t distribution s o n reductiv e p-adi c group s / Harish-Chandr a ; note s b y

Stephen DeBacke r an d Pau l J . Sally , Jr . p. cm . — (Universit y lectur e series , ISS N 1047-399 8 ; v. 16 )

Includes bibliographica l reference s an d index . ISBN 0-8218-2025- 7 (alk . paper ) 1. p-adi c groups . 2 . Distributio n (Probabilit y theory) . I . DeBacker , Stephen , 1968 - .

II. Sally , Paul . III . Title . IV . Series : Universit y lectur e serie s (Providence , R.I. ) ; 16 . QA174.2.H37 199 9 512'.74—dc21 99-3101 2

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Contents

Preface i x

Introduction 1

Part I . Fourie r transform s o n th e Li e algebra 5

1. Th e mappin g / »- > 0? 5 2. Som e result s abou t neighborhood s o f semisimple element s 1 3 3. Proo f o f Theore m 3. 1 1 7 4. Som e consequences o f Theorem 3. 1 3 0 5. Proo f o f Theore m 5.1 1 3 2 6. Applicatio n o f the inductio n hypothesi s 3 8 7. Reformulatio n o f the proble m an d completio n o f the proo f 4 2 8. Som e result s o n Shalika' s germ s 4 8 9. Proo f o f Theore m 9. 6 5 1

Part II . A n extensio n an d proo f o f Howe' s Theore m 5 5

10. Som e specia l subset s o f Q 5 5 11. A n extensio n o f Howe's Theore m 5 8 12. Firs t ste p i n the proo f o f Howe' s Theore m 6 2 13. Completio n o f the proo f o f Howe' s Theore m 6 3

Part III . Theor y o n th e grou p 7 1 14. Representation s o f compact group s 7 1

15. Admissibl e distribution s 7 4 16. Statemen t o f the mai n result s 7 4 17. Recapitulatio n o f Howe's theory 7 5 18. Applicatio n t o admissibl e invarian t distribution s 7 6 19. Firs t ste p o f the reductio n fro m G t o M 7 9 20. Secon d ste p 8 2 21. Completio n o f the proo f 8 4 22. Forma l degre e o f a supercuspida l representatio n 8 6

Bibliography 9 1

List o f Symbol s 9 5

Index 9 7

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Preface

Harish-Chandra firs t presente d thes e note s o n admissibl e distribution s i n lec -tures a t the Institute fo r Advanced Stud y during 1973 . I n this preface , w e provide a brief guide to the content of Harish-Chandra's notes and discuss the advances in this area o f mathematic s sinc e thes e lecture s wer e delivered . O f course , an y suc h dis -cussion will necessarily overlap Harish-Chandra's ow n introductory remark s (whic h begin o n pag e 1) .

A sketc h o f thi s materia l wa s publishe d b y Harish-Chandr a i n hi s Queen' s notes [17] . Ever y statemen t i n Harish-Chandra' s Queen' s note s als o occur s here . Therefore, whe n w e mak e a statemen t whic h occur s a s a n enumerate d statemen t in th e Queen' s notes , w e provid e i n parenthese s th e statemen t numbe r appearin g there (see , for example , the statemen t o f Theore m 5.11) .

A numbe r o f year s ago , Harish-Chandr a aske d on e o f u s (Sally ) t o produc e a detailed versio n o f hi s Queen' s note s base d o n hi s ow n lectur e notes . A s wa s hi s custom, Harish-Chandr a produce d severa l version s o f hi s lectur e notes . W e hav e made onl y mino r change s to these , an d mos t o f these change s were with respec t t o the ordering . Th e two of us (DeBacke r an d Sally ) carefull y worke d through Harish -Chandra's notes , an d th e versio n include d her e wa s type d b y DeBacker . W e tak e full responsibilit y fo r an y errors .

The mai n results . Withou t furthe r commen t w e adopt th e terminology use d by Harish-Chandr a i n [20] .

Let 0 b e ap-adic field of characteristic zero with ring of integers R. Le t G be the group o f fi-rational point s o f a connected , reductiv e fJ-group . Th e grou p G , wit h its usual topology , i s a locally compact, totall y disconnected , unimodula r group . I n particular, i t ha s a neighborhoo d basi s o f th e identit y consistin g o f compac t ope n subgroups. Le t dx denot e th e Haa r measur e o n G an d le t Q denot e th e se t o f regular element s i n G.

A comple x representatio n (n, V) o f G i s smooth if , fo r eac h v £ V, ther e i s an ope n subgrou p K v o f G whic h fixe s v (i.e. , 7r(k)v = v fo r al l k £ K v). Th e representation (7r , V) i s admissible i f

(1) 7 r is smooth , an d

(2) fo r ever y compac t ope n subgrou p K o f G , th e spac e o f X-fixe d vector s ha s finite dimension .

ix

x PREFAC E

Every irreducibl e an d smoot h representatio n i s admissibl e [29] . Le t (TT,V) b e a n irreducible smoot h representatio n o f G . Denot e b y C£°(G) th e spac e o f locall y constant, compactl y supported , complex-value d function s o n G . Fo r / G C^°(G),

the operato r

*"(/) = / f( x) -n(x)dx JG

is an operator o f finite rank. Consequently , i t make s sense to define th e distribution

character o f TT b y

e w ( / ) = tr7r(/ )

for a l l / G G C ° ° ( G ) .

Motivated b y th e cas e o f rea l reductiv e groups , w e ma y as k i f ther e exist s a locally summabl e functio n F n o n G which i s locally constan t o n G' suc h tha t

©*(/)= [ f(x)-F n(x)dx JG

for al l / G C%°(G). I t i s the mai n purpos e o f these note s t o provid e a n affirmativ e answer t o thi s question . I f F i s a n arbitrar y nonarchimedea n loca l field, the n fo r the grou p GL n(F) thi s resul t wa s establishe d i n th e "tame " cas e b y Rodie r [59 ] and i n th e remainin g case s b y Lemair e [39] . I n general , fo r th e F-rationa l point s of a connecte d reductiv e grou p define d ove r F , th e mos t w e ca n sa y i s tha t th e distribution characte r o f a n irreducibl e smoot h representatio n i s represente d b y a locally constan t functio n o n th e se t o f regula r element s [22 ] (se e also Howe [25]) .

One o f th e majo r result s o f thes e note s i s a descriptio n o f th e behavio r o f ©^ nea r a semisimpl e poin t 7 o f G (se e Theore m 16.2) . Thi s i s accomplishe d b y deriving a n asymptoti c expansio n fo r G ^ i n a neighborhoo d o f 7 . Whe n 7 i s th e identity elemen t i n G , w e refer t o thi s asymptoti c expansio n a s th e local character

expansion o f TT. W e nee d som e definitions an d notatio n befor e describin g th e loca l character expansion .

Let g denot e th e Li e algebr a o f G . Le t C^°(g) denot e th e spac e o f complex -valued, locall y constant , compactl y supporte d function s o n g. Le t B b e a n Q-

valued, non-degenerate , symmetric , G-invarian t bilinea r for m o n g. Fi x a non -trivial additiv e characte r x o n f i . Le t dX denot e the Haa r measur e on the additiv e group of g and , fo r / G C£°(g), se t

f(Y)= ff(X)- X{B(X,Y))dX J9

for Y G g. Th e ma p / 1— • / i s a linea r bijectio n o f C^°(g) ont o itself . I f T i s a distributio n o n g (i.e. , a linea r functiona l o n C^(g)), w e defin e th e Fourie r transform T o f T b y

T(f) = T(f)

for/€Cc°°(fl).

PREFACE XI

If O i s a G-orbi t i n Q (under th e adjoin t action) , the n O carrie s a G-invarian t measure whic h w e denot e b y no [55] , I t wil l follo w fro m Theore m 4. 4 tha t th e Fourier transfor m o f the distributio n

f *-> Vo(f)

for / G C£°(fj) i s represented b y a locally summabl e functio n o n Q which i s locall y constant o n g' , th e se t o f regula r element s o f g. W e denot e thi s functio n b y ftQ.

Since ft ha s characteristi c zero , the se t o f nilpotent orbits , which we denote b y O(0), ha s finit e cardinality . W e ca n no w stat e th e loca l characte r expansio n (se e Theorem 16.2) :

THEOREM. Let IT be an irreducible smooth representation ofG. We can choose

complex numbers CQ(TT), indexed by O G O(0), such that

en(exVY)= ] T co(7r).fc(Y) oeO(o)

for all F G g' sufficiently near zero.

This remarkabl e theorem , whic h wa s firs t prove d b y How e [23 ] fo r th e genera l linear group , i s a qualitative resul t tha t leave s man y unresolve d quantitativ e ques -tions. Fo r example, almos t n o results exist abou t th e quantitative natur e o f the CQS

and th e ySe>s . Moreover , outsid e o f som e stunnin g wor k o f Waldspurge r [72 , 73 ] and a conjectur e o f Hales , Moy , an d Prasa d [43 ] w e have onl y limite d informatio n about th e precis e rang e i n which th e equalit y holds .

Quantitatively, thi s i s what w e know abou t th e CQS an d ftos. Fo r th e genera l linear group , How e [23 ] observed tha t th e function s JTQ have a ver y nic e form (se e also [41] ) an d showe d tha t CO(TT) is an intege r fo r ever y irreducibl e supercuspida l representation TT an d ever y nilpoten t orbi t O. B y usin g result s o f Kazhda n [31] , Assem [1 ] determine d th e function s jlo fo r SL^(H ) wit h £ a prime . Finally , b y using a resul t late r prove d i n genera l b y Huntsinge r [27] , DeBacke r an d Sall y [8 ] and Murnagha n [46 ] evaluate d a n integra l formul a t o obtai n value s fo r th e £LQS in the case s SL2(fi ) an d GSp 4(fl).

In Theore m 22. 3 Harish-Chandra derive s a formul a fo r th e leadin g ter m CQ(TT)

in th e loca l characte r expansio n o f a n irreducibl e supercuspida l representatio n TT

of G . Strengthenin g a conjecture o f Shalik a [66] , Harish-Chandra conjecture s tha t this formul a ough t t o hol d fo r al l irreducibl e discret e serie s representation s o f G . Rogawski prove d thi s i n [61] . Moreover , Huntsinge r [28 ] use d som e work o f Kazh -dan [30 ] to sho w tha t fo r a n irreducibl e tempere d representatio n TT, CQ(TT) i s zero if and onl y i f TT i s not a discret e serie s representation .

At th e othe r extreme , Rodie r [60 ] showe d (fo r spli t G ) tha t a n irreducibl e ad -missible representatio n TT ha s a Whittake r mode l i f an d onl y i f ther e i s a regula r nilpotent orbi t O suc h tha t CQ(TT) is no t zero . Moegli n an d Waldspurge r [41 ] re -fined thi s result . The y showe d tha t i f O i s maxima l amon g thos e nilpoten t orbit s for whic h CQ(TT) is nonzero , the n th e valu e o f CQ(TT) is related t o th e dimensio n o f

Xll PREFACE

a degenerat e o r generalize d Whittake r model . Ther e hav e been man y application s of thes e results . Fo r classica l group s Moegli n [40 ] showe d tha t i f O i s maxima l among thos e nilpoten t orbit s fo r whic h CQ{TT) i s nonzero , the n th e orbi t O i s spe -cial. Savi n [65 ] showe d that , fo r th e representation s constructe d b y Kazhda n an d Savin i n [32] , the loca l characte r expansio n involve s onl y th e trivia l orbi t an d th e minimal nilpoten t orbits . A representation wit h thi s propert y i s calle d a minima l representation. Thi s wor k wa s extende d b y Rumelhar t [63 ] an d Torass o [69] . A version o f Rodier' s resul t fo r coverin g group s o f GL n(fi) i s provide d an d use d b y Flicker an d Kazhda n i n [10] .

In general, the remaining CQS have been calculated explicitly in only a few cases, most notabl y i n the work of Assem [1] , Barbasch and Mo y [2] , and Murnagha n [45 , 46, 49 , 50 , 51] . I n [13 ] Hale s showe d tha t mos t o f th e basi c object s o f harmoni c analysis—including character s an d th e ftos —are non-elementary . Tha t is , a t som e point, thei r value s ca n b e calculate d b y countin g point s o n hyperellipti c curve s over finit e fields. Perhap s thi s i s why thes e object s hav e bee n s o hard t o quantif y explicitly.

A guid e t o thes e notes . Th e Li e algebr a 9 is a vector spac e ove r S I of finite dimension, an d G operate s o n 9 by th e adjoin t representation , denote d Ad . Le t T be a distributio n o n 9 . Then , fo r x G G, th e distributio n X T i s defined b y

xT{f)=T(fx)

for / e C c°°(s) wher e

fx(X) = f(Ad(x)X)

for X e 9 . Th e distributio n T i s sai d t o b e G-invarian t i f X T — T fo r al l x G G. Let J denot e th e spac e o f al l G-invarian t distribution s o n 9 .

For u) C 9 , le t J{UJ) denot e th e spac e o f al l G-invarian t distribution s T suc h that th e suppor t o f T i s containe d i n th e closur e o f Ad(G)cj . I f L i s a lattic e i n 9 (i.e. , a compac t ope n i^-submodul e o f 9 ) an d T i s a distributio n o n 9 , le t JLT

denote th e restrictio n o f T t o C c{g/L). Th e followin g theorem , whic h wa s first conjectured b y How e in [26] , makes nearl y everythin g i n thes e note s possible .

THEOREM 12. 1 (Theore m 2) . Let UJ be a compact set in 9 and L a lattice in

9. Then

dim JLJ(U) < (X) .

Although How e [23 ] prove d Theore m 12. 1 onl y fo r th e genera l linea r group , Harish-Chandra [17 ] attribute s thi s theore m t o him . Consequently , Theore m 12. 1 is often referre d t o a s Howe's Theorem i n the literature . Althoug h Theore m 12. 1 is used throughou t Par t I , the mos t significan t application s ca n b e foun d i n §1.1 , §4, and §5 . I n Par t I I o f thes e note s Harish-Chandr a state s an d prove s a n extensio n of Howe' s Theore m whic h i s use d i n Par t III , §21 . Fo r th e genera l linea r group ,

PREFACE xii i

Howe [23 ] wa s the first t o prov e thi s extension . Waldspurge r [76 ] als o proved thi s extension o f Howe' s Theore m i n th e contex t o f weighte d orbita l integrals . Wald -spurger's proo f include s th e situatio n unde r consideratio n i n thes e notes .

The ke y t o understandin g element s o f J i s Theore m 3.1 . Thi s theore m say s that th e regula r semisimpl e orbita l integral s ar e dens e i n J (i.e. , i f / G C£°(Q) an d ^o(f) = 0 fo r al l G-orbit s O whic h ar e containe d i n &', then T(f) — 0 fo r al l T G J ). Thi s result , combine d wit h a n understandin g o f fto fo r a regula r orbi t O

(§3) an d Howe' s Theorem , allow s Harish-Chandr a t o prov e tha t £LO is represente d by a locally summable functio n o n g which i s locally constant o n g' (Theore m 4.4) . Waldspurger [76 ] showe d tha t fa r fro m zero , th e functio n fto ha s a particularl y nice form. Anothe r applicatio n o f Howe's Theore m an d som e understanding o f th e geometry o f ope n an d close d G-invarian t neighborhood s o f zer o permit s Harish -Chandra to write down an asymptotic expansion of T fo r T € J(w) wit h UJ compact. Finally, i n § 7 Harish-Chandra derive s a n explici t integra l formul a fo r th e Fourie r transform o f a regula r orbita l integral . Thi s formul a let s hi m se e that th e functio n representing \r]\ 1//2 • \±o is locally bounde d o n Q (here r\ is the usua l discriminant) .

The technique s o f § 7 were extended b y Hun t singer [27 ] to sho w tha t th e func -tion representin g a compactly supporte d distributio n o n g has a n integra l formula . Rader an d Silberge r [54 ] extende d a resul t o f Harish-Chandr a [21 ] t o sho w tha t the character o f an irreducible discrete series representation ha s an integra l formul a which i s remarkably simila r t o th e integra l formul a fo r th e Fourie r transfor m o f a regular orbita l integra l obtaine d i n §7 . Murnagha n [47 , 48 , 50 , 51 ] showe d tha t this i s no t a n acciden t an d tha t i n som e case s th e characte r o f a supercuspida l representation ca n be related t o the Fourie r transfor m o f a regular orbita l integral . Murnaghan's wor k was extended b y Cunningha m [6 ] and DeBacke r [7] .

Following Shalika [66] , in § 8 Harish-Chandra develop s the theor y o f what hav e become known as Shalika germs. I n [56 , 57, 58] Repka explicitly computed the Sha-lika germs corresponding t o the regula r an d subregula r unipoten t orbit s o f GL n(fi) and SL n(fi) o n the se t o f regular ellipti c elements . (Ki m [33 , 34, 35 ] and Ki m an d So [36 ] partiall y compute d th e regula r an d subregula r Shalik a germ s fo r Sp 4(fi).) For regula r unipoten t orbits , thes e result s wer e extende d t o al l group s b y Shel -stad [67] , an d fo r subregula r unipoten t orbits , the y wer e extende d t o al l group s by Hale s [15] . Fo r GL n(fi), Rogawsk i [62 ] state d an d prove d (i n som e cases ) a conjecture abou t th e value s o f th e Shalik a germ s evaluate d a t certai n ellipti c ele -ments. Rogawski' s conjectur e fo r th e Shalik a germ s o f GL n(fi) wa s confirme d b y Waldspurger i n [74] . Thi s wor k wa s use d b y Murnagha n an d Repk a [52 ] t o in -vestigate whic h Shalik a germ s contribut e t o expansion s abou t singula r ellipti c ele -ments. Fo r GL n(fi), Waldspurge r [75 ] provided an algorithm fo r computing Shalik a germs which significantly extende d th e result s o f his earlier pape r [74] . Courte s [5 ] extended thi s wor k o f Waldspurger' s t o th e grou p SL n(fi). A fe w group s hav e had thei r Shalki a germ s nearl y completel y worke d out : Sall y an d Shalik a [64 , 66 ]

XIV PREFACE

computed the m fo r SL2(^ ) o n th e ellipti c set , Langland s an d Shelsta d [38 ] calcu -lated mos t o f them fo r SU(3 , fi), and Hale s [14 , 16 ] worked the m ou t fo r GSp 4(f£) and Sp 4(Q). Ther e ar e man y interestin g question s surroundin g Shalik a germ s an d the theor y o f endoscop y whic h ar e beyon d th e scop e o f thi s discussio n (se e [37 ] and [71]) .

Finally, i n Par t II I Harish-Chandra studie s admissibl e distributions o n G. Th e goal of this sectio n i s to provid e a way to transfe r th e result s o f Par t I and Par t II , which wer e concerne d wit h G-invarian t distribution s o n g , t o admissibl e distribu -tions o n G . Suppos e tha t w e ar e onl y concerne d wit h th e behavio r o f ou r distri -bution o n G nea r th e identity . Th e definitio n o f a n admissibl e distribution , whic h Harish-Chandra attribute s t o th e wor k o f How e (se e [20 , §16] , and [24 , 25 , 26]) , combined with some results derived from Howe' s "Kirillo v theory" ([24 , 25]) , allows him t o deriv e fro m a n admissibl e distributio n o n G a distribution o n £ which

(1) satisfie s th e hypothesi s o f the extensio n o f Howe's Theore m an d

(2) i s related t o th e origina l distributio n o n G vi a th e exponentia l map .

Harish-Chandra i s then abl e t o conclud e tha t nea r th e identity , 0 ^ i s represente d

by a locall y summabl e functio n o n G.

There have been some generalizations of these notes to different settings . I n [4], Clozel showe d tha t th e mai n result s o f these note s hol d fo r non-connecte d groups . Clozel's pape r als o include s almos t al l o f Par t II I o f these notes . I n [11 ] an d [12] , Hakim extended the content of these notes to certain symmetric spaces. However , in general, no t everythin g i n these note s ca n be extended t o symmetri c spaces . Rade r and Ralli s [53 ] generalize d som e o f wha t ca n b e carrie d ove r an d provide d coun -terexamples fo r thos e result s whic h canno t b e extende d t o th e symmetri c spac e setting. Fo r furthe r analysi s o f th e symmetri c spac e situation , se e th e wor k o f Bosman [3 ] an d Flicke r [9] . Finally , Vignera s [70 ] explore d th e situatio n fo r mod -ular representations .

We than k J . Adler , J . Boiler , R . Huntsinger , D . Joyner , an d M.-F . Vignera s for thei r extensiv e correction s an d comment s o n earlie r version s of these notes . W e also than k J . Hakim , T . Hales , R . Kottwitz , G . Muic , F . Murnaghan , G . Savin , and th e reviewe r fo r thei r comment s o n earlie r draft s o f thi s opening .

Stephen DeBacke r Paul J . Sally , Jr .

The Universit y o f Chicago , 1999 .

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List o f Symbol s

ft, 1

O.T, 1

G\ 1

q, 1

0 , 1 T>, 1

J, 1 J(w), 2

fl, 2

T L , 2

J L , 2

£ , 2

X, 2

/ , 2

4 2

*7g,2

0 ' , 2 e>, 2 CG (Xo) , 2

Me>, 3 Af, 3

co(T) , 3

C^(TT), 3

CO(TT), 3

d(7T), 3

0 / , 5

£/> 6

/ P , 7

17g/m. 8

r?m, 8

w(A2\A1)i 9

Ai - « A2 , 9

Ai x A 2 , 9

0b. 1 1 °D, 1 2

d(O), 1 3

r(O), 1 3

Cfl(X), 1 3

0 (0) , 1 3

|X|, 1 4

0 ( 7 ) , 1 6 £>o, 17

Mo(/) , 1 7 A/d, 1 9

3,23

W^, 2 3

[SI 2 5

Jo, 2 7

0 ~ 0 , 32 c 5 , 3 2

R', 3 5

0e, 3 8

w, 4 2

5, 4 2

r 0 , 4 8

f)", 5 0

^ o , 5 1

4 , 5 1 4g,57

X(A), 5 8

L*, 5 8

0W, 5 8

/ x , 5 8

J(V,*,L), 5 9

C(V,t,L), 5 9

0a , 5 9 J(V,oo,L) , 5 9

Jo, 5 9

Jo(V,t ,L) , 5 9

J 0 (V,oo,L) , 5 9

5(g), 6 0

/(0), 6 0 C(L), 6 2

£(K) , 7 1

id, 71

d{d), 7 1

£ F , 7 1

A(F), 7 1

96 LIS T O F SYMBOL S

A(d), 7 1 \d: 6], 7 1 [ F i : F 2 ] , 7 1

£x,Fit 7 3

1 K 0 , 7 4

DG, 7 5 K 1 / 2 ) 7 5

£ 1 / 2 j 7 5

Od, 7 6 C / M , 7 7

e F , 7 8

m0, 7 9

rc, 8 0

L„, 8 0

KU1 8 0

xi/2, so 3V, 8 0

in so Oo, 8 2

0(6), 8 2

F / , 8 6

*4(TT), 8 7

ra^L), 8 8

Index

admissible

admissible distribution , 7 4

(G, Xo)-admissible, 7 4

G-admissible, 7 4

admissible representation , i x

admissible subset , 5 7

(G, g)-admissible G-domain , 5 8

Condition C{L), 6 2

Condition C(V,t,L), 5 9

cusp form , 1 2

distribution, x

distribution character , x

G-invariant distribution , 1

elliptic

elliptic Carta n subalgebra , 5

elliptic Carta n subgroup , 8 6

elliptic orbit , 3 8

exp, 5 5

Fourier transfor m

of a distribution , 2

of a function , 2

G-domain, 3 , 1 3

germ

O-germ, 4 8

Shalika germ , 4 8

homogeneity

of nilpoten t orbita l integrals , 1 8

of Shalik a germs , 4 8

Howe's Theorem , xii , 2 , 6 2

dual lattice , 5 8

s-lattice, 7 5

local characte r expansion , x

locally summable , 1

log, 5 5

nilpotent

nilpotent element , 3

nilpotent orbit , 3

orbit, 2

dimension of , 1 3

elliptic orbit , 3 8

G-orbit, 1 3

rank of , 1 3

regular orbit , 3 0

regular

regular elemen t o f g , xi , 2

regular elemen t o f G , ix , 1

regular orbit , 3 0

semisimple element , 3

smooth representation , i x

unipotent element , 3

interact, 7 1

intertwines, 7 1

lattice, 2

adapted lattice , 5 8

well adapte d lattice , 5 8

97