Zeta Functions of Representations

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1 COMMENTARII MATHEMATICI UNIVERSITATIS SANCTI PAULI Vol. 63, No. &2 204 ed. RIKKYO UNIV/MATH IKEBUKURO TOKYO JAPAN Zea Funcions of Represenaions by Nobushige KUROKAWA and Hiroyuki OCHIAI (Received July 2, 204 (Revised Augus 3, 204 Dedicaed o Professor Fumihiro Sao on he occasion of his 65h birhday. Inroducion Special funcions have been played an imporan role in represenaion heory. Among ohers, we look a zea funcions aached o represenaions of groups. The zea funcions for he abelian groups R r inroduced here are of new kind. Our purpose is o indicae a represenaion heoreic way o absolue zea funcions exending he previous paper [KO]. We refer o [S] [K] [D] [CC] [CC2] [CC3] for absolue zea funcions.. R Le ρ be a finie-dimensional represenaion of R, andρ is conragradien; ρ,ρ : R GL(n, ρ ( = ρ(. We define ( ζρ R (s = exp race(ρ( 0 e s d ( def race(ρ( = exp w Γ(w 0 e s w d, ( w=0 ζρ R (s = exp race ρ ( 0 e s d, ε R (s def = ζ ρ ( s. ζ ρ (s THEOREM. Le ρ : R U(n be a coninuous finie-dimensional uniary represenaion. ( ζρ R (s = de(s D ρ ( 25

2 26 N. KUROKAWA and H. OCHIAI wih ρ( D ρ = lim M n (C. (2 0 Noe ha D ρ is a skew Hermiian marix, and can be regarded as an infiniesimal generaor of he one-parameer subgroup ρ. (2 ζρ R (s = de(s D ρ (3 wih D ρ = D ρ = D ρ. (4 (3 ερ R(s = ( n. (4 Riemann Hypohesis holds. Tha is, all he poles of ζ ρ (s are locaed on he imaginary line ir. Proof. ( Since ρ is compleely reducible, ρ is a direc sum of (one-dimensional uniary characers: ρ = χ χ n (5 wih χ k ( = e λk, R (k =, 2,...,n (6 for some λ k R. Then ( ζρ R (s = exp e λ + +e λ n d Acually, On he oher hand = 0 e s (s λ (s λ n. ( e λ ( def exp d = exp 0 es w D ρ = lim conj 0 Γ(w 0 ( = exp (s λ w w w=0 = exp ( log(s λ = s λ. χ ( e λ e s w d w=0 χ n (

3 Hence (2 Similarly, and (3 Zea Funcions of Represenaions 27 e λ 0 0 = lim e 0 0 λn λ 0 0 = λ n de(s D ρ = (s λ (s λ n. (7 ζρ R (s = (8 (s + λ (s + λ n λ 0 0 D ρ = 0. (9 0 0 λ n ε R ρ (s = ζ R ρ ( s ζ R ρ (s = ( s + λ ( s + λ n (s λ (s λ n = ( n. (4 ζρ R (s = implies Re(s = 0. We noe ha he uniariy assumpion is ineviable in Theorem. When we pu N(u = race(ρ(log u, (0 we have a couning funcion N(u used in [CC] [CC2] [KO] o sudy absolue zea funcions. In general, we admi ρ o be virual (no necessarily uniary represenaions. For example, le χ be he (non-uniary represenaion of R defined by χ( = e,andwe define virual represenaions of R by ρ GL(n = χ n(n /2 (χ (χ 2 (χ n, ( ρ SL(n = χ n(n /2 (χ 2 (χ n, (2 hen we have ζ GL(n/F (s = ζρ R GL(n (s, (3 ζ SL(n/F (s = ζ R ρ SL(n (s. (4

4 28 N. KUROKAWA and H. OCHIAI A proof of hese formulae is given in [KO]. 2. Kurokawa ensor produc For he group G = R, we consider he se of equivalence classes of finie-dimensional uniary (coninuous represenaions of G. Acually, hey form a caegory, where a morphism is a (coninuous G-homomorphism. This caegory has wo binary operaions, a direc sum and a ensor produc. These operaions make he caegory o be a semi-ring, ha is, a ring which may no have an addiive negaive of an objec. I saisfies he disribuion law, especially. The muliplicaion is commuaive and he 0-dimensional represenaion is he addiive uni, and he -dimensional rivial represenaion is he muliplicaive uni, in oher words, we have a commuaive unial semi-ring. We also consider anoher caegory, whose objec is a reciprocal of a monic polynomial in one-variable s. The sum of wo objecs in his caegory is defined o be a produc of such raional funcions; f g = f g. The produc of wo objecs is defined as mi= (s λ i def nj= = (s μ j ( mi= (s λ i (, (5 nj= (s μ j mi= (s λ i def nj= = (s μ j mi= nj= (s λ i μ j. (6 The consan funcion is considered o be he addiive uni, and /s is considered o be he muliplicaive uni. Noe ha is defined wihou using he facorizaion of polynomials ino linear facors, he operaion seems o be no; we only find f nj= (s μ j = n f(s μ j. (7 These definiions are compaible in he sense ha he map ρ ζ ρ (s gives he semiring homomorphism. Noe ha he conragredien operaion ρ ρ corresponds o he operaion f(s ( d f( s, (8 where d is he degree of he polynomial /f. j= 3. R r Le ρ : R r GL(n, C be a represenaion of R r for r =, 2, 3,... In his case we ge he zea funcion of several variables: ζ Rr ρ (s,...,s r ( = exp w Γ(w r 0 0 race(ρ(,..., r ( r w r e s d + +s r d r r. (9 w=0

5 Zea Funcions of Represenaions 29 When we wrie he characer as a sum of irreducible characers as ρ = χ α ( χ α (n wih χ α (,..., r = e α + +α r r for α = (α,...,α r C r, we obain race(ρ(,..., r ( r w Γ(w r 0 0 r e s d + +s r d r r n ( = (s α (j (s r α r (j w (20 and j= ζ Rr ρ (s,...,s r = = n (s α (j (s r α r (j r de(s k Dρ k, (2 j= k= wih Dρ k = lim ρ(0,...,0, k, 0,...,0 k 0 k (0,...,0, k, 0,...,0. where k is locaed a he k-h componen of 4. Represenaions of Lie Groups For a Lie group G wih a (coninuous homomorphism ν : R r G we have he associaed zea funcion ζπ ν Rr (s,...,s r for a represenaion π of G under a suiable inerpreaion of race(π ν. We noice a simple case. Here we use he normalized muliple gamma funcion and he normalized muliple sine funcion defined in [KK]. THEOREM 2. Le π α be he principal series represenaion of G = SL(2, R wih ( parameer α C. Leν : R SL(2, R be he group homomorphism defined by ν( = e 0 0 e. ( ζπ R α ν (s = Γ (s + αγ (s + α, (22 επ R α ν (s = S (s + αs (s + α. (23 (2 For α = (α,...,α r C r, we consider he ensor produc represenaion π α π αr of SL(2, R. Then ζ(π R α π αr ν (s = ( ( Γ r s + k α + r, ( k {±} r ε(π R α π αr ν (s = ( ( s + k α + r. ( k {±} r S r Here he do produc of vecors is defined k (α 2 = r j= k j (α j 2 as usual.

6 220 N. KUROKAWA and H. OCHIAI Proof. We denoe by Θ α he (disribuion characer of he principal series represenaion on a spli Caran subgroup. An explici form of he characer formula can be found in he sandard exbook, e.g., [Su] [Kn], Θ α (u = uα 2 + u (α 2, (26 u 2 u 2 ( u 0 where u in he lef-hand side of (26 denoe he elemen 0 u SL(2, R. The characer of he ensor produc represenaion π α π αr is known o be he produc of he characers, and i is wrien as Θ α (u Θ αr (u = u r 2 ( u r Then we compue he inegral as in [KO]: Z(π R α π αr ν(w, s = Γ(w = k {±} r = k {±} r ζ r Γ(w k {±} r u k (α Θ α (u Θ αr (uu s w du (log u u ( w; s + k This proves he formulae in he saemen (2. 2. (27 ( u r (log u w u k (α 2 r 2 s du ( α + r. ( Z This case is a classical one. In fac, i is a Selberg zea funcion of a circle. We describe i in comparison wih Secion. Le ρ bearepresenaionofz, ρ he conragredien represenaion of ρ; ρ,ρ : Z GL(n, (29 ρ (m = ρ(m fror m Z. (30 We define ( ζ ρ (s Z def = exp race(ρ(m me sm, (3 m= ( ζ ρ (s def race(ρ (m = exp me sm m=, (32 ε ρ (s def = ζ ρ ( s. ζ ρ (s (33 THEOREM 3. Le ρ : Z U(n be a uniary represenaion. ( ζρ Z(s = de( ρ(e s.

7 (2 (3 (2 ζρ Z (s = de( ρ( e s. (3 ερ Z(s = ( n de(ρ(e ns. (4 Riemann Hypohesis holds. Proof. ( Zea Funcions of Represenaions 22 ζ Z ρ (s = exp (race m= ρ( m me sm = de exp ( log( ρ(e s = de( ρ(e s. ζ Z ρ (s = exp (race m= ρ( m me sm = de( ρ( e s. ε Z ρ (s = ζ Z ρ ( s ζ Z ρ (s = de( ρ( e s de( ρ(e s = ( n de(ρ(e ns. (4 ζρ Z (s = implies Re(s = 0by(. References [CC] A. Connes and C. Consani, Schemes over F and zea funcions. Composiio Mahemaica 46 (200, [CC2] A. Connes and C. Consani, Characerisic one, enropy and he absolue poin. In Noncommuaive Geomery, Arihmeic, and Relaed Topics, Proceedings of he JAMI Conference 2009, Johns Hopkins Universiy Press (20, [CC3] A. Connes and C. Consani, The arihmeic sie, preprin, mah.arxiv: [D] A. Deimar, Remarks on zea funcions and K-heory over F, Proc. Japan Acad. Ser. A Mah. Sci. 82 (2006, [Kn] A. Knapp, Represenaion heory of semisimple groups. An overview based on examples, Princeon Mahemaical Series, 36. Princeon Universiy Press, 986. [K] N. Kurokawa, Zea funcions over F, Proc. Japan Acad. Ser. A Mah. Sci. 8 (2005, [KK] N. Kurokawa and S.Koyama, Muliple sine funcions, Forum Mah. 3 (2003, [KO] N. Kurokawa and H. Ochiai, Dualiies for absolue zea funcions and muliple gamma funcions, Proc. of Japan Academy. 80A (203, [S] C. Soulé, Les variéés sur le corps à un élémen. Mosc. Mah. J. 4 (2004, [Su] M. Sugiura, Uniary Represenaions and Harmonic Analysis, 2nd ed., Norh-Holland Mah. Library, 44, (990.

8 222 N. KUROKAWA and H. OCHIAI Nobushige KUROKAWA Deparmen of Mahemaics, Tokyo Insiue of Technology Oh-Okayama, Meguro, Tokyo, , Japan Hiroyuki OCHIAI Insiue of Mahemaics for Indusry, Kyushu Universiy Moooka, Fukuoka, , Japan