Quantization and Analysis on Symmetric Spaces

Proceedings of the ESI workshops

Quantization, Complex and Harmonic Analysis Complex Analysis and Operator Theory on Symmetric Spaces

Harald Upmeier (editor)

Contents

Preface 3 Harald Upmeier

Holomorphic Function Spaces and Operator Theory

Index Theory for Wiener-Hopf Operators on Convex Cones 4 Alexander Alldridge

Qp-spaces on bounded symmetric domains 17 Miroslav Engliˇs

Commutative C∗-algebras of Toeplitz Operators on the Unit Disk 33 Nikolai Vasilevski

Quantization and Mathematical Physics

Berezin–Toeplitz quantization of the moduli space of flat SU(n) connec- tions 39 Martin Schlichenmaier

Kontsevich Quantization and Duflo Isomorphism 48 Micha Pevzner

Quantization Restrictions for Diffeomorphism Invariant Gauge Theories 55 Christian Fleischhack

1 2

Geometry of Symmetric and Homogeneous Domains

Orbits of triples in the Shilov boundary of a bounded symmetric domain 64 Jean-Louis Clerc and Karl-Hermann Neeb

On a certain 8-dimensional non-symmetric homogenous convex cone 70 Takaaki Nomura

A characterization of symmetric tube domains by convexity of Cayley transform images 74 Chifune Kai

Complex and Harmonic Analysis

The asymptotic expansion of Bergman kernels on symplectic manifolds 79 George Marinescu

Hua operators and Poisson transform for non-tube bounded symmetric domains 90 Khalid Koufany and Genkai Zhang

Unitarizability of holomorphically induced representations of a split solv- able Lie group 96 Hideyuki Ishi Preface

Harald Upmeier

As part of the special program “Complex Analysis and Applications”, Fall 2005, at the Erwin Schr¨odinger Institute, organized by F. Haslinger, E. Straube and H. Upmeier, two mini-workshops

Quantization, Complex and Harmonic Analysis and

Complex Analysis and Operator Theory on Symmetric Spaces were held during September 22–23, 2005 and on October 12, 2005, resp. These workshops were devoted to interactions between complex analysis on hermitian symmetric spaces and other areas of mathematics and mathematical physics, notably harmonic analysis on semisimple Lie groups and quantization theory on K¨ahler manifols. This research area has been developing considerably during the past few years, and has now reached a certain maturity from where new research directions, such as infi- nite dimensional Hilbert symmetric manifolds or analysis on vector-valued holomorphic functions and more general discrete series representations can be actively pursued. The talks given at the two workshops provide an overall picture of the current status of the field, and it was agreed to collect short expository articles from the participants as (informal) Proceedings. In addition, several participants have published separate research papers in the ESI preprint series. The financial support by the Erwin Schr¨odinger Institute is gratefully acknowledged.

Harald Upmeier – Fachbereich Mathematik und Informatik, Philipps-Universitat¨ Marburg, Hans-Meerwein-Straße, 35032 Marburg, Germany [email protected]

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NOg7GKeˆJXHILz“Uj_XG&p+_XLD~'pqUVQ‘†uLDH}F’UjGKQ7GKH„ƒB yLDJa†LDJMGKH€pqNILDHI[}NILYZHILAR pqQ7_‘NOgXHOUVzoGZl GKN `7[y~'pqToG1NOgXUj[

Ñ

~&LDHIG JXHIG\^KUj[IGZl

Uj[NOg7GfJXHIG\[€^KHOUVJXNOUjLDQLq†NOg7G†uLDdVdjLRfUVQXY”_7pqN€p Q i r3djL9^ApqdVdVeP^\LD~ŽJ;pD^KNA]“~&GApD[O`XHIG\_M¢YZHILD`XJLDUj_ Ð|½u¼E½VÐi½

G

r«{ Sq€l¯hg7Gf[€GKNI[ Lq†E`XQXUVNI[]ZNOg7Gd?pqNONIGKHUVQDb„G\^KNIG\_PUVQ“NIL}NOg7G)†uLDHO~&GKHw“e”NOg7G~'pqJ

Lq†pqHOHILR¯[$pqQ7_ (0)

NIL–NOgXUj[~'pqJi]v_XGŠ‰7QXUVQXY,@TJM=ON ;.pqQ7_/.>E¦0(@21; G NBR)LˆdjGŠ†uNUVQzoGKHI[IG\[G (0)

x idx - r, s :

]¢_XGŠ‰7Q7G\_¤LDQ¦NOg7GŽ[IGKN Lq†^\LD~ŽJLZ[€pqwXdjGPJ;pqUVHI[ p.^\LD~ŽJLZ[OUVNOUjLDQ Lq†fpqdVd‡pqHOHILAR¯[7→ G(2)→ G

- : G

]M[I`7^Ig•NOg;pqN ] ]MNOg7G Rfg7GKHIG ◦ G →

(γ1, γ2) r(γ1) = s(γ2) r(γ2 γ1) = r(γ2) s(γ2 γ1) = s(γ1)

pqHIG `XQXUVNI[†>LDH ] Uj[—pD[I[IL9^KU?pqNOUVzoGZ]7pqQ7_–pqQ“e-pqHOHILR Uj[fUVQ“zoGKHONOUVwXdjGZl `XQXUVNI[ ◦ ◦ id

x ◦ ◦

t`XN¯~&LDHIG[O`7^\^KUVQ7^KNOdVeo] Uj[—pP[I~'pqdVd¢^ApqNIGKYoLDHOe.UVQ-RfgXUj^Ig‘pqdVdvpqHOHILAR¯[ypqHIG}Uj[€LD~&LDHOJXgXUj[O~&[Al

G

*V“€l i djL9^ApqdVdVe+^\LD~ŽJ;pD^KNyNILDJMLDdjLDYZe†uLDHfRfgXUj^€g pqQ7_ pqHIG ^\LDQ“NOUVQ`7LD`7[Al r«{ xy 1

◦ −

r«{*W“€l i CFJ [email protected]¢;59P]7RfgXUj^€g•Uj[ypŽR)GApqT“dVec^\LDQNOUVQ`7LD`7[}~'pqJ NIL'NOg7G=[IGKN

λ : x λx †uHILD~ (0)

]M[O`7^Ig‘NOg;pqN †uLDH}pqdVd pqQ7_ Lq†JLZ[OUVNOUVzoG1kypD_XLDQ‘~&GApD[O`XHIG\[}LDQ x 7→1 G (0)

supp λ = r− (x) x l †>LDH—pqdVd G ∈ G γ(λs(γ)) = λr(γ) γ

∈ G

+yQ7Gˆ[Og7LD`Xdj_ Q7LDNIG-NOg;pqN'NOg7G–GŠ‹9Uj[ONIGKQ7^\GcLq† p ¨pZpqH'[Ie9[ONIGK~°_XL“G\['Q7LDNŽ†uLDdVdjLR҆uHILD~ ^\LDQ7_aUVNOUjLDQ7[

pqQ7_ pqHIGˆLDJGKQ LDQNIL Sq”pqQ7_ r«{V“€l"ÔBQ7_XG\G\_|]^\LDQ7_aUVNOUjLDQ r«{W“UV~ŽJXdVUjG\[&NOg;pqN&NOg7G.~'pqJ7[

r«{ r s

7cLDHIG\LzoGKHA]EGKzoGKQ‘UÀ†‡p& }pZpqHy[Oea[ONIGK~ GŠ‹9Uj[ONI[]7UVN—~'pAe-wMG†>pqH—†3HILD~Ó`XQXUj˜`7GZlfԐQˆ†«pD^KNA]7UÀ† Uj[

(0) l Y Uj[yUVQp'`XQXUj˜`7G”†«pD[OgXUjLDQ¦p&YZHILD`XJLDUj_|]|[O`7^Ig•NOg;pqN}NOg7G GpqQe–djL“^ApqdVdVe–^\LD~ŽJ;pD^KN[IJ;pD^\GZ]ENOg7GKQ Y Y

JXHILqbOG\^KNOUjLDQ7[yLDQ“NIL&NOg7G NR$L'†>pD^KNILDHI[¨pqHIG=[ILD`XHI^\G1pqQ7_-H€pqQXYoGZlhg7GKQi]MpqQ“e†u`XdVdVe-[O`XJXJLDHONIG\_–kypD_XLDQ×

~&GApD[O`XHIG _XGŠ‰7Q7G\[ypP }pZpqH¯[Oe9[INIGK~ we.NOg7G JXHIG\[I^KHOUVJXNOUjLDQ l LDQ x

α Y λ = δx α

RfUVNOgc }pZpqH—[Oea[ONIGK~ ];R$G=~'p\e-_XGŠ‰7Q7G=^\LDQ“zoLDdV`XNOUjLDQ‘LDQˆNOg7G”[IGKN {¨UVzoGKQ‘pPYZHILD`XJMLDUj_ ⊗

λ ( ) ]7pD[†uLDdVdjLR¯[ Lq†v^\LD~ŽJ;pD^KNOdVeˆ[I`XJXJMLDHONIG\_ˆ^\LDQ“NOUVQ`7LD`7[—†3`XQ7^KNOUjLDQ7[yLDQG K G G

1 s(γ) ϕ ψ(γ) = ϕ(γτ)ψ(τ − ) dλ (τ) . ∗ Z

¢¡¤£¦¥¨§ © ¤¥ ¢ ¥¡¤¥ !#"$ %¥ &(' )* ¡,+- ¡!./¥¨§0+- ¡¤¥) ]

UVQNIL–p ƒpqdVYoGKwXH€pal&hg7GPd?pqNONIGKH”^ApqQ

h¢LDYoGKNOg7GKH=RfUVNOg¦NOg7G&Q;pqNO`XH€pqdUVQzoLDdV`XNOUjLDQi]vNOgXUj[}~'pqToG\[ ( )

^\LD~ŽJXdjGKNIG\_NIL”pFcfpqQ;pD^Ig ƒpqdVYoGKwXH€p lÔNI[`XQXUVzoGKHI[€pqdGKQzoGKdjLDJXUVQXY' ƒpqdVYoGKwXH€p wMG 1 1 K G ∗

L L ( ) ∗ C∗( )

FNM@TE¦0IG(EM9¡ £¢ AJaLbNO;P>@TJ.Lq† l Uj[fNOg7G ∗ G G

∗

{¨HILD`XJLDUj_X[1GŠ‹aUj[ON1UVQ pqwX`XQ7_7pqQ7^\GZlhg7G†3`XQ7_7pq~&GKQ“N€pqd¯GŠ‹Xpq~ŽJXdjG+Uj[1NOg7G 7<@JM= .2HEM@ 9J¦7<9EM=

¥¤i½

Ð|½u¼;½ G

0G(E 9¦ Rfg7GKHIG Uj[p djL9^ApqdVdVe'^\LD~ŽJ;pD^KN[OJ;pD^\G}pD^KNIG\_'`XJLDQ&†uHILD~ NOg7GyHOUVYZgN)we'NOg7G

NM@E = Y o G Y

l-^9GKN‡NOg7G\LDHIGKNOUj^ApqdVdVeo]XUVN$Uj[‡NOg7G¨_aUVHIG\^KN)JXHIL“_a`7^KN lhg7GyYZHILD`XJLDUj_ djL“^ApqdVdVe'^\LD~ŽJ;pD^KN)YZHILD`XJG G Y G [ONOHO`7^KNO`XHIG Uj[fYZUVzoGKQˆw“e ×

r(y, g) = y Y = (0) , s(y, g) = y.g , (y, g) (y.g, h) = (y, gh) pqQ7_ λy = δ µ

∈ G ◦ y ⊗

Uj[ }pZpqHv~&GApD[I`XHIGfLDQ lj^“JG\^KU?pqdX^ApD[IG\[ºUVQ7^KdV`7_XG r ¢pqQ7_ r €l

Rfg7GKHIG µ G = Y G = 1 = G Y = pt

Uj[1[ILD~&G&djL9^ApqdVdVe¤^KdjLZ[IG\_W[O`Xw7[IGKNA]vNOg7GKQ ]ºNOg7G+[IGKN”Lq† Ô«† (0) G 1 G1

U U = r− (U) s− (U)

]XUj[¨pPdjL“^ApqdVdVeˆ^\LD~ŽJ;pD^KN¨[OJ;pD^\G=[5pqNOUj[„†3e9UVQXY&^\LDQ7_aUVNOUjLDQ7[Žr«{ Sq pqHOHILAR¯[ywMGKYZUVQXQXUVQXY-pqQ7_ˆGKQ7_aUVQXY'UVQ⊂ G G| ∩

U

r«{*V“€lPhg7G&^KUVHI^K`X~&[ON€pqQ7^\G\[=`XQ7_XGKH RfgXUj^IgS^\LDQ7_aUVNOUjLDQ’r«{*W“¯g7LDdj_X[}†>LDH NOg7GPHIG\[ONOHOUj^KNOUjLDQ¤Lq†

pqQ7_ λ Lq†pŽNOH€pqQ7[„†uLDHO~'pqNOUjLDQ‘YZHILD`XJLDUj_|]7Uj[¯Uj[

pqHIG=˜“`XUVNIG_XGKdVUj^ApqNIG=UVQ–YoGKQ7GKH€pqd«lf¾7LDH¯NOg7G=^ApD[€G = Y o G

¨§'^KUjGKQNyNOg;pqN¯NOg7G}†>LDdVdjLARfUVQXY&g7LDdj_X[—NOHO`7G& §yUj^AÇ ]9]XJXHILDJil SZl WÉÈ¢Q

Q7G\^\G\[I[€pqHOeˆpqQ7_-[O` G

R)G1g;p\zoG Uj[y†3`XdVdVe‘[O`XJXJLDHONIG\_‘LDQ †uLDHpqdVd FSq€l¯¾7LDH ruk 1 1 1 y− U = g G y.g U µ y− U y− U l { ∈ | ∈ } y U

∈

rukdV“€l¯hg7G1~'pqJ _XGKQ7LDNOUVQXY-NOg7G1^€g;pqH€pD^Šƒ Uj[}R)GApqT“dVe‘^\LDQNOUVQ“`7LD`7[A] 1

U L∞(G) : y 1y− U 1A l NIGKHOUj[ONOUj^¨†u`XQ7^KNOUjLDQ–Lq†→ 7→

A

F’UVNOg–NOg7G\[€G=_XGŠ‰7QXUVNOUjLDQ7[}pqN—g;pqQ7_|]EdjGKN—`7[—^\LDQ7[IUj_XGKH—NOg7G †>LDdVdjLARfUVQXY.GŠ‹7pq~ŽJXdjGZlfhg7GzoG\^KNILDH

Ð|½u¼;½ ©¢½

[OJ;pD^\G pD^KNI[yLDQ-UVNI[IGKdÀ†vw“e.NOH€pqQ7[Od?pqNOUjLDQ7[Al GKN X Ñ

= (X o X) Ω = (x, y) Ω X y Ω x .

G | ∈ × ∈ −

hgXUj[nUj[p—YZHILD`XJMLDUj_1RfUVNOgŽ }pZpqHv[Oea[ONIGK~ ] _XGKQ7LDNOUVQXY GKwG\[OYZ`7G$~&GApD[O`XHIG x Ñ

λ = δx λX (Ω x ) λX

lFG ^ApqQ–_XGŠ‰7Q7G”p ƒ«HIGKJXHIG\[IGKQN€pqNOUjLDQ•Lq† LDQ w“e LDQ ⊗ | − X C ( ) L2(Ω) ∗ ∗ G

L(ϕ)u(x) = ϕ(x, y x)u(y) dy †>LDH—pqdVd ϕ ( ) , u L2(Ω) , x Ω . − ∈ K G ∈ ∈

ZΩ ^\LDQ7[OUj[ONI[LDQXdVe¦Lq†fF’UjGKQ7GKH„ƒB yLDJa†¯LDJGKH€pqNILDHI[Al' ¨LAR)GKzoGKHA]

ÔBNUj[=GApD[OecNIL–[IG\GŽNOg;pqNNOg7GŽUV~'pqYoGLq† L pqHIG&pqdVd^\LD~ŽJ;pD^KNA]¢[IL.Q7LDN NOg7GŽGKQ“NOUVHIGŽF‚UjGKQ7GKH„ƒB ¨LDJa†¯pqdVYoGKwXH€p-^ApqQ¦wMGPHIGApqdVUj[€G\_•UVQ¦NOgXUj[

NOg7G L(ϕ) g;pD[¯NIL&wG^\LDQ7[ONOHO`7^KNIG\_cRfgXUj^IgcUj[

†>pD[OgXUjLDQilÔQ–LDHI_XGKH—NILpD^IgXUjGKzoG”NOgXUj[A]EpŽ^\LD~ŽJ;pD^KNOUÀ‰;^ApqNOUjLDQLq† X

iywGKdVU?pqQ-YZHILD`XJ l

G\˜`XUVzZpqHOU?pqQN—†uLDHfNOg7G=pD^KNOUjLDQcLq†vNOg7G X

1º£21 ®°­«¦;±‡¦7²D³\´¤µi¶$·=§aµ ’¶‡ª|§9»o­ §aªM»Z­Bµ¢±

GKN wMG–NOg7Gc[€GKN'Lq†”pqdVdy^KdjLZ[IG\_‚[O`Xw7[IGKNI[+Lq† l hg7GKHIG‘GŠ‹9Uj[ONI[.p¦`XQXUj˜“`7Gˆ~&GKNOHOUj^

Ð|½jÐ|½u¼;½fÑ F(X) X

NILDJMLDdjLDYZe–LDQ [I`7^Ig‘NOg;pqN¨^\LDQ“zoGKHOYoGKQ7^\G”Uj[¨^Ig;pqH€pD^KNIGKHOUj[€G\_cw“eˆNOg7G=G\˜“`;pqdVUVNBe

X limk Ak = limk Ak

Rfg7GKHIG Uj['NOg7Gc[IGKN.Lq†dVUV~ŽUVNI[ †uLDH pqQ7_ Uj['NOg7G‘[IGKN+Lq†

limk Ak a = limk ak ak Ak limk Ak

] _XGKQ7LDNOUVQXYWp¦[O`Xw7[€G\˜`7GKQ7^\GˆLq† NOg7G-Uj_XGKQ“NOUVNBeol †uLDH ∈

a = limk aα(k) aα(k) Aα(k) α : N N

eJM9D=¦L¥>A¦0(@J 75E . 9 1EM=/;>@DN ;>= 1;Kl)F’UVNOg-NOgXUj[NILDJMLDdjLDYZeo] Uj[ hgXUj[¯^\LDQzoGKHOYoGKQ7^\G Uj[f^ApqdVdjG\_ ∈ →

F(X)

^\LD~ŽJ;pD^KN¨pqQ7_ˆ[IGKJ;pqH€pqwXdjGZl

pD^KNI[”^\LDQNOUVQ“`7LD`7[OdVeSLDQ ]pqQ7_NOg7G&~'pqJ Uj[

hg7G&YZHILD`XJ X F(X) X F(X) : x x Ω

^\LDQNOUVQ`7LD`7[]fUVQDb„G\^KNOUVzoG‘pqQ7_ G\˜“`XUVzDpqHOU?pqQ“NAl‚Ô _XGKQNOUÀ†ue RfUVNOg‚NOg7GKUVH&UV~'pqYoG\[&`XQ7_XGKHNOgXUj[ pqQ7_ → 7→ − X Ω

Ç ¢¡¥¨§R& ¡¤£¦¥¢ £¡¤¡£ ¨£¦¥¦¥

pqQ7_ l)ԐN¯Uj[}pPQ7LDQaƒ«NOHOUVz9U?pqd¢†«pD^KNy_a`7G

GK~=wG\_X_aUVQXY-pqQ7_–_XGKQ7LDNIG”NOg7GKUVH—HIG\[OJG\^KNOUVzoG=^KdjLZ[O`XHIG\[ywe X Ω

NIL §—Uj^Ap‘ §yUj^AÇ ]9]MÈNOg;pqN NOg7G1HIG\[ONOHOUj^KNIG\_•NOH€pqQ7[„†>LDHO~'pqNOUjLDQYZHILD`XJMLDUj_ g;pD[}NOg7G

Ω = (X o X) Ω

lFG ^ApqdVd NOg7G 8:9<;>=?;>@BADCFEBGIHdNM@TE¦0IG(EM9¡ Zl }pZpqH¯[Oe9[INIGK~ A 1 W |

λ = δA λX (A− Ω) Ω

‘LDHIG\LAzoGKHA]1R$GWg;pAzoG¨NOg7G¤†>LDdVdjLARfUVQXY NOg7G\LDHIGK~  §yUj^AÇ ]a]”JXHILDJil Val Õ7l SŠÈB]. § \ZÆa]”NOgil}ÔIÔIԊl SKÕ7]

7 ⊗ | W

NOgilXÔ f1l SMSŠÈBl

GKN wG}NOg7G}F‚UjGKQ7GKH„ƒB ¨LDJa†YZHILD`XJMLDUj_|l

Ê=ËnÌÍEÎÉÌ9Ï Ð|½jÐi½VÐi½Ñ

Ω

 ƒpqdVYoGKwXH€p g;pD[-p¦†«pqUVNOga†3`Xd ƒ«HIGKJXHIG\[IGKQ“N€pqNOUjLDQ LDQ ] r3Uu€lÒhg7G‘YZHILD`XJLDUj_ W 2 ∗ C∗( Ω) L L (Ω) YZUVzoGKQˆw“e W ∗

L(ϕ)u(x) = ϕ(x Ω, y x)u(y) dy †>LDH—pqdVd ϕ ( ) , u L2(Ω) , x Ω . − − ∈ K WΩ ∈ ∈

ZΩ

] ]7R$Gg;p\zoG l hg7GKQ-†>LDH ϕ˜(A, y) = ϕ( y) ϕ (X) L(ϕ˜) = W

− ∈ K ϕ

Uj[¯JXHIG\^KUj[IGKdVe.NOg7G}F‚UjGKQ7GKH„ƒB ¨LDJa†‡pqdVYoGKwXH€p l r3UVUu€lÒhg7G UV~'pqYoG=Lq† L (Ω)

W

^\LDQ“N€pqUVQ7[NOg7Gy^\LD~ŽJ;pD^KNfLDJGKH€pqNILDHI[ lxyQaƒ

r3UVUVUu€lÒhg7G¨F’UjGKQ7GKH„ƒB yLDJa†vpqdVYoGKwXH€p (Ω) K(L2(Ω))

]|NOgXUj[}Uj_XGApqd^\LDHOHIG\[OJLDQ7_X[ NIL.NOg7GŽ[O`Xw;pqdVYoGKwXH€p Lq† YoGKQ7GKH€pqNIG\_¦w“ecNOg7G _XGKH W

L C∗( Ω Ω) C∗( Ω)

RfUVNOg–^\LD~ŽJ;pD^KN¨[O`XJXJMLDHONfUVQ l ^\LDQNOUVQ“`7LD`7[¯†u`XQ7^KNOUjLDQ7[ W | W ϕ Ω = (X o X) Ω

WΩ| |

hg7G}NOg7G\LDHIGK~ ^\LDQN€pqUVQ7[—NOg7G}w;pD[OUj^ JXgXUVdjLZ[ILDJXg“e.Lq†ºNOg7G}YZHILD`XJLDUj_–pqJXJXHILopD^IgcNILPNOg7G [ONO`7_ae

Ð|½jÐi½ ¤¢½

Lq†F’ LDJGKH€pqNILDHI[¨UVQ‘pPQ“`XNI[Og7GKdVd5Qfhg7GF pqdVYoGKwXH€pŽUj[yz“UjGKR)G\_‘pD[—NOg7G=[IGKN¨Lq††3`XQ7^KNOUjLDQ7[¨LDQ–NOg7G

Q7LDQaƒB^\LD~Ž~=`XN€pqNOUVzoGP[OJ;pD^\Gr3U«l GZlV]XNOg7G YZHILD`XJLDUj_M)LDwXN€pqUVQ7G\_ˆw“e.YoG\LD~&GKNOHOUj^”^\LD~ŽJ;pD^KNOUÀ‰;^ApqNOUjLDQc†3HILD~

UVNI[fUj_XGApqdj[¯^\LDHOHIG\[IJMLDQ7_-NIL'^\GKHON€pqUVQ @T; ¦0 1 7<9DEM= .”Lq†vNOgXUj[—YZHILD`XJMLDUj_|l

Ω -

‘LDHIG}JXHIG\^KUj[IGKdVeo]7NOgXUj[—djGApD_X[¯NILPNOg7G Q7LDNOUjLDQ‘Lq† 9D=IJ @ 9JM= 7 .80¦P .;57fLq†NOg7G `XQXUVNy[OJ;pD^\G Lq†p

7 (0)

l i¨djL9^ApqdVdVe^KdjLZ[IG\_1[O`Xw7[IGKN Uj[n^ApqdVdjG\_.r3HOUVYZgNK|UVQ“zDpqHOU?pqQ“NA]ZUÀ† UV~ŽJXdVUjG\[ YZHILD`XJLDUj_ G

U (0) r(γ) U

Lq†—pˆ }pZpqH”[Ie9[ONIGK~ÃLDQ pq`XNILD~'pqNOUj^ApqdVdVe Gl-ÔBN”Uj[=^KdjGApqH1NOg;pqN=NOg7GHIG\[ONOHOUj^KNOUjLDQSNIL⊂ G ∈

s(γ) U U Uj[¨UVQzZpqHOU?pqQNAl ÔB† [€pqNOUj[„‰;G\[yNOg7G1UVQzZpqHOU?pqQ7^\GŽpqQ7_‘[I`XJXJMLDHON}^\LDQ7_aUVNOUjLDQ7[ LDQp' }pZpqH¨[Ie9[ONIGK~ˆ]EUÀ†∈ G| G

U U

Uj[A]M~&LDHIG\LAzoGKHA]nLDJMGKQi]NOg7GKQ¦NOgXUj[}HIG\[INOHOUj^KNOUjLDQ¦Uj[yUVQ•†«pD^KN p }pZpqH[Oea[ONIGK~ˆlԐN—†>LDdVdjLAR¯[NOg;pqN¨UVQ•NOgXUj[

^ApD[IGZ] Uj[fQ;pqNO`XH€pqdVdVeˆpqQ-Uj_XGApqd¢Lq† l

C∗( U) C∗( ) ԐQˆNOg7G=^ApD[€G=Lq†NOg7G=F’UjGKQ7GKH„ƒB ¨LDJa†YZHILD`XJMLDUj_|] G| G (0) Uj[}pqQcLDJGKQ–UVQzZpqHOU?pqQN}[O`Xw7[IGKNAl

Ω Ω = Ω

Uj[Pp-HIGKYZ`Xd?pqHP^\LD~ŽJ;pD^KNOUÀ‰;^ApqNOUjLDQ Lq†  § \ZÆa] NOg7G'Q7LDQaƒ«NOHOUVz“U?pqd$†«pD^KN”NOg;pqN hgXUj[=†uLDdVdjLR¯[†uHILD~ ⊂ W

Ω Ω

NOgilvÔIÔ5l SMSŠÈBl 7‘LDHIG\LAzoGKHA]UVQ¨NOgXUj[”†uH€pq~&GKR$LDHOT]NOg7G+wMG\[ON1JLZ[I[OUVwXdjG+_XG\[I^KHOUVJXNOUjLDQ"Lq†¨p–^\LD~ŽJLZ[OUVNOUjLDQ

wG.NIL•‰7Q7_ p•[O`XUVN€pqwXdVeW‰7Q7G.‰7dVNOH€pqNOUjLDQ’Lq† w“e"LDJMGKQ UVQ“zDpqHOU?pqQ“N[O`Xw7[€GKNI[Al hgXUj[

[IGKHOUjG\[ŽR)LD`Xdj_ Ω

HIG\˜“`XUVHIG\[—p1wMGKNONIGKH—`XQ7_XGKHI[ON€pqQ7_aUVQXYLq†ºNOg7G F ^\LD~ŽJ;pD^KNOUÀ‰;^ApqNOUjLDQil

1º£¢¡ £ ­B±‡¦W¬É»Z²¹$§9»Z¹‡²o¦Wµi· »o¥¦W®°­B¦;±¦7²Z³K´Sµi¶‡·=§aµ ‚¶$ª|§9»Z­ )§aªM»o­«µ¢±

^ApqQ•wMG ԐNyUj[}GApD[OeˆNIL+wGKdVUjGKzoG-r3NOg7LD`XYZg¦djG\[I[}GApD[OeˆNILJXHILAzoGq¯NOg;pqN}NOg7GP_XG\[I^KHOUVJXNOUjLDQLq†

¥¤¢½3¼E½

Ð|½ X

HIG\_a`7^\G\_¨NIL•NOg;pqN&Lq† UVQ7_XG\G\_|]pqQeWQ7LDQaƒ«zoLDUj_ GKdjGK~&GKQN'Lq† dVUjG\[ŽLDQ"NOg7G ƒBLDHOwXUVN'Lq†}[ILD~&G

Ω - X X

 § \ZÆa]7djGK~ˆlXÔIÔ5l SAÇÉÈBlhg“`7[A]XR)G^\LDQ7^\GKQNOH€pqNIG1LDQˆNOg7G _XG\[I^KHOUVJXNOUjLDQ‘Lq† l

GKdjGK~&GKQN¨Lq† Ω Ω

GKN `7[ ^\LDQ7[IUj_XGKHA]¢pD[pqQGŠ‹Xpq~ŽJXdjGZ]iNOg7GŽ˜`;pqHONIGKH JXd?pqQ7G l=hgXUj[^\LDQ7G

Ñ Ω = [0, 1[2 R2 = X

_XGKQNOUÀ†ue9UVQXY¦pcJLDUVQN RfUVNOg¨NOg7G.[IGKN ]ºR$G.[IG\G+NOg;pqN Uj[P[IGKdÀ†3ƒB_a`;pqdpqQ7_¨[OUV~ŽJXdVUj^KU?pqd«lÔ ⊂

x Ω x Ω

^ApqQ¨^\LDQNOHOUVwX`XNIG-NIL UVQWNBR)L¦_aUj[ONOUVQ7^KN1†>pD[IgXUjLDQ7[AlcŒUVNOg7GKHA]LDQ7G.Lq† dVUV~ŽUVNI[PLq†y[IG\˜“`7GKQ7^\G\[ ∈ −

xk Ω Ω

HIGK~'pqUVQ7[wLD`XQ7_XG\_ UVQ.NOgXUj[f^ApD[IGZ]aNOg7G¨dVUV~ŽUVNfJMLDUVQ“NfRfUVdVdwMG pqQˆp§&Q7G}g;pqdÀ† NOg7G}^\LD~ŽJMLDQ7GKQ“NI[¯Lq† \

xk -

[OJ;pD^\G¨Q7LDN¯^\LD~ŽJXdjGKNIGKdVe-^\LDQN€pqUVQXUVQXY l +yHA]awLDNOgˆ^\LD~ŽJLDQ7GKQNI[¯NIGKQ7_.NILPUVQa‰7QXUVNe UVQ+RfgXUj^€g–^ApD[IGZ]

Ω - NOg7G}dVUV~ŽUVN—[Og;pqdVd¢wMG}NOg7G GKQ“NOUVHIG=[OJ;pD^\G X lhgXUj[fUj[fUVdVdV`7[ONOH€pqNIG\_ˆwMGKdjLRl

¢¡¤£¦¥¨§ © ¤¥ ¢ ¥¡¤¥ !#"$ %¥ &(' )* ¡,+- ¡!./¥¨§0+- ¡¤¥) \

− ∗ = − ∗

¤¥¤

¦¥¦ ¡ x F X 0 0

Ω Ω

− Ω − Ω

© £

¢ x x

§¥§ ¨¥¨ 0

F

KHJ 1;'Lq† Uj[p+[O`XwM^\LDQ7GPLq† [O`7^Ig¦NOg;pqN}†>LDHpqQe•[IGKYZ~&GKQN [I`7^Ig

k—G\^ApqdVdNOg;pqN=p Ω F [x, y] Ω

UVQ“NIGKHI[IG\^KNI[ ]aR$G g;pAzoG lj7‘LDHIG\LAzoGKHA];_XGKQ7LDNIG NOg;pqN ⊂

]x, y[ F [x, y] F C∗ = x X (x : C) > 0

lºhg7GKQ+pqQe'JMLDUVQ“N‡UVQ Uj[‡Lq†iNOg7G¯†uLDHO~ ]“Rfg7GKHIG Uj[$[ILD~&G¯†>pD^\GZlºhgXUj[$Uj[ †uLDHpqQe ⊂ { ∈ | }

C X Ω x F ∗ F

pqQ7_ lnhg7GJLDUVQNI[ºUVQ dVUjGfpqwMLzoG lvtºpD[I[OUVQXY UVdVdV`7[ONOH€pqNIG\_&pqwMLzoG)†uLDH⊂ −

F =]0, ] 0 F = 0 Ω F = Ω∗

NOg7G}GŠ‹7pq~ŽJXdjG}NIL”NOg7G}YoGKQ7GKH€pqdi^ApD[€GZ]aR$G}g;p\zoG}NOg7Gy†>LDdVdjLARfUVQXY&NOg7G\LDHIGK~! §yUj^AÇ ]a]9JXHILDJilaÕ7l [al VÉÈBl

†3HILD~ ∞ ×

wMG=NOg7G”Lq††«pD^\G\[—Lq† l U}GKQ7LDNIG=w“e NOg7G”_a`;pqdv^\LDQ7G GKN

Ê=ËnÌÍ;Îq̓Ï

¥¤i½jÐi½|Ñ Ð|½ ~

P Ω∗ F = F ∗ F

HIGKd?pqNOUVzoG=NILPUVNI[¯[OJ;pqQ lhg7GKQ–NOg7G}†uLDdVdjLRfUVQXY'~'pqJ-Uj[—pŽR$GKdVdÀƒB_XGŠ‰7Q7G\_ˆUVQDbOG\^KNOUjLDQi] Lq† ∩ h i F F h i ~ ν : F F Ω : (F, x) x F ∗ . F P ∈ { } × → 7→ −

[

hg7GfJLDUVQN‡Lq†NOg7GNOg7G\LDHIGK~ Uj[NOg;pqN‡NOg7GH€pqQXYoG¯Lq† Uj[UVQP†«pD^KN^\LDQN€pqUVQ7G\_'UVQ lºhg7GJXHIL“Lq†

¥¤i½¥¤i½

Ð|½ ν Ω

HIGKdVUjG\[)LDQ+NOg7G—GK~”wMG\_X_aUVQXYŽLq†i†«pD^\G\[$UVQ^Ig;pqUVQ7[fLq†iHIGKd?pqNOUVzoGKdVeGŠ‹9JLZ[IG\_Ž†«pD^\G\[Al §yUj^ApgXUV~&[IGKdÀ†iYZUVzoG\[fp

]7Q;pq~&GKdVeo]MNOg7G=†uLD`XH„ƒB_aUV~&GKQ7[IUjLDQ;pqdº^\LDQ7G RfUVNOgcNOg7G ^\LD`XQNIGKHIGŠ‹7pq~ŽJXdjG1†uLDHyNOg7G”[O`XHubOG\^KNOUVz9UVNBeˆLq†

ν Ω∗ †uLDdVdjLRfUVQXY'w;pD[IGZl

non-extreme point

extreme point

ÔBNUj[)GKz9Uj_XGKQN‡NOg;pqN$NOg7G—[IGKN)Lq†iGŠ‹9NOHIGK~&G—H€p\ea[$Lq† Uj[‡Q7LDQaƒB^\LD~ŽJ;pD^KNAl ¨LAR$GKzoGKHA]9NOgXUj[$Uj[Q7G\^\G\[I[5pqHOe

Ω∗

†uLDHfNOg7G[I`XHub„G\^KNOUVz9UVNBe-Lq†§—Uj^Ap l [f~'pqJi]EpD[fR$G[Ig;pqdVdi[IG\G JXHIG\[IGKQ“NOdVeol

hg7G–[IGKN Uj[+^\LDQN€pqUVQ7G\_’UVQ ]¯wX`XN+[OUVQ7^\G–NOg7G–_a`;pqd}^\LDQ7G–~'pqJ Uj[+pqQ

Ð|½¥¤i½ ©¢½

P F(X) C C∗

pqdj[ILy^ApqHOHOUjG\[nNOg7G$[O`Xw7[OJ;pD^\G$NILDJMLDdjLDYZe LDQ”NOg7G$[O`Xw7[IGKNnLq†;^KdjLZ[€G\_=^\LDQzoGŠ‹1^\LDQ7G\[A] g7LD~&G\LD~&LDHOJXgXUj[O~ 7→

P

lvÔ«† Rfg7GKHIG pqQ7_ ] `XQ7_XGKHnNOg7G)GK~=wG\_X_aUVQXY ~

P Ω : F F ∗ Fk∗ x F ∗ Fk, F P x F

lhg“`7[A]XUÀ† Uj[¯[O`XHubOG\^KNOUVzoGZ]XNOg7GKQ Uj[¯^KdjLZ[€G\_|];pqQ7_.g7GKQ7^\G=^\LD~ŽJ;pD^KNAl NOg7GKQ → 7→ − − → − ∈ ∈

x = 0 ν P

GKN wG.NOg7G-[IGKN'Lq†y†>pD^\Gˆ_aUV~&GKQ7[IUjLDQ7[A]$LDHI_XGKHIG\_ UVQaƒ Ñ

n0 < < nd = dim F F P

7‘LDHIG\LAzoGKHA]MdjGKN l¯hg7G”_aUV~&GKQ7[OUjLDQc†3`XQ7^KNOUjLDQ‘Uj[—GApD[OUVdVe ^KHIGApD[OUVQXYZdVeol{ · · · } { | ∈ }

Pj = F P dim F = nd j

Uj[f^KdjLZ[IG\_+UVQ lhg`7[]aUVN[€G\GK~&[HIGApD[ILDQ;pqwXdjG NIL [IG\GKQ-NIL1wMG¨djLAR)GKH—[IGK~ŽUÀƒB^\LDQ“NOUVQ`7LD`7[A]E[IL{ ∈ |d − }

i=j Pi P

pD[I[O`X~&GNOg;pqN pqdVd¢NOg7G pqHIG”^\LD~ŽJ;pD^KNAl—ԐQˆ†«pD^KNA];UVNyNO`XHOQ7[yLD`XN¨NOg;pqN—NOgXUj[¨^\LDQ7_aUVNOUjLDQ•Uj[y[I`¨§'^KUjGKQN

Pj l †uLDHfNOg7G[I`XHub„G\^KNOUVz9UVNBe-Lq† S

ν

U}GŠ‰7Q7G †>LDH pqQ7_ l

Ê=ËnÌÍ;Îq̓Ï

¥¤i½ i½ Ð|½ ~

π(x F ∗) = F F P x F

Uj[¯^\LD~ŽJ;pD^KNA];NOg7GKQ Uj[¯^\LDQ“NOUVQ`7LD`7[Al r3Uu€l¯ÔB† − ∈ ∈ P π : π 1(P ) P

j − j → j

Uj[¯^\LD~ŽJ;pD^KNA];NOg7GKQˆUVNfUj[—p”‰7QXUVNIGŠƒB_aUV~&GKQ7[OUjLDQ;pqdv~&GKNOHOUj^[OJ;pD^\GZl r3UVUu€l¯ÔB†

Pj

r3UVUVUu€l¯ÔB†ºpqdVdiNOg7G ] ];pqHIG^\LD~ŽJ;pD^KNA]7NOg7GKQ Uj[¯[O`XHubOG\^KNOUVzoGZl Pj j = 0, . . . , d ν

Sh ¢¡¥¨§R& ¡¤£¦¥¢ £¡¤¡£ ¨£¦¥¦¥

pD[[ONOH€pqNOUÀ‰;G\_.w“eŽ‰7wXHIG¨wX`XQ7_adjG\[A] hg7G¨`XJ7[Og7LDNfLq†¢NOg7G¨NOg7G\LDHIGK~ Uj[)NOg;pqN)R)G}^ApqQ+NOgXUVQXTLq†

¥¤¢½¡ ¢½

Ð|½ Ω

pD[¯[IL9LDQˆpD[fNOg7G[IGKNI[ Lq†n†«pD^\G\[¯Lq†v‰X‹aG\_-_aUV~&GKQ7[OUjLDQ•pqHIG ^\LD~ŽJ;pD^KNAl

Pj

tºpqHONPr3UuLq†NOg7GNOg7G\LDHIGK~ Uj[yGApD[Ieo]EpqQ7_cJ;pqHON1r3UVUu)†uLDdVdjLR¯[—w“eˆGK~=wG\_X_aUVQXY UVQNIL.p&[O`XUVN€pqwXdjG

Pj

{¨H€pD[I[I~'pqQXQXU?pqQ¤zZpqHOUjGKNBeol.hg7GŽJXHIL9Lq†fLq†=r3UVUVUu¨Uj[ djGKQXYZNOgeo]ºNOg7G&G\[€[IGKQNOU?pqd)[ONIGKJwGKUVQXY–NOg7GP†>LDdVdjLARƒ

UVQXY¦NIG\^IgXQXUj^Apqd}[ON€pqNIGK~&GKQ“NQ–ÔB† pqQ7_ Rfg7GKHIG ]$NOg7GKQ

xk Fk∗ C Fk F dim Fk = dim F

Uj[ `XQ“wMLD`XQ7_XG\_|l+ ¨GKHIGZ] Uj[ NOg7G Rfg7GKQ7GKzoGKH − → →

dim dom σC < dim F (xk) σC (x) = supy C (x : y)

pqQ7_ Uj[¯NOg7G [IGKN¯Rfg7GKHIG UVN—pqNON€pqUVQ7[¯‰7QXUVNIG zDpqdV`7G\[l [O`XJXJLDHON†3`XQ7^KNOUjLDQ;pqdnLq† ∈

C dom σC

ԐN_XL“G\[ºQ7LDN‡[€G\GK~ NIL}wMG^KdjGApqHRfg7GKNOg7GKHNOg7Gf^\LDQ7_aUVNOUjLDQ'NOg;pqNNOg7G wMGf^\LD~ŽJ;pD^KNUj[ºQ7G\^\G\[€[€pqHOe

Pj

†>LDH NIL.wGP[O`XHub„G\^KNOUVzoGZlŽ$djGApqHOdVeo]iNOg7GŽ^\LD~ŽJ;pD^KNOQ7G\[€[=Lq† _XL9G\[}Q7LDN UV~ŽJXdVe‘NOg;pqNLq†$NOg7G ]¢pD[

ν P Pj

^ApqQwGŽ[IG\GKQw“e•^\LDQ7[IUj_XGKHOUVQXY•p+NOgXHIG\GŠƒB_aUV~&GKQ7[OUjLDQ;pqdf^\LDQ7GŽRfg7LZ[€G&^\LD~ŽJ;pD^KN”w;pD[€GŽUj[ NOg7G&^\LDQ“zoGŠ‹

g`XdVdvLq†NOg7G=[IGKN¨Lq†JMLDUVQ“NI[¨LDQ•p&^KUVHI^KdjG=Rfg7LZ[IG1pqQXYZdjG\[ywGKdjLDQXY'NILp+pqQNILDH [IGKNAleiydj[ILX]7NOg7G†>pD^KU?pqd

GŠ‹aJMLZ[IG\_aQ7G\[€[¨Lq† Uj[pqJXJ;pqHIGKQNOdVe•`XQXHIGKd?pqNIG\_•NIL.NOg7GŽ^\LD~ŽJ;pD^KNOQ7G\[I[Lq†)NOg7G ^\LDQ7_aUVNOUjLDQil1hg7G

Ω∗ Pj

`7[O`;pqdjk J;pqHOT9UVQXY&H€pq~ŽJ lV]XRfgXUj^€g–Uj[fQ7LDNf†«pD^KU?pqdVdVe.GŠ‹9JLZ[IG\_|]ag;pD[ ^\LD~ŽJ;pD^KN—†uLDH—pqdVd l

Pj j

+yQˆNOg7GJMLZ[OUVNOUVzoG”[OUj_XGZ]XJLDdVe“g7G\_aH€pqdv^\LDQ7G\[¯g;pAzoG=^\LD~ŽJ;pD^KN lZ7‘LDHIG\LAzoGKHA]M[ILŽ_XL&UVHOHIG\_a`7^KUVwXdjG

Pj

[Oe9~Ž~&GKNOHOUj^•^\LDQ7G\[Al hg7G¦^Kd?pD[I[.Lq†P^\LDQ7G\[.RfUVNOgs^\LD~ŽJ;pD^KN.†>pD^\G¦[OJ;pD^\G\[+Uj[-^KdjLZ[IG\_ `XQ7_XGKH+‰7QXUVNIG

JXHIL9_a`7^KNI[Al—tgXUVdjLZ[€LDJXgXUj^ApqdVdVeo]ipqdVdv[IGKQ7[IUVwXdjG=^\LDQ7G\[A]|pqQ7_‘[ILD~&G=LDNOg7GKHI[A]Mg;pAzoG”^\LD~ŽJ;pD^KN¨†«pD^\G=[OJ;pD^\G\[ l

Pj

ԐQ-NOg7G}HIGK~'pqUVQ7_XGKHyLq†ºNOgXUj[fJ;pqJMGKHA]7R$G[Og;pqdVdvpqdVR)p\ea[ypD[€[O`X~&G}NOg7G NILPwG^\LD~ŽJ;pD^KNAl

Pj

1º££¢ ¤µ¢±$¬É»D²“¹‡§9»o­«µ¢± µi·ª’§aµ ’¶fµ¢¬D­>»o­«µn± ¬q¦7²­B¦;¬

Q;pqNO`XH€pqdVdVe YZUVzoGWHOUj[IGSNIL‚NOg7GWLDJGKQ UVQzZpqHOU?pqQN[O`Xw7[€GKNI[ hg7G¨[O`Xw7[IGKNI[

Ð|½ ©n½3¼E½

Pj P Uj =

]ELq† lhg7G †«pD^KN¯NOg;pqN—NOg7G\[€G=[O`Xw7[IGKNI[¨pqHIGUVQ7_XG\G\_cLDJMGKQ 1 j 1 ] ⊂ (0)

π− i=1− j = 0, . . . , d + 1 Ω = Ω

NOg7G&djLAR)GKH1[€GK~ŽUÀƒB^\LDQNOUVQ`XUVNeWLq† l+FG&JLDUVQNPLD`XN=NOg;pqN ] †>LDdVdjLAR¯[1_aUVHIG\^KNOdVe•†3HILD~ W

dim π U0 = ∅ l S pqQ7_
◦

U1 = Ω Ud+1 = Ω

pqdVdEg;pAzoG¨ }pZpqH[Oe9[INIGK~&[AlºÔBQ†>pD^KNA] ^ApqQ+wMG hg7GKQ+NOg7G¨YZHILD`XJMLDUj_X[

Ω Uj Ij = C∗( Ω Uj)

ƒ«Uj_XGApqdLq† lFG=g;pAzoG&pqdVHIGApD_ae–Q7LDNOUj^\G\_‘NOg;pqN ^\LDHOHIG\[IJMLDQ7_X[ ^\LDQ7[OUj_XGKHIG\_¦pD[ p+^KdjLZ[IG\_ W W

C∗( Ω) I1

NIL `XQ7_XGKH NOg7GPF HIGKJXHIG\[IGKQ“N€pqNOUjLDQ †uHILD~ NOg7G\LDHIGK~ Val Val Val GKN `7[ _XGKNIGKHO~ŽUVQ7GŽNOg7G ∗ W Ñ

K(L2(Ω)) L

LDNOg7GKH¯Uj_XGApqdj[Al

]ip'djL9^ApqdVdVe–^KdjLZ[IG\_‘UVQzZpqHOU?pqQN[O`Xw7[€GKN¨Lq† l^“UVQ7^\G 9GKN ^ 1 (0)

Yj = Uj+1 Uj = π− (Pj) Ω = Ω

g;pD[fp= }pZpqH[Oe9[INIGK~ˆl 7‘LDHIG\LAzoGKHA]aw“e– k—GKQ;ÇMh7]“^€gil9ԀÔ5]JXHILDJil“Õ7l ÆÉÈB] Uj[$UVQzZpqHOU?pqQNA]EpqYpqUVQ\ W

Yj Ω Yj

Uj[}UVQ7_a`7^\G\_•we‘HIG\[ONOHOUj^KNOUjLDQ Lq† W]|Rfg7GKHIGPNOg7G1Uj[ILD~&LDHOJXgXUj[I~

Ij+1/Ij = C∗( Ω Yj) ϕ ϕ Yj

LDQ l ^\LD~ŽJ;pD^KNOdVe-[O`XJXJLDHONIG\_ˆ^\LDQNOUVQ“`7LD`7[¯†u`XQ7^KNOUjLDQ7[∼ W 7→

ϕ Ω Uj+1

Uj[ [IN€pqwXdVecUj[ILD~&LDHOJXgXUj^&NILˆp.^\LD~Ž~=`XN€pqNOUVzoG NOg;pqN=NOg7GŽ˜`7LDNOUjGKQ“N FGPUVQ“NIGKQ7_¦NIL-[Og7LAR W

Ij+1/Ij

ƒpqdVYoGKwXH€pal‡ÔQˆLDHI_XGKH¯NILŽ_XLŽNOgXUj[A]aR)G=[Og7LRsNOg;pqNfNOg7G YZHILD`XJLDUj_ Uj[—p1‰7wXHIGwX`XQ7_adjGZl 

∗ Ω Yj

GKN wG-NOg7G k Q7LDHO~'pqd¨wX`XQ7_adjGMl)Lq† l

Ê=ËnÌÍEÎÉÌ9Ï

©n½VÐi½fÑ Ð|½ W

Σj = (F, v) F Pj , v F ⊥ Pj

Uj[pzoG\^KNILDH}wX`XQ7_adjGŽLAzoGKH ]ipqQ7_ Uj[}pdjL9^ApqdVdVe–NOHOUVz9U?pqd^\LDQNOUVQ“`7LD`7[ †«pq~ŽUVdVecLq† hg7GKQ ∈ ∈

Σj Pj Ω Yj ]aRfUVNOgˆdjL9^ApqdiNOHOUVz9U?pqdVUj[€pqNOUjLDQ7[¯YZUVzoGKQˆw“e YZHILD`XJLDUj_X[¯LAzoGKH  W | Σj

1 ~

Ω π− (UE) Σj UE E~ E :

7→

Uj[”pqQ¤LDJGKQ¦Q7GKUVYZgwLD`XHOg7L9L“_WLq† ]npqQ7_ Uj[=p.wXUÀƒ UVJ7[I^IgXUVNO¿ ¨GKHIG
Ñ UE Pj E ψEF : F E

~'pqJ-RfgXUj^Ig‘[€pqNOUj[„‰;G\[⊂ h i → h i

†>LDH—pal GZl ~ ~ pqQ7_ ψEF (F ) = E det ψE0 F (x) = 1 x .

¢¡¤£¦¥¨§ © ¤¥ ¢ ¥¡¤¥ !#"$ %¥ &(' )* ¡,+- ¡!./¥¨§0+- ¡¤¥) SMS

Uj[ŽNOg7GKHIGKwe"Uj[ILD~&LDHOJXgXUj^–NIL•NOg7Gˆ[IG\^KNOUjLDQ‚pqdVYoGKwXH€p Lq†}NOg7Gˆ^\LDQaƒ 7cLDHIG\LzoGKHA]

C∗( Ω Yj) K( j)

ƒpqdVYoGKwXH€pD[ pD[I[IL9^KU?pqNIG\_¤NIL‘NOg7G NOUVQ`7LD`7[1‰;GKdj_¨Lq†yGKdjGK~&GKQN€pqHOe"W ~ 2 ~ E

∗ C∗( E~ E ) = K(L (E )) l ^\LDQNOUVQ`7LD`7[y‰;GKdj_ˆLq† yUVdVwMGKHON¯[OJ;pD^\G\[ 2 W~ j = L (F ) (F,y) Σ

E ∈ j

hg7GP~'pqJ7[ pqHIGŽ^\LDQ7[ONOHO`7^KNIG\_¤we•_XGŠ†uLDHO~ŽUVQXYcG\˜`;pqdÀƒB_aUV~&GKQ7[OUjLDQ;pqd)^\LDQ7G\[ UVQNIL–GApD^Ig ¢¡¤£¥£¤¦¨§

ψEF

LDNOg7GKHAlj^“`7^€gp¨_XGŠ†uLDHO~'pqNOUjLDQ^ApqQŽwMG)HIGApqdVUj[IG\_Pw“e1^\LDQ7[OUj_XGKHOUVQXY NOg7Ge7cUVQXToLAR¯[IT“UXYpq`XYoG)†u`XQ7^KNOUjLDQ;pqdj[

†uLDH¯^\LD~ŽJ;pD^KNyw;pD[IG\[¯Lq†ºNOg7G^\LDHOHIG\[OJLDQ7_aUVQXY'^\LDQ7G\[Al

hg7G”G\[€[IGKQNOU?pqdvJMLDUVQ“N—Uj[yNIL'^\LD~ŽJX`XNIG1NOg7G=_XGKHOUVzZpqNOUVzoG\[}Lq†NOg7G\[IG=Ypq`XYoG†3`XQ7^KNOUjLDQ;pqdj[]M[Ig7LARfUVQXY

NOg;pqNNOg7GKe”g;pAzoGfJMLZ[IUVNOUVzoGfmpD^\LDwXU?pqQilhg7GJXHILDJMGKHONe pal GZlDUj[N€pqQN€pq~&LD`XQN$UVQŽJXHILAz9UVQXY

det ψEF = 1

NOg;pqN  ƒpqdVYoGKwXH€pD[ Lq†NOg7G1LDwXN€pqUVQ7G\_‘YZHILD`XJMLDUj_X[ pqHIG”Uj[ILD~&LDHOJXgXUj^Z]UVQ7^\G=UVN}[Og7LR¯[yNOg;pqN¨NOg7G1 ¨pZpqH

∗

G\˜“`;pqdj[ Uj[¯Q7LAR [ON€pqQ7_7pqHI_|l [Oe9[INIGK~&[¯^\LDHOHIG\[OJLDQ7_|lhg7G JXHIL9Lq†vNOg;pqN C ( Y ) K( ) 

∗ WΩ j Ej

Uj[)py‰7QXUVNIGŠƒB_aUV~&GKQ7[OUjLDQ;pqd~&GKNOHOUj^—[OJ;pD^\GZ]NOg7G‰;GKdj_ Uj[‡NOHOUVz“U?pqd«lº ¨GKQ7^\GZ] ^“UVQ7^\G

Ð|½ ©¢½¥¤i½

Σj j K( d) =

UÀ† lhg“`7[A]XR$Gg;p\zoG NOg7G}†>LDdVdjLARfUVQXYNOg7G\LDHIGK~ˆl ];pqQ7_ E E (X) K( ) = (Σ ) K j < d

C0 Ej C0 j ⊗

hg7G †uLDHO~ pqQ‘pD[I^\GKQ7_aUVQXY'^Ig;pqUVQ‘Lq†vUj_XGApqdj[—Lq† lFG g;pAzoG

Ê=ËnÌÍ;Îq̓Ï

©¢½ ©n½ Ð|½ I C ( ) j ∗ WΩ

0(X) j = d , Ij+1/Ij ∼= C ( 0(Σj) K 0 6 j < d .

C ⊗

FGJMLDUVQ“N¨LD`XN¯NOg;pqNyNOg7G”pqwLAzoG=Uj[ILD~&LDHOJXgXUj[O~&[}pqHIGHIGApqdVUj[IG\_–we-NOg7G}†>LDdVdjLARfUVQXY+HIGKJXHIG\[IGKQaƒ

Ð|½ ©¢½ |½

Lq† LDQ ] N€pqNOUjLDQ F,y σj = (L )(F,y) Σ C∗( Ω) j ∈ j W E

F,y i(y:w2) L (ϕ)h(v) = ϕ(F, v, w1 + w2 v)e− h(w1) dw1 dw2 ~ −

ZF ⊥ ZF

] ] ] l †uLDH—pqdVd ϕ ( ) (F, y) Σ h L2(F ~) v F ~

∈ K WΩ ∈ j ∈ ∈

GKN'`7[ŽwXHOUjG ©7e¨HIGKz9UjGKR LD`XH&NOg7G\LDHIGK~°UVQ NOg7Gˆ^ApD[IGˆLq†}NOg7Gˆ˜“`;pqHONIGKH&JXd?pqQ7G Ñ 2

Ω = Ω∗ = [0, [

Val Wal SZlFGg;p\zoG ] ^\LDQ7[OUj_XGKHIG\_ˆUVQ ∞

d = 2 pqQ7_ P = Ω∗ , P = [0, [ 0, 0 [0, [ P = 0 . 0 { } 1 ∞ × × ∞ 2 { }

hg`7[A] 

Σ = pt , Σ = 2 R pqQ7_ Σ = X = R2 ,

0 1 · 2

_XGKQ7LDNIG\[¯NOg7G NILDJLDdjLDYZUj^Apqd¢[O`X~ÓLq† ^\LDJXUjG\[yLq† lhg7G NOg7G\LDHIGK~ YZUVzoG\[ Rfg7GKHIG 2 R 2 R · I = 0 , I = K , I /I = (R) C2 K pqQ7_ I /I = (R2) .

0 1 2 1 ∼ C0 ⊗ ⊗ 3 2 C0

Uj[¯NOg7G^\LD~Ž~=`XN€pqNOUVzoGP ƒpqdVYoGKwXH€p&Lq†v†3`XQ7^KNOUjLDQ7[¯LDQ JMLDUVQ“NI[Al yGKHIGZ] 2

C ∗ 2

7cLDHIGYoGKQ7GKH€pqdVdVeo]X†>LDHfNOg7G [OUV~ŽJXdVUj^KU?pqdn^\LDQ7G Ω = Ω = [0, [n ]XR$GYoGKN ∗ ∞

n †>LDH—pqdVd n pqQ7_ d = n , Σ = Rj I /I = (Rj) C(j ) K j < n .

j j · j+1 j C0 ⊗ ⊗

—†^\LD`XHI[IGZ]NOg7G\[IGHIG\[O`XdVNI[‡pqdj[€L¨†uLDdVdjLR"†3HILD~ NOg7GR)LDHOTŽLq†¤7c`XgXdVeƒBkyGKQ;pq`XdVN¯Âb7ck¨ÇMVÉÈ;LDQŽJMLDdVe9g7G\_aH€pqd

+
^\LDQ7G\[Al

SV ¢¡¥¨§R& ¡¤£¦¥¢ £¡¤¡£ ¨£¦¥¦¥

¡X¡£¢¥¤ › §¦

¡º£B¢ ¤µ¢±$¬É»D²“¹‡§9»o­«µ¢± µi·ª|±©¨¦±‡ª|© &º»o­B§aª|© q± (¦

hg7Gc^\LD~ŽJLZ[OUVNOUjLDQ [IGKHOUjG\[.^\LDQ7[ONOHO`7^KNIG\_’UVQ NOg7G\LDHIGK~ Val Õ7l ÕWYZUVzoG\[HOUj[€GˆNILW^\LD~Ž~=`XN€pqNOUVzoG

¤i½u¼E½3¼E½ _aU?pqYZH€pq~&[

/ / σj / / 0 Ij/Ij 1 Ij+1/Ij 1 K( j) 0 − − E σj 1 σj 1 τj − −   / K( ) / ( ) / /K( ) / 0 j 1 j 1 πj 1 j 1 0

E − L E − − L E −

¨GKHIGZ] Uj[ŽNOg7G– ƒpqdVYoGKwXH€p¦Lq† pD_qbOLDUVQN€pqwXdjGcLDJMGKH€pqNILDHI['Lq†¨NOg7G- yUVdVwMGKHON

( j 1) = M(K( j 1)) ∗

lvFGyJMLDUVQ“N)LD`XN)NOg;pqN Uj[pqQ+G\[€[IGKQNOU?pqdMUj_XGApqdMLq† ]“[€LNOg;pqN UVQaƒ L Eƒ«~&L“_a`XdjG− E −

0(Σj 1) j 1 Ij Ij+1 Ij+1

]pqQ7_PNOg`7[ ]ZRfgXUj^Ig'^\LDUVQ7^KUj_XG\[RfUVNOgŽNOg7Gf[ONOHOUj^KNGŠ‹9NIGKQ7[IUjLDQ&Lq†EUVNI[ºHIG\[ONOHOUj^KNOUjLDQ bOG\^KNI[ºUVQNILC − E −

M(Ij) σj 1

]aN€pqToG\[¯zDpqdV`7G\[—UVQ Rfg7GKQ–GKzZpqdV`;pqNIG\_–LDQ l NIL −

Ij ( j 1) Ij+1

Uj[NOg7G£0!.P549D=IJM@B9JM= 7‡Lq†nNOg7G¨GŠ‹aNIGKQ7[OUjLDQ Lq† we lÔBN hg7G¨~'pqJ L E −

τj Ij+1/Ij 1 Ij/Ij 1 K( j)

]9Rfg7GKQ7GKzoGKH Uj[fpP^\LD~ŽJXdjGKNIGKdVeJLZ[OUVNOUVzoG [IG\^KNOUjLDQˆLq† l ~'p\ewMG}^\LD~ŽJX`XNIG\_–pD[ − − E

πj 1 σj 1 %j %j σj

“`7^Ig GŠ‹aUj[ON¯[OUVQ7^\G Uj[fQ“`7^KdjGApqHA]7wGKUVQXY'Lq†ºNBe9JMG Ԋl ^ − ◦ − ◦

%j Ij+1/Ij 1

yQe&[Og7LDHONGŠ‹7pD^KN)[IG\˜“`7GKQ7^\G Lq† ƒpqdVYoGKwXH€pD[f_XGŠ‰7Q7G\[A]9we&NOg7G¨[OUÀ‹9ƒ i − / / π / /

0 J A B 0 ∗

ƒ«NOg7G\LDHOeo]ip~'pqJ ]|LDQXdVec_XGKJGKQ7_aUVQXYˆLDQ lFi NIGKHO~ GŠ‹XpD^KN [IG\˜“`7GKQ7^\G1UVQ

K ∂ : K1(B) K0(J) π

ƒ«NOg7G\LDHOeŽ^ApqQŽwMGYZUVzoGKQpD[v†>LDdVdjLAR¯[]^Š†OlXÂď)LDQ \qÕ;]oÔIԊlbc}l ]DdjGK~ˆlO[ÉÈBl [OUV~ŽJXdjGf_XG\[I^KHOUVJXNOUjLDQ'UVQ&NIGKHO~&[Lq† →

E γ

GKN ] ]MwMG1p&^\LDQ“NOUVQ`7LD`7[=pqJXJXHILA‹9UV~'pqNIG1`XQXUVNyLq† ]E˜`;pD[OUÀƒB^\GKQ“NOH€pqdv†uLDH ] Ñ

0 6 ut 6 1 t [1, [ J A

pŽ[IG\^KNOUjLDQˆLq† r3Q7LDN¯Q7G\^\G\[I[€pqHOUVdVe.^\LD~ŽJXdjGKNIGKdVe-JLZ[OUVNOUVzoGq€lhg7GKQ pqQ7_ ∈ ∞ % : B A π → ϕ (f b) = f(u )%(b) †>LDH—pqdVd f (]0, 1[) , b B , t [1, [ ,

t ⊗ t ∈ C0 ∈ ∈ ∞

_XGŠ‰7Q7G\[pqQpD[Oe9~ŽJXNILDNOUj^ ƒ«~&LDHOJXgXUj[I~ †3HILD~ NIL l=hg7GP~'pqJ¦pD[€[IL“^KU?pqNIG\_

SB = 0(]0, 1[) B J

UVQ ƒ«NOg7G\LDHOe r3YZUVzoGKQ’we ^\LD~ŽJLZ[OUVNOUjLDQ‚UVQ ƒ«NOg7G\LDHOeE NIL ∗ C ⊗

(ϕt) K K1(B) = K0(SB) K0(J) E l ^\LDUVQ7^KUj_XG\[¯RfUVNOg →

∂

U}GKQ7LDNIG”NOg7G=~'pqJ YZUVzoGKQcw“eˆNOgXUj[y^\LDQ7[INOHO`7^KNOUjLDQ‘w“e ldc)GŠƒ K1(K( J )) K0(K( j 1)) Indj ^Apq`7[IG Lq†ºNOg7G Q;pqNO`XH€pqdVUVNBeˆLq†^\LDQXQ7G\^KNOUVQXY'g7LD~&LD~&LDHOJXgXUj[I~&[A]E → E −

†uLDH—pqdVd + Indj([u]) = ∂[τj(u)] [u] K1(K( j)) = π0 GL (K( j) )

∈ E ∞ E Uj[fNOg7G^\LDQXQ7G\^KNOUVQXYg7LD~&LD~&LDHOJXgXUj[O~ˆl Rfg7GKHIG
∂ : K1( /K( j 1)) K0(K( j 1))

− −

pqdVd9pqQGKdjGK~&GKQN A @¨;  EaL 9P]DUÀ†aNOg7GKHIGGŠ‹aUj[ONI[|[ILD~&G ] i½u¼E½VÐi½ ¤ L E → E

a MN (C∗( Ω)) j b MN (C∗( Ω))

lÓhg7GKQ pqQ“e ƒ«NOg7G\LDHOe HIGKJXHIG\[IGKQ“N€pqNOUVzoG [O`7^€gsNOg;pqN ∈ W ∈ W

ab 1 ba 1 0 (mod MN (IJ )) K

ƒB¾XHIG\_ag7LDdV~ GKdjGK~&GKQ“N l ÔBN}^ApqQ‘wMG1[Og7LARfQ•NOg;pqN − dVUÀ†uNI[¨NIL+p≡ − ≡

u GLN (K( j)) j %j(u) σj 1(%j(u))

¾XHIG\_ag7LDdV~ ~'pqNOHOUj^\G\['LDQ NOg7Gc^\LDQNOUVQ`7LD`7[†«pq~ŽUVdVe Lq† Uj['p¦^\LDQ“NOUVQ`7LD`7[†«pq~ŽUVdVeWLq†∈ E −

N N j 1

r3NILDJMLDdjLDYZUj^Apqd ƒ«NOg7G\LDHOe.RfUVNOg yUVdVwMGKHON¯[OJ;pD^\G\[lxy[OUVQXY'NOg7G}Uj_XGKQNOUÀ‰;^ApqNOUjLDQ× 1 E −

K1(K( j)) = Kc (Σj) K Uj[$NOHOUVz9U?pqd|LD`XNI[OUj_XG ^\LD~ŽJ;pD^KNf[O`XJXJLDHONI[K€]9R$G}^ApqQi]a~&LDHIG\LAzoGKHA]XJXHILzoG¨NOg;pqN)NOg7Gy†«pq~ŽUVdVeE

σj 1(%j(u))

l=hg“`7[A]|NOg7GiNOUVepqgaƒBmqQXUj^€g†«pq~ŽUVdVe–UVQ7_XGŠ‹ p+^\LD~ŽJ;pD^KN[O`Xw7[IGKN Lq† −

Σj 1 IndexΣj 1 σj 1(%j(u)) lºFGg;p\zoG NOg7G}†>LDdVdjLARfUVQXYNOg7G\LDHIGK~ˆl ~'pqToG\[¯[IGKQ7[€GpD[—pqQ–GKdjGK~&GKQN¨Lq†− 0 − − Kc (Σj 1)

−

GKN l‡hg7GKQ

Ê=ËnÌÍEÎÉÌ9Ï

i½u¼E½ ¤¢½Ñ ¤ [u] K1(Σ ) = K (K( )) ∈ c j 1 Ej 0 Indj([u]) = IndexΣj 1 σj 1(%j(u)) Kc (Σj 1) = K0(K( j 1)) . − − ∈ − E −

¢¡¤£¦¥¨§ © ¤¥ ¢ ¥¡¤¥ !#"$ %¥ &(' )* ¡,+- ¡!./¥¨§0+- ¡¤¥) SW

¡º£21 ¤µ¢±‡¬É»Z²¹$§9»Z­Bµ¢± µi·ª’¤yµ¢¶fµ¢©«µ 6n­B§aª|© q± (¦

hg7G}Q7GŠ‹9N[ONIGKJ.UVQ.JXHILAz9UVQXY'pqQ.UVQ7_XGŠ‹NOg7G\LDHIGK~ Uj[NOg7G}_XGŠ‰7QXUVNOUjLDQˆLq†ºp=NILDJMLDdjLDYZUj^Apqd¢UVQ7_XGŠ‹l

¤i½jÐ|½u¼;½

Uj[ˆpS^\LD~ŽJ;pD^KNˆ[OJ;pD^\G•UVQ NOg7G h¢L¨NOg;pqN-GKQ7_|]—Q7LDNIG‘NOg;pqN j = (E, F ) Pj 1 Pj E F

P { ∈ − × | ⊃ }

lˆhg7G'JXHILqbOG\^KNOUjLDQ7[ NO`XHOQ UVQNIL•p.‰7wXHIG NILDJMLDdjLDYZeUVQ7_a`7^\G\_Swe o ξ η /

Pj 1 Pj Pj 1 j Pj j

HIG\[OJil ]9djG\[ON wX`XQ7_adjG LAzoGKHNOg7GKUVHfHIG\[OJG\^KNOUVzoG¨^\LD~ŽJ;pD^KNUV~'pqYoG\[=r3RfgXUj^€g-Q7G\G\_+Q7LDN)wG}pqdVdLq†− × − P P

Pj 1 Pj

wMG'~&L“_a`Xd?pqHK€l^“UVQ7^\G&NOg7G&Q;pqNO`XH€pqd)~'pqJ Uj[JXHILDJGKHA]vR$GYoGKN1p NOg7GŽ†«pD^\G&d?pqNONOUj^\G −

P η∗Σj Σj ƒ«NOg7G\LDHOeo] ~'pqJˆUVQ → K 1 1 0 η∗ : K (Σ ) K (η∗Σ ) = K (η∗Σ R) .

c j → c j c j ×

†uLDH g;pD[—p1Q;pqNO`XH€pqdvŒ`7^KdVUj_XGApqQcGK~”wMG\_X_aUVQXY7]MpD[†uLDdVdjLR¯[Al Œ‡pD^Ig‘Lq†vNOg7G}‰7wXHIG\[ 1

ξ− (E) E ξ( j)

lF¦HOUVNIG ] ];pqQ7_ GKN

ÎÉÍ¢¡ͤ£¦¥¨§©¥3Í i½jÐ|½jÐi½MÑ ¤ ∈ P (E, F ) E (F ) = F E (F ) = F E~ ∈ Pj 0 h i 1 h ⊥ ∩ i E (F ) = E (F )⊥ E (F )⊥ E .

1/2 0 ∩ 1 ∩ h i

FG g;p\zoG

†>LDH—p1`XQXUj˜“`7G ] l r3Uu€l E (F ) = R e e E~ e = 1

1 · F F ∈ k F k

Uj[$p ^\LD~ŽJ;pD^KN ƒB[O`XwX~'pqQXUÀ†>LDdj_'Lq† ]9pqQ7_ r3UVUu€l¯hg7G—[€GKN (1)

S1(E) = eF F Pj , F E X

Uj[ l UVNI[fN€pqQXYoGKQN¨[OJ;pD^\G=pqN { | ∈ ⊂ } ⊂ C

eF E1/2(F ) ]XUj[—pqQ-g7LD~&G\LD~&LDHOJXgXUj[I~ˆl r3UVUVUu€l¯hg7G ~'pqJ ξ 1(E) S (E) : (E, F ) e

− → 1 7→ F

]¢NOg7G hg“`7[A]vUVNUj[Q;pqNO`XH€pqd$NILˆ_XGŠ‰7Q7G

¤i½jÐ|½¥¤i½

T j = (E, F, u) (E, F ) j , u E1/2(F )

pqdjLDQXY'NOg7G}‰7wXHIG\[¯Lq† l%7‘LDHIG\LAzoGKHA];^\LDQ7[OUj_XGKH N€pqQXYoGKQNy[IJ;pD^\GLq† P ∈ P ∈

j ξ % : Σj 1 Pj 1 : (E, v)

ƒB_aUV~&GKQ7[OUjLDQ;pqdVUVNe–Lq† UV~ŽJXdVUjG\[¯NOg;pqN—NOg7G}~'pqJ lhg7G P  − → − 7→ E 1 E1(F )

η∗Σj %∗T j : (E, F, y) E, p (y), F, p (y)

→ P 7→ E⊥ E1/2(F )

NIL=p¨HIGApqdEdVUVQ7GfwX`XQ7_adjG¯LzoGKH lº ¨GKHIGZ]oR)GfQ7LDNIG l NO`XHOQ7[

η∗Σj %∗T j F ⊥ = E⊥ E1/2(F ) E1(F )

Uj[)p ^\LD~ŽJXdjGŠ‹ŽdVUVQ7G¯wX`XQ7_adjGyLAzoGKH ]pqQ7_'R$GfYoGKN$NOg7G¯^\LDHOHIG\[OJLDQ7_aUVQXY”hg7LD~ hg`7[A] P ⊕ ⊕

η∗Σj R %∗T j l Uj[ILD~&LDHOJXgXUj[O~ × 0 0 P 1

β : Kc (%∗T j) Kc (η∗Σj R) = Kc (η∗Σj)

ÔBQ.LDHI_XGKH)NIL1^\LDQ7[INOHO`7^KNfpqQ.UVQ7_XGŠ‹'~'pqJ ]“R)G¨`7[IG—NOg7G}_XGKz9Uj^\GyLq†

i½jÐ|½ ©¢½ ¤ P → × 0 0

Kc (%∗T j) Kc (Σj 1)

‰7wXHIGŠƒ«RfUj[€Gq¢75JM=ONO;= 7 NM@TE¦0IG(EM9¡ Z]7_a`7G¨NIL&)LDQXQ7G\[Âď)LDQ \qÕ;]aÔIԊl ÆÉÈB]apqQ7_-p1^\GKHON€pqUVQc ƒpqdVYoGKwXH€pqUj^ NOg7GPr P → −

∗

_XGŠ†uLDHO~'pqNOUjLDQ ˜`;pqQ“NOUj[€pqNOUjLDQil GKN wG–p¦‰7wXHIGˆwX`XQ7_adjGˆRfUVNOg ‰7wXHIG\[•r3NOg7G

Ñ ξ : M B C1 C1 wG [ONOHO`7^KNO`XHIG‡_XGKJGKQ7_aUVQXY—^\LDQNOUVQ“`7LD`7[OdVe=LDQ NOg7Gw;pD[€GJMLDUVQ“NI[K€]DpqQ7_ djGKN→ 1

T M = b B b T ξ− (b) ]o^ApqdVdjG\_ŽNOg7G)‰7wXHIGKRfUj[€GfN€pqQXYoGKQNYZHILD`XJLDUj_|] NOg7G)‰7wXHIGKRfUj[IGfN€pqQXYoGKQN‡wX`XQ7_adjGZlhg7GKQpyYZHILD`XJMLDUj_ ∈ { }× M

^ApqQˆwMG ^\LDQ7[INOHO`7^KNIG\_ˆUVQ-NOg7G}†>LDdVdjLARfUVQXY~'pqQXQ7GKHAl G S

^9GKN„ƒ«NOg7G\LDHIGKNOUj^ApqdVdVeo]idjGKN lŒQ7_XLAR NOgXUj[ [OJ;pD^\G”RfUVNOg‘NOg7G

M = T M 0 (M B M) ]0, 1]

pqQ7_.NOg7G ~'pqJ7[ Rfg7GKHIG NILDJMLDdjLDYZe.YoGKQ7GKH€pqNIG\_ˆw“e G × ∪ × × (M M) ]0, 1] φ : M R [0, 1] ×B × h GM → × ×

pqQ7_ h(x ) h(x ) φ (x, X, 0) = (x, dh (X), 0) φ (x , x , ε) = x , 1 − 2 , ε , h x h 1 2 1 ε

 

Uj['pqQ“e¨^\LDQ“NOUVQ`7LD`7[+~'pqJ RfgXUj^€g Uj[ pqdjLDQXY¤NOg7G.‰7wXHIG\[Lq† l hg7GKQ pqQ7_ 1

h : M R ξ M w“e&^\LDQ7[OUj_XGKHOUVQXY wMG\^\LD~&G\[p djL9^ApqdVdVe&^\LD~ŽJ;pD^KNYZHILD`XJMLDUj_'RfUVNOg`XQXUVN)[OJ;pD^\G→ C (0) G

M = M [0, 1] T M

k YZHILD`XJwX`XQ7_adjGMl“LzoGKH ]“pqQ7_ pD[)p [O`XwXYZHILD`XJLDUj_'Lq†|NOg7G¯J;pqUVH$YZHILD`XJMLDUj_ l pD[$p G ×

M M B M M M

~&LDHIG_XGKdVUj^ApqNIG=~'pqNONIGKH¯Uj[fNOg7GGŠ‹aUj[ONIGKQ7^\G Lq†pŽ }pZpqH¯[Oe9[INIGK~ÒLDQ l i × ×

M

pD[p^\LDQ“NOUVQ`7LD`7[)†«pq~ŽUVdVeŽYZHILD`XJMLDUj_•ruLq†¢^Kd?pD[I[  h¢L”NOg;pqN)GKQ7_|]“UVN‡Uj[$`7[IGŠ†u`Xd;NIL”^\LDQ7[OUj_XGKH G 1,0

M

 tvpqNhMh7] Wa]_XGŠ†„l¤WÉÈBl¨Ô5l GZlV]MR$G1Q7G\G\_cNIL.[Og7LAR NOg;pqN UVQcNOg7GP[IGKQ7[IG”Lq†$tºpqNIGKHI[ILDQ G C r, s : (0) GM → GM

SKÕ ¢¡¥¨§R& ¡¤£¦¥¢ £¡¤¡£ ¨£¦¥¦¥

UVQNILPp ^\LDQ“NOUVQ`7LD`7[$†>pq~ŽUVdVeŽLq† ~'pqQXUÀ†uLDdj_X[]“pqQ7_&NOg;pqN$UVQzoGKHI[OUjLDQ.pqQ7_'^\LD~ŽJLZ[OUVNOUjLDQ+pqHIG NO`XHOQ 1

M NILŽ_XGŠ‰7Q7G”pqQcpqNOd?pD[Al

^\LD~ŽJ;pqNOUVwXdjGRfUVNOg-NOgXUj[y[ONOHO`7^KNO`XHIGZlFG UVQ7_aUj^ApqNIGg7LARG C

Uj[$p pqNOd?pD[)Lq† ]odjGKN ]XpqQ7_'_XGŠ‰7Q7G F’g7GKQ7GKzoGKH 1,0

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¥¥ 1  324.5 6(879;: <9= § Qp-spaces on bounded symmetric domains

Miroslav Engliˇs

Abstract Qp spaces on the unit disc were introduced, and their basic properties established, in 1995 by Aulaskari, Xiao and Zhao. Later some of these results were extended also to the unit ball or even to strictly pseudoconvex domains in the complex n-space. We briefly review the theory of bounded symmetric domains, of which the disc and the ball are the simplest examples, and then discuss the Qp spaces in this setting. It turns out that some new phenomena appear, most notably concerning the relationships of these spaces to the various kinds of Bloch spaces on symmetric domains.

1 Introduction

The Qp spaces on the unit disc D were introduced in 1995 by Aulaskari, Xiao and Zhao [AXZ] by 2 p f Qp sup f 0(z) g(z, a) dz < , ∈ ⇐⇒ a D D | | ∞ ∈ Z the square root of the right-hand side being, by definition, the (semi)norm in Qp. Here g(z, a) stands for the Green function z a g(z, a) = log − , 1 az

− and dz denotes the Lebesgue area measure. It is not difficult to see that one gets the same spaces, with equivalent seminorms, upon replacing the Green function by the function z a log 1 −az which has (for each fixed a) the same boundary behaviour: | − | 2 p 2 z a f Qp sup f 0(z) 1 − dz < . (1.1) ∈ ⇐⇒ a D D | | − 1 az ∞ ∈ Z   − We will adhere to this latter definition throughout the sequel.

The most notable feature of the Qp spaces is that they are M¨obius invariant. Indeed, a z any M¨obius map (i.e. a biholomorphic self-map of D) is of the form φ(z) =  − , with 1 az  = 1 and a D. Thus the right-hand side of (1.1) can be rewritten as − | | ∈ 2 2 p sup f 0(z) (1 φ(z) ) dz φ Aut(D) D | | − | | ∈ Z = sup ∆ f 2(z) (1 φ(z) 2)p dz φ Aut(D) D | | − | | ∈ Z = sup (∆ f 2)(z) (1 φ(z) 2)p dµ(z) φ Aut(D) D | | − | | ∈ Z = sup ∆e f φ(z) 2 (1 z 2)p dµ(z), φ Aut(D) D | ◦ | − | | ∈ Z e

17 18 Miroslav Engliˇs

∂2 dz where ∆ = (1 z 2)2 and dµ(z) = are the invariant Laplacian and the − | | ∂z∂z (1 z 2)2 − | | invariant measure on D, respectively. From the last formula it is apparent that f Qp e ∈ implies f φ Q and f and f φ have the same norm in Q , for all φ Aut(D). ◦ ∈ p ◦ p ∈ It was shown in [AXZ] that

p > 1 = Q = , the Bloch space, ⇒ p B p = 1 = Q = BMOA, ⇒ p 0 p < p 1 = Q ( Q , ≤ 1 2 ≤ ⇒ p1 p2 p = 0 = Q = , the Dirichlet space, ⇒ p D p < 0 = Q = const . ⇒ p { }

Thus the Qp spaces provide a whole range of M¨obius-invariant function spaces on D lying strictly between the Dirichlet space on the one hand, and BMOA and the Bloch space on the other. The Qp spaces subsequently attracted a lot of attention; see e.g. the book by Xiao [X] and the references therein. They were generalized to the unit ball Bd Cd in 1998 by ⊂ Ouyang, Yang and Zhao [OYZ]:

2 p f Qp sup ∆ f(z) G(z, a) dµ(z) < ∈ ⇐⇒ a Bd Bd | | ∞ ∈ Z sup e ∆ f φ 2 G(z, 0)p dµ(z) < , ⇐⇒ φ Aut(Bd) Bd | ◦ | ∞ ∈ Z e where ∆, dµ and G(z, a) denote the invariant Laplacian, the invariant measure and the Green function of ∆ on Bd, respectively. Again, these spaces are M¨obius invariant, and e d ep , = Q = const , ≥ d 1 ⇒ p { } −d 1 < p < = Q = (Bd), the Bloch space, d 1 ⇒ p B − p = 1 = Q = BMOA(Bd), ⇒ p d 1 − < p < p 1 = Q ( Q , d 1 2 ≤ ⇒ p1 p2 d 1 p − = Q = const . ≤ d ⇒ p { }

d The cut-off at p = d 1 turns out to due to the pole of G(z, a) at z = a, and disappears − z a 2 d if we replace (as we did for the disc) the Green function by (1 − ) , i.e. upon − k 1 a,z k setting −h i 2 2 pd f Qp sup ∆ f φ (z) (1 z ) dµ(z) < . ∈ ⇐⇒ φ Aut(Bd) Bd | ◦ | − k k ∞ ∈ Z Then Q = (Bd) p > 1, while the othere cases remain unchanged. (We again stick to p B ∀ this latter definition in the sequel.) Note that, in contrast to the disc, for d > 1 the Dirichlet space does not turn up as one of the Qp’s, though in all other cases the situation is the same as for D. Qp-spaces on bounded symmetric domains 19

Other generalizations include Qp spaces on smoothly bounded strictly pseudoconvex domains [AC] or the F (p, q, s) spaces of R¨atty¨a and Zhao [R],[Z]. In this talk, we will consider generalization in another direction, suggested by the appearance of the invariant Laplacians ∆, the invariant measures dµ, and the invariance of the spaces under M¨obius maps — namely, the generalization to bounded symmetric domains. e

2 Bounded symmetric domains

Recall that a bounded domain Ω Cd is called symmetric if x Ω there exists s ⊂ ∀ ∈ x ∈ Aut(Ω) such that s s = id and x is an isolated fixed-point of s . One calls s the x ◦ x x x geodesic symmetry at x. The motivating example behind this is, of course, the complex n n-space Ω = C with sx(z) = 2x z (except that this is not a bounded domain). Another − x z example is the unit disc D with s (z) = z and s = φ s φ , where φ (z) = − 0 − x x ◦ 0 ◦ x x 1 xz is the geodesic symmetry interchanging 0 and x. A more general example is the unit−ball Ir R of r R complex matrices (R, r 1), again with s0(z) = z and sx = φx s0 φx, × × ≥ − ◦ ◦ the geodesic symmetry φx interchanging 0 and x being now given by

1/2 1 1/2 s (z) = (I xx∗)− (x z)(I x∗z)− (I x∗x) . x r − − R − R −

Note that this includes the unit ball Bd Cd as the special case I . ⊂ 1d It turns out that symmetry implies homogeneity: the group Aut(Ω) := G acts tran- sitively on Ω. (In fact, already the symmetries sx do.) It is a semisimple Lie group. A bounded symmetric domain is called irreducible if it is not biholomorphic to a Cartesian product of two other bounded symmetric domains. Irreducible bounded symmetric domains were completely classified by E. Cartan. There are four infinite series of such domains plus two exceptional domains in C16 and C27:

Domain Description r R IrR Z C × : Z CR Cr < 1 R r 1 ∈ k k → ≥ ≥ II Z I , Z = Zt r 2 r ∈ rr ≥ III Z I , Z = Zt m 5 m ∈ mm − ≥ IV Z Cn 1, ZtZ < 1, 1 + ZtZ 2 2Z Z > 0 n 5 n ∈ × | | | | − ∗ ≥ V Z O1 2, Z < 1 ∈ × k k VI Z O3 3, Z = Z , Z < 1 ∈ × ∗ k k

The restrictions on R, r, m, n stem from a few isomorphisms in low dimensions:

IV = III = II = I (= D), IV = II , IV = D D, 1 ∼ 2 ∼ 1 ∼ 1,1 ∼ 3 ∼ 2 2 ∼ × IV4 ∼= I2,2, III3 ∼= I1,3, III4 ∼= IV6, IrR ∼= IRr.

Up to biholomorphic equivalence, any irreducible bounded symmetric domain is uniquely determined by three integers, namely its rank r and its characteristic multiplicities a, b. 20 Miroslav Engliˇs

domain r a b d p IrR (r R) r 2 R r rR r + R ≤ − 1 IIr r 1 0 2 r(r + 1) r + 1 III ,  0, 1 r 4 2 r(2r + 2 1) 4r + 2 2 2r+ ∈ { } − − IV 2 n 2 0 n n n − V 2 6 4 16 12 VI 3 8 0 27 18

The two other important quantities given in the table, the genus p and the dimension d are related to a, b and r by

r(r 1) p = (r 1)a + b + 2, d = − a + rb + r. − 2 The domains with b = 0 are in some respects “simpler” than others and are called tube domains. Thus, for instance, IrR is tube r = R. d ⇐⇒ The unit balls B = I1d are the only bounded symmetric domains of rank 1, and the only bounded symmetric domains with smooth boundary. The domains in the list above are called Cartan domains. Clearly, any Cartan domain is convex, contains the origin, and is circular with respect to it. From now on, we will suppose (unless explicitly stated otherwise) that Ω is a Cartan domain, and for each x Ω we denote by φ the (unique) geodesic symmetry interchang- ∈ x ing 0 and x. We further denote by K the stabilizer of the origin in G,

K := k G : k0 = 0 . { ∈ } It is a consequence of one of Cartan’s theorems that any k K is automatically a unitary ∈ linear map on Cd. Note that from the definition of K it is immediate that any φ G is of the form ∈ φ = φ k, where k K, x Ω. (In fact x = φ(0).) x ∈ ∈ Having recalled the definition of bounded symmetric domains, we can turn to the Qp spaces on them. We have seen in the Introduction that their definition involves three ingredients — namely, the invariant Laplacian ∆, the invariant measure dµ, and powers of the function 1 z 2 (or, for the ball, 1 z 2). Let us now clarify what are the − | | − k k counterparts of these on a general Cartan domain.e

3 Invariant differential operators

A differential operator L on a Cartan domain Ω is called invariant if

L(f φ) = (Lf) φ φ G = Aut(Ω). ◦ ◦ ∀ ∈ It is well known that on the unit disc, invariant operators are precisely the polynomials of the invariant Laplacian ∆ = (1 z 2)2∆. The same is true for Bd. For a general bounded −| | symmetric domain, the situation is more complicated: namely, the algebra of all invariant differential operators consistse of all polynomials in r commuting differential operators ∆1, . . . , ∆r, of orders 2, 4, . . . , 2r, respectively, where r is the rank. In particular, the Qp-spaces on bounded symmetric domains 21

n1 nr monomials ∆1 . . . ∆r form a linear basis of all invariant differential operators. However, often it is much more convenient to use another basis, the construction of which we now describe. For any invariant differential operator L, let L0 be the (non-invariant) linear differ- ential operator obtain upon freezing the coefficients of L at the origin, that is, Lf(0) =: L0f(0). From the invariance of L it follows that k G, k0 = 0 = L (f k) = (L f) k ∈ ⇒ 0 ◦ 0 ◦ (i.e. L0 is K-invariant) and Lf(z) = L (f φ )(0). (3.1) 0 ◦ z Conversely, if L0 is a K-invariant constant-coefficient differential operator, then the recipe (3.1) clearly defines an invariant differential operator L on Ω. Thus there is a 1-to-1 correspondence between (G-)invariant linear differential operators on Ω and K-invariant linear constant-coefficients differential operators on Cd. Further, any constant-coefficient linear differential operator L0 can be written in the form L = p(∂, ∂) for some polynomial p on Cd Cd. It is not difficult to see that such 0 × operator is K-invariant if and only if the polynomial p is K-invariant in the sense that p(x, y) = p(kx, ky) x, y Cd k K. ∀ ∈ ∀ ∈ Combining this with the observation in the preceding paragraph, we thus see that invariant differential operators are in 1-to-1 correspondence with K-invariant polynomials. Example. Since K consists of unitary maps, the simplest K-invariant polynomial (apart from the constants) is p(x, y) = x, y . The corresponding invariant differential operator is h i Lf(a) = ∆(f φ )(0). ◦ a This operator is called the invariant Laplacian of Ω; it coincides with the Laplace-Beltrami operator with respect to the Bergman metric on Ω. Note that for f holomorphic,

d 2 2 ∂(f φa)(0) 2 L f (a) = ◦ = ∂(f φa)(0) | | ∂zj k ◦ k j=1 X is what we might call the invariant holomorphic gradient of f. In a moment, we will see that there exists a very natural basis for K-invariant poly- nomials (which will thus yield the sought basis for invariant differential operators). Prior to that, however, we review some facts about Bergman kernels on Cartan domains.

4 Bergman spaces and kernels

The function h(x, y) := 1 xy on D is noteworthy in a number of ways. First of all, − 1 h 2 is, up to a constant factor, the Bergman kernel K(x, y) = . Second, − π(1 xy)2 dz − dµ(z) = is the invariant measure on D. Finally, for any α > 1, the Bergman h(z, z)2 − 2 α kernel of Lhol(D, h(z, z) dz) is given by const K (x, y) = . α h(x, y)α+2 22 Miroslav Engliˇs

The same properties are also possessed by the function h(x, y) = 1 x, y on the − h i ball, only in all three formulas 2 must be replaced by d + 1. It is a notable fact that the same phenomenon persists for a general Cartan domain. Namely, the Bergman kernel of a Cartan domain has the form

1 p K(x, y) = h(x, y)− , vol Ω where h(x, y) is an irreducible polynomial, analytic in x and y, and such that h(0, z) = h(z, 0) = 1 h(z, z) 0 z Ω. The degree of h is equal to the rank, r, and p is the ≥ ≥ ∀ ∈ genus (this is always an integer 2). Finally, the Bergman kernel of L2 (Ω, h(z, z)α dz) ≥ hol equals const K (x, y) = , α h(x, y)α+p for any α > 1, and − dz dµ(z) := h(z, z)p is an invariant volume element on Ω. For the domains I and II in Cartan’s list, h is given by h(X, Y ) = det(I XY ); for − ∗ domains of type III, the determinant gets replaced by the Pfaffian. Explicit formulas are known also for the types IV–VI. Comparing all the facts above with the situations for the disc and the ball, we see that we should define the Q spaces on a Cartan domain for any ν R and any invariant p ∈ differential operator L as follows:

2 ν f Qν,L sup L f φ (z) h(z, z) dµ(z) < . (4.1) ∈ ⇐⇒ φ G Ω | ◦ | ∞ ∈ Z (Here we have started using the subscript ν instead of p since the letter p is already reserved for the genus.) Clearly, this reduces to the original definitions for Ω = D (or Bd) and L the invariant Laplacian. A small catch here is, however, that in order to have the square-root of the right-hand side for a seminorm, we need this right-hand side to be nonnegative for all holomorphic functions f. It is precisely at this point that the promised linear basis for invariant differential operators comes to the rescue; so let us exhibit it without further delay.

5 Peter-Weyl decomposition

Let denote the vector space of all (holomorphic) polynomials on Cd. We endow with P P the Fock inner product

f, g : = f(∂) g∗ (0), where g∗(z) := g(z), h iF d z 2 = π− f(z)g(z) e−k k dz. Cd Z This makes into a pre-Hilbert space, and the action P f f k, k K, 7→ ◦ ∈ Qp-spaces on bounded symmetric domains 23

is a unitary representation of K on . It is a deep result of W. Schmidt that this P representation has a multiplicity-free decomposition into irreducibles

= ⊕ m P m P X where m ranges over all signatures, i.e. r-tuples m = (m , m , . . . , m ) Zr satisfying 1 2 r ∈ m m m 0. Polynomials in m are homogeneous of degree m := 1 ≥ 2 ≥ · · · ≥ r ≥ P | | m + m + + m ; in particular, are the constants and the linear polynomials. 1 2 · · · r P(0) P(1) Any holomorphic function thus has a decomposition f = m fm, fm m, which refines ∈ P the usual homogeneous expansion. P Since the spaces m are finite dimensional, they automatically possess a reproducing P kernel: there exist functions Km(x, y) on Cd Cd such that for each f m and y Cd, × dim ∈mP ∈ f(y) = f, K( , y) . Explicitly, for any orthonormal basis ψ P of m, Km is h · iF { j}j=1 P given by dim m P Km(x, y) = ψj(x)ψj(y). (5.1) Xj=1 It follows from the definition of the m spaces that the kernels Km(x, y) are K- P invariant. By the discussion in the penultimate section, we therefore know that each Km defines an invariant differential operator

∆mf(a) := Km(∂, ∂)(f φ )(0), a Ω. (5.2) ◦ a ∈ Further, one can show that Km are actually a basis of all K-invariant polynomials, and, consequently, ∆m are a linear basis for invariant differential operators. Further, from (5.1) and (5.2) we see that for any f holomorphic,

2 2 ∆m f (a) = ψ (∂)(f φ )(0) 0. | | | j ◦ a | ≥ Xj What makes the basis ∆m important for our applications to the Qν-spaces is the following converse to the last inequality. Theorem. An invariant differential operator

L = lm∆m m X satisfies L f 2 0 f holomorphic if and only if | | ≥ ∀ lm 0 m. ≥ ∀

6 Bloch spaces and Qν spaces on bounded symmetric do- mains

Thus we see that the invariant differential operators L that can be used in (4.1) are pre- cisely those which are linear combinations of ∆m with nonnegative coefficients. The most basic among such L are evidently the operators ∆m themselves. We are thus lead to the following definitions. 24 Miroslav Engliˇs

Definition. For each signature m, the m-Bloch space is defined by

2 m = f holomorphic on Ω : ∆m f < . B { k | | k∞ ∞}

Definition. For each signature m and ν R, the space Q m is defined by requiring ∈ ν, that, for f holomorphic,

2 ν f Qν,m sup ∆m f φ h dµ < ∈ ⇐⇒ φ G Ω | ◦ | ∞ ∈ Z 2 ν sup ∆m f φa h dµ < ⇐⇒ a Ω Ω | ◦ | ∞ ∈ Z 2 ν sup ∆m f (h φa) dµ < . ⇐⇒ a Ω Ω | | ◦ ∞ ∈ Z Here h(z) h(z, z). (All the three conditions above are equivalent owing to the invariance ≡ of ∆m and dµ.)

Clearly, the above definitions reduce to the usual definitions for Ω = D or Bd and m = (1) (so that ∆m is the invariant Laplacian). In particular, the case of m = (1) gives 6 something new even for the unit disc. Let us work out the simplest special cases of Bloch spaces.

Example. For m = (0) we have ∆m = I, so = H . Also, B(0) ∞

H∞ ν > p 1, Qν,(0) = − ( 0 ν p 1. { } ≤ −

Example. For m = (1) we have ∆(1) = ∆, so that

2 (1) = f : supe ∂(f φa)(0) < . B { a k ◦ k ∞}

This is known as the Timoney Bloch space [T].

Example. Assume that Ω is of tube type and s := d/r Z. Let m = (s, s, . . . , s) (sr). s∈ ≡ It is known that in that case the space r = CN is one-dimensional (for the unit P(s ) disc, N(z) = z; for Irr, N(Z) = det Z), the kernel Km is given (up to a constant factor) by Km(∂, ∂) = N(∂)sN(∂)s, and the so-called Bol’s lemma says that for any f holomorphic, N(∂)s(f φ )(0) = const h(a)sN(∂)sf(a). (For the disc, this reads ◦ a · (f φ ) (0) = (1 a 2) f (a).) Hence, ◦ a 0 − − | | 0 2 p s 2 ∆ r f = h N(∂) f (6.1) (s )| | | | and p s 2 r = f holomorphic: h N(∂) f is bounded . B(s ) { | | } We might call this the Arazy Bloch space [A]. Qp-spaces on bounded symmetric domains 25

7 Example: the polydisc

For clarity, let us also see what is the situation for the bidisc Ω = D2. This is definitely NOT a Cartan domain (it is not irreducible), but in many respects it behaves like a Cartan domain with the rank, dimension and genus r = p = d = 2, multiplicities a = b = 0, and h(x, y) = (1 x y )(1 x y ). Namely, the invariant measure is h(z, z) 2 dz; the invariant − 1 1 − 2 2 − differential operators are precisely the symmetric polynomials in ∆1, ∆2, where ∆j := 2 2 m1 m2 m2 m1 (1 zj ) ∂j∂j; the Peter-Weyl spaces are given by m = Cz1 z2 + Cz1 z2 , for m = − | | P m1 m2 m2 m1 (m1, m2); and, up to a constant factor, Km(x, y) = (x1y1) (x2y2) e+(xe1y1) (x2ye2) . It follows that ∆(1,0) = ∆1 + ∆2, ∆(1,1) = ∆1∆2. Thus the Timoney Bloch space consists of all f holomorphic on D2 for which B(1e,0) e e e 2 2 2 2 ∂f 2 2 ∂f (1 z1 ) + (1 z2 ) is bounded, − | | ∂z1 − | | ∂z2

while the Arazy Bloch space consists of all f holomorphic on D2 for which B(1,1) 2 2 2 2 2 2 ∂ f (1 z1 ) (1 z2 ) is bounded. − | | − | | ∂z1∂z2

We thus see that the Timoney Bloch space is contained in the Arazy Bloch space, and is a proper subset thereof: any holomorphic function of the form f(z1, z2) = g(z1) belongs to , but does not belong to unless g belongs to the Bloch space on the disc. B(1,1) B(1,0) We also see that the Timoney-Bloch norm vanishes precisely on the constants, while the Arazy-Bloch norm vanishes precisely on functions of the form f(z1) + g(z2). Similar situation can be seen to prevail for a general polydisc Dn: there are n Bloch spaces, of which Timoney is the smallest, and Arazy the largest. Remark. The big Hankel operator H is compact f belongs to the Timoney Bloch f ⇐⇒ space.

8 Composition series

The phenomenon that we have observed for the polydiscs is connected with the existence of the composition series. Let us explain this concept on the example of the unit disc D. There the following assertion holds. Claim. Let E be any topological vector space of holomorphic functions on D which is M¨obius invariant and on which the group of rotations acts strongly-continuously. Then either E = 0 , or E = constants , or E contains all polynomials. { } { } k Proof. Let E f = ∞ f z ; then by rotation invariance, 3 k=0 k 2π P iθ miθ dθ m f(e z) e− = f z E. (8.1) 2π m ∈ Z0 m a z m Thus if f = 0 for some m, then z E; hence, by invariance, ( − ) E a D. m 6 ∈ 1 az ∈ ∀ ∈ Taking this for the f in (8.1) and 0 for the m in (8.1), we get am1 E−; thus the constants ∈ are in E. 26 Miroslav Engliˇs

If even fm = 0 for some m 1, then, applying (8.1) to the same function again but 6 ≥ a z m m 2 2 this time taking 1 for the m in (8.1) and noting that ( 1 −az ) = a m(1 a )z +O(z ), a z − 2 − j −j | | we see that z E; thus by invariance − = a (1 a )z ∞ a z belongs to E, ∈ 1 az − − | | j=1 for any a D. Taking the last function for− the f in (8.1) shows that z j E j, i.e. all ∈ P ∈ ∀ polynomials are in E. This completes the proof.

The last theorem admits the following reformulation. Denote = all holomorphic M1 { functions , 0 = constants , 1 = 0 . Then } M { } M− { } E j 1 = = j E. \ M − 6 ∅ ⇒ P ∩ M ⊂ It turns out that, in a sense, precisely the same thing holds for a general Cartan domain. Namely, let (x) := x(x + 1) . . . (x + k 1) k − denote the familiar Pochhammer symbol, and for a signature m = (m1, m2, . . . , mr), consider the function a r 1 x (x) (x ) (x a) . . . (x − a) , x C. 7→ m1 − 2 m2 − m3 − 2 mr ∈ Let q(m) be the multiplicity of zero of this function at x = 0: j 1 q(m) := card j : m > − a Z . { j 2 ∈ } Also denote by q the maximum possible value of q(m), i.e.

r a even, q = r + 1  a odd.  2 h i For 1 j q, let − ≤ ≤  j = f = fm holomorphic : fm = 0 if q(m) > j . M { m } X Thus, in particular,

1 0 1 q, M− ⊂ M ⊂ M ⊂ · · · ⊂ M 1 = 0 , 0 = constants , q = all holomorphic . M− { } M { } M { } The following deep result is due to Orsted, Faraut and Koranyi. Theorem. (1) Each is G-invariant; Mj (2) for any G-invariant space E of holomorphic functions on which the action of K is strongly continuous,

E j 1 = = j E. \ M − 6 ∅ ⇒ P ∩ M ⊂ Example. For the bidisc D2, q = 2 and

1 = 0 , 0 = constants , M− { } M { } = f(z ) + g(z ), f, g holomorphic on D, M1 1 2 = all holomorphic . M2 { } Qp-spaces on bounded symmetric domains 27

9 Results

After all the preparations, we can finally give our results.

Theorem. If j < q(m), then the Q m-norm vanishes on ; thus is contained ν, Mj Mj in Qν,m in a trivial way. The same is true also for the Bloch space m. B Theorem. If ν > p 1, then m Q m continuously. − B ⊂ ν, Theorem. If q(m) q(n), then Q m n continuously. ≤ ν, ⊂ B Corollary. ν > p 1 = Q m = m, with equivalent norms. − ⇒ ν, B q(m) q(n) = m n continuously. ≤ ⇒ B ⊂ B q(m) = q(n) = m = n, with equivalent norms. ⇒ B B q(m) = q(n), ν > p 1 = Q m = Q m, with equivalent norms. − ⇒ ν, ν, The last corollary exhausts the case ν > p 1 completely. What about ν p 1? − ≤ − Theorem. If ν < 0, then Qν,m = q(m) 1. M − Theorem. For r > 1 and m = (1, 0, . . . , 0), that is,

2 ν f Qν,(1) sup ∆ f φa h dµ, ∈ ⇐⇒ a Ω Ω | ◦ | ∈ Z we have e (1), the Timoney Bloch space ν > p 1, Qν,(1) = B − constants ν p 1. ({ } ≤ − Note that the last theorem means that the situation for r > 1 differs radically from the one for r = 1 (disc, ball): there Q is nontrivial also for p 2 < ν p 1 (for the ν − ≤ − disc, even for p 2 ν p 1). − ≤ ≤ − Theorem. For a tube domain with s = d Z, and m = (sr), r ∈ r , the Arazy Bloch space ν > p 1 B(s ) − Q r = ν = 0 ν,(s ) D   q 1 ν < 0. M − Here is the Dirichlet space D  = f holomorphic: N(∂)sf L2(Ω, dz) . D { ∈ } At the moment, we do not know what are the spaces Q m for ν between 0 and p 1 ν, − and m > 1 — for instance, whether they are properly increasing with ν. We can offer | | somewhat more complete information only for the polydisc: Theorem. For the polydisc Dr (so that p = 2, q(m) = # j : m > 0 , and q = r), { j } m ν > 1, q(m) < r = Qν,(m) = B ⇒ ( q(m) 1 ν 1; M − ≤ Arazy-Bloch ν > 1, q(m) = r = Q m = ν = 0, ⇒ ν,( )  D  q(m) 1 ν < 0. M −  28 Miroslav Engliˇs

All this suggests the following conjecture about the nontriviality of the spaces Qν,m.

Conjecture. For tube domains with s = d Z and q(m) = q, r ∈

Q m ν 0; ν, ⇐⇒ ≥ in all other cases, Q m ν > p 1. ν, ⇐⇒ −

10 Some proofs

We proceed to give some hints about the proofs of the theorems.

Theorem. j < q(m) = Q m and the norm vanishes. Similarly for m. ⇒ Mj ⊂ ν, B Proof. Recall that

2 ν f Qν,m sup ∆m f (h φa) dµ < , ∈ ⇐⇒ a Ω | | ◦ ∞ Z 2 f m ∆m f L∞. ∈ B ⇐⇒ | | ∈ Now

2 2 ∆m f (z) = Km(∂, ∂) f φ (0) | | | ◦ z| = ψ (∂)ψ (∂) f φ 2(0) j j | ◦ z| Xj = ψ (∂)(f φ )(0) 2 | j ◦ z | Xj = f φ , ψ 2 |h ◦ z jiF | Xj 2 = Pm(f φ ) , k ◦ z kF where Pm denotes the projection onto m. P Thus f = f φ = Pm(f φ ) = 0 = ∆m f 2 = 0 = f m ∈ Mj ⇒ ◦ z ∈ Mj ⇒ ◦ z ⇒ | | ⇒ ∈ B and f Q m. ∈ ν, Theorem. ν > p 1 = m Q m continuously. − ⇒ B ⊂ ν, Proof. It is known that for ν > p 1, the measure hν dµ is finite. Thus a Ω, − ∀ ∈

2 ν 2 (∆m f ) φa h dµ cν ∆m f Ω | | ◦ ≤ k | | k∞ Z 2 = cν f m . k kB

Theorem. q(m) q(n) = Q m n continuously. ≤ ⇒ ν, ⊂ B Qp-spaces on bounded symmetric domains 29

Proof. By the K-invariance of ∆m and h, the integral

ν ∆m(fg) h dµ ZΩ is a K-invariant bilinear form of f, g . It follows from the Peter-Weyl decomposition ∈ P of into the m that any such bilinear functional must be of the form P P

cmk fk, gk F , k h i X for some coefficients cmk 0. Suppose we can show that ≥ cmn > 0. (10.1)

Since ∆n f 2(0) = Pnf 2 = fn 2 , it will follow that | | k kF k kF

2 1 2 ν ∆n f (0) ∆m f h dµ. | | ≤ cmn | | ZΩ Replacing f by f φ , this becomes ◦ a 2 1 2 ν ∆n f (a) ∆m f φa h dµ. | | ≤ cmn | ◦ | ZΩ Taking suprema over all a Ω gives the assertion. ∈ It remains to prove (10.1). But by the properties of the composition series,

2 ν cmn = 0 ∆m fn h dµ = 0 fn n ⇐⇒ Ω | | ∀ ∈ P Z 2 ∆m fn (z) = 0 z fn ⇐⇒ | | ∀ ∀ 2 Pm(fn φ ) = 0 z fn ⇐⇒ k ◦ z kF ∀ ∀ Pm(fn φ ) = 0 z fn ⇐⇒ ◦ z ∀ ∀ Pm n = 0 ⇐⇒ Mq( ) q(m) > q(n). ⇐⇒

Theorem. ν < 0 = Qν,m = q(m) 1. ⇒ M − Proof. From the composition series we know that

q(m) 1 ( Qν,m = q(m) Qν,m M − ⇒ P ∩ M ⊂ = m Q m ⇒ P ⊂ ν, 2 ν = sup ∆m f (h φ ) dµ < f m. ⇒ | | ◦ a ∞ ∀ ∈ P a ZΩ Since Km(z, z) = ψ (z) 2 for any basis ψ of m, we can continue by j | j | { j } P P ν = sup ∆mKm (h φ ) dµ < ⇒ · ◦ a ∞ a ZΩ 30 Miroslav Engliˇs

where Km = Km(z, z). It can be shown that

m m 0 : ∆mKm c h . ∃  ≥ Thus we can continue by

= sup hm (h φ )ν dµ < . (10.2) ⇒ ◦ a ∞ a ZΩ Forelli-Rudin inequalities show that this happens iff ν 0. ≥ Theorem. For rank r > 1 and m = (1, 0, . . . , 0),

(1), the Timoney Bloch space ν > p 1, Qν,(1) = B − constants ν p 1. ({ } ≤ − Proof. As above,

constants ( Q sup ∆ z 2 (h φ )ν dµ < . { } ν,(1) ⇐⇒ k k ◦ a ∞ a ZΩ The fact that e h2 Ω = D, ∆ z 2 h Ω = Bd, k k ≈  1 r > 1 e and (10.2) again yield the conclusion. 

Theorem. For a tube domain with s = d Z, and m = (sr), r ∈

r , the Arazy Bloch space ν > p 1 B(s ) − Q r = ν = 0 ν,(s ) D  q 1 ν < 0. M −  Proof. As mentioned before, in this case

p s s ∆m = h N(∂) N(∂) for a certain polynomial N (the Jordan norm). Hence

p s 2 ν f Q m sup h N(∂) f (h φ ) dµ < ∈ ν, ⇐⇒ | | ◦ a ∞ a ZΩ sup N(∂)sf 2 (h φ )ν dz < . ⇐⇒ | | ◦ a ∞ a ZΩ Thus for ν 0, all polynomials belong to Q m. ≥ ν, For ν = 0, this coincides with the definition of the Dirichlet space. The case ν < 0 was settled by the previous theorem. Qp-spaces on bounded symmetric domains 31

Theorem. For the polydisc Dr (so that p = 2, q(m) = # j : m > 0 , and q = r), { j } m ν > 1, q(m) < r = Qν,(m) = B ⇒ ( q(m) 1 ν 1; M − ≤ Arazy-Bloch ν > 1, q(m) = r = Q m = ν = 0, ⇒ ν,( ) D   q(m) 1 ν < 0. M − Proof. Using explicit formulas for Km, ∆metc. given in one of the preceding sections, this is easily reduced to explicit calculations on the disc.

11 Open problems

We conclude the paper by a list of open problems. (1) The first of them is, of course, to determine when Qν,m is nontrivial — we repeat here the conjecture stated above:

Conjecture. Qν,m nontrivial iff ν 0 (for tube domain with d Z and q(m) = q) ≥ r ∈ ν > p 1 (otherwise). −

(2) If q(m) = q(n), is Qν,m = Qν,n? (We have seen that this holds for the Bloch spaces, hence also for ν > p 1; the case of ν p 1 remains unresolved.) − ≤ −

(3) If ν1 < ν2 and Qν1,m, Qν2,m are nontrivial, is Qν1,m ( Qν2,m? (For D, this was proved in [AXZ], and for Bd in [AC].)

(4) In principle, one can define Q and for any invariant differential operator L, ν,L BL even when the right-hand side in (3) is not nonnegative, by f L f 2 is bounded, ∈ BL ⇐⇒ | | 2 ν f Qν,L sup L f φa h dµ < . ∈ ⇐⇒ a Ω | ◦ | ∞ Z If L is such that L f 2 0 for all f holomorphic f, i.e. if | | ≥ L = lm ∆m, lm 0, (11.1) m ≥ X then it is easy to see that

Qν,L = Qν,m, m :\lm>0 = m where q(m) = min q(k) : lk > 0 . BL B { } What happens for operators L not satisfying (11.1)? For instance, does the space of all holomorphic f on D for which

2 2 2 ν 2 sup ∆ f(φa(z) (1 z ) − dz < a D D | | − | | ∞ ∈ Z coincide with the Bloch space fore ν > 1? 32 Miroslav Engliˇs

References

[AC] M. Andersson, H. Carlsson: Qp spaces in strictly pseudoconvex domains, J. Anal. Math. 84 (2001), 335–359.

[A] J. Arazy: Boundedness and compactness of generalized Hankel operators on bounded symmetric domains, J. Funct. Anal. 137 (1996), 97–151.

[AXZ] R. Aulaskari, J. Xiao, R. Zhao: On subspaces and subsets of BMOA and UBC, Analysis 15 (1995), 101–121.

[OYZ] C. Ouyang, W. Yang, R. Zhao: M¨obius invariant Qp spaces associated with the Green’s function on the unit ball of C n, Pacific J. Math. 182 (1998), 69–99.

[R] J. R¨atty¨a: On some complex function spaces and classes, Ann. Acad. Sci. Fenn. Math. Diss. 124 (2001), 73 pp.

[T] R.M. Timoney: Bloch functions in several complex variables I, Bull. London Math. Soc. 12 (1980), 241–267; II, J. reine angew. Math. 319 (1980), 1–22.

[X] J. Xiao: Holomorphic Q classes, Lecture Notes in Mathematics 1767, Springer- Verlag, Berlin, 2001.

[Z] R. Zhao: On a general family of function spaces, Ann. Acad. Sci. Fenn. Math. Diss. 105 (1996), 56 pp.

Joint work with J. Arazy (Haifa)

Miroslav Engliˇs – Academy of Sciences, Prague [email protected] Commutative C∗-algebras of Toeplitz Operators on the Unit Disk Nikolai Vasilevski

We give a complete characterization of commutative C ∗-algebras of Toeplitz operators acting on weighted Bergman spaces over the unit disk. This note is a short version of the paper [7], where all proofs and details can be found.

1 Commutative algebras and hyperbolic geometry

We introduce the following M¨obius invariant normalized measure on the unit disk D 1 dx dy 1 dz dz dµ(z) = ∧ = ∧ . π (1 (x2 + y2))2 2πi (1 z 2)2 − − | | For h (0, 1), the weighted Bergman space 2 (D) on the unit disk (see, for example, [3]) ∈ Ah is the space of analytic functions in L2(D, dµh), where

1 2 1 dµ (z) = ( 1)(1 z ) h dµ(z), h h − − | | and 1 2 2 f h = f(z) dµh(z) . k k D | | Z  Note, that for h = 1 we have the classical weightless Bergman space 2(D) (with nor- 2 A malized measure). The orthogonal Bergman projection from L2(D, dµh) onto the weighted Bergman space has the form (see, for example, [3]):

1 h (h) f(ζ) 1 1 ζζ (BD f)(z) = 1 dµh(ζ) = ( 1) f(ζ) − dµ(ζ). D (1 zζ) h h − D 1 zζ Z − Z  −  (h) Given a function a(z) L (D), the Toeplitz operator Ta with symbol a is defined ∈ ∞ on 2 (D) as follows Ah T (h) : ϕ 2 (D) B(h)(aϕ) 2 (D). a ∈ Ah 7−→ D ∈ Ah It was recently shown ([11, 12]) that apart from the known case of radial symbols in the classical (weightless) Bergman space 2(D) there exists a rich family of commutative C - A ∗ algebras of Toeplitz operators. Moreover, surprisingly it turns out that these commutative properties of Toeplitz operators do not depend at all on smoothness properties of the symbols: the corresponding symbols can be merely measurable. Furthermore it turns out ([4, 5, 6]) that the above classes of symbols generate commutative C ∗-algebras of Toeplitz

33 34 Nikolai Vasilevski

operators on each weighted Bergman space 2 (D). The prime cause here appears to be Ah the geometric configuration of level lines of symbols. In this context it is useful to consider the unit disk D as the hyperbolic plane equipped with the standard hyperbolic metric

1 dx2 + dy2 ds2 = . π (1 (x2 + y2))2 − Recall that a geodesic, or a hyperbolic straight line, on D is a part of an Euclidean circle or of a straight line orthogonal to the boundary of D. Each pair of geodesics, say L1 and L2, determines (see, for example, [1]) a geomet- rically defined object, a one-parameter family of geodesics, which is called the pencil P defined by L and L . Each pencil has an associated family of lines, called cycles, which 1 2 C are the orthogonal trajectories to geodesics forming the pencil. The pencil defined by L and L is called P 1 2 1. parabolic if L and L are parallel (and tend to the same point z ∂D), in this 1 2 0 ∈ case is the set of all geodesics parallel to L and L , and the cycles are called P 1 2 horocycles;

2. elliptic if L and L are intersecting (at a point z D), in this case is the set of 1 2 0 ∈ P all geodesics passing through the common point of L1 and L2;

3. hyperbolic if L and L are disjoint, in this case is the set of all geodesics orthog- 1 2 P onal to the unique common orthogonal geodesic (with endpoints z , z ∂D) of L 1 2 ∈ 1 and L2, and the cycles are called hypercycles.

Figure 1. Parabolic, elliptic and hyperbolic pencils.

In Figure 1, illustrating possible pencils, the cycles are drawn in bold lines. The following main theorem has been proved in [11, 12] for the classical (weightless) Bergman space, and in [4, 5, 6] for all weighted Bergman spaces.

Theorem 1.1. Given a pencil of geodesics, consider the set of L -symbols which are P ∞ constant on corresponding cycles. The C ∗-algebra generated by Toeplitz operators with such symbols is commutative on each weighted Bergman space 2 (D). Ah Commutative C∗-algebras of Toeplitz Operators on the Unit Disk 35

2 Three-term asymptotic expansion formula

To get an inverse statement to Theorem 1.1 we will use the familiar Berezin quantization procedure on the unit disk (see, for example, [2, 3]). (h) For each function a = a(z) C ∞(D) consider the family of Toeplitz operators Ta ∈ 2 with (anti-Wick) symbol a acting on h(D), for h (0, 1). The Wick symbols of the (h) A ∈ Toeplitz operator Ta has the form

1 1 (1 z 2)(1 ζ 2) h ah(z, z) = ( 1) a(ζ) − | | − | | dµ(ζ), h − D (1 zζ)(1 ζz) Z  − −  and the star proeduct of Wick symbols is defined as follows

1 1 (1 z 2)(1 ζ 2) h (ah ? bh)(z, z) = ( 1) ah(z, ζ) bh(ζ, z) − | | − | | dµ(ζ). h − D (1 zζ)(1 ζz) Z  − −  To acehievee our goal we need thee three-terme asymptotic expansion formula of the commutator of two Wick symbols.

Theorem 2.1. For any pair a = a(z, z) and b = b(z, z) of six times continuously differ- entiable functions the following three-term asymptotic expansion formula holds

~2 a ? b b ? a = i~ a, b + i (∆ a, b + a, ∆b + ∆a, b + 8π a, b ) h h − h h { } 4 { } { } { } { } ~3 e e e e +i ∆a, ∆b + a, ∆2b + ∆2a, b + ∆2 a, b 24 { } { } { } { } +∆ a, ∆b + ∆ ∆a, b + 28π (∆ a, b + a, ∆b + ∆a, b ) { } { } { } { } { } + 96π2 a, b + o(~3), { } h  where ~ = 2π , and the Poisson bracket and the Laplace-Beltrami operator are given by ∂a ∂b ∂a ∂b a, b = 2πi(1 zz)2 , { } − ∂z ∂z − ∂z ∂z   ∂2 ∆ = 4π(1 zz)2 . − ∂z∂z Corollary 2.2. Let (D) be a subalgebra of C (D) such that for each h (0, 1) the A ∞ ∈ Toeplitz operator algebra ( (D)) is commutative. Then for all a, b (D) we have Th A ∈ A a, b = 0, (2.1) { } a, ∆b + ∆a, b = 0, (2.2) { } { } ∆a, ∆b + a, ∆2b + ∆2a, b = 0. (2.3) { } { } { } 3 Consequences of (2.1), (2.2), and (2.3)

Discussing commutative C ∗-algebras of Toeplitz operators we will always assume that the corresponding generating class of symbols is a linear space. To underline the geometric 36 Nikolai Vasilevski

nature of symbol classes which generate the commutative C ∗-algebras of Toeplitz opera- tors we have considered bounded measurable symbols in Theorem 1.1. This also agrees with the desire for such (commutative) algebras to be, in a sense, maximal. Note that the arguments used in the proof do not require any assumption on smoothness properties of symbols. The same result (commutativity of Toeplitz operator C ∗-algebra) remains valid for any linear subspace of L -symbols (constant on cycles). Moreover, we can start with ∞ a much more restricted set of symbols (say, smooth symbols only) and extend them fur- thermore to all L -symbols by means of uniform and strong operator limits of sequences ∞ of Toeplitz operators. As we consider the C∗-algebra generated by Toeplitz operators, we can always assume, without loss of generality, that our set of symbols is closed under complex conjugation and contains the function e(z) 1. ≡ Let (D) be a linear space of (smooth) functions. Denote by ( (D)) = ( (D)) A T A {Th A }h the family of C -algebras ( (D)) generated by Toeplitz operators with symbols from ∗ Th A (D) and acting on the weighted Bergman spaces 2 (D). A Ah To introduce our symbol classes we need the notion of the jet of a function (see, for example, [9, 10]). Given two complex valued smooth functions f and g defined in a neighborhood of a point z D, we say that they have the same jet of order k at z if their ∈ real partial derivatives at z up to order k are equal. It is easy to see that such relation does not depend on the coordinate system and that it defines an equivalence relation. k The corresponding equivalence class of a function f at z is denoted by jz (f) and is called the k-th order jet of f at z. Furthermore, given a complex vector space (D) of smooth A functions, we denote with J k( (D)) the space of k-jets at z of the elements in (D). We z A A observe that J k( (D)) is a finite dimensional complex vector space. z A In what follows, for a differentiable function f : D C we will say that z D is a → ∈ nonsingular point of f if df = 0. z 6 The symbol classes that we are considering are given in the next definition. Definition 3.1. Let (D) be a complex vector space of smooth functions. We will say A that (D) is k-rich if it is closed under complex conjugation and the following conditions A are satisfied: (i) there is a finite set S such that for every z D S at least one element of (D) is ∈ \ A nonsingular at z, (ii) for every point z D S and l = 0, . . . , k, the space of jets J l ( (D)) has complex ∈ \ z A dimension at least l + 1. As the set (D) is closed under the complex conjugation, it is sufficient to consider A the conditions (2.1), (2.2), and (2.3) for real valued functions only. Recall that each real valued function a (D), nonsingular in some open set, has in this set two systems of ∈ A mutually orthogonal smooth lines, the system of level lines and the system of gradient lines. The geometric information contained in the first term of asymptotic expansion of a commutator, or equivalently in the condition (2.1), is given by the next lemma. Lemma 3.2. Let (D) be a 2-rich space of smooth functions which generates for each A h (0, 1) the commutative C -algebra ( (D)) of Toeplitz operators. Then all real ∈ ∗ Th A valued functions in (D) have (globally) the same set of level lines and the same set of A gradient lines. Commutative C∗-algebras of Toeplitz Operators on the Unit Disk 37

Vanishing of the second term of asymptotic in a commutator, or equivalently the condition (2.2), leads to the following theorem.

Theorem 3.3. Let (D) be a 2-rich space of smooth functions which generates for each A h (0, 1) the commutative C -algebra ( (D)) of Toeplitz operators. Then the common ∈ ∗ Th A gradient lines of all real valued functions in (D) are geodesics in the hyperbolic geometry A of the unit disk D.

Vanishing of the third term of asymptotic in a commutator, or equivalently the con- dition (2.3), implies the following theorem.

Theorem 3.4. Let (D) be a 3-rich vector space of smooth functions (D) which gener- A A ates for each h (0, 1) the commutative C -algebra ( (D)) of Toeplitz operators. Then ∈ ∗ Th A the common level lines of all real valued functions in (D) are cycles. A The next theorem provides a geometric characterization of the real valued functions on D whose gradient lines define a pencil of geodesics.

Theorem 3.5. A nonconstant C 3 real valued function a in D defines a pencil if and only if the following two conditions are satisfied:

(i) The gradient lines of a are geodesics.

(ii) Each level line of a is a cycle.

Lemma 3.2, Theorem, 3.3, Corollary 3.4, and Theorem 3.5 lead directly to the follow- ing result.

Corollary 3.6. Let (D) be a 3-rich vector space of smooth functions such that ( (D)) A Th A is commutative for each h (0, 1). Then there exists a pencil of geodesics in D such ∈ P that all functions in (D) are constant on the cycles of . A P Now the main result of the paper reads as follows.

Corollary 3.7. Let (D) be a 3-rich vector space of smooth functions. Then the following A three statements are equivalent:

(i) there is a pencil of geodesics in D such that all functions in (D) are constant P A on the cycles of ; P (ii) the C -algebra generated by Toeplitz operators with (D)-symbols is commutative ∗ A on each weighted Bergman space 2 (D), h (0, 1). Ah ∈ References

[1] A. F. Beardon. The Geometry of Discrete Groups. Springer-Verlag, Berlin etc., 1983.

[2] F. A. Berezin. General concept of quantization. Commun. Math. Phys., 40:135–174, 1975.

[3] F. A. Berezin. Method of Second Quantization. “Nauka”, Moscow, 1988. 38 Nikolai Vasilevski

[4] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Hyperbolic case. Bol. Soc. Mat. Mexicana, 10:119–138, 2004.

[5] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators on the upper half-plane: Parabolic case. J. Operator Theory, 52(1):185– 204, 2004.

[6] S. Grudsky, A. Karapetyants, and N. Vasilevski. Dynamics of properties of Toeplitz operators with radial symbols. Integr. Equat. Oper. Th., 20(2):217–253, 2004.

[7] S. Grudsky, R. Quiroga-Barranco, N. Vasilevski. Commutative C*-algebras of Toeplitz Operators and Quantization on the Unit Disk. Reporte Interno # 361, Departamento de Matematicas, CINVESTAV del I.P.N., Mexico, 2005, 50 p. (to appear in J. Func. Analysis)

[8] S. Grudsky and N. Vasilevski. Bergman-Toeplitz operators: Radial component in- fluence. Integr. Equat. Oper. Th., 40(1):16–33, 2001.

[9] I. Kol´aˇr, P. Michor, and J. Slov´ak. Natural Operations in Differential Geometry. Springer-Verlag, 1993.

[10] D. J. Saunders. The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series, no 142. Cambridge University Press, Cambridge, 1989.

[11] N. L. Vasilevski. Toeplitz operators on the Bergman spaces: Inside-the-domain effects. Contemp. Math., 289:79–146, 2001.

[12] N. L. Vasilevski. Bergman space structure, commutative algebras of Toeplitz oper- ators and hyperbolic geometry. Integr. Equat. Oper. Th., 46:235–251, 2003.

This work was partially supported by CONACYT Project 46936, M´exico and by the International Erwin Schr¨odinger Institute for Mathematical Physics (ESI), Vienna, Aus- tria.

Nikolai Vasilevski – Departamento de Matematicas,´ CINVESTAV Apartado Postal 14-740, 07000 M´exico, D.F., M´exico [email protected] Berezin–Toeplitz quantization of the moduli space of flat SU(n) connections

Martin Schlichenmaier

1 Introduction

This is a condensed write-up of a talk presented at the program “Complex Analysis, Operator Theory, and Applications to Mathematical Physics” organized by F. Haslinger, E. Straube, and H. Upmeier at the Erwin-Schr¨odinger-Institute in Vienna in September 2005, and at the “Conference on Poisson Geometry”, organized by T. Ratiu, A. Weinstein, and N.T. Zung at the ICTP, Trieste, in 2005. First, we recall the basics of the Berezin-Toeplitz quantization (operator and formal deformation quantization). Then we discuss the moduli space of flat SU(n) connections on a fixed Riemann surface in its different guises. Finally, we present recent results obtained by Jørgen Andersen showing the asymptotic faithfulness of the representations of the mapping class group (MCG, Teichmuller¨ group) on the covariantly constant sections of the projectivized Verlinde bundle. In his approach he uses the Toeplitz operators and results on their correct semiclassical behavior as they will be presented in the first part. As far as the Berezin-Toeplitz quantization is concerned the results are results ob- tained partly in joint works with E. Meinrenken, and M. Bordemann resp. with A. Karabegov [3], [9], [10], [11], [8]. Quite a number of mathematician (and physicists) were involved in the study of the moduli space of connections and the mapping class group. Instead giving references here, let me refer to the recent overviews by L. Jeffrey [5] and G. Masbaum [6]. The beautiful results on the asymptotic faithfulness presented are entirely due to Andersen [1]. For similar results for the U(1) obtained by him see also [2].

2 BT quantization of compact K¨ahler manifolds

2.1 K¨ahler manifolds Let (M, ω) be a K¨ahler manifold, i.e. M a complex manifold, and ω a a closed (1, 1)-form on M which is positive (a K¨ahler form). Examples: 1. Cn, ω = i n dz dz¯ , i=1 i ∧ i 2. P1, ω = i dz dz¯, (1+Pzz¯)2 ∧ 3. every Riemann surface carries a K¨ahler form, 4. every (complex) torus of arbitrary dimension with the standard K¨ahler form on Cn (see 1.),

39 40 Martin Schlichenmaier

5. every (quasi-)projective manifold, i.e. every non-empty open subset of a projective variety without singularities, with the restriction of the Fubini-Study K¨ahler form of the projective space, 6. very often moduli spaces in the algebraic or analytic context carry a natural K¨ahler structure coming from their construction.

2.2 Quantizable K¨ahler manifolds Definition 2.1. ( Quantization condition) A K¨ahler manifold (M, ω) is called quantiz- able, if there exists an associated quantum line bundle (L, h, ), i.e. a holomorphic line ∇ bundle L over M, with hermitian metric h on L, and compatible connection , fulfilling ∇

curvL, = i ω. ∇ − Note: Not all K¨ahler manifolds are quantizable. For example only such tori are quan- tizable which have enough theta functions, i.e. which can be embedded holomorphically into projective space. They are called abelian varieties. For the rest of this write-up we assume that M is a compact K¨ahler manifold. We fix m m (m) a quantum line bundle L and consider L := L⊗ , with metric h , and take Γ (M, Lm) the space of smooth global sections, and ∞ m 0 m Γhol(M, L ) = H (M, L ) the subspace of global holomorphic sections. Due to the compactness of M, the latter is finite-dimensional. On these spaces a scalar product is defined via 1 ϕ, ψ := h(m)(ϕ, ψ) Ω, Ω := ω ω ω . h i n! ∧ · · · ∧ ZM n We will need the projector | {z }

Π(m) : L2(M, Lm) Γ (M, Lm). −→ hol 2.3 Berezin-Toeplitz operator quantization Fix f C (M), and let s Γ (M, Lm) then the following map ∈ ∞ ∈ hol s T (m)(s) := Π(m)(f s) 7→ f · defines the Toeplitz operator of level m

T (m) : Γ (M, Lm) Γ (M, Lm). f hol → hol The Berezin-Toeplitz (BT) operator quantization is the map

(m) f Tf . 7→ m N0   ∈

The reason to call it a quantization is, that it has the correct semi-classical behavior as expressed in the following theorem. BT quantization of the moduli space of flat SU(n) connections 41

Theorem 2.1. (Bordemann, Meinrenken, and Schlichenmaier (BMS) [3]) (a) lim T (m) = f , (2.1) m f ∞ →∞ || || | | (b) (m) (m) (m) mi [Tf , Tg ] T f,g = O(1/m), (2.2) || − { }|| (c) (m) (m) (m) Tf Tg Tf g = O(1/m). (2.3) || − · || The proofs of (b) and (c) are based on the symbol calculus of generalized Toeplitz operators developed by Boutet de Monvel and Guillemin [4].

2.4 Deformation quantization Theorem 2.1. (BMS, Schl., Karabegov and Schl.) [3], [9], [10], [11], [8]. There exists a unique differential star product, the BT star product,

k f ?BT g = ν Ck(f, g), (2.4) X such that k (m) (m) ∞ 1 (m) T Tg T , (m ). (2.5) f ∼ m Ck(f,g) → ∞ Xk=0   This star product is of “separation of variables” type, and has classifying Deligne-Fedosov class 1 1  ( [ω] ), (2.6) i λ − 2 and Karabegov form 1 − ω + ω , (2.7) λ can

A star product is a differential star product if the Ck(., .) are bidifferential operators in their function arguments. Such a differential star product is of “separation of variables type” if the first argument is only differentiated in holomorphic directions and the second argument only in anti-holomorphic directions (resp. the opposite directions depending on the convention chosen). This notion is due to Karabegov [7], and corresponds to the fact that the star product respects the complex structure. Such star products are classified by their formal Karabegov form. Above λ is used as formal variable for the forms and the formal forms are formal power series in λ if we ignore 1/λ which comes with the fixed ω. In particular, for the BT star product no higher formal powers of λ occur. The 2 Deligne-Fedosov class is a formal HdeRahm class which classifies the star product up to equivalence. The form ωcan is the curvature form of the canonical (holomorphic) line bundle with fibre metric coming from the Liouville form. Note also that the asymptotic formula (2.5) is a short-hand notation for a very precise and strong asymptotic behaviour of the norms of the involved operators. See the cited references for the precise statement. 42 Martin Schlichenmaier

3 The moduli space of flat SU(n) connections

3.1 Its symplectic structure Let X be an oriented compact surface, and p X a fixed point. We denote by G the ∈ group SU(n). Let be the set of flat SU(n) connections over X p with holonomy ξ of finite AF,ξ \ { } order d around p. We fix for the center of SU(n) a generator and identify it with Z/nZ. Then ξ corresponds to d mod n. The group of maps X G from the surface X to the group G with pointwise multi- → plication in G, is the gauge group . It acts on the connection via gauge transformations G g 1 1 A := g− dg + g− Ag. (3.1)

The moduli space of connections is the quotient of the set of connections modulo these gauge transformations

:= / = Hom (π˜ (X), G)/G (3.2) M AF,ξ G ∼ d 1 The latter equivalence is the fact that this moduli space can be identified with the space of those group homomorphisms of the central extension π˜1(X) of the fundamental group π1(X) defined by 0 Z π˜ (X) π (X) 0 −→ −→ 1 −→ 1 −→ with values in G, for which the generator 1 Z in the central extension is mapped to ∈ d mod n in the center of G, where the homomorphism are identified modulo conjugation in G. Let be the moduli space of irreducible flat connections (this corresponds to irre- Ms ducible representations). It is a manifold, carries a natural symplectic structure ω, and an associated hermitian line bundle which is a quantum line bundle with respect to L the symplectic structure. It is constructed from the WZW cocycle of the Chern-Simons action. See the appendix for more details and [5] for references and further information.

3.2 Its complex structure We choose a complex structure σ on X. This structure will induce complex structures on all introduced objects.

1. X = Xσ is now a (compact) Riemann surface, ⇒ 2. ( , ω) = ( σ, ωσ) is now a K¨ahler manifold, Ms ⇒ Ms 3. = σ becomes a hermitian holomorphic line bundle, in fact, it is a quantum L ⇒ L line bundle with respect to ωσ.

Hence, σ is a quantizable K¨ahler manifold with quantum line bundle σ. But what Ms L is the geometry of σ? Is it compact? To study these questions we discuss another Ms description of the moduli space. BT quantization of the moduli space of flat SU(n) connections 43

4 Holomorphic rank n bundles E over smooth projective curves

4.1 The moduli space Recall that the compact Riemann surface X σ can be identified with a smooth projective curve C over C. In the following we consider holomorphic vector bundles over C. First we define for every rank n holomorphic vector bundle E, its determinant line bundle as det E := n E, and its degree as deg(E) := deg(det E). The question is: Does there exist a moduli space of isomorphy classes of such bundles? The answer is: In generally not! V We need to restrict our considerations to the subset of isomorphy classes of (Mumford) stable bundles, resp. S-equivalence classes of semi-stable bundles. A bundle E is stable (resp. semi-stable) iff for every non-trivial subbundle F of E one has deg(F )/rk(F ) < deg(E)/rk(E) (resp. ). For the S-equivalence relation two semi-stable (but not stable) ≤ bundles are identified if certain associated graded objects are isomorphic. Let T be a line bundle and n N. We use the following notation for the moduli space of ∈ bundles

Us(n, d), rk (E) = n, deg(E) = d, E stable, Us(n, T ), rk (E) = n, det(E) = T , E stable, U(n, d), rk (E) = n, deg(E) = d, E semi-stable, U(n, T ), rk (E) = n, det(E) = T , E semi-stable. In the following let [p] be the line bundle corresponding to the divisor p, i.e. the line bundle which has a non-trivial section with exactly a zero of order one at p and which is non-vanishing elsewhere. Furthermore let d[p] be its d-tensor power. In particular, deg d[p] = d.

We have the following properties: 1. M := U(n, d[p]) is always projective algebraic (hence compact),

2. Ms := Us(n, d[p]) is Zariski open and smooth in M, hence a smooth manifold,

3. if gcd(n, d) = 1 then M = Ms, and hence Ms is a compact K¨ahler manifold, 4. the singularities of M are rather mild,

5. for the Picard group of isomorphy classes of line bundles we have P ic(Ms) = P ic(M) = Z [L], where L is a special ample line bundle, · m m 6. Γhol(Ms, L ) = Γhol(M, L ), | 7. if g = 2 and n = 2 then M is always smooth.

The fundamental result is σ = U (n, d[p]) = M Ms ∼ s s as complex manifold and as K¨ahler manifolds, and σ = L L ∼ as holomorphic line bundles. A few names of people involved are Narasimhan, Seshadri, Weil, Mumford, ..... 44 Martin Schlichenmaier

4.2 The Verlinde bundle 0 m m The Verlinde spaces are the vector spaces H (M, L ) =Γhol(M, L ) and the dimension formula (as function of m) is called the Verlinde formula.

These Verlinde spaces are the quantum spaces, and the BT operators T (m) : H0(M, Lm) H0(M, Lm) f → are the quantum operators. We can apply Theorem 2.1 (BMS) und use the natural deformation quantization ?BT of Theorem 2.1 (at least without modification, if M = Ms, resp. if M is smooth). We have to go one step further. If we consider the following diagram we see that the first line does not depend on the complex structure σ, but the second does. m m X ( s, ) Γ ( s, ) −−−−→ M L −−−−→ ∞ M L choose σ Πσ,(m) σ σ  σ m m  m X ( s = Ms, ( ) = L ) Γhol(Ms, L ) y −−−−→ M yL −−−−→ y If we vary our σ over the Teichmuller¨ space , (i.e. the space of all complex structures on T X modulo a certain equivalence relation) the first line will give trivial families of objects, the second line nontrivial families over . T In particular, over there is the trivial (infinite dimensional) bundle with fibre m T m Γ ( s, ) which contains the subbundle m with fibre Γhol(Ms, L ). The bundle ∞ M L V is the called the Verlinde bundle over Vm T Given f C ( ) its Toeplitz operator depends on the complex structure. Hence, ∈ ∞ Ms (m) Tf,σ σ ∈T   (m) is a family of operators on the Verlinde bundle. In other words Tf, . is a section of End( ). Vm 5 The mapping class group (MCG) action

Over Teichmuller¨ space we have the bundles and End( ). We will discuss the T Vm Vm following points: 1. There exists a naturally defined projectively flat connection on , it is the ∇ Vm Axelrod, della Pietra, Witten – Hitchin connection. 2. The MCG operates on the covariantly constant sections of P( ). Vm 3. J. Andersen showed that this action of the MCG is asymptotically faithful (i.e. given an element of the MCG, there is an m such that the element operates non-trivially). Recall that the mapping class group(MCG) is defined as + Γ := MCG := Diff (X)/Diff0(X), here X is the surface of genus g, Diff +(X) the group of orientation preserving dif- feomorphisms and Diff0(X) the subgroup of diffeomorphisms which are isotop to the identity. BT quantization of the moduli space of flat SU(n) connections 45

1. By definition Γ operates on the surface X.

2. It operates on the Teichmuller¨ space. In fact the moduli space of isomorphism Mg classes of compact genus g Riemann surfaces (resp. smooth projective curves of genus g) is the quotient /Γ. T

3. It operates on the fundamental group π1(X), and on Homd(π˜1(X), G)/G.

4. And furthermore it operates on σ = M , the moduli spaces of irreducible con- Ms ∼ s nections, resp. stable bundles.

5.1 Andersen’s result Let be the Axelrod-della Pietra-Witten – Hitchin (AdPW-H) connection on which ∇ Vm is projectively flat. It induces a flat connection end on End( ). We denote by P(W ), ∇ Vm m the space of covariantly constant sections of P( ) with respect to . Then the MCG Vm ∇ operates also on P(Wm): ρ : Γ Aut(P(W )). m → m

Theorem 5.1. (Andersen, [1]) For g 3 the map ρ is asymptotically faithful. ≥ m More precisely,

1, g > 2, or g = 2, n > 2, or ∞ ker(ρm) =  g = 2, n = 2, d odd, (5.1) m=1  1, H , g = 2, n = 2, d even, \ { } where H is the hyperelliptic involution.

5.2 Importance The assignment X V (X) = H0(M , Lm) −→ s corresponds to a Topological Quantum Field Theory (TQFT). It should be independent of the complex structure chosen. The projectively flat connection gives locally a natural identification. Globally the choice reduces to on action of the mapping class group Γ — (which is also a topological invariant). Hence, this action gives invariants of the TQFT in question.

5.3 The relation to BT Note in the following that f C ( ), i.e. f is a smooth function on the moduli space ∈ ∞ Ms of connections, resp. bundles.

Proposition 5.1. (Andersen, [1]) For σ , σ , denote by P end the parallel transport 0 1 ∈ T σ0,σ1 from σ to σ in End( ), then 0 1 Vm P end T (m) T (m) = O(1/m). (5.2) || σ0,σ1 f,σ0 − f,σ1 || 46 Martin Schlichenmaier

He uses Theorem 2.1 (BMS), the deformation quantization of Theorem 2.1 and carries out further ingenious hard work. Proposition 5.2. (Andersen, [1]) Let φ Γ, such that φ kerρ , then ∈ ∈ m (m) end (m) Tf,σ = Pφ(σ),σTf φ,φ(σ) (5.3) ◦ Theorem 5.3. (Andersen, [1]) Let φ Γ, such that φ N ker ρ , then φ induces ∈ ∈ m m the identity on . ∈ Ms T Proof. By Proposition 5.2 and the linearity in the function argument of the Toeplitz operators we have (m) (m) (m) end (m) (m) Tf f φ,σ = Tf,σ Tf φ,σ = Pφ(σ),σTf φ,φ(σ) Tf φ,σ. − ◦ − ◦ ◦ − ◦ We take the norm of this expression and use Proposition 5.1: (m) Tf f φ,σ = O(1/m). || − ◦ || Or, lim T (m) = 0. m f f φ,σ →∞ || − ◦ || This implies f f φ = 0 by Theorem 2.1, Part a, for all f, hence φ = id considered | − ◦ |∞ as element acting on the moduli space. Theorem 5.1 follows from known results which elements of the mapping class group act trivially on the moduli space of connections.

6 Appendix: Symplectic form on M Let be the affine space of all flat SU(n) connections, and g = su(n). The tangent AF vectors at A can be given as α, β Ω1(X) g. On this space ∈ AF ∈ ⊗ i Ω (α, β) = Tr(α β) A 2π ∧ ZX is a skew-symmetric form which is invariant under the gauge group and hence descends to = / . If we restrict the situation to the irreducible connections, then the quotient M AF G = s / is a manifold and Ω descends to a symplectic form on Ms AF G Ms To define the bundle one uses the Chern-Simons(CS) action. Let N be a 3-manifold with boundary ∂N = X. For any connection A˜ on N 1 2 CS(A˜) := Tr(A˜ dA˜ + A˜ A˜ A˜). 4π ∧ 3 ∧ ∧ ZN For a connection on X we take any extension A˜ to N. Also for a gauge transformation g we take any extension g˜ : N G. Then ∈ G → θ(A, g) := exp(i(CS(A˜g˜) CS(A˜)) − is a U(1)-valued well-defined cocycle (the WZW cocycle). It is used to construct the bundle over as quotient L Ms := ( s C)/ s / = L AF × ∼ → AF G Ms where (A, z) (Ag, θ(A, g)z) . The one form η(α) = 1 Tr(A α) on induces a ∼ 4π X ∧ AF unitary connection on , whose curvature is essentially equal to the symplectic form. L R BT quantization of the moduli space of flat SU(n) connections 47

References

[1] J. Andersen, Asymptotic faithfulness of the quantum SU(n) representations of the mapping class group, math.QA/0204084, (to appear in Ann. Math.).

[2] J. Andersen, Deformation quantization and geometric quantization of abelian moduli spaces, Comm. Math. Phys. 255 (2005), 727–745.

[3] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of K¨ahler manifolds and gl(n), n limits, Commun. Math. Phys. 165 (1994), 281– → ∞ 296.

[4] L. Boutet de Monvel and V. Guillemin, The spectral theory of Toeplitz operators. Ann. Math. Studies, Nr.99, Princeton University Press, Princeton, 1981.

[5] L. Jeffrey, Flat connections on oriented 2-manifolds, Bull. London Math. Soc. 37 (2005), 1–14.

[6] G. Masbaum, Quantum representations of mapping class groups, Journ´ee annuelle 2003, 19–36, SMF.

[7] A. Karabegov, Deformation quantization with separation of variables on a K¨ahler manifold, Commun. Math. Phys. 180 (1996), 745–755.

[8] A. Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, J. reine angew. Math. 540(2001), 49–76, math.QA/0006063.

[9] M. Schlichenmaier, Berezin-Toeplitz quantization of compact K¨ahler manifolds, (in) Quantization, Coherent States and Poisson Structures, Proc. XIV’th Workshop on Geometric Methods in Physics (Bialo wieza,˙ Poland, 9-15 July 1995) (A. Strasburger, et.al. eds.), Polish Scientific Publisher PWN, 1998, q-alg/9601016, pp. 101–115.

[10] M. Schlichenmaier, Deformation quantization of compact K¨ahler manifolds by Berezin-Toeplitz quantization, Conference Mosh´e Flato 1999 (September 1999, Di- jon, France) (G. Dito and D. Sternheimer, eds.), Kluwer, 2000, math.QA/9910137, Vol. 2, pp. 289–306.

[11] Martin Schlichenmaier, Berezin-Toeplitz quantization and Berezin transform, in “Long time behaviour of classical and quantum systems, Proc. of the Bologna AP- TEX International Conference”, September 13-17, 1999, (eds. S. Graffi, A. Martinez), World Scientific, 2001, pp. 271–287. math/0009219.

Martin Schlichenmaier – Universit´e du Luxembourg, Laboratoire de Mathematiques, Campus Limpertsberg, 162 A, Avenue de la Fa¨ıencerie, L-1511 Luxembourg [email protected] Kontsevich Quantization and Duflo Isomorphism

Micha Pevzner

The aim of this expository note is to explain the relationship between the Kontse- vich’s Formality theorem and the problem of local solvability of bi-invariant differen- tial operators on finite-dimensional real Lie groups. Main references to this subject are [5, 1, 2, 3, 6, 7]. Let G be a connected real finite dimensional Lie group and g be its Lie algebra. The symmetric algebra S(g) of g can be seen as the algebra of differential operators with constant coefficients on g as well as the algebra (with respect to the convolution on the vector space g) of distributions on g supported at the origin, it is always commutative. On the other hand side the universal enveloping algebra U(g) of g can be seen as the algebra of left-invariant differential operators on G as well as the algebra (with respect to the convolution on G) of distributions on G supported at the identity element. The algebra U(g) is neither commutative (except when g is commutative) nor graded. However U(g) is filtered by the order of differential operators and the Poincar´e-Birkhoff-Witt theorem ensures that its associated graded algebra grU(g) is isomorphic to S(g). This isomorphism, called symmetrization, is given by β(X ...X ) = 1 X ...X . 1 n n! σ Sn σ(1) σ(n) Obviously S(g) and U(g) are not isomorphic as algebras so one cannot∈”transform” the P convolution of distributions on the Lie algebra into the convolution of distributions on the corresponding Lie group and thus to reduce the solvability of left-invariant differential operators on G to the solvability of differential operators with constant coefficients on g. However, it turns out that the set of ad(g)-invariants in S(g) is isomorphic as an algebra to the center Z(g) of U(g). This remarkable fact was described for reductive Lie algebras by Harish-Chandra and for nilpotent ones by Dixmier. The validity of this fact for arbitrary real finite-dimensional Lie algebras was established by Duflo [4] in 1979. A highly non trivial proof of this result was based on the orbit method that relates coadjoint orbits of G (parametrized by invariant symmetric tensors) with irreducible representations of G (whose infinitesimal characters are elements of Z(g)). More precisely, let

1 sinh( adx ) 2 q(x) = det 2 , (0.1) g adx 2 ! be a formal power series on g∗ and ∂q be the differential operator of infinite order on g with symbol q. Then, the so-called Duflo map β ∂ : S(g) U(g) is a vector space ◦ q → isomorphism that becomes algebra isomorphism when restricted to invariants. Therefore the convolution of invariant distributions on G can be recovered form the convolution of their invariant pull-backs on g and thus every non-zero bi-invariant differential operator on G admits a local fundamental solution. We shall explain how does this theorem follow from the formality theorem and how can it be extended in cohomology.

48 Kontsevich Quantization and Duflo Isomorphism 49

1 Formality theorem and its consequences

Let X be a smooth manifold. One associates to X two graded differential Lie algebras (GDLA). The first GDLA g1 = Tpoly(X) is the graded algebra of poly-vector fields on X: T n (X) := Γ(X, Λn+1T X), n 1, equipped with the Schouten-Nijenhuis bracket poly ≥ − [ , ]SN and the differential d := 0. Recall that the Schouten-Nijenhuis bracket is given for all k, l 0 , ξ , η Γ(X, T X) ≥ i j ∈ by:

k l [ξ ξ , η η ] = ( 1)i+j[ξ , η ] 0 ∧ · · · ∧ k 0 ∧ · · · ∧ l SN − i j ∧ Xi=0 Xj=0 ξ0 ξi 1 ξi+1 . . . ξk η0 ηj 1 ηj+1 ηl ∧ · · · ∧ − ∧ ∧ ∧ ∧ ∧ · · · ∧ − ∧ ∧ · · · ∧ And for k 0 and h Γ(X, ), ξ Γ(X, T X): ≥ ∈ OX i ∈ k i [ξ0 ξk, h]SN = ( 1) ξi(h) (ξ0 ξi 1 ξi+1 . . . ξk) . ∧ · · · ∧ − · ∧ · · · ∧ − ∧ ∧ ∧ Xi=0

Where [ξi, ηj] is the usal vector fields bracket i.e. the Lie derivative Lξi (ηj).

The second GDLA associated to X is the algebra of poly-differential operators g2 = Dpoly(X) seen as a sub-algebra of the shifted Hochschild complex of functions algebra of X. The grading on D (X) is given by A = m 1 where A D (X) is a poly | | − ∈ poly m differential operator. The composition of two operators A Dm1 (X) and A − 1 ∈ poly 2 ∈ Dm2 (X) is given for f by: poly i ∈ OX m1 (m2 1)(j 1) (A1 A2)(f1, . . . , fm1+m2 1) = ( 1) − − A1(f1, . . . , fj 1, ◦ − − − Xj=1 A2(fj, . . . , fj+m2 1), fj+m2 , . . . , fm1+m2 1). − − One defines the Gerstenhaber bracket: [A , A ] := A A ( 1) A1 A2 A A . Thus 1 2 G 1 ◦ 2 − − | || | 2 ◦ 1 the differential on D (X) is given by dA = [µ, A] , where µ is the bi-differential poly − G operator of multiplication: µ(f1, f2) = f1f2. Certainly these GDLA are not isomorphic, however the map (0) : T D U1 poly 7→ poly given by

n sgn(σ) (0) : (ξ ξ ) f f ξ (f ) (1.1) U1 0 ∧ · · · ∧ n 7→  0 ⊗ · · · ⊗ n → (n + 1)! σ(i) i  σ Sn+1 i=0 ∈X Y   for n 0 and by f (1 f) for f Γ(X, ) is a quasi-isomorphism of complexes. ≥ 7→ → ∈ OX This statement is a version of the Kostant-Hochschild-Rosenberg theorem. It turns out, see [5], that one can extend this map to an application that is a GDLA-morphism up to homotopy. Consider shifted algebras g1[1] and g2[1] and the coalgebras without unit

+ n S gi[1] = S gi[1] , i = 1, 2 n 0
M≥
Both of them have co-derivations Qi of degree 1 defined by the GDLA structure. 50 Micha Pevzner

Theorem 1. There exists a L -quasi-isomorphism between shifted GDLA g1[1] and g2[1], ∞ i.e. a co-algebra morphism

: S+ g [1] S+ g [1] U 1 → 2 such that

Q1 = Q2 U ◦ ◦ U and such that the restriction to g [1] S1(g [1]) is the quasi-isomorphism of co-chain U 1 ' 1 complex (0) given by (1.1). U1 This construction is based on an explicit realization of the L -quasi-isomorphism ∞ U in the flat case. We shall now recall it, see [5]. Let X be the vector space Rd. Then one shows that such a L -quasi-isomorphism ∞ is determined by its ”Taylor coefficients” : Sk(g [1]) g [1], with k 1 that one U Uk 1 → 2 ≥ gets composing with the canonical projection π : S +(g ) g . One denotes this U 2 → 2 U composition. Coefficients are described in terms of graphs and their weights. Un Let Gn,m be the set of labeled oriented graphs with n verices of the first kind and m vertices of the second kind such that:

1. All edges start from vertices of the first kind. 2. There are no loops. 3. There are no double edges.

One says that such a graph is admissible. By labeling an admissible graph Γ one under- stands a total order on the set EΓ of edges of Γ compatible with the order on the set of vertices. Consider Γ G and denote by s the number of edges starting from the vertex of ∈ n,m k the first kind with label k. To any n-uplet (α1, . . . , αn) of poly-vector fields on X such that for all k = 1, . . . , n the element αk is a sk-vector field, one associates, a m-differential operator B (α α ), given by the following construction : let e1 , . . . , esk be an Γ 1 ⊗ · · · ⊗ n { k k } ordered sub-set of EΓ of edges starting from the vertex of the first kind k. To every map I : E 1, . . . , d and every vertex x of Γ one associates the differential operator with Γ → { } constant coefficients : DI(x) = e=( ,x) ∂I(e), where for all i 1, . . . , d one denotes by − ∈ { } ∂i the partial derivative with respect to the i-th coordinate. The product is taken for all I Q edges ending at x. Let αk be the coefficient :

1 sk I I(e ) I(e ) k ··· k s s αk = αk = αk, dxI(e1 ) dxI(e k ) = αk, dxI(e1 ) dxI(e k ) . h k ∧ · · · ∧ k i h k ⊗ · · · ⊗ k i One set : n m B (α α )(f f ) = D αI D f . Γ 1 ⊗ · · · ⊗ n 1 ⊗ · · · ⊗ m I(k) k I(l) l I:E 1,...,d k=1 l=1 Γ→{X } Y Y The Taylor coefficient is given then by : Un (α , . . . , α ) = w B (α α ), (1.2) Un 1 n Γ Γ 1 ⊗ · · · ⊗ n Γ Gn,m ∈X Kontsevich Quantization and Duflo Isomorphism 51

where the sum is taken over all admissible graphs Γ such that the corresponding operator B (α α ) is well defined and the integer m is defined by m 2 = n s 2n. Γ 1 ⊗ · · · ⊗ n − k=1 k − The coefficient w is a certain weight associated to every graph Γ. Then (α , . . . , α ) Γ UPn 1 n is a m differential operator. − The weight w is zero unless the number of edges E of Γ is equal to 2n + m 2. Γ | Γ| − It is given by integration of a closed form Ω of degree E on a connected component Γ | Γ| of the Fulton-McPherson compactification of a configuration space whose dimension is precisely 2n + m 2, see [5]. This weight does depend on the order chosen on the set of − edges, but the product w B does not. Γ · Γ More precisely one denotes Conf n,m the set of (p1, . . . , pn, q1, . . . , qm) where the pj are distinct points in the Poincar´e upper half-plane : H = z C, Im z > 0 , and q + { ∈ } j are distinct points in R seen as the boundary of H . The group : G = z az + + { 7→ b with (a, b) R and a > 0 acts freely on Conf . The coset : C = Conf /G is a ∈ } n,m n,m n,m 2n + m 2-dimensional manifold. Kontsevich described compactifications C of these − n,m configuration manifolds. They are 2n + m 2-dimensional manifolds with corners such − that the boundary components in C C correspond to various degenerations of the n,m \ n,m configurations of points. Consider, for example, the space

C = (p , p ) H2 p = p /G1. 2,0 { 1 2 ∈ + | 1 6 2} For each point c C we can choose a unique representative of the form (√ 1, z) ∈ 2,0 − ∈ Conf . Thus C is homeomorphic to H √ 1 . 2,0 2,0 + \ { − } Similarly, it is easy to see that

C S1, C (0, 1), C 0, 1 . 2 ' 1,1 ' 0,2 ' { } and C = C , C = C C = [0, 1]. 2 2 1,1 1,1 t 0,2 Of particular interest is the space C = C (C C C C ), which can be 2,0 2,0 t 0,2 t 1,1 t 1,1 t 2 drawn as “the Eye”. . .

Figure 1: C2,0 (the Eye)

The circle C2 represents two points coming close together in the interior of H+, and two arcs C C represent the first point (or the second point) coming close to the real 1,1 t 1,1 line. Finally, the two corners C0,2 correspond to both points approaching the real line. For every graph Γ G one defines an angular function : Φ : C (R/2πZ) EΓ ∈ n,m Γ n,m −→ | | in the following way : one draws the graph in H+ joining vertices by geodesics with re- spect to the hyperbolic metric, and to every edge e = (p, q) one associates the angle q p ϕe = Arg q−p¯ formed by the vertical line passing through p and the edge e (see [2]). −   52 Micha Pevzner

ϕ e

Figure 2: Angular function

Choosing an order on edges this defines ΦΓ on Cn,m and one checks that this map can be extended to the compactification. Let ΩΓ be the differential form ΦΓ∗ (dv) on Cn,m + EΓ where dv is the normalized volume form on (R/2πZ)| |. Let Cn,m be the connected + component of Cn,m. The set Cn,mhas natural orientation and one defines the weight wΓ by : w = Ω . (1.3) Γ + Γ ZCn,m 1.1 Tangent quasi-isomorphism and homotopy d Let γ Tpoly(R )[1] be a 2 vector field such that γ[ 1] satisfies the Maurer-Cartan ∈ d − − equation in Tpoly(R ). Thus it is a Poisson 2 vector field. The Kontsevich’s L -quasi- − ∞ isomorphism starting with ~γ gives rise to a star-product ?~γ : ~n ?~ = µ + (~γ) = µ + (γ, . . . , γ), (1.4) γ U n! Un n 1 X≥ where ~ is a formal parameter. Indeed, a star-product is associative if, as an element of g , it satisfies a Maurer-Cartan type equation, so relies two solutions of Maurer-Cartan 2 U equations in corresponding GDLA’s. Notice that the associativity results from the Stokes theorem applied to the integrals defining the weights wΓ. Example. Let γ be the constant Poisson 2-vector field given by a constant symplectic structure Λ on R2n. Then the only non-vanishing graphs are those whose vertices of the first kind do not receive incoming edges. Thus the corresponding star-product is precisely the Moyal star-product:

rs f ?M g (z) = exp(~ Λ ∂xr ∂xs )(f(x)g(y)) x=y=z . |

Identifying GDLA’s g1 and g2 with their tangent spaces one defines the derivate map d : T (Rd)[1] D (Rd)[1][[~]] at the point ~γ : U poly → poly n 1 ~ − .n d ~ (δ) := (δ.γ ). (1.5) U γ (n 1)!Un n>0 X − Tangent spaces to GDLA’s g1 and g2 inherit differential structures in such a way that the co-boundary operator of the tangent co-chain complex of Tpoly(X)[1] is given by Kontsevich Quantization and Duflo Isomorphism 53

Q~γ = [~γ, ] . this is a graded derivative of the exterior product of poly-vector − − SN ∧ fields. This exterior product induces an associative and commutative product ( called ∪ cup-product) on the cohomology space H~γ of the first tangent space. On the second tangent space one introduces an associative graded product :

(A A )(f f ) = 1 ∪ 2 1 ⊗ · · · ⊗ m1+m2 A (f f ) ?~ A (f f ) (1.6) 1 1 ⊗ · · · ⊗ m1 γ 2 m1+1 ⊗ · · · ⊗ m2 for every m1-differential operator A1 and m2-differential operator A2. This operation is compatible with the co-boundary [ , ?] of the second tangent complex and it induces a − G cup-product on the cohomology space H (~γ). U The following theorem is due to Kontsevich and was carefully proven by Manchon and Torossian, see [5] 8 and [6] Th´eor`eme 1.2. § Theorem 2. Let X = Rd and be the Kontsevich’s L -quasi-isomorphism. The differ- U ∞ ential d ~ induces an algebra isomorphism from the cohomology space H~ of the tangent U γ γ space T~γ (g1[1]) onto the cohomology space H (~γ) of the tangent space T (~γ)(g2[1]). U U I.e. for every pair (α, β) of poly-vector fields such that [α, γ]SN = [β, γ]SN = 0 one has d 0(α β) = d 0(α) d 0(β) + D, (1.7) U ∪ U ∪ U where D is the Hochschild co-boundary of the algebra (C ∞(X)[[~]], ?γ ) given by D = [? , d 1(α, β)] . − γ U G 2 Quantization of the Kirillov-Kostant-Poisson bracket

In the case when the manifold X is the dual of a finite dimensional Lie algebra g coef- ficients of the canonical Kirillov-Kostant-Poisson 2-vector field γ are linear functions on g . Let e , . . . , e be a basis of g and (e , . . . , e ) be its dual basis, then the associated ∗ { 1 d} 1∗ d∗ Poisson 2-vector field is given by: 1 γ = [e , e ]e∗ e∗. 2 i j i ∧ j Xi,j Therefore on can considerably simplify the expression of poly-differential operators BΓ that occur in the definition of the star-product ?γ (1.4). Because of the linearity of coefficients of γ the only remaining graphs are those whose vertices of the first kind receive at most one incoming edge. Let f1 and f2 be two polynomials on g∗ with deg(f1) = l1, deg(f2) = l2. Then we remark that the graphs contributing to the star-product formula for f1 ? f2 can have no more than l + l vertices of the first type. Indeed, for any Γ A the corresponding 1 2 ∈ n bi-differential operator BΓ,γ contains exactly 2n differentiations. When 2n > n + l1 + l2, BΓ is obviously 0 (because f1 can be differentiated at most l1 times, f2 at most l2 times and each of the coefficients of γ corresponding to the remaining vertices at most once). Hence l1+l2 ~n f ? f = w B (f , f ). 1 2 n! Γ Γ 1 2 n=0 Γ An X X∈ 54 Micha Pevzner

This sum is finite, and if we set ~ = 1, we obtain a polynomial on g∗ of degree l1 +l2. Therefore, Kontsevich’s ?-product descends to an actual product on the algebra of polynomial functions on g∗, which can be naturally identified with the symmetric algebra (g). S In [5] 8.4, Kontsevich denotes by I the algebra isomorphism between (S(g), ? ) § alg γ and U(g)and shows that the identification of (S(g), ?γ ) with U(g) is given precisely by the Duflo isomorphism of vector spaces. Notice now that the zero tangent cohomology of g1 which is in this case the zero Poisson cohomology of the symmetric algebra S(g) is precisely the set of ad(g)-invariants in S(g) and on the other hand side that the zero tangent cohomology of g2 is the zero Hochshild cohomology of g with coefficients in U(g) that is nothing else but the cen- ter Z(g) of the universal enveloping algebra. Therefore, according to the theorem 2 the differential d ~γ of the Kontsevich’s L -quasi-isomorphism gives rise to an algebra U ∞ isomorphism between these to set which is precisely the Duflo map (0.1). The fact that the Duflo map extends to all tangent cohomology groups was shown in [7].

References

[1] M.Andler, A. Dvorsky, S. Sahi, Kontsevich quantization and invariant distributions on Lie groups. Ann. Sci. cole Norm. Sup. (4) 35, (2002), no. 3, 371–390. [2] D.Arnal, D. Manchon, M. Masmoudi, Choix des signes pour la formalit de M. Kontsevich. Pacific J. Math. 203, (2002), no. 1, 23–66. [3] M.Andler, S. Sahi, Ch. Torossian, Convolution of invariant distributions on Lie groups, e-print, math.QA/0104100. [4] M. Duflo, Op´erateurs diff´erentiels bi-invariants sur un groupe de Lie. Ann. Sci. Ecole Norm. Sup. (4) 10 (1977), pp.265–288. [5] M. Kontsevich, Deformation quantization of Poisson manifolds I, e-print, QA/9709040. [6] D. Manchon, Ch. Torossian, Cohomologie tangente et cup-produit pour la quantification de Kontse- vich. Ann. Math. Blaise Pascal 10, (2003), no. 1, 75–106. [7] M. Pevzner, Ch. Torossian, Isomorphisme de Duflo et la cohomologie tangentielle. J. Geom. Phys. 51 (2004), 487–506.

Micha Pevzner – Universit´e de Reims [email protected] en.wikipedia.o

Quantization Restrictions for Diffeomorphism Invariant Gauge Theories

Christian Fleischhack

Abstract The classical Stone-von Neumann theorem guarantees the uniqueness of the kine- matical framework of quantum mechanics. In this short article we review the cor- responding situation for diffeomorphism invariant field theories like gravity. Within the loop quantization framework, we first describe the Weyl algebra and sketch then a fundamental uniqueness result for its representations.

1 Introduction

The quantization of a classical system can be performed in different ways. Nevertheless, there are typical features common to many of them. Often, the major step consists of three choices: First, one selects some set of classical variables to be “quantized”. V Second, one picks out some Hilbert space H. Finally, one chooses some assignment between the selected classical variables and some operators on H. It is a fundamental question – both mathematically and physically –, how far these three choices are unique or may be made freely. Unfortunately, in general, this question cannot be answered adequately. First of all, the selection of is obviously highly non-unique. At the same V time, it very much constrains the two latter steps. In fact, if is too small, we may loose, V e.g., information about some physical degrees of freedom; if, however, is too large, van V Hove arguments may show that there is no (nontrivial) quantization at all. Therefore, we will restrict ourselves to the two latter steps, i.e., the “representation theory” of some given . V The focal point of the present paper is to study the situation in gauge field theories that incorporate diffeomorphism invariance. This is of enormous relevance, in particular, for the quantization of gravity. There are several approaches to this issue. We will consider here the loop quantization. This mainly means to include nonlocal parallel transports along paths (or loops) instead of the connections themselves into the set of V classical variables to be quantized. We are going to review under which assumptions the kinematical framework used there is indeed unique. Before, however, we will motivate our considerations by the celebrated Stone-von Neumann theorem in quantum mechanics. Proven some 75 years ago, it is responsible for the (to a large extent) uniqueness of quantization of classical mechanics. In classical mechanics in, for brevity, one dimension, the configuration space equals C R. The wave functions of quantum mechanics then are L2 functions over R w.r.t. the Lebesgue measure dx on R. In the Schr¨odinger representation, the position and mo- mentum variables x and p turn into self-adjoint multiplication and derivation operators

55 56 Christian Fleischhack

fulfilling [x, p] = i. They generate weakly continuous one-parameter subgroups of uni- taries: multiplication operators eiσx = eiσx and pull-backs eiλp = L of translation b · b λ∗ operators on = R. b bC Although we presented the quantization procedure above like an unrevocable fact, there are many other options to quantize classical mechanics. The easiest way to see this is to exchange the rˆoles of x and p. There we end up with the Heisenberg picture of quantum mechanics with p being a multiplication and x a differential operator. Neverthe- less, the Heisenberg and Schr¨odinger pictures are still unitarily equivalent, i.e. physically indistinguishable. Indeed,beven more, the Stone-von Neumannb theorem tells us that as- suming continuity and irreducibility, all “pictures” of quantum mechanics are equivalent: Each pair (U, V ) of unitary representations of R on some Hilbert space H satisfying the commutation relations U(σ)V (λ) = eiσλ V (λ)U(σ) (1.1) for all σ, λ R, is equivalent to multiples of the Schr¨odinger representation above [14]. ∈ Hence, assuming irreducibility, we get the desired uniqueness. It should be emphazised that the commutation relations (1.1) directly follow from U(σ) = eiσX , V (λ) = eiλP and [X, P ] = i, i.e., they do not use any relations induced by special representations. Finally, however, note that dropping the continuity assumption admits other, non- equivalent representations. One of them is given by almost-periodic functions leading to the Bohr compactification of the real line. The Hilbert space basis is given by x {| i | x R , and the operators U and V act by U(σ) x = eiσx x and V (λ) x = x + λ . Of ∈ } | i | i | i | i course, V is not continuous. Hence, p is not defined, but the operator V (λ) corresponding to eiλp only. We remark that this representation reappears in loop quantum cosmology yielding to a resolution of the big bangb singularity [7].

2 Configuration Space of Quantum Geometry

Let us now focus to (pure) gauge field theories. There the configuration space consists of C all smooth connections (modulo gauge transforms) in a principal fibre bundle P (M, G) with M being some manifold and G being some structure Lie group. Here, we will assume that M is at least two-dimensional and that G is connected and compact. For canonical gravity using Ashtekar variables [1], e.g., M is some Cauchy slice and G equals SU(2). Aiming at a functional-integral description of quantum theory, one needs some measure1 on . In general, however, the structure of is too complicated to allow for a C C rigorous measure theory describing non-pathological measures. Therefore, in parallel to the experiences known from the Wiener integral, it is reasonable to compactify . C 2.1 Compactification Ashtekar et al. [2, 4, 3] successfully implemented the compactification strategy. Their crucial idea was to use parallel transports instead of connections. In fact, a connection is uniquely determined by its parallel transports along all (sufficiently smooth) paths in M. Now, one considers the holonomy algebra2 , which is a subset of the bounded HA 1From now on, all measures are assumed to be normalized, regular, and Borel. 2Note that in [2] the holonomy algebra denotes the algebra generated by all Wilson loop variables, giving the gauge invariant functions only. Here, we also include gauge variant functions. Quantization Restrictions for Diffeomorphism Invariant Gauge Theories 57

functions on the space of smooth connections. is the (unital) -subalgebra A ≡ C HA ∗ of C ( ) generated by all matrix elements T := φ(h )m of parallel transports bound A γ,φ,m,n γ n hγ , where γ runs over all paths in M, φ runs over all (equivalence classes of) irreducible representations of G, and m and n over all the corresponding matrix indices. The space of generalized (or, distributional) connections is now defined to be the spectrum of the A completion of the holonomy algebra. Since is unital abelian, is compact Hausdorff. HA A Equivalently, can be described using projective limits. First, observe that the A elements of are one-to-one with the homomorphisms from the groupoid3 of paths A P in M to the structure group G. In fact, each h Hom( , G) defines a multiplicative ∈m P functional hA on via hA(Tγ,φ,m,n) := φ(h(γ))n implying hA . Now, any finite HA ∈ A #γ graph γ in M defines a continuous projection πγ : Hom( γ , G) = G via A −→ P ∼

πγ(h ) := h γ = h (γ) h (γ1), . . . , h (γ#γ ) , A A|P A ≡ A A where γ denotes the paths in γ. Note that the edges γ1, . . . , γ#γ of γ
freely generate P δ γ . Using the natural subgraph relationb and defining π : Hom( δ, G) Hom( γ , G) P γ P −→ P for γ δ again by restriction, we get a projective system over the set of all finite graphs ≤ in M, whose projective limit lim Hom( γ , G) is again. γ P A ←− 2.2 Ashtekar-Lewandowski Measure Compactness opens the door to many far-reaching theorems in measure theory, in particu- lar, on projective limits. In general, given a measure µ on a projective limit X := lim X , a a we may always push-forward this to all Xa using the canonical projections πa : X ←− Xa. −→ b Of course, there are certain compatibility relations among different constituents: If πa b projects Xb to Xa then (πa) (πb) µ equals (πb) µ. For a directed projective limit of com- ∗ ∗ ∗ pact Hausdorff spaces, however, the Riesz-Markov theorem establishes also the other way round. For each sequence µa of measures on Xa that fulfill the compatibility relations b (πa) µb = µa for all a b, there is a unique measure µ on X with (πa) µ = µa for all a. ∗ ≤ ∗ Using this general theorem, it is rather easy to define measures on . The most obvi- A ous choice gives the Ashtekar-Lewandowski measure µ0 [4] whose relevance will become #γ clear below. Here, one simply demands that the measure on Hom( γ , G) = G is the P ∼ Haar measure for each γ. If the set of all finite graphs is directed, the compatibility con- ditions are fulfilled and guarantee the existence and uniqueness of µ0. The directedness is given if we assume all paths and graphs to be piecewise analytic. It is no longer given for smooth paths. There it may happen that two graphs have infinitely many intersection points without sharing a full segment, whence there is no third graph containing both graphs. Although it is in principle possible to circumvent this problem [9], we will restrict ourselves to piecewise analytic paths from now on.

2.3 Spin Networks Having now H := L ( , µ ) as a candidate for the kinematical Hilbert space, we still 0 2 A 0 have to look for a basis or, at least, some reasonable generating system of this space. As in the case of measures, the problem is solved by focussing first on the graph level n and then lifting it to the continuum. In fact, bases for L2(G , µHaar) are given by the

3The groupoid structure is induced by the standard concatenation of paths modulo reparametrization and deletion/insertion of immediate retracings. 58 Christian Fleischhack

Peter-Weyl theorem. Each one contains just all tensor products of matrix elements of irreducible representations of the n group factors. Combining this with the projections 4 πγ, we get the spin network functions [5] T (φ )mk π : C. k γk,φk,mk,nk ≡ k k nk ◦ γk A −→ Here, γ = γ , . . . , γ γ is a graph, each φ a nontrivial irreducible representation of G, { 1 N# } Nk and each mk and each nk a matrix index. The set of all spin network functions generates H0. However, it does not form a basis, since two spin network functions that correspond, e.g., to graphs where one graph is a refinement of the other, need not be orthogonal. But, at least, the trivial spin network function, i.e., the constant function, is orthogonal to all the others.

2.4 Semianalytic Diffeomorphisms Until now, only general gauge theory ingredients have been implemented. Now, we are going to consider the most important difference between quantum gauge field theories and quantum geometry – the diffeomorphism invariance. Fortunately, within the loop formal- ism, this task is straightforward. The action of diffeomorphisms on M lifts naturally to an action on the sets of paths and graphs, whence to an action on and finally on C( ) as A A well. For instance, let there be given a cylindrical function, i.e. a function f C( ) which ∈ A equals fγ πγ for some graph γ and some fγ C(πγ( )). Then α (fγ πγ ) = fγ π , ◦ ∈ A ϕ ◦ ◦ ϕ(γ) with α denoting the action of the diffeomorphism ϕ on C( ). One shows easily that µ ϕ A 0 is diffeomorphism invariant, whence the action of diffeomorphisms is even unitary on H0. Of course, we can only consider diffeomorphisms that preserve the piecewise ana- lyticity of paths and graphs. Smooth diffeomorphisms, in general, do not meet this requirement. Analytic ones, on the other hand, are too restrictive from the physical point of view. In fact, gravity is a local theory, meaning it is invariant w.r.t. diffeo- morphisms being the identity outside some subset of M. But, analyticity is nonlocal: Changing an analytic object locally, modifies it globally. In other words, we are looking for some intermediate kind of diffeomorphisms reconciling locality and analyticity. This is indeed possible working in the semianalytic category [12, 13, 6] what we will always do in the following. Recall that semianalytic sets are stratified by analytic manifolds, whence semianalytic diffeomorphisms only need to be analytic on each single stratum.

3 Weyl Algebra of Quantum Geometry

In quantum mechanics, the Schr¨odinger representation assigns multiplication operators to the exponentiated position operators and pull-backs of translation operators to the exponentiated momentum operators. In quantum geometry, since we are dealing with fields, we do not only exponentiate the operators, but also smear them. In fact, parallel transports are exponentiated connections smeared along paths being one-dimensional ob- jects. The corresponding momenta in quantum geometry (or, more specific, in canonical gravity) are densitized dreibein fields. They are now exponentiated after getting smeared along one-codimensional objects, namely (semi-)analytic submanifolds. While the paral- lel transports (or, to be precise, their matrix elements) act naturally by multiplication

4Note again that, originally, spin network functions have been defined a little bit differently in order to implement gauge invariance. Quantization Restrictions for Diffeomorphism Invariant Gauge Theories 59

operators on H0, the exponentiated and smeared dreibein field operators will turn into unitary pull-backs of translation operators on . A 3.1 Weyl Operators Let now S be some oriented analytic hypersurface and let g G. We define the intersec- ∈ tion function σ (γ) to be +1 ( 1) if the path γ starts at S, such that γ˙ is non-tangent S − to S and such that some initial path of γ lies fully above (below) S; otherwise σS(γ) is 0. Above and below, of course, refer to the orientation of S. Next, observe that due to the semianalyticity of paths and hypersurfaces, each path can be decomposed into a finite number of paths whose interior is either fully contained in S (internal path) or disjoint to S (external path). One now checks quite easily [8] that there is a unique map ΘS : , such that5 g A −→ A 1 σS (γ) σS (γ− ) h S (γ) = g h (γ) g− Θg (A) A for all external and all internal γ and for all A . The map ΘS is a homeo- ∈ P ∈ A g morphism. It even preserves the Ashtekar-Lewandowski measure, due to the translation invariance of the Haar measure. The Weyl operator wS is now the pull-back of ΘS. It becomes an isometry on C( ) g g A and, by µ -invariance, a unitary operator on H = L ( , µ ). The subset of (H ) 0 0 2 A 0 B 0 generated by all Weyl operators wS is denoted by . We remark that the Weyl operator { g } W of the disjoint union of hypersurfaces equals the product of the (mutually commuting) Weyl operators of the single hypersurfaces. This way, it is natural to extend the notion of Weyl operators even to semianalytic submanifolds S having at least codimension 1. For instance, the Weyl operator for an equator (i.e., a one-dimensional circle in a three- dimensional space) can be seen as the Weyl operator of the full sphere times the inverses of the Weyl operators corresponding to the upper and lower hemisphere. Therefore, w.l.o.g., is always assumed to contain also the semianalytic subsets of M having at W least codimension 1. S 1 S ϕ(S) The diffeomorphisms act covariantly on via α w α α (w ) = wg , as W ϕ ◦ g ◦ ϕ− ≡ ϕ g one checks immediately.

3.2 Weyl Algebra The C -subalgebra A of (L ( , µ )), generated by C( ) and , is called Weyl algebra ∗ B 2 A 0 A W of quantum geometry. Its natural representation on L ( , µ ) will be denoted by π . 2 A 0 0 Sometimes, we will consider the C -subalgebra of (L ( , µ )) generated by the Weyl ∗ B 2 A 0 algebra A and the diffeomorphism group . It will be denoted by A . One immediately D Diff sees that acts covariantly on A. D 3.3 Irreducibility In this subsection, we are going to prove the irreducibility of A. For this, let f A . ∈ 0 First, by C( ) A, we have A0 C( )0 = L ( , µ0). Second, by unitarity of Weyl A ⊆ ⊆ A ∞ A operators w, we have f = w f w = w (f). Therefore, we have T, f = T, w (f) = ∗ ◦ ◦ ∗ h i h ∗ i 5Usually, instead of g being just an element of G, it denotes a function from S to G, encoding the smearing of the flux. We will not use non-constant smearings here; we skip this possibility. 60 Christian Fleischhack

w(T ), f for every spin network function T . Each nontrivial T may be decomposed into h i T = (Tγ,φ,m,n) T 0 with some edge γ and nontrivial φ, where T 0 is a (possibly trivial) spin network function.

If φ is abelian, choose some hypersurface S intersecting γ, but no edge used for T . • 0 Then wS (T ) = φ(g2) T for all g G, whence T, f = wS (T ), f = φ(g2) T, f . g ∈ h i h g i h i Since φ is nontrivial, there is some g G with φ(g2) = 1. Hence, T, f = 0. ∈ 6 h i If φ is nonabelian, then tr φ has a zero [11]. Since square roots exist in any compact • connected Lie group, there is a g G with tr φ(g2) = 0. Choose now infinitely ∈ many mutually disjoint surfaces Si intersecting γ, but no edge used for T 0. A straightforward calculation yields for i = j 6 2 2 Si Sj tr φ(g ) w (T ), wg (T ) = = 0. h g i dim φ

S Sj Now, w i (T ), f = T, f = wg (T ), f implies T, f = 0 again. h g i h i h i h i

Altogether, f is constant. Hence, A0 consists of scalars only.

4 Uniqueness Theorem

The natural representation π0 of A is not only irreducible, but also regular, i.e., it is weakly continuous w.r.t. the Weyl operator smearings g. Moreover, π0 is diffeomorphism invariant, i.e., there is a diffeomorphism invariant vector in H0 (the constant function) and the diffeomorphisms act covariantly on A. The fundamental question now is how far these properties already determine π0 among the C∗-algebra representations of A. Fairly uniquely, as we will learn from the following theorem.

4.1 Theorem Let dim M 3, let G be nontrivial, and let all the paths, hypersurfaces, and diffeo- ≥ morphisms be semianalytic. Assume that all the hypersurfaces used for the definition of are widely6 triangulizable. Let now π : A (H) be some regular C -algebra W Diff −→ B ∗ representation of ADiff on some Hilbert space H, which has some diffeomorphism invari- 7 ant vector being cyclic for π A. If the diffeomorphisms even act naturally , then π A is | | unitarily equivalent to π0.

4.2 Sketch of Proof

The proof will consist of three main steps: First, by general C ∗-algebra arguments, we have π = π , where each π is the canonical representation of C( ) on L ( , µ ) |C( ) ν ν ν A 2 A ν for some Ameasure µ . Second, we prove that π equals π for some ν using diffeo- L ν ν 0|C( ) morphism invariance and regularity. Finally, from naturality, Adiffeomorphism invariance and cyclicity we deduce that the directed sum above consists of just a single component

6A triangulation (K, f) is called wide iff for every σ ∈ K there is some open chart in M containing the closure of f(σ) and mapping it to a simplex in that chart. 7The definition of naturality will be provided in the sketched proof below. Quantization Restrictions for Diffeomorphism Invariant Gauge Theories 61

and that π A = π . Let us now sketch the final steps 2 and 3. The full proof may be | 0 found in [10]. For step 2, let us assume for simplicity that G is abelian. Fix ε > 0, and let f = h ( )n T be some nontrivial spin network function, i.e., γ and the graph underlying T form γ · a graph again and n = 0. Assume 1 , π(f)1 H = 0 for some (cyclic and) diffeomorphism 6 h 0 0i 6 invariant vector 1 H. Define for some “cubic” hypersurface S 0 ∈ 2m 1 w := wS and v := α (w ). t eit/2 t 2m ϕk t Xk=1 Here, each ϕk is a diffeomorphism winding γ, such that it has exactly m punctures with S. Each k corresponds to a sequence of m signs + or . These signs correspond to the − relative orientations of S and ϕk(γ) at the m punctures. Then eint + e int m v (f) = − f t 2 · and   1 m(nt)2 4 (vt e− 2 ) f O(m(nt) ) f . k − k∞ ≤ k k∞ Hence, for any small t there is some m = m(nt, ε) with 2 4 ε < O(m(nt) ) 10, π(f)10 H O(m(nt) ) f |h i | − k k∞ 1 m(nt)2 1 m(nt)2 (1 e− 2 ) 1 , π(f)1 H 1 , π[(v e− 2 )f]1 H ≤ | − h 0 0i | − |h 0 t − 0i | 1 , π[(v 1)f]1 H . ≤ |h 0 t − 0i | Now, for each small t there is a diffeomorphism ϕ with

ε 1 , π[(w 1)(α (f))]1 H ≤ |h 0 t − ϕ 0i | 2 10 H (π(wt) 1)10 H π(αϕ(f)) (H) ≤ k k k − k k kB = 2 10 H (π(wt) 1)10 H f , k k k − k k k∞ by diffeomorphism invariance. The final term, however, does not depend on ϕ, whence by regularity it goes to zero for t 0 giving a contradiction. Hence, 1 , π(f)1 H = 0 → h 0 0i for n = 0 implying π = π . 6 |C( ) ∼ 0 For step 3, let w be Asome Weyl operator assigned to an open ball or simplex, possibly having higher codimension. Observe that, for w commuting with αϕ, we have

π(α (f))1 , π(w)1 H = π(f)1 , π(w)1 H. h ϕ 0 0i h 0 0i Next, for nonconstant spin-network functions f, choose diffeomorphisms ϕi, commuting with w, such that

δ = α (f), α (f) H π(α (f))1 , π(α (f))1 H. ij h ϕi ϕj i 0 ≡ h ϕi 0 ϕj 0i (This, however, is not always possible causing some technical difficulties that will, never- theless, be ignored in the present article.) Consequently, we have

0 = π(f)1 , π(w)1 H = f, P π(w)1 H , h 0 0i h 0 0i 0 with P being the canonical projection from H to π(C( )) 1 = L ( , µ ) H . Now, 0 A 0 ∼ 2 A 0 ≡ 0 P π(w)1 = c(w) 1 with c(w) C, whence (1 P )π(w)1 generates L ( , µ ). The 0 0 0 ∈ − 0 0 2 A 0 naturality of π w.r.t. the action of diffeomorphisms implies that π(w)10 is diffeoinvariant 62 Christian Fleischhack

itself.8 Since S is assumed to be a ball or simplex (with lower dimension than M), there is a (semianalytic) diffeomorphism mapping S to itself, but inverting its orientation. Now, we have 2 π(w) 10 = π(w)π(αϕ)π(w)∗π(αϕ)∗10 = 10 implying π(w)10 = 10 by taking the square root of the smearing. The proof furnishes using cyclicity and triangulizability.

5 Acknowledgements

The author is very grateful to Friedrich Haslinger, Emil Straube and Harald Upmeier for the kind invitation to the Erwin-Schr¨odinger-Institut in Vienna and for the organiza- tion of the inspiring workshop “Complex Analysis, Operator Theory and Applications to Mathematical Physics”.

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8A representation π is called natural w.r.t. the action of diffeomorphisms iff, for each decomposition

Lν πν into cyclic components of π|C( ), the diffeomorphism invariance of the C(A)-cyclic vector for πν1 A implies that for πν2 , provided the measures µ1 and µ2 underlying these representations coincide. Quantization Restrictions for Diffeomorphism Invariant Gauge Theories 63

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Christian Fleischhack – Max-Planck-Institut fur¨ Mathematik in den Naturwissenschaften Inselstraße 22–26 04103 Leipzig, Germany chfl@mis.mpg.de Orbits of triples in the Shilov boundary of a bounded symmetric domain

Jean-Louis Clerc and Karl-Hermann Neeb

Abstract Let be a bounded symmetric domain of tube type, S its Shilov boundary, and G the neutralD component of its group of biholomorphic transforms. We classify the orbits of G in the set S S S. × × 1 The problem

It is well-known that any bounded symmetric domain in a finite-dimensional complex D vector space can be realized as the open unit ball

(1) = z V z < 1 D { ∈ k k } of a normed complex vector space V . Such a unit ball is symmetric if and only if the group Aut( ) of biholomorphic transforms of acts transitively, i.e., if is homogeneous. Let D D D G := Aut( ) be the identity component of Aut( ) and S be the Shilov boundary of . D 0 D D The action of any element of G extends to a neighborhood of , and hence G acts on S. D It is well known that this action is transitive. The main result we are reporting on is a classification of the G-orbits in the set S S S of triples in S, when is of tube type. × × D The action of G on S S can be easily studied as an application of Bruhat theory, and × the description of the orbits is the same, whether is of tube type or not. But for triples D there is a drastic difference between tube type domains and non tube type domains. In the first case, there is a finite number of orbits in S S S, whereas there are an infinite × × number of orbits for a non tube type domain.

2 Jordan triples, Jordan frames, and polydiscs

In the following we shall always assume that is realized as a unit ball as in (1). Then D is said to be reducible if V = V V with the norm on V satisfying (v , v ) = D 1 × 2 k 1 2 k max( v , v ), so that = , where is the unit ball in V . k 1k k 2k D D1 × D2 Dj j Any can be written as a product or irreducible domains, and since G and S de- D compose accordingly as products, we assume in the following that is irreducible. D Then G is a finite-dimensional simple real Lie group acting transitively on , the D stabilizer K of 0 is maximal compact and = G/K is a non-compact Riemannian ∈ D D ∼ Keywords: bounded symmetric domain, tube type domain, Shilov boundary, face, Maslov index, flag manifold, Jordan triples Subject classification: 32M15, 53D12, 22F30

64 Orbits of triples in the Shilov boundary of a bounded symmetric domain 65

symmetric space of hermitian type. We consider the Lie algebra g of G as realized by vector fields on . Since all these vector fields are polynomial of degree 2, they extend D ≤ to all of V . Moreover, g has a Cartan decomposition g = k p, where k = L(K) consists ⊕ of linear vector fields and p consists of even vector fields. Since the evaluation map p V = T ( ), X X(0) is a linear isomorphism, each element of p can be written as → ∼ 0 D 7→

Xv(z) = v + Q(z)(v), where Q(z)(v) is quadratic in z and real linear in v. Polarization now leads to the triple product , , V 3 V {· · ·} → which is uniquely determined by

Q(z)(v) = z, v, z and a, b, c = c, b, a for a, b, c, z, v V. { } { } { } ∈ By definition, the triple product is linear in the first and third argument. It is antilinear in the second argument. An element e V is called a tripotent if e, e, e = e and two tripotents e, f are said ∈ { } to be orthogonal if e, e, f = 0, which turns out to define a symmetric relation. The sum { } e + f of two non-zero orthogonal tripotents is a tripotent, called decomposable. A Jordan frame is a maximal tuple (c1, . . . , cr) of pairwise orthogonal indecomposable tripotents. Their number r is called the rank of . D It is a well-known fact that for each Jordan frame (c , . . . , c ) with E := span c , 1 r { 1 . . . , c the intersection r} r E = ζ c ζ < 1, 1 j r = ∆r D ∩ j j | j| ≤ ≤ ∼ n Xj=1 o is an r-dimensional polydisc, i.e., the unit ball in (C r, ). The converse is less obvious k·k∞ ([CN05]):

Theorem 2.1. If E V is an r-dimensional subspace for which E is a polydisc, ⊆ ∩ D then there exists a Jordan frame (c1, . . . , cr) in V spanning E.

3 Faces and orbits

It is easy to describe the G-orbit structure on in terms of a Jordan frame: There are D r + 1 orbits which are represented by the tripotents

ek := c1 + . . . + ck, k = 0, . . . , r.

Here e = 0, G.e = , and G.e = S is the Shilov boundary of . We may therefore 0 0 D k D define a rank function rank 0, . . . , r , g.e k, D → { } k 7→ classifying the G-orbits in . D Our next goal is to extend this rank function to tuples of elements in , which is done D by using faces of the compact convex set . For a subset M we write Face(M) D ⊆ D ⊆ D 66 Jean-Louis Clerc and Karl-Hermann Neeb

for the face generated by M, i.e., the intersection of all faces of containing it. Using the D result that the faces of coincide with the closures of the holomorphic arc components D of , we see that the group G acts on the set ( ) of faces of . There are precisely D F D D r + 1 orbits, represented by the faces Face(e ), k = 0, . . . , r. As for , this leads to a k D G-invariant rank function

rank ( ) 0, . . . , r , g. Face(e ) = Face(g.e ) k, F D → { } k k 7→ classifying the G-orbits. This picture provides a nice connection between the convex geometry of and its D complex geometric properties. In particular, the Shilov boundary S coincides with the set of extreme points, i.e., the one-point faces of . D From the rank function for faces, we immediately obtain integral invariants for the m diagonal G-action on the product sets : D m 0, . . . , r , (z , . . . , z ) rank(Face(z , . . . , z )). D → { } 1 m 7→ 1 m 4 The Diagonalization Theorem and its consequences

2 The main result of Section 1 in [CN05] is a classification of the G-orbits in the set of D pairs which are transversal in the sense that rank Face(x, y) = 0, i.e., x and y do not lie in a common proper face of . The main tool for the classification of G-orbits in S S S D × × (for tube type domains) is the characterization of this relation in Jordan theoretic terms: it is equivalent to quasi-invertibility ([CN05, Th. 2.6]). This characterization is also valid for non tube type domains. This fact is used to setup an inductive proof of the

Theorem 4.1 (Diagonalization Theorem). If is a bounded symmetric domain of D tube type and ∆r is a polydisc of maximal rank r (hence given by a Jordan frame), ⊆ D then every triple in S is conjugate to a triple in the Shilov boundary T of ∆r.

The assumption that is of tube type is essential in the preceding theorem because D it is invalid for all non-tube type domains. This reduces the classification of G-orbits in S S S to the description of intersections × × of these orbits with the 3r-dimensional torus T 3. This is fully achieved by assigning a 5-tuple (r0, r1, r2, r3, ι) of integer invariants to each orbit and by showing that triples with the same invariant lie in the same orbit. The first four components of this 5-tuple are

(r0, . . . , r3) =

(rank Face(x1, x2, x3), rank Face(x1, x2), rank Face(x2, x3), rank Face(x1, x3)). (4.1)

The fifth component is the Maslov index ι(x1, x2, x3), an integer invariant taking val- ues in r, . . . , r (cf. [CØ01], [Cl04a/b]). If all pairs (x , x ), (x , x ) and (x , x ) are {− } 1 2 2 3 3 1 transversal, then all ri vanish, which implies that the G-orbits in the set of transversal triples are classified by the Maslov index.

Theorem 4.2 (Classification of orbits). Let be a bounded symmetric domain of D tube type. Then the five integers (r , r , r , r , ι) separate the orbits of G in S S S, 0 1 2 3 × × and a 5-tuple (r , r , r , r , ι) Z5 arises from some orbit if and only if 0 1 2 3 ∈ Orbits of triples in the Shilov boundary of a bounded symmetric domain 67

(P1) 0 r r , r , r r. ≤ 0 ≤ 1 2 3 ≤ (P2) r + r + r r + 2r . 1 2 3 ≤ 0 (P3) ι r + 2r (r + r + r ). | | ≤ 0 − 1 2 3 (P4) ι r + r + r + r mod 2. ≡ 1 2 3 A special case of this theorem was known before: If is the Siegel domain (the D unit ball in the space of complex symmetric matrices Symr(C)), then the group G is the projective symplectic group PSp (R) := Sp (R)/ 1 , and the Shilov boundary of 2r 2r { } D can be identified with the Lagrangian manifold (the set of Lagrangian subspaces of R2r). Then the orbits of triples of Lagrangians have been described (see [KS90, p.492]), using linear symplectic algebra techniques. As a byproduct of Theorem 2, we also obtain the following axiomatic characterization of the Maslov index, which actually was our original motivation for the project:

Theorem 4.3. The Maslov index is characterized by the following properties: (M1) It is invariant under the group G. (M2) It is an alternating function with respect to any permutation of the three arguments. (M3) It is additive in the sense that if = , so that S = S S , then D D1 × D2 1 × 2

ιS(x, y, z) = ιS((x1, x2), (y1, y2), (z1, z2)) = ιS1 (x1, y1, z1) + ιS2 (x2, y2, z2) .

(M4) If Φ : is an equivariant holomorphic embedding of bounded symmetric D1 −→ D2 domains of tube type of equal rank, then ι Φ = ι . S2 ◦ S1 (M5) It is normalized by ιT(1, 1, i) = 1 for the Shilov boundary T of the unit disc ∆. − − 5 More on orbits on products of flag manifolds

The Shilov boundary S of a bounded domain is in particular a generalized flag manifold of G, i.e. of the form G/P , where P is a parabolic subgroup of G. A nice description of P is obtained after performing a Cayley transform. The domain is transformed to an D unbounded domain C which is a Siegel domain of type II and the group P is the group D of all affine transformations preserving C . The group P has some specific properties: D it is a maximal parabolic subgroup of G, conjugate to its opposite. Moreover, one can show that the domain is of tube type if and only if the unipotent radical U of P is D abelian. A natural question arises to which extent results similar to Theorem 3 could be valid for other generalized flag manifolds. The natural background for this problem is the following. If P1, . . . , Pk are parabolic subgroups of a connected semisimple group G0, then the product manifold

M := G0/P . . . G0/P 1 × × k is called a multiple flag manifold of finite type if the diagonal action of G0 on M has only finitely many orbits. For k = 1 we always have only one orbit, and for k = 2 the finiteness of the set of orbits follows from the Bruhat decomposition of G0. For G0 = GLn(K) or G0 = Sp2n(K) and K an algebraically closed field of characteristic zero, it has been shown in [MWZ99/00] that finite type implies k 3, and for k = 3 ≤ the triples of parabolics leading to multiple flag manifolds of finite type are described and the G0-orbits in these manifolds classified. The main technique to achieve these 68 Jean-Louis Clerc and Karl-Hermann Neeb

classifications was the representation theory of quivers. In [Li94], Littelmann considers general simple algebraic groups over K and describes all multiple flag manifolds of finite type for k = 3 under the assumption that P1 is a Borel subgroup and P2, P3 are maximal parabolics. Actually Littelmann considers the condition that B = P1 has a dense orbit in G /P G /P , but the results in [Vi86] show that this implies the finiteness of the number 0 2 × 0 3 of B-orbits and hence the finiteness of the number of G -orbits in G /B G /P G /P . 0 0 × 0 2 × 0 3 From Littelmann’s classification one can easily read off that for a maximal parabolic P 3 in G0 the triple product (G0/P ) is of finite type if and only if the unipotent radical U of P is abelian, and in two exceptional situations. If U is abelian, then P is the maximal parabolic defined by a 3-grading of g0 = L(G0), so that G0/P is the conformal completion of a Jordan triple (cf. [BN05] for a discussion of such completions in an abstract setting). This case was also studied in [RRS92]. The first exceptional case, where U is not abelian, 2n corresponds to G0 = Sp2n(K), where G0/P = 2n 1(K) is the projective space of K , U is ¶ − the (2n 1)-dimensional Heisenberg group and the Levi complement is Sp2n 2(K) K×. − − × In the other exceptional case G0 = SO2n(K), and G0/P is the highest weight orbit in the n ˜ 2 -dimensional spin representation of the covering group G0 = Spin2n(K) of G0. Here U = Λ2(Kn) Kn also is a 2-step nilpotent group and the Levi complement acts like ∼ ⊕ GLn(K) on this group. A classification of all spaces G/P (G reductive algebraic over an algebraically closed field) for which G has open orbits in (G/P )3 has been obtained recently by V. Popov in [Po05]. It seems that the positive finiteness results have a good chance to carry over to the split forms of groups over more general fields and in particular to K = R. Suppose that G is a real reductive groups and GC its complexification, and that GC acts with finitely many orbits on (GC /PC )3. Then, for each GC-orbit M (GC /PC )3 meeting the ⊆ totally real submanifold (G/P )3, the intersection M (G/P )3 is totally real in M, hence ∩ a real form of M, and [BS64, Cor. 6.4] implies that G has only finitely many orbits in M (G/P )3. ∩

6 References

[BN05] Bertram, W., and K.-H. Neeb, Projective completions of Jordan pairs, Part II: Manifold structures and symmetric spaces, Geom. Dedicata 112:1 (2005), 75–115. [BS64] Borel, A., and J.-P. Serre, Th´eor`emes de finitude en cohomologie galoisi- enne, Comment. Math. Helv. 39 (1964), 111–164. [Cl04a] Clerc, J.-L., The Maslov triple index on the Shilov boundary of a classical domain, J. Geom. Physics 49:1 (2004), 21–51. [Cl04b] —, L’indice the Maslov g´en´eralis´e, Journal de Math. Pure et Appl. 83 (2004), 99–114. [CN05] Clerc, J.-L., and K.-H. Neeb, Orbits of triples in the Shilov boundary of a bounded symmetric domain, Transformation Groups, to appear. [CØ01] Clerc, J-L., and B. Ørsted, The Maslov index revisited, Transformation Groups 6 (2001), 303–320. [KS90] Kashiwara, M., and P. Schapira, “Sheaves on Manifolds”, Grundlehren der math. Wiss. 292, Springer-Verlag, 1990. Orbits of triples in the Shilov boundary of a bounded symmetric domain 69

[Li94] Littelmann, P., On spherical double cones, J. Algebra 166 (1994), 142– 157. [MWZ99] Magyar, P., J. Weyman, and A. Zelevinsky, Multiple flag varieties of finite type, Advances in Math. 141 (1999), 97–118. [MWZ00] —, Symplectic multiple flag varieties of finite type, J. Algebra 230:1 (2000), 245–265. [Po05] Popov, V. L., Generically multiple transitive algebraic group actions, in “Algebraic Groups and Homogeneous Spaces”, Proc. Internat. Col- loquium Jan. 2004, Tata Inst. Fund. Research, Bombay, to appear. [RRS92] Richardson, R., G. R¨ohrle, and R. Steinberg, Parabolic subgroups with Abelian unipotent radical, Invent. Math. 110 (1992), 649–671. [Vi86] Vinberg, E. B., Complexity of the actions of reductive groups, Functional Anal. Appl. 20 (1986), 1–13.

The present note is an extended abstract of a talk given by the second author at the ESI. Complete proofs and details appear in [CN05]. We also refer to [CN05] for more detailed references.

Jean-Louis Clerc – Institut Elie Cartan Facult´e des Sciences, Universit´e Nancy I B.P. 239 F - 54506 Vandœuvre-l`es-Nancy Cedex France [email protected]

Karl-Hermann Neeb – Fachbereich Mathematik Technische Universitat¨ Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt Germany [email protected] On a certain 8-dimensional non-symmetric homogenous convex cone Takaaki Nomura

This work started with a discussion with Simon Gindikin when I visited Rutgers University last year. The idea of considering the present 8-dimensional cone comes from just looking through a list of basic relative invariants associated to homogeneous convex cones given by Yusuke Watanabe who is currently preparing his master thesis. Discussions with Hideyuki Ishi also contribute to the contents. Let us begin with some facts about matrices. Consider the following subgroups AC and NC of GL(r, C):

AC := a = diag[a , . . . , a ] ; a C×, . . . , a C× , 1 r 1 ∈ r ∈ 1 0 0 0  · · · n21 1 0 0   . ·.· · .   NC := n = . .. . ; nji C (j > i) .    ∈   nr 1,1 nr 1,2 1 0    − −   nr1 nr2 nr,r 1 1    · · · −       We denote by V the vector space of real r r symmetric matrices: V := Sym (r, R). In × V , we have the open convex cone Ω of positive definite matrices. Then the tube domain Ω + iV in the complexified vector space VC is contained in the orbit of the triangular subgroup NCAC through the r r unit matrix E Ω: × ∈ Ω + iV NCAC E. (0.1) ⊂ · For every real or complex r r matrix w = (w ), we denote by ∆ (w) the k-th principal × ij k minor of w (k = 1, . . . , r): w w 11 · · · 1k . . ∆k(w) = det  . .  , w w  k1 · · · kk   and we set ∆ (w) 1. By (0.1) we see that if w Ω + iV , then we have ∆ (w) = 0 for 0 ≡ ∈ k 6 any k = 1, . . . r. Lemma 0.1. Let w Sym(r, C) and suppose that Re w Ω. If one writes w = natn ∈ ∈ with a = diag[a , . . . , a ] AC and n N according to (0.1), then one has 1 r ∈ ∈ C ∆k(w) ak = (k = 1, . . . , r). ∆k 1(w) − Subject classification: Primary 32M10; Secondary 32A07, 52A20 Keywords: homogeneous convex cone, tube domain, principal minor, basic relative invariant

70 On a certain homogeneous convex cone 71

Lemma 0.2. Suppose that Re(natn) is positive definite for a = diag[a , . . . , a ] AC 1 r ∈ and n NC. Then Re a > 0, . . . , Re a > 0. ∈ 1 r From these two lemmas we have the following proposition.

Proposition 0.3. Let w Sym(r, C) and suppose that Re w Ω. Then ∈ ∈ ∆ (w) Re k > 0 (k = 1, . . . , r). ∆k 1(w) − By using the framework of Euclidean Jordan algebra, it is not hard to generalize Proposition 0.3 to the case where Ω is a general symmetric cone. For this we just think of ∆k(w) as the Jordan algebra principal minors which are described in the book of Faraut– Kor´anyi [1]. In view of the fact that Lemma 0.2 can be proved by making use of the stability of the Jordan algebra inverse map w w 1 for symmetric tube domains, and 7→ − then of the fact that this stability characterizes symmetric tube domains (cf. Kai–Nomura [4]), it would be quite natural to have the following question:

Question 0.4. Is Proposition 0.3 characteristic of symmetric cones?

Here if one would like to generalize Proposition 0.3 to any homogenous open convex cone (we say homogenous cone in what follows for simplicity), then it is necessary to generalize ∆k to such a case. Ishi has done this in [3], and we take ∆k(w) as the basic relative invariants associated to homogeneous cones following [3]. These are polynomial functions on the ambient vector space V of the homogeneous cone Ω under considera- tion, so that they are naturally continued to holomorphic polynomial functions on the complexification VC. Of course, if the cone is symmetric, these basic relative invariants coincide with the principal minors. Now Question 0.4 can be formulated in the following way:

Conjecture 0.5. With the above notation, the implication for w VC that ∈ ∆ (w) Re w Ω = Re k > 0 (k = 1, . . . , r) ( ) ∈ ⇒ ∆k 1(w) ∗ − is equivalent to the symmetry of Ω.

The purpose of this note is to present a counterexample to this conjecture. In other words, we show that there is a non-symmetric homogeneous cone for which we have the above implication ( ) for elements w VC. The cone is 8-dimensional as mentioned in ∗ ∈ the title. From now on, let I be the 2 2 unit matrix, and V will denote the 8-dimensional real × vector space described in the following manner:

x11I x21I y y1 2 z1 2 V := x = x21I x22I z ; y = R , z = R , xij R .   t t  y2 ∈ z2 ∈ ∈   y z x33        We note V Sym(5, R). As for an open convex cone Ω, we take the positive definite ⊂ ones in V : Ω := x V ; x 0 . (0.2) { ∈  } 72 Takaaki Nomura

Let A := a = diag[a I, a I, a ] ; a > 0, a > 0, a > 0 , { 1 2 3 1 2 3 } I 0 0 2 2 (0.3) N := n = ξI I 0 ; ξ R, n1 R , n2 R .  t t  ∈ ∈ ∈   n1 n2 1    Then we see withoutdifficulty that the semidirect product group H :=N n A acts on Ω by H Ω (h, x) hxth Ω simply transitively. Indeed, if x Ω is expressed as in the × 3 7→ ∈ ∈ definition of V , then the equation x = natn with x A and n N as in (0.3) is solved as ∈ ∈ ∆ (x) ∆ (x) a = ∆ (x), a = 2 , a = 3 , 1 1 2 ∆ (x) 3 ∆ (x) 1 2 (0.4) x21 y x11z x21y ξ = , n1 = , n2 = − . ∆1(x) ∆1(x) ∆2(x)

Here, ∆1, ∆2, ∆3 are the polynomial functions on V given by

∆1(x) = x11, ∆ (x) = x x x2 ,  2 11 22 − 21 ∆ (x) = x x x + 2x y z x x2 x y 2 x z 2, 3 11 22 33 21 · − 33 21 − 22k k − 11k k  with y z the canonical inner product in R2 and the corresponding norm. Moreover · k · k these ∆k(x) (k = 1, 2, 3) are the basic relative invariants associated to the current homo- geneous cone Ω. We note here that, if δk(x) (k = 1, . . . , 5) stands for the k-th principal minors of the 5 5 matrix x V , then × ∈ 2 δ1(x) = ∆1(x), δ2(x) = ∆1(x) , δ3(x) = ∆1(x)∆2(x), 2 δ4(x) = ∆2(x) , δ5(x) = ∆2(x)∆3(x).

Therefore x Ω is equivalent to ∆ (x) > 0 for any k = 1, 2, 3. Of course this is also seen ∈ k from the general case treated in Ishi [3]. Now, extending the canonical inner product y z in R2 to a complex bilinear form · on C2 which we denote by the same symbol, and writing ν(y) := y y instead of y 2 · k k (and similarly for ν(z)), we have the obvious analytic continuations of ∆k(x) (k = 1, 2, 3) to holomorphic polynomial functions on VC. Let AC and NC be the complexifications of A and N, respectively. As shown in Nomura [6] for the general case, the tube domain Ω + iV is contained in the orbit of NCA through the 5 5 identity matrix E Ω, that c × ∈ is, Ω + iV NCAC E. In particular, none of ∆ vanishes on Ω + iV . ⊂ · k Proposition 0.6. Suppose that w VC satisfies Re w Ω. Then ∈ ∈ ∆ (w) Re k > 0 (k = 1, 2, 3). ∆k 1(w) − Proposition 0.6 is almost clear from the complex version of (0.4) together with Lemma 0.2 applied to the case r = 5, our cone (0.2) being evidently a subset of the cone of 5 5 × real positive definite symmetric matrices. On a certain homogeneous convex cone 73

References

[1] J. Faraut and A. Kor´anyi, Analysis on symmetric cones, Clarendon Press, Oxford, 1994.

[2] S. Gindikin, Tube domains and the Cauchy problem, Transl. Math. Monogr., 111, Amer. Math. Soc., Providence, R. I., 1992.

[3] H. Ishi, Basic relative invariants associated to homogeneous cones and applications, J. Lie Theory, 11 (2001), 155–171.

[4] C. Kai and T. Nomura, A characterization of symmetric cones through pseudoinverse maps, J. Math. Soc. Japan, 57 (2005), 195–215.

[5] S. Kaneyuki and T. Tsuji, Classification of homogeneous bounded domains of lower dimension, Nagoya Math. J., 53 (1974), 1–46.

[6] T. Nomura, On Penney’s Cayley transform of a homogeneous Siegel domain, J. Lie Theory, 11 (2001), 185–206.

Takaaki Nomura – Faculty of Mathematics, Kyushu University, Hakozaki, Higashi-ku 812- 8581, Fukuoka, Japan [email protected] A characterization of symmetric tube domains by convexity of Cayley transform images Chifune Kai

Abstract We show that a homogeneous tube domain is symmetric if and only if its Cayley transform image is convex. Moreover this convexity forces the parameter of the Cayley transform to be a specific one, so that the Cayley transform coincides with the standard one defined in terms of the Jordan algebra structure associated with the domain.

1 Introduction

A homogeneous bounded domain is an important geometric and analytic object. It is holo- morphically equivalent to a homogeneous Siegel domain, which is a higher dimensional analogue of the right half-plane in C and is affine homogeneous. Among homogeneous Siegel domains, there is a special subclass consisting of symmetric ones. In [3] we charac- terized symmetric Siegel domains by the simple geometric condition that the image of the naturally defined Cayley transform is convex. We have this convexity as follows. Since a symmetric Siegel domain is a Hermitian symmetric space of non-compact type, it has a canonical bounded realization, the Harish-Chandra realization. In [6] Kor´anyi and Wolf introduced (the inverses of) the Cayley transforms which map a symmetric Siegel domain to its Harish-Chandra realization. Since the Harish-Chandra realization is known to be the open unit ball for a certain norm, the image of the Cayley transform is convex. Be- fore proceeding, we would like to mention that it is shown in [7] that the Harish-Chandra realization of a symmetric Siegel domain is characterized essentially among bounded re- alizations by its convexity. In other words, the Cayley transform is essentially the only bounded convex realization of a symmetric Siegel domain. In this article we deal with homogeneous tube domains (homogeneous Siegel domains of type I) for simplicity. We present an example. We put V := Sym(n, R), the real vector space of symmetric matrices of order n and set W := VC = Sym(n, C). We define an open convex cone Ω by

Ω := X V X 0 (positive definite) . { ∈ |  } Then the tube domain Ω + iV W is symmetric, which is called the Siegel upper half- ⊂ plane (though it is a right half-plane here). The Cayley transform for Ω+iV is introduced by

1 (w) := (w e)(w + e)− (w W ), C − ∈ Subject classification: 32M15, 43A85 Keywords: Homogeneous cone, Homogeneous tube domain, Cayley transform, Jordan algebra, clan

74 Characterization of symmetric tube domains 75

where e is the unit matrix of order n. We put z := (w). By an easy computation we C see that

1 1 Re w = (e z)− (e zz∗) (e z)− ∗. − − − Hence we have

(Ω + iV ) = z W e zz∗ 0 . C { ∈ | −  } The right hand side is the open unit ball for a certain norm and is convex. For every symmetric tube domain, we can define the Cayley transform in a natural way using the structure of the associated Jordan algebra, and the image of the Cayley transform is convex (see 3 for details). We shall see that this convexity characterizes symmetric tube § domains among homogeneous ones. In this article we deal with the parametrized family of Cayley transforms for homo- geneous Siegel domains defined by Nomura [11], which is specialized to tube domains. This family includes Penney’s Cayley transform [13], and Nomura’s one associated with the Bergman kernel (resp. Szeg¨o kernel) of the domain appearing in [8], [9] and [10] (resp. [12]). Though there is no essential difference between these three Cayley transforms in the case of tube domains, the parametrization is significant when we use the main theorem of this article to prove [3, Theorem 3.1]. If the domain is symmetric, the parametrized family of Cayley transforms includes the above-mentioned Cayley transform defined by means of the Jordan algebra structure. Our theorem states also that the convexity of Cayley transform image is characteristic of that Cayley transform.

2 Homogeneous tube domains

Let V be a finite-dimensional vector space over R. An open convex cone Ω V is called ⊂ a homogeneous convex cone, if the linear automorphism group

G(Ω) := g GL(V ) g(Ω) = Ω { ∈ | } acts transitively on Ω. We put W := VC, the complexification of V , and denote by w w 7→ ∗ the complex conjugation of W with respect to the real form V . For a homogeneous convex cone Ω V , we call the domain Ω + iV W a homogeneous tube domain. It is ⊂ ⊂ homogeneous, in particular, affine homogeneous.

2.1 Admissible relative invariants on the cone By [14, Theorem 1], there exists a split solvable subgroup H of G(Ω) acting simply transitively on Ω. A function ∆ : Ω R is called a relative invariant if there exists → + a character (one-dimensional representation) of H such that ∆(hx) = χ(h)∆(x) (h ∈ H, x Ω). We take any E Ω and fix it throughout this article. For a relative invariant ∈ ∈ ∆ on Ω, we define a bilinear form on V by

x y := D D log ∆(E) (x, y V ), h | i∆ x y ∈ where for a C function f on V , v V and x Ω, we define ∞ ∈ ∈ d Dvf(x) := dt f(x + tv) t=0 .

76 Chifune Kai

If the bilinear form defines a positive definite inner product on V , we say that ∆ is h·|·i∆ admissible (only in this article). We know that one of the admissible relative invariants is given by ∆ (h) := det h 1 (h H). Det − ∈ 3 Cayley transforms for symmetric tube domains

We suppose that Ω + iV is symmetric in this section. Then Ω is a symmetric cone and the associated Jordan algebra structure is introduced in the following way. We define a commutative product on V by ◦ x y z = 1 D D D log ∆ (E) (x, y, z V ). (3.1) h ◦ | i∆Det − 2 x y z Det ∈ We know that E is the unit element. Since Ω + iV is symmetric, we see that V with the product is a Jordan algebra. This means that in addition to the commutativity we ◦ have for all x, y V , ∈ x2 (x y) = x (x2 y). ◦ ◦ ◦ ◦ Moreover this Jordan algebra is Euclidean in the sense of [1]. In fact, by (3.1) we have

x y z = x y z (x, y, z V ). h ◦ | i∆Det h | ◦ i∆Det ∈ We extend the product to W by complex bilinearity. Then W is a semisimple complex ◦ Jordan algebra. We introduce the Cayley transform for Ω + iV by CJ 1 (w) := (w E) (w + E)− (w W ). CJ − ◦ ∈ Remark 3. We note that the above definition is rewritten as

1 (w) := E 2(w + E)− (w W ). CJ − ∈ For an invertible v V , the Jordan algebra inverse v 1 is characterized as ∈ − 1 v− x = D log ∆ (v) (x V ). h | i∆Det − x Det ∈ Let us describe the Cayley transform image (Ω + iV ). For w W , we denote by CJ ∈ L(w) the multiplication operator by w: L(w)v := w v (v W ). For x, y W , we define ◦ ∈ ∈ a complex linear operator xy on W by

xy := L(x y) + [L(x), L(y)]. ◦ For w W , we set w := ww 1/2, where the operator norm is computed through the ∈ | | k ∗k norm associated with . By [1, Proposition X.4.1], we know that is a norm on h·|·i∆Det | · | W , which is called the spectral norm. Proposition 3.1. We have

(Ω + iV ) = w W w < 1 , CJ { ∈ | | | } so that (Ω + iV ) is convex. CJ Characterization of symmetric tube domains 77

4 Cayley transforms for homogeneous tube domains

Now we proceed to the case of homogeneous tube domains. Let Ω+iV be the homogeneous tube domain defined in 2. Let ∆ be any admissible relative invariant on Ω. For x Ω, § ∈ the pseudoinverse (x) of x is defined by I∆ (x) y = D log ∆(x) (y V ). hI∆ | i∆ − y ∈ We call : Ω V the pseudoinverse map. Let us present the key properties of : I∆ → I∆ We denote by Ω∆ the dual cone of Ω realized in V by means of the inner product • . We see that gives a bijection from Ω onto Ω∆. h·|·i∆ I∆ One has (E) = E. • I∆ is analytically continued to a birational map W W . Let HC be the com- • I∆ → plexification of H. We extend to a complex bilinear form on W . Then is h·|·i∆ I∆ HC-equivariant: (hx) = ∆h 1 (x) (h HC), where ∆h stands for the transpose I∆ − I∆ ∈ of h with respect to . h·|·i∆ is holomorphic on Ω + iV . • I∆ We suppose that Ω+iV is symmetric and ∆ = ∆ p for some p > 0. We introduce • Det the Jordan algebra structure with the unit element E as we did in 3. Then § I∆ coincides with the Jordan algebra inverse map.

We define the Cayley transform for Ω + iV by C∆ (w) := E 2 (w + E) (w W ). C∆ − I∆ ∈ It is shown that maps Ω+iV biholomorphically onto a bounded domain. Thus we have C∆ the Cayley transform for every admissible relative invariant on Ω. Our characterization theorem for symmetric tube domains is stated as follows:

Theorem 4. Let Ω + iV be a homogeneous tube domain and ∆ an admissible relative invariant on Ω. Then ∆(Ω + iV ) is convex if and only if Ω + iV is symmetric and p C ∆ = ∆Det for some p > 0.

5 Acknowledgment

The author would like to thank the Erwin Schr¨odinger International Institute for Mathe- matical Physics for its support during his stay at Vienna. He is also grateful to Professor Harald Upmeier for organizing the workshop “Quantization, Complex and Harmonic Analysis”.

References

[1] J. Faraut, A. Kor´anyi, Analysis on Symmetric Cones, Clarendon Press, Oxford, 1994. 78 Chifune Kai

[2] C. Kai, A symmetry characterization of quasisymmetric Siegel domains by convexity of Cayley transform images, J. Lie Theory 16 (2006), 47–56.

[3] C. Kai, A characterization of symmetric Siegel domains by convexity of Cayley transform images, preprint.

[4] C. Kai, T. Nomura, A Characterization of Symmetric Cones through Pseudoinverse Maps, J. Math. Soc. Japan 57 (2005), 195–215.

[5] C. Kai, T. Nomura, A characterization of symmetric tube domains by convexity of Cayley transform images, Diff. Geom. Appl. 23 (2005), 38–54.

[6] A. Kor´anyi, J. A. Wolf, Realization of Hermitian symmetric spaces as generalized half-planes, Ann. of Math. 81 (1965), 265–288.

[7] N. Mok, I-H. Tsai, Rigidity of convex realizations of irreducible bounded symmetric domains of rank 2, J. Reine Angew. Math. 431 (1992), 91–122. ≥ [8] T. Nomura, On Penney’s Cayley tranform of a homogeneous Siegel domain, J. Lie Theory 11 (2001), 185–206.

[9] T. Nomura, A characterization of symmetric Siegel domains through a Cayley trans- form, Transform. Groups 6 (2001), 227–260.

[10] T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, Diff. Geom. Appl. 15 (2001), 91–106.

[11] T. Nomura, Family of Cayley transforms of a homogeneous Siegel domain parametrized by admissible linear forms, Diff. Geom. Appl. 18 (2003), 55–78.

[12] T. Nomura, Geometric norm equality related to the harmonicity of the Poisson kernel for homogeneous Siegel domains, J. Funct. Anal. 198 (2003), 229–267.

[13] R. Penney, The Harish-Chandra realization for non-symmetric domains in Cn, in Topics in Geometry in Memory of Joseph D’Atri, Ed. by S. Gindikin, Birkhauser, Boston, 1996, 295–313.

[14] E. B. Vinberg, The theory of convex homogeneous cones, Tr. Mosk. Mat. Ob. 12 (1963), 303–358; Trans. Mosc. Math. Soc. 12 (1963), 340–403.

Partly supported by the Grant-in-Aid for JSPS Fellows, The Ministry of Education, Culture, Sports, Science and Technology, Japan.

Chifune Kai – Department of Mathematics, Faculty of Science, Kyoto University, Sakyo-ku 606-8502, Kyoto, Japan [email protected] The asymptotic expansion of Bergman kernels on symplectic manifolds

George Marinescu

1 Introduction

In this talk, we explain some ideas of our approach to the asymptotic expansion of the Bergman kernel associated to a line bundle. The basic philosophy developed in [8, 15, 18] is that the spectral gap properties for the operators proved in [3, 14] implies the existence of the asymptotic expansion for the corresponding Bergman kernels if the manifold X is compact or not, or singular, or with boundary, by using the analytic localization technique inspired by [2, 11]. The interested readers may find complete references in [8, 15, 17], § and in the forthcoming book [18]. We consider a compact complex manifold (X, J) with complex structure J, and holo- 0,q p n morphic vector bundles L, E on X, with rk L = 1. Let H (X, L E) q=0 be the { ⊗ } p 0, p L E Dolbeault cohomology groups of the Dolbeault complex (Ω •(X, L E), ∂ ⊗ ) := Lp E ⊗ ( Ω0,q(X, Lp E), ∂ ⊗ ). ⊕q ⊗ We fix Hermitian metrics hL, hE on L, E. Let L be the holomorphic Hermitian ∇ connection on (L, hL) with curvature RL and let gT X be a Riemannian metric on X such that

gT X (J , J ) = gT X ( , ). (1.1) · · · · Lp E, Lp E We denote by ∂ ⊗ ∗ the formal adjoint of the Dolbeault operator ∂ ⊗ on the Dol- 0, p 2 beault complex Ω •(X, L E) endowed with the L -scalar product associated to the L E T X ⊗ metrics h , h and g and the Riemannian volume form dvX (x0). Set

Lp E Lp E, Dp = √2 ∂ ⊗ + ∂ ⊗ ∗ . (1.2)

Then 1 D2 is the Kodaira-Laplacian acting on Ω0, (X, Lp
E) and preserves its Z-grading. 2 p • ⊗ By Hodge theory, we know that

2 0,q p Ker D 0,q = Ker D 0,q H (X, L E). (1.3) p|Ω p|Ω ' ⊗ We denote by P the orthogonal projection from Ω0, (X, Lp E) onto Ker D . The p • ⊗ p Bergman kernel P (x, x ), (x, x X) of Lp E is the smooth kernel of P with respect p 0 0 ∈ ⊗ p to the Riemannian volume form dvX (x0). In this setting, we are interested to understand the asymptotic expansion of Pp(x, x0) as p . If RL is positive, it is studied in [22, 20, 26, 7, 4, 21, 13, 23, 12] in various → ∞ generalities. Moreover, the coefficients in the diagonal asymptotic expansion encode geometric information about the underlying complex projective manifolds. This diagonal

79 80 George Marinescu

asymptotic expansion plays a crucial role in the recent work of Donaldson [10] where the existence of K¨ahler metrics with constant scalar curvature is shown to be closely related to Chow-Mumford stability. In the symplectic setting, Dai, Liu and Ma [8] studied the asymptotic expansion of the Bergman kernel of the spinc Dirac operator associated to a positive line bundle on compact symplectic manifolds, and related it to that of the corresponding heat kernel. This approach is inspired by local Index Theory, especially by the analytic localization techniques of Bismut-Lebeau [2, 11]. In [8] they also focused on the full off-diagonal § asymptotic expansion [8, Theorem 4.18] which is needed to study the Bergman kernel on orbifolds. By exhibiting the spectral gap properties of the corresponding operators, in [17], we also explained that without changing any step in the proof of [8, Theorem 4.18], the full off-diagonal asymptotic expansion still holds in the complex or symplectic setting if the curvature RL of L is only non-degenerate. Note that Berman and Sj¨ostrand [1] recently also studied the asymptotic expansion in the complex setting when the curvauture RL of L is only non-degenerate. 1 2 Along with 2 Dp there is another geometrically defined generalization to symplectic manifolds of the Kodaira-Laplace operator, namely the renormalized Bochner-Laplacian. In this talk, we explain the asymptotic expansion of the generalized Bergman kernels of the renormalized Bochner-Laplacian on high tensor powers of a positive line bundle on compact symplectic manifolds. In this situation the operators have small eigenvalues when the power p (the only small eigenvalue is zero in [8], thus we have the key → ∞ equation [8, (3.89)]) and we are interested in obtaining Theorem 8, that is, the near diagonal expansion of the generalized Bergman kernels. There are three steps: In Step 1, by using the finite propagation speed of solutions of hyperbolic equations, we can localize our problem if the spectral gap properties holds. In Step 2, we work on T X R2n, and extend the bundles and connections from a x0 ' neighborhood of 0 to all of Tx0 X such that the curvature of the line bundle L is uniformly non-degenerate on Tx0 X. We combine the Sobolev norm estimates from [8] and a formal power series method to obtain the asymptotic expansion. In Step 3 we compute the coefficients by using the formal power series method from Step 2. Actually, in [17], we compute also some coefficients in the asymptotic expansion of the Bergman kernel associated to the spinc Dirac operators in [8] by using the formal power series method here.

2 Main results

Let (X, ω) be a compact symplectic manifold of real dimension 2n. Assume that there exists a Hermitian line bundle L over X endowed with a Hermitian connection L with √ 1 L L L 2 L ∇ E the property that − R = ω, where R = ( ) is the curvature of (L, ). Let (E, h ) 2π ∇ ∇ be a Hermitian vector bundle on X with Hermitian connection E and its curvature RE. ∇ Let gT X be a Riemannian metric on X. Let T X be the Levi-Civita connection T X T X ∇ X on (T X, g ) with its curvature R and its scalar curvature r . Let dvX be the Riemannian volume form of (T X, gT X ). The scalar product on the space C (X, Lp E) ∞ ⊗ of smooth sections of Lp E is given by ⊗

s1, s2 = s1(x), s2(x) Lp E dvX (x) . h i h i ⊗ ZX The asymptotic expansion of Bergman kernels on symplectic manifolds 81

Let J : T X T X be the skew–adjoint linear map which satisfies the relation −→ ω(u, v) = gT X (Ju, v) (2.1) for u, v T X. Let J be an almost complex structure which is (separately) compatible ∈ with gT X and ω, especially, ω( , J ) defines a metric on T X. Then J commutes also with J. X · · T X Lp E Let J T ∗X End(T X) be the covariant derivative of J induced by . Let ⊗ ∇ ∈ ⊗ p L E ∇ ∇ be the connection on L E induced by and . Let ei i be an orthonormal frame ⊗ ∇ ∇ p { } p p of (T X, gT X ). Set X J 2 = ( X J)e 2. Let ∆L E = [( L E)2 L E ] ij ei j ⊗ i ei ⊗ T⊗X e |∇ | | ∇ | − ∇ − ∇∇ei i be the induced Bochner-Laplacian acting on C (X, Lp E). We fix a smooth hermitian P ∞ ⊗ P section Φ of End(E) on X. Set τ(x) = π Tr T X [JJ], and − | Lp E ∆ = ∆ ⊗ pτ + Φ. (2.2) p,Φ − By [14, Cor. 1.2] (cf. also [11, Theorem 2]) there exist µ0, CL > 0 independent of p such that the spectrum of ∆p,Φ satisfies Spec ∆ [ C , C ] [2pµ C , + [ . (2.3) p,Φ ⊂ − L L ∪ 0 − L ∞ This is the spectral gap property which plays an essential role in our approach. In the first place, it indicates a natural space of sections which replace the space of holomorphic sections from the complex case. Let P be the orthogonal projection from (C (X, Lp E), , ) onto the eigenspace 0,p ∞ ⊗ h · · i of ∆p,Φ with eigenvalues in [ CL, CL]. If the complex case (i.e. J is integrable and Φ = √ 1 E − − R (e , Je )) the interval [ C , C ] contains for p large enough only the eigenvalue − 2 j j − L L 0 whose eigenspace consists of holomorphic sections. For the computation of the spectral density function we need more general kernels. Namely, we define Pq,p(x, x0), q > 0 q 0 as the smooth kernels of the operators Pq,p = (∆p,Φ) P0,p (we set (∆p,Φ) = 1) with respect to dvX (x0). They are called the generalized Bergman kernels of the renormalized Bochner-Laplacian ∆ . Let det J be the determinant function of J End(T X). p,Φ x ∈ x Theorem 5. There exist smooth coefficients b (x) End(E) which are polynomials in q,r ∈ x RT X , RE (and RL, Φ) and their derivatives of order 6 2(r + q) 1 (resp. 2(r + q)), and − reciprocals of linear combinations of eigenvalues of J at x, and 1/2 b0,0 = (det J) IdE, (2.4) such that for any k, l N, there exists C > 0 such that for any x X, p N, ∈ k, l ∈ ∈ k 1 r k 1 Pq,p(x, x) bq,r(x)p− 6 Ck, l p− − . (2.5) pn − C l r=0 X Moreover, the expansion is uniform in that for any k, l N, there is an integer s such ∈ that if all data (gT X , hL, L, hE, E, J and Φ) run over a bounded set in the C s- norm T X ∇ ∇ T X l and g stays bounded below, the constant Ck, l is independent of g ; and the C -norm in (2.5) includes also the derivatives on the parameters. Theorem 6. If J = J, then for q > 1, 1 1 b = rX + X J 2 + 2√ 1RE(e , Je ) , (2.6) 0,1 8π 4|∇ | − j j 1h √ 1 q i b = X J 2 + − RE(e , Je ) + Φ . (2.7) q,0 24|∇ | 2 j j   82 George Marinescu

Theorem 5 for q = 0 and (2.6) generalize the results of [7], [26], [13] and [23] to the symplectic case. The term rX + 1 X J 2 in (2.6) is called the Hermitian scalar curvature 4 |∇ | in the literature and is a natural substitute for the Riemannian scalar curvature in the almost-K¨ahler case. It was used by Donaldson [9] to define the moment map on the space of compatible almost-complex structures. We can view (2.7) as an extension and refinement of the results of [11, 5] about the density of states function of ∆ , (2.7) § p,Φ implies also a correction of a formula in [6]. Now, we try to explain the near-diagonal expansion of Pq,p(x, x0). Let aX be the injectivity radius of (X, gT X ). We fix ε ]0, aX /4[. We denote by X TxX ∈ B (x, ε) and B (0, ε) the open balls in X and TxX with center x and radius ε. We identify BTxX (0, ε) with BX(x, ε) by using the exponential map of (X, gT X ). Tx0 X p We fix x0 X. For Z B (0, ε) we identify LZ , EZ and (L E)Z to Lx0 , Ex0 ∈ ∈ ⊗ p and (Lp E) by parallel transport with respect to the connections L, E and L E ⊗ x0 ∇ ∇ ∇ ⊗ along the curve γ : [0, 1] u expX (uZ). Then under our identification, P (Z, Z ) is Z 3 → x0 q,p 0 a section of End(E) on Z, Z T X, Z , Z ε, we denote it by P (Z, Z ). Let x0 0 ∈ x0 | | | 0| ≤ q,p,x0 0 π : T X T X X be the natural projection from the fiberwise product of T X on X. ×X → Then we can view P (Z, Z ) as a smooth section of π End(E) on T X T X (which q,p,x0 0 ∗ ×X is defined for Z , Z0 ε) by identifying a section S C ∞(T X X T X, π∗ End(E)) with | | | | ≤ C s ∈ × the family (Sx)x X . We denote by C s(X) a norm on it for the parameter x0 X. ∈ N | | ∈ We will define the function P (Z, Z 0) in (4.5). Theorem 7. There exist J (Z, Z ) End(E) polynomials in Z, Z with the same q,r 0 ∈ x0 0 parity as r and deg J (Z, Z ) 3r, whose coefficients are polynomials in RT X , RE q,r 0 ≤ ( and RL, Φ) and their derivatives of order 6 r 1 ( resp. r), and reciprocals of linear − combinations of eigenvalues of J at x0 , such that if we denote by N Fq,r(Z, Z 0) = Jq,r(Z, Z 0)P (Z, Z 0), (2.8) then for k, m, m N, σ > 0, there exists C > 0 such that if t ]0, 1], Z, Z T X, 0 ∈ ∈ 0 ∈ x0 Z , Z 0 6 σ/√p, | | | | k α + α0 ∂| | | | F r/2 sup α Pq,p(Z, Z 0) q,r(√pZ, √pZ 0)p− α 0 − C m0 (X) α + α0 6m ∂Z ∂Z 0 r=2q | | | |  X  (k m+1)/2 6 Cp− − . (2.9)

3 Idea of the proofs

3.1 Localization First, (2.3) and the finite propagation speed for hyperbolic equations, allows us to localize the problem. In particular, the asymptotics of P (x , x ) as p are localized on a q,p 0 0 → ∞ neighborhood of x . Thus we can translate our analysis from X to the manifold R2n 0 ' Tx0 X =: X0. Let f : R [0, 1] be a smooth even function such that f(v) = 1 for v 6 ε/2, and → | | f(v) = 0 for v > ε. Set | | + 1 + F (a) = ∞ f(v)dv − ∞ eivaf(v)dv. (3.1)  Z−∞  Z−∞ The asymptotic expansion of Bergman kernels on symplectic manifolds 83

Then F (a) is an even function and lies in the Schwartz space (R) and F (0) = 1. Let S F be the holomorphic function on C such that F (a2) = F (a). The restriction of F to R lies in the Schwartz space (R). Then there exists c such that for any k N, the S { j }j∞=1 ∈ functione e e k F (a) = F (a) c ajF (a), (3.2) k − j Xj=1 e e verifies (i) Fk (0) = 0 for any 0 < i 6 k. (3.3) Proposition 3.1. For any k, m N, there exists C > 0 such that for p > 1 ∈ k,m 1 k +2(2m+2n+1) Fk ∆p,Φ (x, x0) P0,p(x, x0) 6 Ck,mp− 2 . (3.4) √p − C m(X X) × m
L E L E T X Here the C norm is induced by , , h , h and g . ∇ ∇ Using (3.1), (3.2) and the finite propagation speed of solutions of hyperbolic equations, it is clear that for x, x X, F 1 ∆ (x, ) only depends on the restriction of ∆ 0 ∈ k √p p,Φ · p,Φ 1 1 1 X 4 > 4 to B (x, εp− ), and Fk √p ∆p,Φ(x, x0) =
0, if d(x, x0) εp− . This means that the q O C m asymptotic of ∆ P p (x, ) when p + , modulo (p−∞) (i.e. terms whose norm p,Φ H ·
→ ∞ l X 1 is O(p ) for any l, m N), only depends on the restriction of ∆ to B (x, εp− 4 ). − ∈ p,Φ 3.2 Uniform estimate of the generalized Bergman kernels We will work on the normal coordinate for x X. We identify the fibers of (L, hL), 0 ∈ E Lx0 Ex0 (E, h ) with (Lx0 , h ), (Ex0 , h ) respectively, in a neighborhood of x0, by using the parallel transport with respect to L, E along the radial direction. ∇ ∇ We then extend the bundles and connections from a neighborhood of 0 to all of T X. In particular, we can extend L (resp. E) to a Hermitian connection L0 on x0 ∇ ∇ ∇ L0 Lx0 E0 E0 Ex0 (L0, h ) = (X0 Lx0 , h ) (resp. on (E0, h ) = (X0 Ex0 , h ) ) on Tx0 X in × ∇ L0 × L0 L such a way so that we still have positive curvature R ; in addition R = Rx0 outside a T X0 compact set. We also extend the metric g , the almost complex structure J0, and the smooth section Φ , (resp. the connection E0 ) in such a way that they coincide with their 0 ∇ values at 0 (resp. the trivial connection) outside a compact set. Moreover, using a fixed unit vector S L and the above discussion, we construct an isometry E Lp E . L ∈ x0 0 ⊗ 0 ' x0 Let ∆X0 be the renormalized Bochner-Laplacian on X associated to the above data by p,Φ0 0 a formula analogous to (2.2). Then (2.3) still holds for ∆X0 with µ replaced by 4µ /5. p,Φ0 0 0 Tx0 X Let dvT X be the Riemannian volume form on (Tx0 X, g ) and κ(Z) be the smooth positive function defined by the equation dvX0 (Z) = κ(Z)dvT X (Z), with k(0) = 1. For s C (R2n, E ), Z R2n and t = 1/√p , set ∈ ∞ x0 ∈ 2 2 s 0 = s(Z) Ex0 dvT X (Z), k k R2n | |h Z and consider

1 1 1 2 X0 L = S− t κ 2 ∆ κ 2 S , where (S s)(Z) = s(Z/t) . (3.5) t t p,Φ0 − t t 84 George Marinescu

L Then t is a family of self-adjoint differential operators with coefficients in End(E)x0 . We denote by : (C (X , E ), ) (C (X , E ), ) the spectral projection P0,t ∞ 0 x0 k k0 → ∞ 0 x0 k k0 of L corresponding to the interval [ C t2, C t2]. Let (Z, Z ) = (Z, Z ), t − L0 L0 Pq,t 0 Pq,t,x0 0 (Z, Z X , q > 0) be the smooth kernel of = (L )q with respect to dv (Z ). 0 ∈ 0 Pq,t t P0,t T X 0 We can view (Z, Z ) as a smooth section of π End(E) over T X T X, where Pq,t,x 0 ∗ ×X π : T X T X X. Let δ be the counterclockwise oriented circle in C of center 0 and ×X → radius µ0/4. By (2.3),

1 q 1 = λ (λ L )− dλ. (3.6) Pq,t 2πi − t Zδ From (2.3) and (3.6) we can apply the techniques in [8], which are inspired by [2, 11], § to get the following key estimate. Theorem 8. There exist smooth sections F C (T X T X, π End(E)) such that q,r ∈ ∞ ×X ∗ for k, m, m N, σ > 0, there exists C > 0 such that if t ]0, 1], Z, Z T X, Z , Z 6 0 ∈ ∈ 0 ∈ x0 | | | 0| σ,

k α + α0 ∂| | | | r k sup q,t Fq,rt (Z, Z 0) 6 Ct . (3.7) α α0 C m α , α 6m ∂Z ∂Z 0 P − 0 (X) | | | 0|  r=0  X Let P (Z, Z ) End(E ) (Z, Z X ) be the analogue of P (x, x ). By (3.5), 0,q,p 0 ∈ x0 0 ∈ 0 q,p 0 for Z, Z R2n, 0 ∈ 2n 2q 1 1 P (Z, Z 0) = t− − κ− 2 (Z) (Z/t, Z 0/t) κ− 2 (Z 0). (3.8) 0, q, p Pq,t By Proposition 3.1, we know that O P0,q,p(Z, Z 0) = Pq,p,x0 (Z, Z 0) + (p−∞), (3.9) uniformly for Z, Z T X, Z , Z ε/2. 0 ∈ x0 | | | 0| ≤ To complete the proof the Theorem 2.1, we finally prove Fq,r = 0 for r < 2q. In fact, (3.7) and (3.8) yield

bq,r(x0) = Fq,2r+2q(0, 0). (3.10)

4 Evaluation of Fq,r

The almost complex structure J induces a splitting TRX R C = T (1,0)X T (0,1)X, where ⊗ ⊕ T (1,0)X and T (0,1)X are the eigenbundles of J corresponding to the eigenvalues √ 1 and n (1,0) − √ 1, respectively. We choose w to be an orthonormal basis of Tx X, such that − − { i}i=1 0 2π√ 1J = diag(a , , a ) End(T (1,0)X). (4.1) − − x0 1 · · · n ∈ x0 1 √ 1 We use the orthonormal basis e2j 1 = (wj +wj) and e2j = − (wj wj) , j = 1, . . . , n − √2 √2 − of Tx0 X to introduce the normal coordinates as in Section 3. In what follows we will use the complex coordinates z = (z , , z ), thus Z = z + z, and w = √2 ∂ , w = √2 ∂ . 1 n i ∂zi i ∂zi · · · + It is very useful to introduce the creation and annihilation operators bi, bi ,

∂ 1 + ∂ 1 bi = 2 + aizi , b = 2 + aizi , b = (b1, , bn) . (4.2) − ∂zi 2 i ∂zi 2 · · · The asymptotic expansion of Bergman kernels on symplectic manifolds 85

Now there are second order differential operators whose coefficients are polynomials Or in Z with coefficients being polynomials in RT X , Rdet, RE, RL and their derivatives at x0, such that

∞ L = L + tr, with L = b b+ . (4.3) t 0 Or 0 i i r=1 X Xi 2 2n Theorem 9. The spectrum of the restriction of L0 to L (R ) is given by n 2 α a : α N i i i ∈ i=1  X n and an orthogonal basis of the eigenspace of 2 i=1 αiai is given by 1 P bα zβ exp a z 2 , with β Nn . (4.4) −4 i| i| ∈ i X

Let N be the orthogonal space of N = Ker L in (L2(R2n, E ), ). Let P N , ⊥ 0 x0 k k0 N ⊥ 2 2n P be the orthogonal projections from L (R , Ex0 ) onto N, N ⊥, respectively. Let N N P (Z, Z 0) be the smooth kernel of the operator P with respect to dvT X (Z 0). From (4.4), we get

n N 1 1 2 2 P (Z, Z 0) = a exp a z + z0 2z z0 . (4.5) (2π)n i − 4 i | i| | i| − i i i=1 i Y  X
 Now for λ δ, we solve for the following formal power series on t, with g (λ) ∈ r ∈ End(L2(R2n, E ), N), f (λ) End(L2(R2n, E ), N ), x0 r⊥ ∈ x0 ⊥ L ∞ r (λ t) gr(λ) + f ⊥(λ) t = IdL2(R2n,E ) . (4.6) − r x0 r=0 X   From (3.6), (4.6), we claim that

1 q 1 q F = λ g (λ)dλ + λ f ⊥(λ)dλ. (4.7) q,r 2πi r 2πi r Zδ Zδ N N From Theorem 9, (4.7), the key observation that P 1P = 0, and the residue 1 N O N formula, we can get F by using the operators L − , P , P ⊥ , , (i 6 r). This gives q,r 0 Oi us a method to compute bq,r in view of Theorem 9 and (3.10). Especially, for q > 0, r < 2q, N F0,0 = P , Fq,r = 0, (4.8) N N N 1 N N q N F = (P P P L − P ⊥ P ) P , q,2q O2 − O1 0 O1 1 N 1 N N 1 N N F = L − P ⊥ L − P ⊥ P L − P ⊥ P 0,2 0 O1 0 O1 − 0 O2 N 1 N 1 N N 1 N + P L − P ⊥ L − P ⊥ P L − P ⊥ O1 0 O1 0 − O2 0 N 1 N 1 N N 2 N N + P ⊥ L − P L − P ⊥ P L − P ⊥ P . 0 O1 O1 0 − O1 0 O1 In fact L and are formal adjoints with respect to ; thus in F we only need 0 Or k k0 0,2 to compute the first two terms, as the last two terms are their adjoints. This simplifies the computation in Theorem 6. 86 George Marinescu

5 Generalizations to non-compact manifolds

In this section we come back to the case of complex manifolds, which was briefly discussed in the introduction, but focus on non-compact manifolds. Let (X, Θ) be a Hermitian manifold of dimension n, where Θ is the (1, 1) form associated to a hermitian metric on X. Given a Hermitian holomorphic bundles L and E on X with rk L = 1, we consider the space of L2 holomorphic sections H 0 (X, Lp E). Let P be the orthogonal projection (2) ⊗ p from the space L2(X, Lp E) of L2 sections of Lp E onto H0 (X, Lp E). By generalizing ⊗ ⊗ (2) ⊗ the definition from Section 1, we define the Bergman kernel P (x, x ), (x, x X) to be the p 0 0 ∈ Schwartz kernel of Pp with respect to the Riemannian volume form dvX (x0) associated to (X, Θ). By the ellipticity of the Kodaira-Laplacian and Schwartz kernel theorem, we know P (x, x ) is C . Choose an orthonormal basis (Sp)dp (d N ) of H0 (X, Lp E). p 0 ∞ i i=1 p ∈ ∪{∞} (2) ⊗ The Bergman kernel can then be expressed as

dp p p p p Pp(x, x0) = S (x) (S (x0))∗ (L E)x (L E)∗ . i ⊗ i ∈ ⊗ ⊗ ⊗ x0 Xi=1 (1,0) det Let KX = det(T ∗ X) be the canonical line bundle of X and R be the curvature of KX∗ relative to the metric induced by Θ. The line bundle L is supposed to be positive √ 1 L and we set ω = 2−π R . T X X We denote by gω the Riemannian metric associated to ω and by rω the scalar T X curvature of gω . Moreover, let α1, . . . , αn be the eigenvalues of ω with respect to Θ T X (αj = aj/(2π), j = 1, . . . , n where a1, . . . , an are defined by (4.1) and (2.1) with g the Riemannian metric associated to Θ). The torsion of Θ is T = [i(Θ), ∂Θ], where i(Θ) = (Θ ) is the interior multiplication with Θ. ∧ · ∗ Theorem 10 ([15]). Assume that (X, Θ) is a complete Hermitian manifold of dimension n. Suppose that there exist ε > 0 , C > 0 such that

√ 1RL > εΘ , √ 1Rdet > CΘ , √ 1RE > CΘ , T 6 CΘ . (5.1) − − − − − | |

Then the kernel Pp(x, x0) has a full off–diagonal asymptotic expansion uniformly on com- pact sets of X X and P (x, x) has an asymptotic expansion analogous to (2.5) uniformly × p on compact sets of X. Moreover, b = α α Id and 0 1 · · · n E n α α b = 1 · · · n rX Id 2∆ log(α α ) Id +4 RE(w , w ) , 1 8π ω E − ω 1 · · · n E ω,j ω,j h   Xj=1 i where w is an orthonormal basis of (T (1,0)X, gT X ). { ω,j} ω By full off-diagonal expansion we mean an expansion analogous to (2.9) where we allow Z , Z 6 σ. | | | 0| Let us remark that if L = KX , the first two conditions in (5.1) are to be replaced by

hL is induced by Θ and √ 1Rdet < εΘ. (5.2) − − Moreover, if (X, Θ) is K¨ahler, the condition on the torsion T is trivially satisfied. The asymptotic expansion of Bergman kernels on symplectic manifolds 87

1 2 The proof is based on the observation that the Kodaira-Laplacian 2p = 2 Dp acting on L2(X, Lp E) has a spectral gap as in (2.3). The proof of Theorem 5 applies then ⊗ and delivers the result. Theorem 10 has several applications e.g. holomorphic Morse inequalities on non- compact manifolds (as the well-known results of Nadel-Tsuji [19], see also [15, 25]) or Berezin-Toeplitz quantization (see [18] or the fothcommimg [16]). We will emphazise in the sequel the Bergman kernel for a singular metric. Let X be a compact complex manifold. A singular K¨ahler metric on X is a closed, strictly positive (1, 1)-current ω. If the cohomology class of ω in H 2(X, R) is integral, there exists a holomorphic line bundle (L, hL), endowed with a singular Hermitian metric, such that √ 1 L L 2−π R = ω in the sense of currents. We call (L, h ) a singular polarization of ω. If we change the metric hL, the curvature of the new metric will be in the same cohomology class as ω. In this case we speak of a polarization of [ω] H 2(X, R). Our ∈ purpose is to define an appropriate notion of polarized section of Lp, possibly by changing the metric of L, and study the associated Bergman kernel. Corollary 11. Let (X, ω) be a compact complex manifold with a singular K¨ahler metric with integral cohomology class. Let (L, hL) be a singular polarization of [ω] with strictly positive curvature current having singular support along a proper analytic set Σ . Then the Bergman kernel of the space of polarized sections

Lp H0 (X r Σ, Lp) = u L0,0(X r Σ, Lp , Θ , hL) : ∂ u = 0 (2) ∈ 2 P ε has the asymptotic expansion asin Theorem 10 for X r Σ, where ΘP is a generalized L Poincar´e metric on X r Σ and hε is a modified Hermitian metric on L. Using an idea of Takayama [24], Corollary 11 gives a proof of the Shiffman-Ji-Bonavero- Takayama criterion, about the characterization of Moishezon manifolds by (1, 1) positive currents. We mention further the Berezin-Toeplitz quantization. Assume that X is a complex C manifold and let const∞ (X) denote the algebra of smooth functions of X which are constant outside a compact set. For any f C (X) we denote for simplicity the operator of ∈ const∞ multiplication with f still by f and consider the linear operator

T : L2(X, Lp) L2(X, Lp) , T = P f P . (5.3) f,p −→ f,p p p

The family (Tf,p)p>1 is called a Toeplitz operator. The following result generalizes [5] to non-compact manifolds. Corollary 12. We assume that (X, Θ) and (L, hL) satisfy the same hypothesis as in Theorem 10 or (5.2). Let f, g C (X). The product of the two corresponding Toeplitz ∈ const∞ operators admits the asymptotic expansion

∞ r O Tf,pTg,p = p− TCr(f,g),p + (p−∞) (5.4) r=0 X where Cr are differential operators. More precisely, 1 C0(f, g) = fg , C1(f, g) C1(g, f) = f, g (5.5) − √ 1{ } − 88 George Marinescu

where the Poisson bracket is taken with respect to the metric 2πω. Therefore 1 O 2 [Tf,p, Tg,p] = p− T 1 f,g ,p + (p− ). (5.6) √ 1 − { } Remark 13. For any f C (X, End(E)) we can consider the linear operator ∈ ∞ T : L2(X, Lp E) L2(X, Lp E) , T = P f P . (5.7) f,p ⊗ −→ ⊗ f,p p p Then (5.4) holds for any f, g C (X, End(E)) which are constant outside some compact ∈ ∞ set. Moreover, (5.5), (5.6) still hold for f, g C (X) C (X, End(E)). ∈ const∞ ⊂ ∞ References

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Xiaonan Ma – Centre de Math´ematiques Laurent Schwartz, UMR 7640 du CNRS, Ecole Polytechnique, 91128 Palaiseau Cedex, France [email protected]

George Marinescu – Fachbereich Mathematik, Johann Wolfgang Goethe-Universitat,¨ Robert- Mayer-Straße 10, 60054, Frankfurt am Main, Germany [email protected] Hua operators and Poisson transform for non-tube bounded symmetric domains Khalid Koufany and Genkai Zhang

1 Introduction

The purpose of the present note is to give an overview of the main results obtained in [9]. Due to the limitation of the space we will be rather brief and descriptive.

Suppose Ω is a bounded symmetric domain in a complex n dimensional space V . Let − S be its Shilov boundary and r its rank. We consider the characterization of the image of the Poisson transform (s C) on the Shilov boundary S. For a specific value of Ps ∈ s (s = 1 in our parameterization) the corresponding Poisson transform := maps P P1 hyperfunctions on S to harmonic functions on Ω. When Ω is a tube domain Johnson and Kor´anyi [7] proved that the image of the Poisson transform is exactly the set of all P Hua-harmonic functions. Lassalle [10] has shown that the Hua system of Johnson and Kor´anyi can be replaced by and equivalent system containing fewer equations. For non- tube domains the characterization of the image of the Poisson transform was done by P Berline and Vergne [1] where certain third-order differential Hua-operator was introduced to characterize the image. In his paper [15] Shimeno considered the Poisson transform Ps on tube domains, it is proved that Poisson transform maps hyperfunctions on the Shilov boundary to certain solution space of the Hua operator. For the Shilov boundary of a non-tube domain the problem is still open. We will construct two Hua operators of third order and use them to give a characterization.

2 Notations

Let Ω = G/K be a bounded symmetric domain of non-tube type in a complex n dimensional − vector space V . Let g = k p be a Cartan decomposition of the Lie algebra of G. It is ⊕ known that V carries a unique Jordan triple structure V V¯ V V : (u, v¯, w) × × → 7→ uv¯w such that p = ξ : v V where ξ (z) = v Q(z)v¯ and Q(z)v¯ = zv¯z . { } { v ∈ } v − { } Let (r, a, b) be the characteristic parameters of the bounded symmetric domain Ω, then r(r 1) r n = rb + r + − a. Fix a Jordan frame c , then a = Rξ is a maximal 2 { j}j=1 ⊕ cj Abelian subspace of p. The restricted roots system Σ = Σ(g, a) consists of the roots 1 1 βj (1 j r) with multiplicity 1, the roots 2 βj 2 βk (1 j = k r) with  ≤ ≤ 1   ≤ 6 ≤ multiplicity a, and the roots 2 βj (1 j r) with multiplicity 2b. The half sum of  r ≤ ≤ b+1+a(j 1) the positive roots, is given by ρ = j=1 ρjβj, where ρj = 2 − , j = 1, . . . , r. Let

Keywords: Bounded symmetric domains,P Shilov boundary, invariant differential operators, eigen- functions, Poisson transform, Hua systems

90 Hua operators on non-tube bounded symmetric domains 91

β n = g and let m = Zk(a) be the centralizer of a in k. Let M, A and N be  β Σ the analytic∈ subgroups of G of Lie algebras m, a and n respectively. The subgroup P − P = Pmin = MAN is parabolic subgroup of G and maximal boundary (the Furstenberg boundary) G/P of Ω can be viewed as K/M. Recall that e = c1 + c2 + . . . + cr is a maximal tripotent of V and the G orbit S = G e is the minimal boundary (the Shilov − · boundary) of Ω.

3 The Poisson transform

Let (Ω)G be the algebra of all invariant differential operators on Ω. Recall the definition D of the Harish-Chandra eλ function : eλ, for λ a∗C is the unique N invariant function r− 2 Pr ∈t (λ +ρ ) − on Ω such that e (exp t ξ 0) = e j=1 j j j . Then e are the eigenfunctions λ j=1 j cj · λ of T (Ω)G and we denote χ (T ) the corresponding eigenvalues. Denote further ∈ D P λ (Ω, λ) = f C (Ω); T f = χ (T )f, T (Ω)G . M { ∈ ∞ λ ∈ D } Corresponding to the minimal parabolic subgroup P there is the Poisson transform on the maximal boundary G/P = K/M. For λ aaC, the Poisson transform is ∈ ∗ Pλ,K/M defined by

1 f(gK) = e (k− g)f(k)dk Pλ,K/M λ ZK on the space (K/M) of hyperfunctions on K/M. B λ,α It is proved by Kashiwara et al. in [8] that for λ aC, if 2 h i / 1, 2, 3, . . . for ∈ ∗ − α,α ∈ { } all α Σ+(g, a), then the Poisson transform is a G-isomorphismh i from (K/M) ∈ Pλ,K/M B onto (Ω, λ). M We now introduce the Poisson transform on the Shilov boundary. Let h(z) be the unique K invariant polynomial on V whose restriction to r Rc is given by − ⊕j=1 j h( r t c ) = r (1 t2). As h is real-valued, we may polarize it to get a polynomial j=1 j j j=1 − j on V V , denoted by h(z, w), holomorphic in z and anti-holomorphic in w such that P × Q n h(z, z) = h(z). The Poisson kernel P (z, u) on Ω S is P (z, u) = h(z, z)/ h(z, u) 2 r . × | | For a complex number s we define the Poisson transform s is defined by  P

( ϕ)(z) = P (z, u)sϕ(u)dσ(u) Ps ZS on the space (S) of hyperfunctions on S. B The kernel P (z, u)s, for u = e is a special case of the e function. The Poisson λ− transform on S can be viewed as a restriction of the Poisson transform . For Ps Pλ,K/M s C, we choose a corresponding λ = λ that satisfies the (non-integer) condition of the ∈ s Kashiwara et al. theorem. More precisely we let λs = ρ+n(s 1)ξc∗ where ξc∗ a such that r − ∈ ξc∗(ξc) = 1 and ξc∗(ξc⊥) = 0 with ξc = j=1 ξcj . Then we have an equivalent (non-integer) condition ([9, (1)]) for s. Moreover, f(z) = f(z) where f on S is viewed as PPs Pλs,K/M a function on K and thus on K/M. Thus (S) (K/M) (Ω, λ ) and PsB ⊂ Pλs,K/M B ⊂ M s when s satisfies ([9, (1)]), (K/M) = (Ω, λ ). Pλs,K/M B M s 92 Khalid Koufany and Genkai Zhang

4 Hua operators

We define some Hua operators of second and third orders by using the covariant Cauchy- Riemann operator studied in [2], [17] and [19]. Recall briefly that the covariant Cauchy- Riemann operator can be defined on any holomorphic Hermitian vector bundle over a K¨ahler manifold. Trivializing the sections of a homogeneous vector bundle E on the bounded symmetric space Ω as the space C ∞(Ω, E) of E-valued functions on Ω, where E is a holomorphic representation of KC, the covariant Cauchy-Riemann operator D¯ is defined by D¯ f = b(z, z¯)∂¯f, (4.1) where b(z, z¯) is the inverse of the Bergman metric, also called Bergman operator of Ω, given by b(z, w¯) = 1 D(z, w¯)+Q(z)Q(w¯). The operator D¯ maps C (Ω, E) to C (Ω, V − ∞ ∞ ⊗ E), with V viewed as the holomorphic tangent space.

4.1 The second order Hua operator

Consider the space C∞(Ω) of C∞-functions on Ω as the sections of the trivial line bundle. The operator ∂ is then well-defined on C ∞(Ω) and it maps C∞(Ω) to C∞(Ω, V 0) = C∞(Ω, V¯ ) with the later identified as the space of sections of the holomorphic cotangent bundle. We can then define the differential operator ¯ ¯ AdV V¯ (D ∂) : C∞(Ω) C∞(Ω, kC), f AdV V¯ (D ∂f) ⊗ ⊗ → 7→ ⊗ ⊗ ¯ + with AdV V¯ : V V = p p− kC being the Lie bracket, u v D(u, v). So ⊗ ⊗ ⊗ → ⊗ → by the covariant property of ∂ and D¯ (see [19]) we have Ad ¯ (D¯ ∂)(f(gz)) = V V ⊗ 1 ¯ (1,0) (1,0)⊗ dg(z)− AdV V¯ (D ∂f)(gz), where dg(z) : V = Tz Tgz is the differential of ⊗ ⊗ → the mapping g, which further is dg(z) = Ad(dg(z)) the adjoint action of dg(z) KC on ∈ kC. It follows easily that this operator agree with the Hua operator introduced by H Johsnon and Kor`anyi [7], ¯ AdV V¯ (D ∂) = . ⊗ ⊗ H Symbolically we may write = D(b(z, z¯)∂¯, ∂). H 4.2 Third-order Hua operators Let again E be a homogeneous holomorphic vector bundle on Ω. On E there is a Her- mitian structure defined by using the Bergman operator b(z, z¯) as an element in KC and thus there exists a unique Hermitian connection : C (Ω, E) C (Ω, T E), com- ∇ ∞ → ∞ 0 ⊗ patible with the complex structure of the cotangent bundle T 0. Under the decomposition (1,0) (0,1) (1,0) T = (T )z + (T )z we have = + ∂¯ with z0 0 0 ∇ ∇ (1,0) (1,0) : C∞(Ω, E) C∞(Ω, (T 0) E) = C∞(Ω, p− E), ∇ → ⊗ ⊗ (1,0) using our identification that (T 0)z = p−. Note that on the space C ∞(Ω) of sections of the trivial bundle we have (1,0) = ∂. ∇ We now define two covariant third-order Hua operators and on C (Ω, E) by W U ∞ ¯ ¯ (1,0) f = Adp+ kC D(Adp+ p (D f)) , W ⊗ ⊗ − ∇  (1,0) ¯ ¯  f = AdkC p+ Adp p+ ( D)Df . U ⊗ −⊗ ∇   Hua operators on non-tube bounded symmetric domains 93

These operators can also be defined by using the enveloping algebra (see [9, Section 7]). For our purpose we consider E to be the trivial representation and the space C ∞(Ω) of smooth functions identified as right K-invariant functions on G. The third-order Hua operator defined by Berline and Vergne in [1] can be viewed, in our context, up to some non-zero constant as

(1,0) ¯ (1,0) = Adp kC p Adp+ p kC (D ) . V −⊗ → − ∇ ⊗ −→ ∇   So it is different from our and . For explicit computations the operators and W U W U are somewhat easier to handle as the operator D¯ has a rather explicit formula (4.1) on different holomophic bundles [2], whereas the formula for (1,0) depends on the metric ∇ on the bundles [19]. Note also that the first (1,0) and the second (1,0) in are different ∇ ∇ V as they are acting on different bundles.

5 Main results

The Hua operator of second-order for a general symmetric domain is defined as a H kC-valued operator. For tube domains it maps the Poisson kernels into the centre of kC, namely the Poisson kernels are its eigenfunctions up to an element in the center Z0, but it is not true for non-tube domains. More precisely we have : Theorem 5.1 ([9, Theorem 5.3]). For u fixed in S, the function z P (z, u)s satisfies 7→ the following differential equation n n P (z, u)s = P (z, u)s ( s)2D(b(z, z¯)(zz¯ uz¯), z¯z u¯z) ( sp)Z , H r − − − r 0 where p is the genus of Ω and wher e xy denotes the quasi-inverse of x with respect to y.

However for type I = Ir,r+b domains of non-tube type, there is a variant of the Hua operator, (1) see [9, Section 6.], by taking the first component of the operator, H (1) (2) since in this case kC = kC + kC is a sum of two irreducible ideals. We prove that the operator (1) has the Poisson kernels as its eigenfunctions and we find the eigenvalues. H We prove further that the eigenfunctions of the Hua operator (1) are also eigenfunctions H of invariant differential operators on Ω. For that purpose we compute the radial part of the Hua operator (1), see [9, Proposition 6.3]. We give eventually the characterization of H the image of the Poisson transform in terms of the Hua operator for type Ir,r+b domains :

Theorem 5.2 ([9, Theorem 6.1]). Suppose s C satisfies the following condition ∈ 4[b + 1 + j + (r + b)(s 1)] / 1, 2, 3, , for j = 0 and 1. − − ∈ { · · · } A smooth function f on I is the Poisson transform (ϕ) of a hyperfunction ϕ on S r,r+b Ps if and only if (1)f = (r + b)2s(s 1)fI . H − r Our method of proving the characterization is the same as that in [10] by proving that the boundary value of the Hua eigenfunctions satisfy certain differential equations and are thus defined only on the Shilov boundary, nevertheless it requires several technically 94 Khalid Koufany and Genkai Zhang

demanding computations. In [9, Section 7] we study the characterization of range of the Poisson transform for general non-tube domains using the third-order Hua-type operators and : U W Theorem 5.3 ([9, Theorem 7.2]). Let Ω be a bounded symmetric non-tube domain of rank r in Cn. Let s C and put σ = n s. If a smooth function f on Ω is the Poisson ∈ r transform of a hyperfunction in (S), then Ps B 2σ2 + 2pσ + c − f = 0. (5.1) U − σ(2σ p b) W  − −  Conversely, suppose s satisfies the condition a n 4[b + 1 + j + (s 1)] / 1, 2, 3, , for j = 0 and 1. − 2 r − ∈ { · · · } Let f be an eigenfunction f (Ω, λ ). If f satisfies (5.1) then it is the Poisson ∈ M s transform (ϕ) of a hyperfunction ϕ on S. Ps After this paper was finished we were informed by Professor T. Oshima that he and N. Shimeno have obtained some similar results about Poisson transforms and Hua operators. The second author would like to thank the organizers of the workshop “Complex Analysis, Operator Theory and Mathematical Physics” at Erwin Schr¨ordinger institute for the invitation and to thank the institute for its support and hospitality.

References

[1] Berline, N.; Vergne, M. Equations´ de Hua et noyau de Poisson. Noncommutative harmonic analysis and Lie groups (Marseille, 1980), 1–51, Lecture Notes in Math., 880, Springer, Berlin-New York, 1981. [2] Englis, M and Peetre, J. Covariant Laplacean operators on K¨ahler manifolds, J. Reine und Angew. Math. 478 (1996), 17–56. [3] Faraut, J.; Kor´anyi, A. Analysis on Symmetric Cones, Oxford Mathematical Mono- graphs, Clarendon Press, Oxford, 1994. [4] Helgason, S. A duality for symmetric spaces with applications to group representa- tions. Advances in Math. 5 (1970), 1–154. [5] Helgason, S. Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions. Pure and Applied Mathematics, 113. Academic Press, Inc., Orlando, FL, 1984. [6] Hua, L. K. Harmonic analysis of functions of several complex variables in the clas- sical domains. American Mathematical Society, Providence, R.I. 1963 iv+164 pp [7] Johnson, K.; Kor´anyi, A. The Hua operators on bounded symmetric domains of tube type. Ann. of Math. (2) 111 (1980), no. 3, 589–608. [8] Kashiwara, M.; Kowata, A.; Minemura, K.; Okamoto, K.; Oshima, T.; Tanaka, M. Eigenfunctions of invariant differential operators on a symmetric space. Ann. of Math. (2) 107 (1978), no. 1, 1–39. Hua operators on non-tube bounded symmetric domains 95

[9] Koufany, K.; Zhang G., Hua operators and Poisson transform for bounded sym- metric domains. Preprint iecn 2005/21. Submitted.

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Research partly supported by European IHP network Harmonic Analysis and Related Problems. Research by Genkai Zhang supported by Swedish Science Council (VR)

Khalid Koufany – Institut Elie´ Cartan, Universit´e Henri Poincar´e, F-54506 Vandœuvre-l`es- Nancy cedex, France [email protected]

Genkai Zhang – Department of Mathematics, CTH/GU, S-41296 G¨oteborg, Sweden [email protected] Unitarizability of holomorphically induced representations of a split solvable Lie group

Hideyuki Ishi

1 Introduction

Let G be a connected and simply connected split solvable Lie group with the Lie algebra g, and h gC a totally complex positive polarization at a linear form λ on g. In this article, ⊂ we consider the unitarizability of a representation of G holomorphically induced from h. The subject is an analogue of the unitarizability of the highest weight representations of a Hermitian Lie group ([4], [11], [16], [17]). We state the settings more precisely. The stabilizer

G := g G ; Ad∗(g)λ = λ λ { ∈ } at λ equals the subgroup exp(h g) of G. Let ν be a unitary character of G given ∩ λ λ by ν (exp X) := ei X,λ (X h g). For X g and φ C (G), we define R(X)φ λ h i ∈ ∩ ∈ ∈ ∞ ∈ C (G) by R(X)φ(a) := d φ(a exp tX) (a G), and for X , X g, define R(X + ∞ dt t=0 ∈ 1 2 ∈ 1 iX )φ := R(X )φ + iR(X )φ. We denote by (G, h, λ) the space of smooth functions 2 1 2
C φ on G satisfying the conditions (C1) φ(xh) = ν (h) 1φ(x) (x G, h G ), and λ − ∈ ∈ λ (C2) R(Z)φ = i Z, λ φ (Z h), where λ is extended complex linearly to gC. Our − h i ∈ holomorphically induced representation τ is defined on (G, h, λ) by the left translation: λ C τ (g)φ(x) := φ(g 1x) (g, x G). Put λ − ∈

2(G, h, λ) := φ (G, h, λ) ; φ 2 := φ(g) 2 dg˙ < + , H ∈ C k k | | ∞ ( ZG/Gλ ) where dg˙ denotes an invariant measure on the coset space G/G . If 2(G, h, λ) is non- λ H trivial, 2(G, h, λ) is a G-invariant Hilbert space and the representation (τ , 2(G, h, λ)) H λ H is unitary. Moreover, 2(G, h, λ) has a reproducing kernel and (τ , 2(G, h, λ)) is irre- H λ H ducible. The non-vanishing condition of 2(G, h, λ) is given by Fujiwara ([5], [6]). Now H we note that, even if 2(G, h, λ) = 0 , there may still exist a non-zero G-invariant H { } subspace (G, h, λ) of (G, h, λ) with a reproducing kernel Hilbert space structure such H C that (τ , (G, h, λ)) is a unitary representation of G. The representation (τ , (G, h, λ)) λ H λ H is necessarily irreducible by [12]. We shall describe the condition for the existence of such (G, h, λ) (Theorem 15), and determine the coadjoint orbit in b corresponding to H ∗ the representation (τ , (G, h, λ)) by the Kirillov-Bernat correspondence [2]. In a certain λ H case, we realize the Hilbert space as a space of holomorphic functions on a Siegel domain (section 4).

96 Unitarizability of holomorphically induced representations 97

2 Normal j-algebras and Siegel domains

Since our study is based on the theory of normal j-algebras established by Piatetskii- Shapiro, we first review his results [14, Chapter 2, Sections 3 and 5] briefly. Let b be a 2 split solvable Lie algebra, j : b b a linear map such that j = idb, and ω a linear → − form on b. The triple (b, j, ω) is called a normal j-algebra if the following are satisfied: (NJA1) [Y , Y ] + j[jY , Y ] + j[Y , jY ] [jY , jY ] = 0 for all Y , Y b, and (NJA2) the 1 2 1 2 1 2 − 1 2 1 2 ∈ bilinear form (Y Y ) := [Y , jY ], ω (Y , Y b) gives a j-invariant inner product on 1| 2 ω h 1 2 i 1 2 ∈ b. Let a be the orthogonal complement of the subspace [b, b] b with respect to ( ) . ⊂ ·|· ω Then a is a commutative subalgebra of b. Put r := dim a, and for a linear form α a , ∈ ∗ set b := Y b; [C, Y ] = C, α Y (C a) . α { ∈ h i ∈ } Proposition 2.1 (Piatetskii-Shapiro). (i) There is a linear basis A , . . . , A of a { 1 r} such that if one puts E := jA , then [A , E ] = δ E (k, l = 1, . . . , r). l − l k l kl l (ii) Let α1, . . . , αr be the basis of a∗ dual to A1, . . . , Ar. Then one has a decomposition b = b(1) b(1/2) b(0) with ⊕ ⊕ r r b(1) := ⊕ RE ⊕ b , b(1/2) := ⊕ b , k ⊕ (αm+αk)/2 αk/2 k=1 1 k 1. (iv) ⊂ { } One has jb(αm α )/2 = b(αm+α )/2 (1 k < m r) and jbα /2 = bα /2 (k = 1, . . . , r). − k k ≤ ≤ k k Let B be the connected and simply connected Lie group corresponding to b. The group B is realized as an affine transformation group acting on a Siegel domain D constructed as follows. The subgroup B(0) := exp b(0) of B acts on b(1) by the adjoint action because of Proposition 2.1 (iii). Let Ω be the B(0)-orbit through E := E + + E in 1 · · · r b(1). Then Ω is a regular open convex cone on which B(0) acts simply transitively. By Proposition 2.1 (iv), the operator j defines a complex structure on b(1/2). We define a b(1)C-valued Hermitian map Q on (b(1/2), j) by Q(u, u ) := ([ju, u ] + i[u, u ])/4 (u, u 0 0 0 0 ∈ b(1/2)). Our Siegel domain D is a complex domain in b(1)C (b(1/2), j) defined by × D := (z, u) ; z Q(u, u) Ω . The group B acts on D simply transitively by { = − ∈ } exp(x + u )h (z, u) 0 0 0 · := (Ad(h0)z + x0 + 2iQ(Ad(h0)u, u0) + iQ(u0, u0), Ad(h0)u + u0) (x b(1), u b(1/2), h B(0), (z, u) D). 0 ∈ 0 ∈ 0 ∈ ∈ Let b be the subspace Y + ijY ; Y b of bC. Then b is a totally complex − { ∈ } − positive polarization at ω b ([15]). Let b be the orthogonal complement of the one − ∈ ∗ 0 dimensional subspace RAr in b, and ω0 the restriction of ω to b0. Put b0 := CEr (b0C − ⊕ ∩ b ) b0C. Then b0 is a totally complex positive polarization at ω0. − ⊂ − − 3 Non-vanishing condition of (G, h, λ) H Now recall the polarization h gC at λ g in section 1. ⊂ ∈ ∗ 98 Hideyuki Ishi

Theorem 14. For the triple (g, h, λ), there exists a normal j-algebra (b, j, ω) and a Lie algebra homomorphism $ : gC bC satisfying either → $(g) = b, $(h) = b , λ = ω $, (3.1) − − ◦ or $(g) = b0, $(h) = b0 , λ = ω0 $. (3.2) − − ◦ This (b, j, ω) is unique up to isomorphisms. Let $˜ : G B the Lie group homomorphism given by $˜ (exp X) := exp $(X) (X → ∈ g). The pull back $˜ ∗ induces an isomorphism from either (B, b , ω) or (B0, b0 , ω0) C − − C − − onto (G, h, λ), where B is the subgroup exp b of B. Thus the existence of a non- C 0 0 zero (G, h, λ) is equivalent of the one of (B, b , ω) or (B0, b0 , ω0), so that we H H − − H − − can describe the non-vanishing condition of (G, h, λ) in terms of the structure of the H normal j-algebra (b, j, ω). For ε = (ε , . . . , ε ) 0, 1 r with ε = 1, we set q (ε) := 1 r ∈ { } r k m>k εm dim b(αm αk)/2 (k = 1, . . . , r) and − P (ε) := s Rr ; s > q (ε)/4 (if ε = 1), s = q (ε)/4 (if ε = 0) . X { ∈ k k k k k k } Let be the disjoint union (ε). X ε X Theorem 15. Put γ := ωF, E (k = 1, . . . , r). Then a non-zero (G, h, λ) exists if k h ki H and only if γ := (γ , . . . , γ ) belongs to . 1 r X r Thanks to Proposition 2.1, any element Y of b is expressed as Y = k=1(ckAk + x E ) + Y with c , x R and Y (a ja) . Then we have Y, ω = r (β c + kk k 0 k kk ∈ 0 ∈ ⊕ ⊥ h i P k=1 k k γ x ), where β := A , ω (k = 1, . . . , r). k kk k h k i P Theorem 16. If γ belongs to (ε), define ω b by Y, ω := r (β c + ε γ x ). X ε ∈ ∗ h εi k=1 k k k k kk Then the irreducible unitary representation (τ , (G, h, λ)) corresponds to the coadjoint λ H P orbit through λ := ω $ g by the Kirillov-Bernat correspondence. ε − ε ◦ ∈ ∗ We remark that λ belongs to the boundary of the coadjoint orbit Ad(G) λ g ε ∗ ⊂ ∗ unless ε = (1, . . . , 1).

4 Function spaces on the Siegel domain D

Rossi and Vergne [15] observed that the spaces (B, b , ω) and 2(B, b , ω) are C − − H − − realized as spaces of holomorphic functions on the Siegel domain D (see also [6, Section 5B]). Thanks to this fact, Theorems 15 and 16 are deduced from results about analysis on Siegel domains [10] in the case (3.1) in Theorem 14. In this section, we give a similar realization of (B0, b0 , ω) and (B0, b0 , ω) as function spaces on D. The results play C − − H − − substantial roles in the study of the case (3.2). We define a one-dimensional representation χ : B C of B in such a way that ω → χ (exp C) = ei C+ijC,ω (C a). Set κ := E , ω . We denote by (D; κ) the space of ω h i ∈ h r i O holomorphic functions F on D with the property F (z + cE , u) = eiκcF (z, u) ((z, u) r ∈ D, c R). For F (D; κ), define a function Ψ F on the group B by Ψ F (b) := ∈ ∈ O ω 0 ω χ (b)F (b p ) (b B ), where p := (iE, 0) D. ω · 0 ∈ 0 0 ∈ Lemma 4.1. The map Ψω gives an isomorphism from (D; κ) onto (B0, b0 , ω0). O C − − Unitarizability of holomorphically induced representations 99

Recalling Theorem 15, we assume that γ (ε). Let (D) be a subspace of (D; κ) ∈ X Hω O with a Hilbert space structure such that Ψ gives a unitary isomorphism from (D) ω Hω onto (B0, b0 , ω0). Put B0(0) := B(0) B0. Then each x Ω is uniquely expressed as H − − ∩ ∈ x = Ad(h)E + cE with h B (0) and c > 1. Let Υ be a function on Ω defined by r ∈ 0 − ω Υ (Ad(h)E + cE ) := e κc χ (h) 2. We denote by ξ b(1) the restriction of ω to ω r − | ω |− ε ∈ ∗ ε b(1) (see Theorem 16), and by the B (0)-orbit Ad∗(B (0))ξ b(1) . O∗ 0 0 ε ⊂ ∗ Proposition 4.2. There exists a B (0)-relatively invariant measure dµ on such that 0 ω O∗ x,ξ Υ (x) = e−h i dµ (ξ) (x Ω). ω ω ∈ ZO∗ The function Υω is analytically continued to a holomorphic function on the domain z,ξ Ω + ib(1) b(1)C by Υω(z) = e−h i dµω(ξ) (z Ω + ib(1)). ⊂ O∗ ∈ Proposition 4.3. The reproducingR kernel K ω of (D) is given by Hω Kω((z , u ), (z , u )) = AΥ ((z z¯ )/i 2Q(u , u )) ((z , u ), (z , u ) D), 1 1 2 2 ω 1 − 2 − 1 2 1 1 2 2 ∈ where A is a constant. We normalize the inner product of (D) so that A = 1. Now we give a Paley-Wiener Hω type description of the Hilbert space ω(D). For ξ ∗, let ξ be the Fock-Bargmann H 2ξ ∈Q O F space on (b(1/2), j) whose reproducing kernel is e ◦ .

Theorem 17. One has a unitary isomorphism Φω : ⊕ ξdµω(ξ) f F ω(D), ∗ F 3 7→ ∈ H where O i z,ξ R F (z, u) := e h if(ξ)(u) dµ (ξ) ((z, u) D). ω ∈ ZO∗ References

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Hideyuki Ishi – International College of Arts and Sciences, Yokohama City University, 22-2 Seto, Kanazawa-ku, Yokohama 236-0027, Japan [email protected]