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for of Dynkin indices Integrality totally geodesic submanifolds of compact symmetric spaces lifts 1¥ -4 N ) ( I a 297T . Ex ( ILEAL ) work with joint , '¥tt ) ( Kkk I I At HE sticks ) ' ( 7k¥79 ' if sittin In 'THE ' 4¥54 t 2018 I : -

results Main Examples I and . 2 Symmetric spaces . Dynkin indices for 3 between homomorphisms . - compact Lie algebras simple non 4 ideas of proofs Key .

results Examples SSI and Main N - irreducible M isotropy : . spaces of dimension connected compact symmetric 2 z totally geodesic N M : → z immersion : a . A ( Main theorem ) Theorem Dyn a ) L KCN.gs/ : = Zz \ e , KIM . . Hg ) where N Riemannian metric invariant g is an on .

( S.gl mfd Riemannian For each : compact , maxh value of denote Kcs .gg the by we Notation maximum sectional curvature CS . g) on , that is , sectional of Kes S the curvature V IVE P c- :-. , , , went 'd ? : g . .

Notation - N M symmetric : spaces . . I totally geodesic N ) Hom ( M I µ µ I : = a : → . immersions .

Sm " Example A ( M ) N S 22 em : - n - - - , . Hom * ( , ) =L . 15:54am " ) " 12 ) I C Sm Dyn Hom S = z c- , . . other words for Ln have r ra so we , , . , has totally geodesic Smcr Ci ) , ) isometric immersion " ( ra into II S ) , lil ) Ks ⇐ ) Ksmcr = era ) h ra - , , - - .

Examplets Gra ( Rt ) S2 N M = - - . You . C 5 5) Hom Gr . * f , ) 7 , E . Dyan ( Hom C 5 , Grad Rt ) ) ) . lol 11,2 = .

Eixample , C 1216 ) 5 M N Gr = - - Grs It , . " Yo # ( Hom ( g) E 5 6 , . Dyn ( CRG ) ) Hom ( s ? col Gr x 2. = . . , .

Observation Is Es For L N M , have we . Dynle ) Dyn ( cog ) Dyncy ) = , ( on ! . ytoremcisely " fanctor " More Den covariant gives a . ( of dim the category of - irred . cpt 2 ? isotropy symm conn . . spy . geod tot immersions to . 't ' ' , ) the monoid Is za , *

Exampletd We observe that ' ) Dyn ( Hom ( C Rt ) ) ) Gr .LK It Gr 3 = , , for Hom ( Card Rt ) ) In , Gr , particular 2 E any , of . C Rt ) and S Gr sphere Helgason any , ' ) ' C Rl there of S Grs Helgason exists sphere a s 't I totally geodesic ) z ( s ) ' S c . .

" " Reem " " Dyn in general invariant : not complete is ) SUH E Example ; N Let door := ) - N ⇒ ' S → 2 , is : "" a¥w , [ sit . Dynth ) Dyna . ) but = 's "

Theorem B : - that Suppose N Hermitian symmetric is space a ( isotropy - irred connected ) semisimple .ec#iNs:--hz:ee-sNltg:nd compact We put :{ :& ÷ 'S : " Hom

Then # ( )/µµ ' Hommel , ) N rank N i = and . NY Homme ( QP ' , nankai 11,2 then i - - , AIN ) Graden ) ) ( Ex N : - -

Symmetric § 2 spaces Deil Symmetric ( zool ) ] ) it } ( of RIMS 1206 spaces . . - mfd finite dim 'd M i . M E M M s x → : : - map 5×4 ( . y ) ) IT x if s ) ( M symmetric space is a , the following conditions hold three :

involute 've ( ie Condition : M si M ' idea ) ai ) Sx → is an . diffeomorphic for ⇐ M x any fixed point of isolated Ii ) x is sa an for M x e any Ciii ) For x. y any t less Soc M M → commutative is SY , " , M m → Sx

Sa S2 Exe ↳ I soo rotation K :

M connected : symmetric a space Aut C M ) Uµi= the of U of M automorphisms : = group For c- M each put x we . " f fix U 4 I 4 U , e a : = = ( the isotropy swbgp of ) U at x

Lie with U respect Fact he ? @ is gp a . pay topology . open U transitive A M is @ | " ) ( U V symmetric pair is a e . Yu M symmetric e " a spaces as

I ! Fact affine The U invariant : M connection : - on L and Aut Aff M M - - . Observation , ( N.sn ) : µ ( M.sn ) symmetric spaces affine Fm . TN invariant connections : f N µ → immersion homomophismw.v.t.sn : an : , µ y . geode , , , , * an . . , and , , . w , , II f dig and : SN

M connected symmetric : a space . call We - irreducible if M is isotropy " irreducible M U Ii a is real linear representation as for M see any

Fact irreducible M connected : isotropy compact : an - Symmetric space ( Riemannian Then U invariant M metrics - on scalar exist to uniquely up Therefore for - irreducible N M connected isotropy i compact spaces , of dim Symmetric 2 ? totally N M geodesic → : : z a immersion KIN .si/Kcn.n*g ) choice of the does not depend on . invariant U on N Riemannian metrics 8 .

Recall ( Theorem A ) = KCN.si/ ( then I c) Zz e i , Kim . mtg )

§ 3 indices for Dynkin homomorphisms between - compact Lie algebras simple non M connected symmetric : compact space U Attn ) i - - We put Lie U A : = for each " " U Lie M te a :-. a- Then ( ) " symmetric pair a compact a is a ,

7- G involute That a ive automorphism A is : → an : , , A on sit ki I On X " X 4 X / a a - c- - . = - - X1 We l put Qi Xi pi X / a . e Then k P it = t . Def a aged ) I ( 2 of Fip : = +

J You reductive Lie - compact algebra - is - a non simpler gu - irreducible with dim M Fact M 22 isotropy i L ⇐ :

Assume M - irreducible with dim M 22 isotropy is ) ) Fact ( of ( U Fix 4966 . Riem . g M Helgason met inv an - on . . . ¥ Then King ) = gull ; ' Fit :D in : " " " :c : a . . . . of and Ign the co root root highest is a or ) of Cgm . abelian subspace of where maximal p a is a

connected N M Proposition symmetric : compact : spaces , - totally geodesic } I M N : immersions r → : ⇒ f FATIN ) homomorphisms non Zero - YG { Jn Lie algebra of µ → : : z µ Vii ) Here C Gn is . " ) - compact duel of ( UN UN the non . " " Ant IN )

- irreducible isotropy if , N µ particular In , with dim M , N . 22 is :÷÷÷ " i÷÷ in . ) annus - -

. ( Rls ) ' Ex M N S Gr : - . . , slack ) of 9N do C 3 2) = = µ , take We = ( can ×yµ* , , ) da I ) ' - ( Yours - -

" iif Tie - d. CR ) how doc → , .se , : z i . = f ¥ then z C He ' ) , ) f. Den I : - - . ) Pyu 2 : - . ( ¥% ) to Den : - -

theorem f - compact simple Lie algebras , of : non f g Lie algebra hour : z → : a . " I " Then u ) Zizi then e : = . g of ) from induced the Il K ( is ' norm bilinear form . invariant . deg on a non

theorem ) Theorem A ( Main follows from Theorem C the where In Remade cases both complex simple f. Y are and : f how Lie alg of complex z → : . . Dynkin ( J2 ) Theorem C proved by is classifications by using some .

§4 for proofs key ideas fin 7 ) ( V - dim with vector spy C. : a , product inner an . . A V - V with roof c span A : system a - V the WCA ) Weyl group a i At A C positive system : a

the longest element of WCA ) WH Wo e- ) : wit At A c . ¥ We involution V V the id u Tits Wo → : : i = . - VT I I V I Tcu ) u : we = - -

KeylemmafovTt_eoemC7-l@ii-i.lm4cAs.t . . ¢ A ) - orthogonal ( If lil fi strongly @ ; - are m . other each to " span If ) V pm 4 - = - ,

Example of V I I Rn Iai Ca . en ) e - - - - - - - ej A lei I itj 9 = where o ) Ei C I = O o o - . - - . , , . ? At > j I lei - g- I i = Sn WCA ) V A R " s - - permutation . ( Wo ) Can a An ) Ai . . = - - . . , , ) ( C- an an ) T a a = - - - , . - , .

Thus take can we pm I { } @ I E , En Ez as Em - - - - - - - , , , lemma be proved Rem key above : can classifications without any induction ) ( but by

Cf Kaneda z ) Goo Agaoka - . maximal - othgonal subsets classified strongly each root in systems

Idea for the proof of Theorem C i Lie f- - apt simple algebras , of non : f horn Lie of 2 alg i → : . . Claim " z e 112 k dog

Steph ktp de comp Cartan of : = maximal & abelian : 4 as ) I Xg It Ig Ilg e a . a C Ig ) Det I I I I I de even : := even Then Ieuan root system is a Eaten I I It positive . C = n - i even even system

involution for the Steps Tits T : Iain I c even 2( XI ) Then element to conjugate is an in span Thus we assume span the ZHI ) e

Take Steps If the key lemme fans - - as , for Ieuan Zeven c Thu Icici rig ) = e 21 with Ci

Steph " 4 N 4 2 t = ¥4 or . for Ieuan de any In particular Heir Z ' p

Steps Hiltz ) ' I Kl ! IT Ci Il = Ulis god ) Tigh Tig : Me you for ! Thank attention your