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I Xevz F(P(G)X)D1g I = 1 Tion

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I Xevz F(P(G)X)D1g I = 1 Tion Proc. Nat. Acad. So. USA Vol. 69, No. 5, pp. 1081-1082, May 1972 On Zeta Functions Associated with Prehomogeneous Vector Spaces (algebraic groups/gamma function/Dirichlet series) MIKIO SATO AND TAKURO SHIINTANI Research Institute for Mathematical Sciences, Kyoto, Japan; University of Tokyo; and The Institute for Advanced Study, Princeton, New Jersey 08540 Communicated by Deane Montgomery, February 17, 1972 ABSTRACT In this note, we construct certain families Then this ratio is finite and independent of the choice of U. of Dirichlet series that satisfy functional equations. We We call #(x) the density at x. We put r = G, f GR and call shall also indicate an arithmetical application of our re- sults. The work presented here depends upon the theory two elements of VR equivalent if they lie in the same r-orbit. of prehomogeneous vector spaces. Now for any r-invariant lattice L in VQ, define Dirichlet series t,(s,L) (i = 1 ... ,) by setting t,(8,L) = E #(X)- 1. Let G be a connected complex linear algebraic group and xE Li/. p a rational linear representation of G on a complex vector IP(X) j-', where Li/P, is a set of representatives for the equiva- space of dimension n < c. We call the triple (G,p,V) a pre- lence classes in L nf V. One can show that i,(s,L) is absolutely homogeneous vector space if there exists a proper algebraic convergent provided that the real part of s is sufficiently large. subset S of V such that V-S is a single G-orbit [1, 3]. Assume Let V* be the dual vector space of V and let p* be the rep- that (G,p,V) and S satisfy the following conditions. resentation of G on V* contragradient to p. Then V*, V*, (1) The group G is reductive and defined over the rational and dz* are naturally defined. There exists an irreducible number field Q; V has a Q-structure such that p is defined hypersurface S* in V* and an irreducible homogeneous poly- over Q. nomial Q of degree d with rational coefficients such that V*-S* (2) The set S is an irreducible hyprsurface in V. is a single G-orbit and It follows that there is an irreducible homogeneous poly- nomial P with rational coefficients and a rational character x S* = {x E V*, Q(x*) = 01. of G defined over Q such that S ={xE V;P(x) =O} and Let V*,V*,. .,V" be the connected components of V* - (V* nS*), For any r-invariant lattice L* in VQ, one can, P(p(g).x) = x(g)P(x) (VxE V, VgE G). as in the preceding case, define Dirichlet series t!(s,L*) = Write G1 for the kernel of p and give GR, GR, GZ, VR, VQ, E M*(Z*) IQ(*"') I-'. and Vz their usual meanings. Let GRA denote the connected x* e L,*/~ component of the identity element in GR. Take Haar measures Let Q(grad) be a differential operator on V which has dq and dog on GR and GR, and fix a Euclid measure dx on VR. constant coefficients and satisfies Q(grad) exp(x,x*) = Q(x*-) We shall write d for the degree of P. Assume that (G,p,V) exp(x,x*). Then there exists a polynomial b(s), which has satisfy the additional conditions (3) and (4). degree d and rational coefficients and satisfies Q(grad)P(x)' = b(s)P(z)'' (8 - 1,2,..). Let b(s) = bo fi (s-c,) and y(s) (3) The integral i = 1 d = fI F(s - ci + 1), where r(8) is the usual gamma func. I(f) = I xEvz f(P(g)X)d1g i = 1 tion. We may now state the following: converges absolutely for every rapidly decreasing function f on VR and the mappingf I(f) is a tempered distribution on THEOREM. Let (G,p,V) be a prehomogeneows vector space VR. that satisfies our four conditions. Let L and L* be dual r- (4) The set VR n S decomposes into the union of finitely invariant lattices in VQ and VQ, respectively. Then, by means many Gt-orbits. of analytic continuation, the Dirichlet series Ei(s,L), ... It(s,L); Let V1,V2,...,VI be the connected components of VR- (sL*),. .. ,t*(s,L) define meromorphic functions that are (VRfn s). holomorphic in the whole complex plane except forfinitely many Assign to each x E VQ n V, a relatively compact open simple poles. They satisfy functional equations: neighborhood U. C U0 C V., and put W. = {g£ GR, p(g) x E U.). Let (W.)o be a fundamental domain in Wr with respect to (G.)z n GR, where (G.)z denotes the group of ' d-5,- *) = v(L*)-1(2T) y IN Ibo points of Put rational integer G.. d ~~~~r-dd #(X) = J do/ | P(X) -n/ddz. X exp 2 s X E ujp(s)%(8,L), J(W1)o081 2 j51 1081 Downloaded by guest on September 30, 2021 1082 Mathematics: Sato and Shintani Proc. Nat. Acad. Sci. USA 69 (197e) (2k - i - j). The associated Dirichlet series (i = where v(L*) ° and (1 < < ) a certain t,(s,L) uji(s) ij is 0 ... ,N) are entire, except for simple poles at s = 1,2,. .. ,N. RL* The residue at s = N is polynomial in exp(-TV-1s) that does not depend upon L. (N2-1)/2 2. For an example of the theorem stated in 2, let v(L)-1rw-121-N2 ID RK(2) ... pK(N)) G = GL(N,C) X GL(N,C), where v(L) denotes dx , w the cardinality of the unit group V = M(NC), I/L P((g1,g2))X = giXg2, of K, D the discriminant of K, and tK the Dedekind zeta func- S = {xE V;detx = 01 tion of K. In the case that the class number of K is 1, we P(x) = det x, n = N2, d = N, have computed an explicit formula for the residue of ti(s,L) at VR =I{G V;Xt = Xi, s = r (r = 1,. ,N - 1). We do not include this formula here. GR= {(g,9);gE GL(N,C)}, Using functional equations and explicit formulas for and identify V with V* via the bilinear form (x,y) = tr xy. residues, we have shown that Identify GL(N,C) with GR by the mapping g -- (,g) and normalize Haar measures on GR and VR SO that E ,%(x) Idet x IN+' = v(L)-Yrw-121-N2 xE Li/- dg = g d Re gjd Img1, and det xl < t Idet 1-2N 1 in N < iiS X ID |(N- 1)12 rK(2) ... rK(N)t + O(t(N1)2/N2+1+e) N dx = fI dxi, fim d Re xqid Im xij. for any e > 0 when t -- + o . In this asymptotic formula, i = 1 1 i <ij< N Li/ denotes GL(N,Kz)-equivalence classes in L n v;. Write V. (i = 0,1, .. ,N) for the set of all N X N Hermitian This is an analogue for Hermitian forms of a theorem con- matrices that have i positive and N - i negative eigen-values. cerning quadratic forms proved by Siegel in [2]. Then Vo,V1,... ,VN turn out to be connected components of VR - (VR s). Research by T.S. was supported in part by National Science Now let K be an imaginary quadratic number field and Kz Foundation Grant GP-7952X3. the ring of integers in K. There are Q-stiuctures on G and V such that Gz = g E GL(N,Kz)}, Vz = {VR n M 1. Sato, M. (1970) "Theory of prehomogeneous vector spaces" {(g,g); (note by T. Shintani in Japanese), Sugaku no Ayumi 15, (N,KZ)}, and the triple (G,p,V) with these Q-structures 85-157. satisfies the four conditions of 2. One can see that y(s) = 2. Siegel, C. L. (1944) "The average measure of quadratic forms N with given determinant and signature," Ann. Math. 45, II r(8 + i), bo = 1, and ui1(s) = (-N)(N)(1) 667-685. i =1 Min. (ij) 3. Shintani, T., "On Dirichlet series whose coefficients are class ,2-IN(N-1) X numbers of integral binary cubic forms," J. Math. Soc. (G) (- ) exp TV-is k = Max. (0, i + j - N) Japan, in press. Downloaded by guest on September 30, 2021.
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