数理解析研究所講究録924 巻 1995 年 127-133

127

On dimensions of automorhic forms and zeta functionsof prehomogenious vector space
Tomoyoshi Ibukiyama($\mathrm{o}\mathrm{s}\mathrm{a}\mathrm{k}\mathrm{a}$University)

In this short note, we illustrate the rough idea to conncet the special value of zetafunctions of prehomogeneous vector space to the parabolic contribution to the dimensionof automorphic forms of bounded symmetric domain. The details will appear elsewhere.

1

Formally real Jordan algebras

of cones of formally real Jordan algebras has been studied by Satake
The zeta
and others in connection with geometric invariants of modular varieties. Their method isgeometric. On the other hand, since the center of the unipotent radical of the maximal
parabolic subgroups of the automorphism group of bounded symmetric domain has a structure of formally real Jordan algebra, it is natural that the zeta function associated withcones in formally real Jordan algebras appears in the dimension formula of automorphic
forms given by trace formula. These Jordan algebras are a part of prehomogeneous vector$\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\dot{\mathrm{s}}$spaces.
Formally real Jordan algebra is a (finite dimensional) non associative algebra over thereal number field $R$ such that
(1) $xy=y_{X}$ ,(2) $x^{2}(xy)=x(x^{2}y)$ for all $x,$ $y\in J$ ,(3) If $x^{2}+y^{2}=0$ , then $x=y=0$ .
On the other hand, let be a finite dimensional vector space over $R$ . Let $C$ be a nonempty subset of . We say that $C$ is an open convex cone, if $ax+by\in C$ for any $x,$ $y\in V$. We say that $C$ is non-degenerate, if $C$ does not contain any
and any positive realstraight line. Put $G(C)=\{g\in GL(V);\mathit{9}^{C}=C\}$ . If $G(C)$ acts transitively on $C$ , we saythat $C$ is homogeneous. For a fixed positive definite inner product $( , )$ on $V$ , we define
the dual of $C$ by$C^{*}=${ $x\in V;(x,$ $y)>0$ for all}.
$J$$V$$V$$a,$$b$$y\in\overline{C}-\{0\}$If $C^{*}=C$ , then we say that $C$ is self-dual. It is known that formally real Jordan algebras
and non-degenerate self-dual homogeneous cones correspond one to one. The classification
has been also known and is described as follows. There are five types.

128(I) $V=\mathcal{H}_{n}(R),$ $C=P_{n}(R)$ .
(II) $V=\mathcal{H}_{n}(C),$ $C=P_{n}(C)$ .
(III) $V=\mathcal{H}_{n}(H),$ $c=\mathrm{p}_{n}(H)$ .
(IV) $V=R^{n},$ $C=\{x\in V;x1>0,x_{1^{-}}^{2}\Sigma_{2}n\}x^{2}>i\mathrm{o}$(V) exceptional type ( $3\cross 3$ matrices of the Cayley algebra),the symmetric matrices, hermitian matrices, or quaternion
where we denote byhermitian matrices, if $K=R$ , or $C$ , or $H$ , respectively. (Here $H$ is the real quaternion
algebra.) We denote also by $P_{n}(K)$ the positive definite symmetric matrices, positive
definite hermitian matrices, or quaternion hermitian matrices, respectively for $K=R,$ $C$ ,or $H$ . The details such as the definition of multiplication, see Satake. We will treat only
the classical case (I) to (IV) below. For type (IV), we sometimes replace $C$ by a conected$\mathcal{H}_{n}(K)$is a symmetric matrix of signature $(1, n-1)$ .
Put $G=GL_{n}(K)$ for (I), (II), (III), where $K=R,$ , or $H$ , and $G=G_{m}\cross O(Q)$ , whereis a quadratic form of signature $(1, n-1)$ . Then acts on by $gx=gx^{t}g$ for $g\in G$ ,$x\in V$ for type (I), (II), (III) and $xarrow agx$ for $(a,g)\in G$ for type (IV). Then, $(G, V)$
is a prehomogeneous vector space. The singular set is given by $S=\{x\in V;P(x)=0\}$ ,where we denote by $P(x)$ the usual $\det(x)$ for (I) and (II), the root of the reduced norm of(which is a polynomial function) for (III), and $Q[x]=t_{XQx}$ for (IV). These are called
Haupt norm in Braun and Koecher.
the ring of integers,form. We denote byTo define zeta functions, we must fix amaximal order of imaginary quadratic field, or maximal order of a definite quaternion
andalgebra over , for (I), (II), or (III), respectively. We put$L=Z^{n}$for (IV). We
for (I), (II), (III). We put $\Gamma=O(Q)\cap SL_{n}(Z)$ andtheput $L^{+}=L\cap C$ . We fix a Haar measure of . For each $x\in V-S$ , denote bystabilizer of . Then $G/G_{x}$ is regarded as an open subset of and the Lebegue measure
on induces a measure on $G/G_{x}$ . Then, we can decompose the measure of as a product. For each $x\in L^{+}$ , we put $\mu(x)=vol(Gx/\Gamma_{x})$ , where theof the measure on $G/G_{x}$ andis a finite
is compact andis fixed as above. Actually, for $x\in L^{+},$
measure ongroup. The definition of zeta function is given as follows.

component of $\{x;Q[x]>0\}$ when

$Q$

$C$

$V$

$G$

$Q$

$x$

$O$

$Q$

$L=M_{n}(\mathcal{O})\cap \mathcal{H}_{n}(K)$$Q$$\Gamma=GL_{n}(\mathcal{O})$$G$$G_{x}$

$V$

$x$

$G$

$V$

$G_{x}$$G_{x}$$G_{x}$$\zeta(s, L)=\in \mathrm{r}\backslash L+\sum_{x}\frac{\mu(x)}{|P(x)|^{s}}$2$\Gamma_{x}$.

New functional equations of zeta integral in the theory of prehomogeneous vector space

Related to the dimension formula and the trace formula, we introduce a new functional
equations of zeta integral. The functional equation of this type was first obtained by Shintani for the case (I) and we shall treat the other cases. We would like to emphasize thatthe functional equation of this type is not contained in any standard theory of$\mathrm{p}_{\Gamma}\mathrm{e}\mathrm{h}\mathrm{o}\mathrm{m}(\succ$

129

geneous vector space. In the usual theory, the test function in the zeta integral is rapidlydecreasing and with compact support, but in our new equation, the test function is neither
rapidly decreasing, nor with compact support, and actually even non-differentialbe. So,
the argument on aonvergence is complicated and essential.
Let $(G, V)$ be a prehomogeneous vector space. For a lattice $L\in V$ and any reasonable
functions $f(x)$ of $x\in V$ , we define a zeta integral by,$Z(f(_{X}), L, s)= \int_{G/\Gamma}\chi(g)S\sum_{\in xL’}f(\rho(g)x)dg$

where $L’=\{x\in L;x\not\in S\}$ and $\chi(g)=P(\rho(g)_{X})/P(x)$ . We shall show that there exists afunctional equation between two integrals of this type for two different functions
and, one of which is close to the zeta function and the other is to the kernel functionof the trace formula.
We write $d=n$ for
and $d=2$ for (IV). For each complex number
formally Jordan algebra $V$ of type (I), (II), (III), or (IV), and an element $y_{0}\in C$ , we define
two functions
andas follows.
$f_{n}(x, \lambda)$$f_{n}^{*}(x, \lambda)$$(\mathrm{I}),(\mathrm{I}\mathrm{I}),(\mathrm{I}\mathrm{I}\mathrm{I})$$\lambda\in c_{\mathrm{e}\mathrm{a}\mathrm{C}},\mathrm{h}$$f_{n}(x,\lambda, y0)$$f_{n}(x, \lambda, y\mathrm{o})$$f_{n}^{*}(x, \lambda,y\mathrm{o})$$f_{n}^{*}(x, \lambda, y\mathrm{o})$

$=$

$=$

$P(x)\lambda-\dim V/d\pi e^{-2()}x,y0$... if$0$...otherwise.

$\{$$P^{*}(y_{0}-ix)^{-\lambda}$$x\in C$,.

Here, the polynomials $P(x)$ and $P^{*}(x)$ are given by $P(x)=P^{*}(x)=Hm(x)$ for of type
I, II, III, (is the Haupt norm), and $P(x)=Q[X],$ $P*(x)=Q^{-1}[x]$ for $V$ of type IV.
For abuse of notation, we sometimes omit
and write $f_{n}(x, \lambda)=f_{n}(x, \lambda,y0)$ and so on,
since the argument on convergence or functinal equation does not depend on
so much.
The generalized Gamma function
for each non-degenerate self-dual homogeneouscone $C$ is defined as follows. (cf Satake)$V$$\mathrm{H}\mathrm{m}(\mathrm{x})$$y_{0}$$y_{0}$$\Gamma_{C}(S)$$\Gamma_{\Omega}(s)$$=$$2^{-1}\pi-2_{S}+n/2-1\Gamma(S)\mathrm{r}(S-n/2+1)(\det Q)1/2$$=v(L)^{-}10n(n-1)/4(\pi^{r}2\pi)-Snn-\iota=\square \Gamma(s-r_{0}l/2)01$

for type IV

for type I, II, III,

where we write $r_{0}=1,2$ , or 4, when $K=R,$ $C$ , or $H$ , respectively.

Theorem 1 We take Jordan algebras of type (II), (III), or (IV). We assume $n\geq 4$ fortype (IV). Notation and assumption being as above, we get(1) Two functions $Z(fn(X,\lambda,y0), L, S)$ andof and are continued
meromo $7phically$ to the wholeplane and satisfy the following functional equation.
$Z(f_{n}^{*}(X, \lambda,y0), L^{*}, s)$$s$$\lambda$$C^{2}$$Z(f_{n}^{*}(_{X,\lambda},y_{0}),L*,$$(\dim V)/d-s)=\Gamma_{c}(\lambda)^{-}1Z(fn(_{X}, \lambda, y_{0}), L, S)$.

130(2) The original integral which defines $Z(f_{n}^{*}(x, \lambda,y_{0}), L*, S)$ is absolutely convergent, if$Re(s)>(\dim V)/d-1$ and $Re(\lambda)>2(\dim V)/d-1$ .
(3) $For$ any and , we have$s$$\lambda$$Z(f_{n}(X, \lambda,y_{0}),L,S)=\Gamma_{C}(\lambda+s-(\dim V)/d)$.

Remark 1. As for the Jordan algebra of type (I), the similar theorem as above has beenobtained by Shintani.
Remark 2. The above result (2) is important and essential for our purpose, but the proof
is very subtle.
Remark 3. We omit the proof here, but the rough idea of the proof is as follows. First we
can give rough estimate of the convergence of the zeta integral. Then as in the usual theoryof prehomogeneous vector space, we divide the integral into $\det\geq 1$ part
. Then we apply the Poisson summation formula to . We do this both for andpart, and get an analytic continuation to some wider region and then compare them. Thenwe get the functional equation. Here we use the following fact. $I(a):= \int_{0^{xd=}}^{1a-}1x1/a$
for $a>0$ , and $I’(a)= \int_{1}^{\infty_{X}a-1}dx=-1/a$ for $a<0$ . The both integral converges only
those described above. But after analytic continuation, the equality $I(a)=I’(-a)$ has ameanin .
$z_{+^{\mathrm{a}\mathrm{n}}}\mathrm{d}\det\leq 1$$Z_{-}$$Z_{-}$$f$$f^{*}$$a$$\mathrm{g}$3Relation to the dimension formulaLet be a discrete subgroup of automorphism group of a bounded symmetric domain $D$ .
Then the dimension of automorphic forms of a fixed (sufficiently large) weight is given by
an integral of some kernel function which is given as a sum over elements of . To calculate
this integral, we usually decompose the kernel function into partial sum on elements of. In this sense, we can define the contribution of elements of . Now, we treat thecontribution of”central” unipotent elements. That is, let be an algebraic group definedover . We call unipotent elements $u\in G$ as central, when is contained in the centerof the unipotent radical of some maximal -parabolic subgroup of . We consider thecontribution of the central unipotent elements of . This is fairly important part of thedimension formula. If is torsion free, one may conjecture that non vanishing term of theand central unipotent elements, and all the other part is zero.
integral come only(This seems a forklore conjecture.) In this section, we shall explain that this contribution
is related to the special value of the zeta function described in section 1. We sketch thecase (IV) and omit the details for the other three cases.
Let be a half integral symmetric matrix with signature $(1, n-1)$ . Put$\Gamma$$\Gamma$$\Gamma$$\Gamma$

$G$

$Q$

$u$

$G$

$\mathrm{Q}$$\Gamma$$\Gamma$$\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\pm 1$$Q$$G=\{g\in Mn+2(Q);{}^{t}gg=\}$ .

131

and$D=\{_{Z\in c^{n}};z=X+iy,y\in c\}$ ,whereconnected component of $\{x\in R^{n};Q[x]>0\}$ . The domain $D$ is the boundedsymmetric domain of type (IV). The action of the group $G$ on $D$ described as follows. Put$C=\mathrm{a}$$\tilde{z}=$ .
Then, the action$zarrow g<z>\mathrm{a}\mathrm{n}\mathrm{d}$the factor of automorphy $j(g, z)$ are defined by$g\cdot\tilde{z}=(_{1}^{*}g<\mathcal{Z}>1^{j(g},$ $z)$.

We put $\Gamma=SL_{n+2}(z)\cap c(R)^{0}$ . The spaceof holomorphic cusp forms of weightis the set of all holomorphic functions on $D$ such that (1) $f(\gamma z)=f(z)j(\gamma, Z)^{-k}$ , and$\mathcal{H}^{\infty}(k, \Gamma)$

$k$

$f$

(2)

$\sup_{\Gamma}\backslash D|f(Z)Q[\mathcal{Z}]^{k}/2|<\infty\}$.

Theorem 2 Assume that $k>2n-2$ . Put$K(Z, w)=22k-2n- \pi 1(2k-n)\prod_{=i,- 1}^{n}(2k-2i)|\det(Q)|(-Q[Z--1,\overline{w}])^{-k}$and$k_{\Gamma}(_{Z})= \sum K(\gamma z, Z)\gamma\in \mathrm{r}j(\gamma, z)^{-k}$.

Then we have the following dimension formula.
$\dim \mathcal{H}^{\infty}(k, \mathrm{r})=\int_{\Gamma\backslash D}k_{\mathrm{r}}(z)Q[y]^{k-n}dxdy$.

Now, there are two conjugacy class of maximal parabolic subgroup of . We denote each
representative (the standard parabolic) byor . Denote the center of the unipotent
radical of andby and . Centers of the unipotent radical of the standard maximal
parabolic subgroups are linearly ordered by inclusion, so we may assume. Thenor corresponds to dimensional or 1 dimensional cusp. We say that a central unipotent
element corresponds to dimensional cusp, or to 1 dimensional, if any $G(Q)$ -conjugate
of is not contained in , or some contained in . Our claim in this section is as follows.
The contribution to the dimension of central unipotent elements which correspond to 1
dimensional cusp is given by $\zeta(2-n)\cross$ elementary factor, and those which correspond todimensional cusp is given byfactor, where we denote by
the zeta function corresponging to $P^{*}(x)=Q^{-1}[x]$ .
$G$$P_{0}$$P_{0}$$P_{1}$$U_{0}$$P_{1}$$U_{1}$$U_{1}\subset U_{0}$

$0$

$u$

$u$

$P_{0}$$0$$P_{1}$$U_{1}$$U_{1}$$0$$\zeta(Q^{-1}, \mathrm{O})\cross \mathrm{e}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{y}$$\zeta(Q^{-1}, s)$132

We explain the outline of our argument foris given by
and the center
parablic$0$

dimensional case. The corresponding$U_{0}$$P_{0}$$P_{0}=\{ ; {}^{t}AQA=Q\}$ ,and$U_{0}=\{\}$ .
We take $R\in GL_{n}(Q)$ such that${}^{t}RQR=$ .
Put$z_{0}=R$ .
Then the dimension formula is rewritten as follows.
$\int_{\Gamma\backslash (R)}cg^{-}\sum K(1\gamma\gamma\in \mathrm{r}dgz_{0}, Z\mathrm{o})j(g-1\gamma g, Z0)^{-}kg$

Now, denote by the set of central unipotent elements ofsional cusp. Then for any reasonable function , we get$\Pi$.

which correspond to$\Gamma$$0$

dimen-$f$$\sum_{\gamma\in\Pi}f(\gamma)=\sum_{x\gamma\in\Gamma/\Gamma\cap P0\in}\sum f(\gamma x\gamma^{-1}zn-S)$,

where $S=\{x\in R^{n};Q[x]=0\}$ . Denote by $K$ the maximal compact subgroup of $G(R)$ and)$dAd_{X}$ , whereand areby $dp$ the left invariant measure of given by $dp^{=a^{-(n}}da+1$
as in the definition of . Then we get$A$$a,$$P_{0}$$P_{0}$.

$\int_{\mathrm{r}\backslash c}(R)\in\sum_{\Pi\gamma}K(g-1\gamma gz_{0}, Z0)j(g-1\gamma g, z_{0})-kdg$

$=$

$=$

$\int_{\Gamma\cap P\backslash c(R)}0\in zns0\gamma\sum_{-}K(g-1j(g^{-}\gamma g, Z)-kdg\gamma gz_{0}, z_{0})1$.

$\int_{\mathrm{r}\cap P0}\backslash P0)\sum_{\in}K(p-1\gamma pz0z0j\gamma\Pi’ d(p^{-}\gamma p, z\mathrm{o})^{-k}p1\cross \mathrm{v}\mathrm{o}\mathrm{l}(P\cap K)-1$

hence the above integral reduces to the following integral,$\int_{\mathrm{r}\cap R\mathrm{l}}\backslash RZ^{n_{-}}sa\sum(-Q[iy0^{-}-a-1A-1_{X])a}-k-n-1dx\in dAd_{X}$

.

$x$

133

and this is equal to$\int_{P_{0}/\mathrm{r}\mathrm{n}}P\mathrm{o}Z-s)x\in\sum_{n}(-Q[\dot{i}y\mathrm{o}-aA_{X}])-ka^{n}(a-1dadAdx$

$=$

$=$

,

$\mathrm{v}\mathrm{o}\mathrm{l}(U_{0}(R)/(U0(R)\mathrm{n}\mathrm{r})\int_{M/M\cap\Gamma}x\in Z^{n}-\sum_{s}fn*(x, k)amn_{d}$volume$\cross Z(f_{n}^{*}(x, k),$$n/2)$,

where $M$ is the Levi part of(as in the definition) and $dm$ is the Haar measure of $M$ .
Hence the final integral is given by easy constant times. So, we get the desiredresult.
Each tube domain of type (I), (II), or (III) corresponds to the Jordan algebra of type(II), (III), or (I) respectively. We can show that the contribution of central unipotent
elements is again given by $\zeta(r_{0}(r-n), L^{*})\cross$ elementary factor, where describes ”rank”(i.e. matrix size of the corresponding center) of corresponding cusps and $r_{0}=1,2,$ $or4$ for
Jordan algebra type (I), (II), or (III), respectively. We omit the details here.
Remark. All the special values which appear above can be calculated explicitly. Thisis one of the main part of our new theory. Hence the above results mean that we can getlarge part (conjecturally ”all” if torsion free) of the dimension formula explicitly.
$P_{0}$$\zeta(s, Q^{-1})$$r$

Reference
1. H. Braun and M. Koecher, Jordan Algebren, Springer Verlag, 1966.

2. I. Satake, Algebraic structures of symmetric domains, Iwanami Shoten and Princeton
University Press, 1980.
3. T. Shintani, On zeta functions associated with the vector space of quadratic forms,
J. Fac. Sci. Univ. Tokyo, Sect.
, 22(1975),.
$\mathrm{I}\mathrm{A}$$2\triangleright 65$