FUNDAMENTAL SOLUTION FOR ( λ z ) ν ON A SYMMETRIC SPACE G/K

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1 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K AMY T. DECELLES Absrac. W drmin a fundamnal soluion for h diffrnial opraor λ ν on h imannian symmric spac G/K, whr G is any complx smi-simpl Li group, and K is a maximal compac subgroup. W dvlop a global onal sphrical Sobolv hory, which nabls us o us h harmonic analysis of sphrical funcions o obain an ingral rprsnaion for h soluion. Thn w obain an xplici xprssion for h fundmnal soluion, which allows rlaivly asy simaion of is bhavior in h ignvalu paramr λ, wih an y owards furhr applicaions o auomorphic forms involving asociad Poincaré sris. 1. Inroducion W drmin a fundamnal soluion for h diffrnial opraor λ ν on h imannian symmric spac G/K, whr G is any complx smi-simpl Li group and K is a maximal compac subgroup. Sinc h dla funcion δ 1 K a h bas poin is a bi-k-invarian, compacly suppord disribuion, a suiabl global onal sphrical Sobolv hory nsurs ha h harmonic analysis of sphrical funcions producs a soluion. In his papr, w firs dvlop h suiabl Sobolv hory, hn driv h fundamnal soluion. To our knowldg, his is h firs consrucion of Sobolv spacs of bi-k-invarian compacly suppord disribuions. Insad of using h xisnc of a fundamnal soluion o prov solvabiliy of a diffrnial opraor, as in, for xampl, [2, 3, 21, 5, 6], w obain an xplici xprssion for h fundamnal soluion, wih y owards furhr applicaions involving h associad Poincaré sris. For xampl, w hav alrady obaind an xplici formula rlaing h numbr of laic poins in an xpanding rgion in a symmric spac o h auomorphic spcrum [7]. In paricular, h prsnc of a complx ignvalu paramr in h diffrnial opraor maks h fundamnal soluion suiabl for furhr applicaions, and h simpl, xplici naur of h fundamnal soluion allows rlaivly asy simaion of is bhavior in h ignvalu paramr, proving convrgnc of h associad Poincaré sris in L 2 and, in fac, in a Sobolv spac sufficin o prov coninuiy [7]. Furhr, his maks i possibl o drmin h vrical growh of h Poincaré sris in h ignvalu paramr. For a drivaion of h fundamnal soluion in h cas G SL 2 C, assuming a suiabl global onal sphrical Sobolv hory, s [11, 13]. Our rsuls for h gnral cas ar skchd in [12]. Afr having submid an iniial vrsion of his papr, i was brough o our anion ha Wallach drivs a similar, hough lss xplici, formula in Scion 4 of [22]. An inroducion o posiivly indxd Sobolv spacs of bi-k-invarian funcions can b found in [4]. Our main rsul is h following horm, whos proof is givn in 3.1. Thorm. L G b a complx smi-simpl Li group wih maximal compac K. Whn G is of odd rank, l ν d + n+1 2, whr d is h numbr of posiiv roos, no couning mulipliciis, and 21 Mahmaics Subjc Classificaion. Primary 43A85; Scondary 43A9, 22E46, 46F12, 33C52, 58J4. Ky words and phrass. fundamnal soluion, onal sphrical funcions, Sobolv spacs. This papr prsns rsuls from h auhor s PhD hsis, compld undr h suprvision of Profssor Paul Garr, whom h auhor hanks warmly. Th auhor would also lik o hank Brian Hall for svral hlpful convrsaions. Th auhor was parially suppord by h Docoral Dissraion Fllowship from h Gradua School of h Univrsiy of Minnsoa and by NSF gran DMS

2 2 AMY T. DECELLES n dim a h rank. Thn h bi-k-invarian fundamnal soluion u for h opraor λ ν on G/K is givn by: u a 1d+n+1/2 π n+1/2 π + ρ Γd + n + 1/2 αlog a 2 sinhαlog a log a Whn G is of vn rank, l ν d + n Thn, wih K n h usual modifid Bssl funcion, u a 1 d+n/2+1 π n/2 π + ρ Γd + n/2 + 1 αlog a 2 sinhαlog a log a K 1 log a 2. Sphrical ransforms, global onal sphrical Sobolv spacs, and diffrnial quaions on G/K 2.1. Sphrical ransform and invrsion. L G b a complx smi-simpl Li group wih fini cnr and K a maximal compac subgroup. L G NAK, g n+a+k b corrsponding Iwasawa dcomposiions. L Σ dno h s of roos of g wih rspc o a, l Σ + dno h subs of posiiv roos for h ordring corrsponding o n, and l ρ 1 2 α Σ m αα, m + α dnoing h mulipliciy of α. L a C dno h s of complx-valud linar funcions on a. L X K\G/K and Ξ a /W a +. Th sphrical ransform of Harish-Chandra and Brin ingras a bi-kinvarian agains a onal sphrical funcion: Ff ξ fg ϕ ρ+iξ g dg G Zonal sphrical funcions ϕ ρ+iξ ar ignfuncions for Casimir rsricd o bi-k-invarian funcions wih ignvalu λ ξ ξ 2 + ρ 2. Th invrs ransform is F 1 f fξ ϕ ρ+iξ cξ 2 dξ Ξ whr cξ is h Harish-Chandra c-funcion and dξ is h usual Lbsgu masur on a n. For brviy, dno L 2 Ξ, cξ 2 by L 2 Ξ. Th Planchrl horm assrs ha h spcral ransform and is invrs ar isomris bwn L 2 X and L 2 Ξ Characriaions of Sobolv spacs. W dfin posiiv indx onal sphrical Sobolv spacs as lf K-invarian subspacs of complions of Cc G/K wih rspc o a opology inducd by sminorms associad o drivaivs from h univrsal nvloping algbra, as follows. L Ug l b h fini dimnsional subspac of h univrsal nvloping algbra Ug consising of lmns of dgr lss han or qual o l. Each α Ug givs a sminorm ν α f αf 2 L 2 G/K on C c G/K. Dfiniion 2.1. Considr h spac of smooh funcions ha ar boundd wih rspc o hs sminorms: {f C G/K : ν α f < for all α Ug l } L H l G/K b h complion of his spac wih rspc o h opology inducd by h family {ν α : α Ug l }. Th global onal sphrical Sobolv spac H l X H l G/K K is h subspac of lf-k-invarian funcions in H l G/K. Proposiion 2.1. Th spac of s funcions C c X is dns in H l X. Proof. W approxima a smooh funcion f H l X by poinwis producs wih smooh cu-off funcions, whos consrucion givn by [1], Lmma is as follows. L σg b h godsic disanc bwn h coss 1 K and g K in G/K. For >, l B dno h ball B {g G : σg < }. L η b a non-ngaiv smooh bi-k-invarian funcion, suppord in B 1/4, such ha ηg ηg 1, for all g G. L char +1/2 dno h characrisic funcion of B +1/2, and l η η char +1/2 η. As shown in [1], η is smooh, bi-k-invarian, aks valus bwn

3 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 3 ro and on, is idnically on on B and idnically ro ousid B +1, and, for any γ Ug, hr is a consan C γ such ha sup γ η g C γ g G W will show ha h poinwis producs η f approach f in h l h Sobolv opology, i.. for any γ Ug l, ν γ η f f as. By dfiniion, ν γ η f f γ η f f L 2 G/K Libni rul implis ha γ η f f is a fini linar combinaion of rms of h form αη 1 βf whr α, β Ug l. Whn dgα, αη 1 βf L2 G/K η 1 βf L 2 G/K β fg 2 dg Ohrwis, αη 1 αη, and αη 1 βf L2 G/K αη βf L2 G/K sup α η g g G β fg 2 dg σg σg σg β fg 2 dg L B b any boundd s conaining all of h finily many β ha appar as a rsul of applying Libni rul. Thn ν γ η f f sup β fg 2 dg β B σg Sinc B is boundd and f H l X, h righ hand sid approachs ro as. Proposiion 2.2. L Ω b h Casimir opraor in h cnr of Ug. Th norm 2l on Cc G/K K givn by f 2 2l f Ω f Ω 2 f Ω l f 2 whr is h usual norm on L 2 G/K, inducs a opology on Cc G/K K ha is quivaln o h opology inducd by h family {ν α : α Ug 2l } of sminorms and wih rspc o which H 2l X is a Hilbr spac. Proof. L {X i } b a basis for g subordina o h Caran dcomposiion g p + k. Thn Ω i X i X i, whr {X i } dnos h dual basis, wih rspc o h Killing form. L Ω p and Ω k dno h subsums corrsponding o p and k rspcivly. Thn Ω p is a non-posiiv opraor, whil Ω k is non-ngaiv. Lmma 2.1. For any non-ngaiv ingr r, l Σ r dno h fini s of possibl K-yps of γ f, for γ Ug r and f Cc G/K K, and l C r b a consan grar han all of h finily many ignvalus λ σ for Ω k on h K-yps σ Σ r. For any ϕ Cc G/K of K-yp σ Σ m and β x 1... x n a monomial in Ug wih x i p, β ϕ, β ϕ Ω + C m+n 1 n ϕ, ϕ whr, is h usual innr produc on L 2 G/K. Proof. W procd by inducion on n dg β. For n 1, β x p. L {X i } b a slf-dual basis for p such ha X 1 x. Thn, xϕ, xϕ X i ϕ, X i ϕ Xi 2 ϕ, ϕ Ω p ϕ, ϕ Ω + Ω k ϕ, ϕ i i Ω + C m ϕ, ϕ Ω + C m+n 1 ϕ, ϕ For n > 1, wri β xγ, whr x x 1 and γ x 2... x n. Thn h K-yp of γϕ lis in Σ m+n 1, and by h abov argumn, x γϕ, x γϕ Ω + C m+n 1 γϕ, γϕ

4 4 AMY T. DECELLES L Cc G/K Σr b h subspac of Cc G/K consising of funcions of K-yp in Σ r and L 2 G/K Σr b h corrsponding subspac of L 2 G/K. For h momn, l Σ Σ m+n 1 and C C m+n 1. Thn, by consrucion, Ω k + C is posiiv on Cc G/K Σ, and hus Ω + C Ω p Ω k + C is a posiiv dnsly dfind symmric opraor on L 2 G/K Σ. Thus, by Fridrichs [8, 9], hr is an vrywhr dfind invrs, which is a posiiv symmric boundd opraor on L 2 G/K Σ, and which, by h spcral hory for boundd symmric opraors, has a posiiv symmric squar roo in h closur of h polynomial algbra C[] in h Banach spac of boundd opraors on L 2 G/K Σ. Thus Ω + C has a symmric posiiv squar roo, namly 1, dfind on Cc G/K Σ, commuing wih all lmns of Ug, and Ω + C γϕ, γϕ γ Ω + C ϕ, γ Ω + C ϕ Now h K-yp of Ω + C ϕ, bing h sam as ha of ϕ, lis in Σ m, so by induciv hypohsis, γ Ω + C ϕ, γ Ω + C ϕ Ω + C m+n 2 n 1 Ω + C ϕ, Ω + C ϕ and his compls h proof of h lmma. Ω + C m+n 2 n 1 Ω + C m+n 1 ϕ, ϕ Ω + C m+n 1 n ϕ, ϕ L α Ug 2l. By h Poincaré-Birkhoff-Wi horm w may assum α is a monomial of h form α x 1... x n y 1... y m whr x i p and y i k. Thn, for any f C c G/K K, ν α f αf, αf L2 G/K x 1... x n f, x 1... x n f L2 G/K x i p By h lmma, hr is a consan C, dpnding on h dgr of α, such ha ν α f Ω + C dg α f, f for all f Cc G/K K. In fac, for bi-k-invarian funcions, Ω + C dg α f Ω p + C dg α f. Sinc Ω p is posiiv smi-dfini, muliplying by a posiiv consan dos no chang h opology. Thus, w may ak C 1. Tha is, h subfamily {ν α : α 1 Ω k, k l} of sminorms on Cc G/K K dominas h family {ν α : α Ug 2l } and hus inducs an quivaln opology. I will b ncssary o hav anohr dscripion of Sobolv spacs. L W 2,l G/K {f L 2 G/K : α f L 2 G/K for all α Ug l } whr h acion of Ug on L 2 G/K is by disribuional diffrniaion. Giv W 2,l G/K h opology inducd by h sminorms ν α f α f 2 L 2 G/K, α Ug l. L W 2,l X b h subspac of lf K-invarians. Proposiion 2.3. Ths spacs ar qual o h corrsponding Sobolv spacs: W 2,l G/K H l G/K and W 2,l X H l X Proof. I suffics o show h dnsiy of s funcions in W 2,l G/K. Sinc G acs coninuously on W 2,l G/K by lf ranslaion, mollificaions ar dns in W 2,l G/K; s 2.5. By Urysohn s Lmma, i suffics o considr mollificaions of coninuous, compacly suppord funcions. L η Cc G and f Cc G/K. Thn, η f is a smooh vcor, and for all α Ug, α η f L α η f. For X g, h acion on η f as a vcor is X η f X ηg g f dg G ηg X g f dg G Now using h fac ha f is a funcion and h group acion on f is by ranslaion, X η f h ηg fg 1 X h dg G η f X h Thus h smoohnss of η f as a vcor implis ha i is a gnuin smooh funcion. Th suppor of η f is conaind in h produc of h compac suppors of η and f. Sinc h produc of wo compac ss is again compac, η f is compacly suppord.

5 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 5 mark 2.1. By Proposiion 2.2, H 2l X W 2,2l X is a Hilbr spac wih norm f 2 2l f Ω f Ω l f 2 whr is h usual norm on L 2 G/K, and 1 Ω k f is a disribuional drivaiv Sphrical ransforms and diffrniaion on Sobolv spacs. Proposiion 2.4. For l, h Laplacian xnds o a coninuous linar map H 2l+2 X H 2l X; h sphrical ransform xnds o a map on H 2l X; and F 1 f 1 λ ξ Ff for all f H 2l+2 X Proof. By h consrucion of h Sobolv opology, h Laplacian is a coninuous map : C G/K H 2l+2 G/K C G/K H 2l G/K Sinc h Laplacian prsrvs bi-k-invarianc, i xnds o a coninuous linar map, also dnod, from H 2l+2 X o H 2l X. Th sphrical ransform, dfind on C c G/K K by h ingral ransform of Harish-Chandra and Brin, xnds by coninuiy o H 2l X. This xnsion agrs wih h xnsion o L 2 X coming from Planchrl. By ingraion by pars, F ϕ λ ξ Fϕ, for ϕ C c G/K K, so, by coninuiy F 1 f 1 λ ξ Ff for allf H 2l+2 X. L µ b h muliplicaion map µvξ 1 λ ξ vξ 1+ ρ 2 + ξ 2 vξ whr ρ is h half sum of posiiv roos. For l Z, h wighd L 2 -spacs V 2l {v masurabl : µ l v L 2 Ξ} wih norms v V 2l µ l v L 2 Ξ 1 + ρ 2 + ξ 2 l vξ 2 cξ 2 dξ Ξ ar Hilbr spacs wih V 2l+2 V 2l for all l. In fac, hs ar dns inclusions, sinc runcaions ar dns in all V 2l -spacs. Th muliplicaion map µ is a Hilbr spac isomorphism µ : V 2l+2 V 2l, sinc for v V 2l+2, µv V 2l µ l+1 v L 2 Ξ v V 2l+2 Th ngaivly indxd spacs ar h Hilbr spac duals of hir posiivly indxd counrpars, by ingraion. Th adjoins o inclusion maps ar gnuin inclusions, sinc V 2l+2 V 2l is dns for all l, and, undr h idnificaion V 2l V 2l h adjoin map µ : V 2l V 2l+2 is h muliplicaion map µ : V 2l V 2l 2. Proposiion 2.5. For l, h sphrical ransform is an isomric isomorphism H 2l X V 2l. Proof. On compacly suppord funcions, h sphrical ransform F and is invrs F 1 ar givn by ingrals, which ar crainly coninuous linar maps. Th Planchrl horm xnds F and F 1 o isomris bwn L 2 X and L 2 Ξ. Thus F on H 2l X L 2 X is a coninuous linar L 2 -isomry ono is imag. L f H 2l X. By Proposiion 2.3, h disribuional drivaivs 1 k f li in L 2 X for all k l. By h Planchrl horm and Proposiion 2.4, 1 l f L 2 X F 1 l f L 2 Ξ 1 λ ξ l Ff L 2 Ξ Thus F H 2l X V 2l. Th following claim shows ha F 1 V 2l H 2l X and finishs h proof. Claim. For v V 2l, h disribuional drivaivs 1 k F 1 v li in L 2 X, for all k l. Proof. For s funcion ϕ, h Planchrl horm implis 1 F 1 v ϕ F 1 v 1 ϕ v F 1 ϕ By Proposiion 2.4 and h Planchrl horm, v F 1 ϕ v 1 λ ξ Fϕ 1 λ ξ v Fϕ F 1 1 λ ξ v ϕ

6 6 AMY T. DECELLES By inducion, w hav h following idniy of disribuions: 1 k F 1 v F 1 1 λ ξ k v. Sinc F is an L 2 -isomry and 1 λ ξ k v L 2 Ξ for all k l, 1 k F 1 v lis in L 2 X for k l. mark 2.2. This Hilbr spac isomorphism F : H 2l V 2l givs a spcral characriaion of h 2l h Sobolv spac, namly h primag of V 2l undr F. H 2l X {f L 2 X : 1 λ ξ l Ffξ L 2 Ξ} 2.4. Ngaivly indxd Sobolv spacs and disribuions. Ngaivly indxd Sobolv spacs allow h us of spcral hory for solving diffrnial quaions involving crain disribuions. Dfiniion 2.2. For l >, h Sobolv spac H l X is h Hilbr spac dual of H l X. mark 2.3. Sinc h spac of s funcions is a dns subspac of H l X wih l >, dualiing givs an inclusion of H l X ino h spac of disribuions. Th adjoins of h dns inclusions H l H l 1 ar inclusions H l+1 X H l X, and h slf-dualiy of H X L 2 X implis ha H l X H l 1 for all l Z. Proposiion 2.6. Th spcral ransform xnds o an isomric isomorphism on ngaivly indxd Sobolv spacs F : H 2l V 2l, and for any u H 2l X, l Z, F1 u 1 λ ξ Fu. Proof. To simplify noaion, for his proof l H 2l H 2l X. Proposiions 2.4 and 2.5 giv h rsul for posiivly indxd Sobolv spacs, xprssd in h following commuaiv diagram,... 1 H 4 1 H 2 1 H... µ F V 4 µ F V 2 µ whr µvξ 1 λ ξ vξ, as abov. Dualiing, w immdialy hav h commuaiviy of h adjoin diagram. F V H 1 H 2 1 H F F F V µ V 2 µ V 4 µ... Th slf-dualiy of L 2 and h Planchrl horm allow h wo diagrams o b conncd H 4 1 H 2 1 H 1 H 2 1 H µ F V 4 µ F V 2 µ F F 1 V µ F V 2 µ F V 4 µ Sinc V 2l+2 is dns in V 2l for all l Z, and H 2l V 2l for all l Z, H 2l+2 is dns in H 2l for all l Z. Thus s funcions ar dns in all h Sobolv spacs. Th adjoin map 1 : H 2l H 2l 2 is h coninuous xnsion of 1 from h spac of s funcions, sinc, for a s funcion ϕ, idnifid wih an lmn of H 2l by ingraion, 1 Λ ϕ f Λϕ 1 f ϕ, 1 f 1 ϕ, f Λ1 ϕ f...

7 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 7 for all f in H 2l+2 by ingraion by pars, whr Λ ϕ is h disribuion associad wih ϕ by ingraion and, dnos h usual innr produc on L 2 G/K. Th map F 1 on H 2l is h coninuous xnsion of F from h spac of s funcions, sinc for a s funcion ϕ, F Λ Fϕ f ΛFϕ Ff Fϕ, Ff V 2l ϕ, f H 2l Λ ϕ f for all f H 2l. Thus, h following diagram commus H 4 1 H 2 1 H 1 H 2 1 H µ F F F F F V 4 µ V 2 µ V µ V 2 µ V 4 µ In ohr words, h rlaion F 1 u 1 λ ξ Fu holds for any u in a Sobolv spac.... call ha, for a smooh manifold M, h posiivly indxd local Sobolv spacs Hloc l M consis of funcions f on M such ha for all poins x M, all opn nighborhoods U of x small nough ha hr is a diffomorphism Φ : U n wih Ω ΦU having compac closur, and all s funcions ϕ wih suppor in U, h funcion f ϕ Φ 1 : Ω C is in h Euclidan Sobolv spac H l Ω. Th Sobolv mbdding horm for local Sobolv spacs sas ha H l+k loc M Ck M for l > dimm/2. Proposiion 2.7. For l > dimg/k/2, H l+k X H l+k G/K C k G/K. Proof. Sinc posiivly indxd global Sobolv spacs on G/K li insid h corrsponding local Sobolv spacs, Hloc l G/K Ck G/K by local Sobolv mbdding. This mbdding of global Sobolv spacs ino C k -spacs is usd o prov ha h ingral dfining spcral invrsion for s funcions can b xndd o sufficinly highly indxd Sobolv spacs, i.. h absrac isomric isomorphism F 1 F : H l X H l X is givn by an ingral ha is convrgn uniformly poinwis, whn l > dimg/k/2, as follows. Proposiion 2.8. For f H s X, s > k + dimg/k/2, f Ffξ ϕ ρ+iξ cξ 2 dξ in H s X and C k X Ξ Proof. L {Ξ n } b a nsd family of compac ss in Ξ whos union is Ξ, χ n b h characrisic funcion of Ξ n, and f n b givn by h following C X-valud Glfand-Pis ingral s 2.5 f n χ n ξ Ffξ ϕ ρ+iξ cξ 2 dξ Ξ Sinc χ n ξ Ffξ is compacly suppord, f n F 1 χ n Ff. Thus, by Proposiions 2.5 and 2.6 f n f m Hs X χ n χ m Ff V s Sinc Ff lis in V s, hs ails crainly approach ro as n, m. Similarly, f n f Hs X χ n 1 Ff V s as n By Proposiion 2.7, f n approachs f in C k X. Th mbdding of global Sobolv spacs ino C k -spacs also implis ha compacly suppord disribuions li in global Sobolv spacs, as follows. Proposiion 2.9. Any compacly suppord disribuion on X lis in a global onal sphrical Sobolv spac. Spcifically, a compacly suppord disribuion of ordr k lis in H s X for all s > k + dimg/k/2.

8 8 AMY T. DECELLES Proof. A compacly suppord disribuion u lis in C G/K. Sinc compacly suppord disribuions ar of fini ordr, u xnds coninuously o C k G/K for som k. Using Proposiion 2.7 and dualiing, u lis in H l+k X, for l > dimg/k/2. mark 2.4. In paricular, his implis ha h Dirac dla disribuion a h bas poin x o 1 K in G/K lis in H l X for all l > dimg/k/2. Proposiion 2.1. For a compacly suppord disribuion u of ordr k, Fu uϕ ρ+iξ in V s whr s > k + dimg/k/2. Proof. By Proposiion 2.9, a compacly suppord disribuion u of ordr k lis in H s for any s > k + dimg/k/2. L f b any lmn in H s X. Thn, Ff, Fu V s V s f, u H s V s uf Sinc h spcral xpansion of f convrgs o i in h H s X opology by Proposiion 2.8, uf u lim Ffξ ϕ ρ+iξ cξ n Ξ 2 dξ lim u Ffξ ϕ ρ+iξ cξ 2 dξ n n Ξ n Sinc h ingral is a C X-valud Glfand-Pis ingral s 2.5 and u is an lmn of C X, u Ffξ ϕ ρ+iξ cξ 2 dξ Ffξ uϕ ρ+iξ cξ 2 dξ Ξ n Ξ n Th limi as n is fini, by comparison wih h original xprssion which surly is fini, and hus Ff, Fu V s V s Ffξ uϕ ρ+iξ cξ 2 dξ Ff, uϕ ρ+iξ V s V s Thus, Fu uϕ ρ+iξ as lmns of V s. Ξ mark 2.5. This implis ha h sphrical ransform of h Dirac dla disribuion is Fδ ϕ ρ+iξ Glfand-Pis ingrals and mollificaion. W dscrib h vcor-valud wak ingrals of Glfand [15] and Pis [19] and summari h ky rsuls; s [14]. For X, µ a masur spac and V a locally convx, quasi-compl opological vcor spac, a Glfand-Pis or wak ingral is a vcor-valud ingral Cc o X, V V dnod f I f such ha, for all α V, αi f α f dµ, X whr his lar ingral is h usual scalar-valud Lbsgu ingral. mark 2.6. Hilbr, Banach, Frch, LF spacs, and hir wak duals ar locally convx, quasicompl opological vcor spacs; s [14]. Thorm 2.1. i Glfand-Pis ingrals xis, ar uniqu, and saisfy h following sima: I f µspf closur of compac hull of fx ii Any coninuous linar opraor bwn locally convx, quasi-compl opological vor spacs T : V W commus wih h Glfand-Pis ingral: T I f I T f. For a locally compac Hausdorff opological group G, wih Haar masur dg, acing coninuously on a locally convx, quasi-compl vcor spac V, h group algbra Cc o G acs on V by avraging: η v ηg g v dg G

9 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 9 Thorm 2.2. i L G b a locally compac Hausdorff opological group acing coninuously on a locally convx, quasi-compl vcor spac V. L {ψ i } b an approxima idniy on G. Thn, for any v V, ψ i v v in h opology of V. ii If G is a Li group and {η i } is a smooh approxima idniy on G, h mollificaions η i v ar smooh. In paricular, for X g, X η v L X η v. Thus h spac V of smooh vcors is dns in V. mark 2.7. For a funcion spac V, h spac of smooh vcors V is no ncssarily h subspac of smooh funcions in V. Thus Thorm 2.2 dos no prov h dnsiy of smooh funcions in V. 3. Fundamnal Soluion for λ ν on G/K W now rurn o prov h main horm, sad a h bginning of h papr, which givs an xplici xprssion for h fundamnal soluion for λ ν on a symmric spac G/K Proof of main horm. As abov, l G b a complx smi-simpl Li group wih fini cnr and K a maximal compac subgroup. L G NAK, g n+a+k b corrsponding Iwasawa dcomposiions. L Σ dno h s of roos of g wih rspc o a, l Σ + dno h subs of m αα, m α dnoing h posiiv roos for h ordring corrsponding o n, and l ρ 1 2 mulipliciy of α. Sinc G is complx, m α 2, for all α Σ +, so ρ α Σ α. L + a C dno h s of complx-valud linar funcions on a. Considr h diffrnial quaion on h symmric spac X G/K: λ ν u δ 1 K whr h Laplacian is h imag of h Casimir opraor for g, λ is 2 ρ 2 for a complx paramr, ν is an ingr, and δ 1 K is Dirac dla a h baspoin x o 1 K G/K. Sinc δ 1 K is also lf-k-invarian, w consruc a lf-k-invarian soluion on G/K using h harmonic analysis of sphrical funcions. Proposiion 3.1. For ingral ν > dimg/k/2, u is a coninuous lf-k-invarian funcion on G/K wih h following spcral xpansion: 1 ν u g ξ ν ϕ ρ+iξg cξ 2 dξ Ξ Proof. Sinc δ 1 K is a compacly suppord disribuion of ordr ro, by Proposiion 2.9, i lis in h global onal sphrical Sobolv spacs H l X for all l > dimg/k/2. Thus hr is an lmn u of H l+2ν X saisfying his quaion. Th soluion u is uniqu in Sobolv spacs, sinc any u saisfying 2 ρ 2 ν u δ 1 K mus ncssarily hav h sam sphrical ransform. For ν > dimg/k/2, h soluion is coninuous by Proposiion 2.7, and by Proposiion 2.8, u g Ξ Fu ξ ϕ ρ+iξ g cξ 2 dξ Ξ 1 ν ξ ν ϕ ρ+iξg cξ 2 dξ For a complx smi-simpl Li group, h onal sphrical funcions ar lmnary. Th sphrical funcion associad wih h principal sris I χ wih χ ρ+iλ, λ a C is ϕ ρ+iλ π+ ρ sgnw i wλ π + iλ sgnw wρ whr h sums ar akn ovr h lmns w of h Wyl group, and h funcion π + is h produc π + µ α> α, µ ovr posiiv roos, wihou mulipliciis. Th raio of π+ ρ o π + iλ is h c-funcion, cλ. Th dnominaor can b rwrin sgnw wρ 2 sinhα w W

10 1 AMY T. DECELLES Proposiion 3.2. Th fundamnal soluion u has h following ingral rprsnaion: 1 ν i d u π + ρ 2 sinh α 1 a λ ν π+ λ iλ dλ Proof. In h cas of complx smi-simpl Li groups, h invrs sphrical ransform has an lmnary form F 1 f fλ ϕ ρ+iλ cλ 2 dλ a /W fλ π+ ρ a /W π + iλ 1 π + ρ 2 sinh α sgnw i wλ sgnw wρ a /W π + 2 iλ π + ρ dλ fλ sgnw i wλ π + iλ dλ Th funcion π + is a homognous polynomial of dgr d, qual o h numbr of posiiv roos, cound wihou mulipliciy, so π + iλ i d π + λ. Also, π + is W -quivarian by h sign characr, so h chang of variabls λ w 1 λ yilds By Proposiion 3.1, F 1 f u i d π + ρ 2 sinh α 1 ν i d π + ρ 2 sinh α w W a fλ π + λ iλ dλ 1 a λ ν π+ λ iλ dλ L Ilog a dno h ingral w nd o compu: 1 Ilog a a λ ν π+ λ i λ,log a dλ Proposiion 3.3. Th ingral Ilog a can b rducd o an ingral ovr h ral lin: Ilog a i d π + Γν d log a πn 1/2 Γν Γν d Γ ν d n 1 iλ1 log a 2 λ dλ ν d n 1/2 1 whr n dim a and d is h numbr of posiiv roos, cound wihou mulipliciy. Proof. Apply h idniy Γs s s d o λ ν in h ingrand of Ilog a: 1 Ilog a Γν ν λ π + λ iλ dλ d a 1 Γν ν 2 2 π + λ i λ,log a dλ d Chang variabls λ λ/. Ilog a 1 Γν 1 Γν ν 2 λ 2 a ν d+n/2 2 a λ d/2 π + λ i λ/, log a π + i λ, log a/ λ dλ λ 2 a n/2 dλ d d

11 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 11 Th polynomial π + is in fac harmonic. S, for xampl, Lmma 2 in [2] or, for a mor dirc proof, Thorm 3.2, blow. Thus h ingral ovr a is h Fourir ransform of h produc of a Gaussian and a harmonic polynomial, and by Hck s idniy, 2 π + λ i λ, log a/ dλ i d π + log a/ log a 2 / a λ urning o h main ingral, Ilog a id π + log a Γν id π + log a Γν placing h Gaussian by is Fourir ransform, Ilog a id π + log a Γν id π + log a Γν id π + log a Γν i d π + log a a i d d/2 π + log a log a 2 / ν d n/2 2 log a 2 / d ν d 2 n/2 log a 2 / d ν d 2 a i λ,log a λ ν d λ d Γν d a λ ν d i λ,log a dλ Γν d a i λ,log a Γν λ dλ ν d 2 dλ d i λ,log a dλ W dno his ingral by J log a, sinc i is is roaion-invarian as a funcion of log a. Wriing λ λ 1,..., λ n, w may assum λ, log a λ 1 log a, and hn J log a a iλ1 log a λ dλ ν d Idnifying a wih n, J log a is 1 Γν d ν d λ iλ1 log a d dλ 1 dλ 2... λ n n 1 Γν d ν d λ iλ1 log a dλ 1 πn 1/2 Γν d ν d n 1/2 πn 1/2 Γν d Γ ν d n 1 2 Thus w hav h dsird conclusion, sinc Ilog a i d π + log a λ 2 n 1 λ iλ1 log a dλ 1 d iλ1 log a λ dλ ν d n 1/2 1 Γν d Γν J log a 2 + +λ2 n d dλ 2... λ n Th rmaining ingral can b valuad by rsidus whn h xponn in h dnominaor is a sufficinly larg ingr, i.. whn G is of odd rank. For h vn rank cas, h ingral can b xprssd in rms of a K-Bssl funcion.

12 12 AMY T. DECELLES Proposiion 3.4. For n odd, l ν d + n+1 2. Thn Ilog a i d π + log a Proof. call from Proposiion 3.3 ha Ilog a is i d π + log a Γν d Γν Taking ν d + n+1 2, iλ1 log a λ dλ ν d n 1/2 1 Thus, π n+1/2 Γd + n + 1/2 πn 1/2 Γν d Γ ν d n 1 2 Ilog a i d π + log a iλ1 log a λ dλ 2 1 2πi s λ 1i π n+1/2 Γd + n + 1/2 Proposiion 3.5. For n vn, l ν d + n Thn, Ilog a i d π + log a Proof. call from Proposiion 3.3 ha Ilog a i d π + log a Γν d Γν Th ingral ovr is iλ1 log a λ dλ 3/2 1 Γn/2 + 1 Γd + n/2 + 1 πn 1/2 Γν d Γ ν d n cosλ 1 log a λ dλ 3/2 1 + π log a Γ3/2 log a iλ1 log a λ dλ ν d n 1/2 1 iλ 1 log a λ log a π n/2 log a K 1 log a Γn/2 + 1 cosλ 1 log a λ dλ 3/2 1 K 1 log a π log a iλ1 log a λ dλ ν d n 1/2 1 i sinλ 1 log a λ /2 dλ 1 whr K 1 is a modifid Bssl funcion K α of h scond kind, which has h following ingral rprsnaion s [1], Thus, K α x Γα α π 1/2 x α I log a i d π + log a cosx α+1/2 d α > 1 2, x >, arg < π 2 Γn/2 + 1 Γd + n/2 + 1 Now w prov h horm sad in h inroducion: π n/2 log a K 1 log a Γn/2 + 1 Thorm 3.1. Whn G is of odd rank, l ν d + n+1 2, whr d is h numbr of posiiv roos, cound wihou mulipliciis, and n is h rank. Thn h fundamnal soluion u for h opraor λ ν on G/K is givn by: u a 1d+n+1/2 π n+1/2 π + ρ Γd + n + 1/2 αlog a 2 sinhαlog a log a

13 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 13 Whn G is of vn rank, l ν d + n Thn, wih K 1 h usual Bssl funcion, u a 1 d+n/2+1 π n/2 π + ρ Γd + n/2 + 1 Proof. call ha h fundamnal soluion is u a αlog a 2 sinhαlog a 1 ν i d π + ρ 2 sinhαlog a By Proposiion 3.4, whn G is of odd rank and ν d + n+1 2, u a log a Ilog a 1 d+n+1/2 i d π + ρ 2 sinhαlog a id π + π n+1/2 log a Γd + n + 1/2 1d+n+1/2 π n+1/2 π + ρ Γd + n + 1/2 π + log a 2 sinhαlog a 1d+n+1/2 π n+1/2 π + ρ Γd + n + 1/2 αlog a 2 sinhαlog a By Proposiion 3.5, whn G is of vn rank and ν d + n 2 + 1, u a 1d+n/2+1 π + ρ 1 d+n/2+1 π n/2 π + ρ Γd + n/2 + 1 log a π + log a Γn/ sinhαlog a Γd + n/2 + 1 αlog a 2 sinhαlog a log a K 1 log a log a log a J log a K 1 log a mark 3.1. For fixd α, larg, and µ 4α 2 s [1], 9.7.2, π K α µ 1 µ 1µ 9 µ 1µ 9µ ! ! arg < 3π 2 Thus in h vn rank cas h fundamnal soluion has h following asympoic: u a 1d+n/2+1 π n+1/2 2 π+ ρ Γd + n/2 + 1 αlog a log a log a 2 sinhαlog a mark 3.2. For any ingr ν > dimg/k/2 n + d/2, h argumn usd in h proof of Proposiion 3.5 givs a formula for h bi-k-invarian fundamnal soluion for λ ν in rms of K-Bssl funcions: 2 1 ν u a π + ργν ν d n/2 αlog a log a 2 sinhlog a K ν d n/2 log a 2 In h odd rank cas, h classical formula for Bssl funcions of half ingr ordr π K m+1/2 2 m m + j! j!m j! 2 j allows us o wri h fundamnal soluion in lmnary rms. L ν m + d + n + 1/2, whr m is any non-ngaiv ingr. Thn u a 1m+d+n+1/2 π n+1/2 m + d + n 1/2! π + ρ log a αlog a 2 sinhαlog a P log a, 1 whr P is a dgr m polynomial in log a and a dgr 2m polynomial in 1. j

14 14 AMY T. DECELLES mark 3.3. call from Proposiion 3.1 ha onal sphrical Sobolv hory nsurs h coninuiy of u for ν chosn as in h horm. For G SL 2 C, h coninuiy is visibl, sinc fundamnal soluion is, up o a consan, u a r r 2 1r 2 1 sinh r whr a r r/2 r/2 mark 3.4. As Brian Hall has poind ou, hr is a simpl rlaion bwn Laplacian on G/K and h Laplacian on p whn G is complx, which allows us o drmin h fundamnal soluion on G/K in rms of h Euclidan fundamnal soluion on p [16], proof of Thorm 2. S also Hlgason s discussion of h wav quaion on G/K in [18]. L J b h funcion on p dfind by fg dg fxpx JX dx G/K Thn, for λ 2 ρ 2, as abov, p G/K λ ν f xp J 1/2 p 2 ν J 1/2 f xp Whn G is complx, h rsricion o a has an lmnary squar roo, J 1/2 H sinhαh αh Indd h ingral w valuad afr applying Hck s idniy is an ingral rprsnaion for h Euclidan fundamnal soluion on a n Th harmonic propry of π +. L G b complx smi-simpl Li group. W will giv a dirc proof ha h funcion π + : a givn by π + µ α> α, µ whr h produc is akn ovr all posiv roos, cound wihou mulipliciy, is harmonic wih rspc o h Laplacian naurally associad o h pairing on a. S also [2], Lmma 2, whr his rsul is obaind as a simpl corollary of h lss rivial fac ha π + divids any polynomial ha is W -quivarian by h sign characr. I is his propry ha nabls us o us Hck s idniy in h compuaions abov. W will us h following lmma. Lmma 3.1. L I b h s of all non-orhogonal pairs of disinc posiiv roos, as funcions on a. Thn π + is harmonic if β,γ β,γ I β γ. Proof. Considring a as a Euclidan spac, is Li algbra can b idnifid wih islf. For any basis {x i } of a, h Casimir opraor Laplacian is i x i x i. For any α, β in a and any λ a α, λ β, λ x i α, x i β, λ + α, λ β, x i i i α, x i β, x i + α, x i β, x i 2 α, β Thus π + i i β> x i x i π + i βx i γ β x i αx i π+ β β> γx i π+ βγ and π + is harmonic if h sum in h samn of h Lmma is ro. β γ β, γ β γ mark 3.5. Whn h Li algbra g is no simpl, bu mrly smi-simpl, i.. g g 1 g 2, any pair β, γ of roos wih β g 1 and γ g 2 will hav β, γ, so i suffics o considr g simpl. π +

15 FUNDAMENTAL SOLUTION FO λ ν ON A SYMMETIC SPACE G/K 15 Proposiion 3.6. For g sl 3, sp 2, or g 2, h following sum ovr all pairs β, γ of disinc posiiv roos is ro: β,γ β γ β γ. Proof. Th posiiv roos in sl 3 ar α, β, and α + β wih α, α 2, β, β 2, α, β 1. In ohr words, h wo simpl roos hav h sam lngh and hav an angl of 2π/3 bwn hm. Th pairs of disinc posiiv roos ar α, β, α, α + β and β, α + β, so h sum o compu is α, β α β + α, α + β α α + β + β, α + β β α + β Claring dnominaors and valuaing h parings, α, β α + β + α, α + β β + β, α + β α α + β + β + α For sp 2, h simpl roos hav lnghs 1 and 2 and hav an angl of 3π/4 bwn hm: α, α 1, β, β 2, α, β 1. Th ohr posiiv roos ar α + β and 2α + β. Th non-orhogonal pairs of disinc posiiv roos ar α, β, α, 2α + β, β, α + β, and α + β, 2α + β. So h sum w mus compu is α, β α β + α, 2α + β α 2α + β + β, α + β β α + β + α + β, 2α + β α + β2α + β Again, claring dnominaors, α, β α + β2α + β + α, 2α + β βα + β + β, α + β α2α + β + α + β, 2α + β αβ and valuaing h pairings, α + β2α + β + βα + β + α2α + β + αβ 2α 2 + 3αβ + β 2 + αβ + β 2 + 2α 2 + αβ + αβ Finally w considr h xcpional Li algbra g 2. Th simpl roos hav lnghs 1 and 3 and hav an angl of 5π/6 bwn hm: α, α 1, β, β 3, α, β 3/2. Th ohr posiiv roos ar α+β, 2α+β, 3α+β, and 3α+2β. Noic ha h roos α and α+β hav h sam lngh and hav an angl of 3π/2 bwn hm. So oghr wih hir sum 2α + β, hy form a copy of h sl 3 roo sysm. Th hr rms corrsponding o h hr pairs of roos among hs roos will cancl, as in h sl 3 cas. Similarly, h roos 3α + β and β hav h sam lngh and hav an angl of 3π/2 bwn hm, so, oghr wih hir sum, 3α + 2β hy form a copy of h sl 3 roo sysm, and h hr rms in h sum corrsponding o h hr pairs among hs roos will also cancl. Th rmaining six pairs of disinc, non-orhogonal posiiv roos ar α, 3α + β, α, β, 3α + β, 2α + β, 2α + β, 3α + 2β, 3α + 2β, α + β, and α + β, β. W shall s ha h six rms corrsponding o hs pairs cancl as a group. Afr claring dnominaors, h rlvan sum is α, β α + β2α + β3α + β3α + 2β + α, 3α + β βα + β2α + β3α + 2β + 3α + β, 2α + β αβα + β3α + 2β + 2α + β, 3α + 2β αβα + β3α + β + 3α + 2β, α + β αβ2α + β3α + β + α + β, β α2α + β3α + β3α + 2β Evaluaing h pairings and facoring ou 3/2, his is α + β2α + β3α + β3α + 2β + βα + β2α + β3α + 2β + αβα + β3α + 2β + αβα + β3α + β + αβ2α + β3α + β + α2α + β3α + β3α + 2β

16 16 AMY T. DECELLES Muliplying ou, This sum is ro. 18α 4 45 α 3 β 4 α 2 β 2 15 αβ 3 2β α 3 β + 13 α 2 β αβ 3 + 2β α 3 β + 5 α 2 β αβ α 3 β + 4 α 2 β 2 + αβ α 3 β + 5 α 2 β 2 + αβ 3 +18α α 3 β + 13 α 2 β αβ 3 Proposiion 3.7. For any complx simpl Li algbra g, h following sum ovr all pairs β, γ of disinc posiiv roos is ro: β,γ β γ β γ. Proof. L I b h indxing s {β, γ} of pairs of disinc, non-orhogonal posiiv roos. For ach β, γ I, l β,γ b h wo-dimnsional roo sysm gnrad by β and γ. For such a roo sysm, l I b h s of pairs of disic, non-orhogonal posiiv roos, whr posiiviy is inhrid from h ambin g. Th collcion J of all such I is a covr of I. W rfin J o a subcovr J of disjoin ss, in h following way. For any pair I and I of ss in J wih non-mpy inrscion, hr is a wo-dimnsional roo sysm such ha I conains I and I. Indd, ling β, γ and β, γ b pairs in I gnraing and rspcivly, h non-mpy inrscion of I and I implis ha hr is a pair β, γ lying in boh I and I. Sinc and ar wo-dimnsional and β and γ ar linarly indpndn, all six roos li in a plan. Sinc all six roos li in h roo sysm for g, hy gnra a wo-dimnsional roo sysm conaining and, and I I, I. Thus w rfin J o a subcovr J : if I in J inrscs any I in J, rplac I and I wih h s I dscribd abov. Th ss I in J ar muually disjoin, and, for any β, γ I, hr is a roo sysm such ha β, γ I J, hus β,γ I β, γ β γ I J β, γ β γ β,γ I By h classificaion of complx simpl Li algbras of rank wo, is isomorphic o h roo sysm of sl 3, sp 2, or g 2. Thus, by Proposiion 3.6, h innr sum ovr I is ro, proving ha h whol sum is ro. No ha h rfinmn is ncssary, as hr ar copis of sl 3 insid g 2. No also ha h only im h roo sysm of g 2 appars is in h cas of g 2 islf, sinc, by h classificaion, g 2 is h only roo sysm conaining roos ha hav an angl of π/6 or 5π/6 bwn hm. mark 3.6. S [17], Lmma 2, for a proof of Proposiion 3.7 whn G is no ncssarily complx. Thorm 3.2. For a complx smi-simpl Li group G, h funcion π + : a givn by π + µ α> α, µ whr h produc is akn ovr all posiv roos, cound wihou mulipliciy, is harmonic wih rspc o h Laplacian naurally associad o h pairing on a. Proof. This follows immdialy from Lmma 3.1, mark 3.5, and Proposiion 3.7. frncs [1] M. Abramowi and I. A. Sgun. Handbook of mahmaical funcions wih formulas, graphs, and mahmaical abls, volum 55 of Naional Burau of Sandards Applid Mahmaics Sris [2] N. B. Andrsn. Invarian fundamnal soluions and solvabiliy for symmric spacs of yp G C /G wih only on conjugacy class of Caran subspacs. Ark. Ma., 362:191 2, [3] N. B. Andrsn. Invarian fundamnal soluions and solvabiliy for GLn, C/Up, q. Mah. Scand., 861:143 16, 2. [4] J-P Ankr. Sharp simas for som funcions of h Laplacian on noncompac symmric spacs. Duk Mah J, 652: , 1992.

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