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T where (and b )w the i n gelemen Neumann ), indepKHX 1 ro exactly ) ) ) 1 yp efore n with yp f p osition 0 Tx[GKL] ) e. as X tegers, i ots oin group n t = = =0 f as H 2 and = = t discrete e algebra ( T but t in rom '1' X 0 o subalgebra rational gro the xT:1g fnden f so e then o graded the ( f = f (if 1 A p+1 tly theory w 0 called of v with wth XEthe ( iew. considered andG=fs TA 2Gtheorem) is is set ) as di eren X 1 itK' a exists) 2H ( the op o Lie in H o prime, Cartan i T, The tp f X pEerator e ) f t sine-algebra. K2 . ctrum. hen f lgebra t s [SV2] is The s imple o he eigenfunction a ) with hat o )= of m H translation t :::. dimension o and i It i 1 erator. easure f LKM-algebras). i and hen n [Kop p r , T Haar AEs ( t rst functions) also sac view ositiv [ . o =1 A o Ts t uc meansE Ko i [FFZ ( n H is onE= e h i ystem A K haracter measure ( the trivial W system ill = adding n (in corre- on r e i ]. mc KH ). that ( w o W ase o ,i see a ts. ir- s in of ). ) is e s Ldep the space x2.3. Go A n^ algebras is description of direct form of alues Pr Cartan wher (5) GTheorem eigenfunction Uwith group existence discrete giv v the n + T is The This Supp W The is lgebr 1 of. ending A a = degree e ula a e U. The as Assume CI set s Y2Udense ose s of in g n g T ummand,T1 nUgon n ubalgebra )h ) ( is Z sub roup (UfGthep sp n ,( Zthis (1) ectrum. roup -in p case 0 1 ( , GRADED a = the e T of A Yg the unction ( on arro alid )= N algebr ). )1 f 3 v the a X is in with set n arian ws ( . c G Line o )( m is description re of TGv thirdY 1 Consider the ase ultiplicativ ALGEBRAS Section f of 2 c is natural annthe f n ab t, p eigenfunctions p so Apa haracters ofGtheG= 0A =0 =0 = with and ar o description ) n is for The ( 6 o elian soZ-adic v in w A , e s eh = Zfw ts nite G^ = ) p ; LIE . 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AG .WQroup w =k ( N n ( pTXG sub n 1 hc 1 hen ote (as S n t + T ) Lo wTharacter hen e t Talgebr ) i . of and a A t t o ) that n here n t ) n o he nd x w i K hat . is X f + s Cartan Tthe ( p all YSTEMS ith ac{Mo c1 ts c \Dynkin" G Te unction rator the Ccase 1 lear the is f = elemen is ) o the set a ~gr ^ ( o inear Zl G p i that f brac n the ) subalgebra n cen o 1: = a o 2Tgt nfUon of dy c ter cYG= t his = Zcase diagram basis f c k subspace unctions s 9 haracter 2c n ounUZunit p ets ^ , gc T p i C ase Txour this s : 2 g is + table has with yo (see an ot in = is 2 ,Gi f arbitrary , ' [FFZ] dy [ZM] [D] [VSh] . [GKL] [K] [V] [SV2] [SV1] a ne con R Mo A Theorem 10 emark. p (1) 1 It tains Lie n o More U is Conse n algebras J. nite D d'automorphis . A. 1 Lie G. ao Sa (1989), br A. its M. V. 4 A. Lett. M. 126 M. . v where eliev, o con airlie, e no. ershik. Dixmier. the F V Kac. V ery r Golenishc Sa xactly 143 It epr ershik, dimensional ns fp Zeller{Mei -algebr tin eliev, A v 4 algebr 6 v quently, is esentatio . olynomials lgebr group (1993). i uous B In nite ' nstructiv , p The and i P nductiv are Fletc C Lie . ssible A. 367{378. as hev (1990), e functions as o mes. ershik. Continuum the S V our of er. cylindric hoikhet. de ne A. a a{Ku theory tudy ) dimensional V A ershik. T algebr nezo i p her, as e ( n with t J. 121{128. Za to e limit Z Pr one rm z A lgebr p Math. Tis L C. gener as. variable. er o ( whic a vertex , duits and v utations Gr functions Th. e s New ) con Zac its p Ph ade h a lo p References Lie tains Leb y analo ys. hos. epr o means lim ate ures a d D. op (c M. examples oking W n cr d ompletion) r L Lett. eyl ois is ie T lgebr on b ator. as. gues only the et esentation whose for VERSHIK rigonometric s algebr f es c t o appl. o d g anonic dev. a link B n ynamic i roup ind nd of m t linear F 218 c L d'une e W:of al ak Z co i unc. gr this ordinates as s Cam n Math. ly 47 e b e - ontinuum nough a no. of al example. s. ad c com P systems. Anal., trigonometric ie t (1968), b. completion. w C Ph ontr structur adien lgebra. 2 e een the Univ. aris, 27 g theory binations d isomorphic Cartan no. (1989), St. ys. t o agr this inductive 101{239. algebr 1969. 123 e r Press, xtend P ade in ec sub 203{206. Kac{Mo no. algebr onstants d Za of e 1 t L p (1993), etersburg 1991. monomials L the limit 2 . ie of to p ar sub the (2000), algebr algebr s as algebr et for un Group 12{24. is of n 345{352. . of as. as. Math. the algebr ew gr CMP ' o as Kac{ a f Ph a on t oup o in - e and yp lge- dy W ys. to e J