# Representations of Clifford algebras and local functional equations (New developments in group representation theory and non-commutative harmonic analysis)

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(1)Title. Author(s). Citation. Issue Date. URL. Representations of Clifford algebras and local functional equations (New developments in group representation theory and non-commutative harmonic analysis) SATO, Fumihiro; KOGISO, Takeyoshi. 数理解析研究所講究録別冊 (2012), B36: 53-66. 2012-12. http://hdl.handle.net/2433/198112. Right. Type. Textversion. Departmental Bulletin Paper. publisher. Kyoto University.

(2) RIMS Kôkyûroku Bessatsu B36 (2012), 53−66. Representations. of Clifford. algebras equations. functional. and local. By. Fumihiro SATO. *. and. KOGISO. Takeyoshi. **. Introduction. Let P and P^{*} be. coefficients. find. a. It is. condition. an. on. homogeneous polynomials. interesting problem. speaking,. (0.1). the Fourier transform of. A beautiful. answer. to this. corresponding. satisfy. a. Namely,. vector space and its. to P and P^{*}. ,. functional equation. Meanwhile,. in. [5],. real) of. simple. algebras.. Jordan. satisfying (0.1),. (see. also Clerc. Received. a. given by. the. if P and P^{*}. which. are. Thus. d with real. and in Number. of. to. roughly. prehomogeneous. relative invariants of. are. theory. \times|P^{*}(y)|^{-n/d-s}. theory. a. vector. regular. and if the characters $\chi$ and. dual, respectively,. the relation. $\chi \chi$^{*}=1 then, ,. $\chi$^{*}. P and P^{*}. (see [9], [10], [6]). Koranyi developed. (0.1), starting. a. method of. not covered. got. by. constructing poly‐. from representations of Euclidean. of Lorentzian type,. we. degree. functional equation,. What is remarkable in their result is. algebras. [4]).. Analysis. Gamma factor. respectively, satisfy. Faraut and. nomials with the property. Jordan. is. variables of. n. they satisfy. |P(x)|^{s}=. problem. spaces due to Mikio Sato.. prehomogeneous. both in. P and P^{*} under which. of the form. in. the. one. can. theory. of. to know that the class of. that,. obtain. a. (formally. from representations series of. prehomogeneous polynomials. polynomials. vector spaces. with the property. September 11, 2009. Accepted December 28, 2009. Subject Classification(s): Primary 11\mathrm{E}45, 11\mathrm{E}88, 11\mathrm{S}41 ; Secondary 15\mathrm{A}63,. 2000 Mathematics. 15\mathrm{A}66.. Key Words:. local zeta. function,. local functional. equation, Clifford algebra, prehomogeneous. vector. space.. The first and second authors. are partially supported by the grant in aid of scientific research of 20540021, respectively Department of Mathematics, Rikkyo University, 3‐34‐1 Nishi‐Ikebukuro, Toshima‐ku, Tokyo, 171‐ 8501, Japan. \mathrm{e} ‐mail: sato@rikkyo.ac.jp Department of Mathematics, Josai University, 1‐1 Keyakidai, Sakado, Saitama, 350‐0295, Japan. \mathrm{e} ‐mail: kogiso@math.josai.ac.jp. JSPS No.20540028 and. *. **. ©. 2012 Research Institute for Mathematical. Sciences, Kyoto University.. All. rights. reserved..

(3) Fumihiro Sato. 54. (0.1). and. Takeyoshi Kogiso. is broader than the class of relative invariants of. regular prehomogeneous. vector. spaces.. [7],. In. (0.1),. the first author gave. construction of. a new. which includes the result of Faraut and. briefly. Suppose. the construction. that. we are. real vector spaces V and its dual V^{*} ,. on a. Koranyi. with the property. polynomials. as a. special. case.. Now. we. given homogeneous polynomials. respectively, satisfying. explain. P and P^{*}. functional equation of. a. (0.1). Further suppose that there exists a non‐degenerate quadratic mapping Q (resp. Q^{*} ) of another real vector space W (resp. W^{*} ) to V (resp. V^{*} ), and Q and Q^{*} are. the form. pullback of the functional equation for P and P^{*} by Q holds; namely, pullbacks \tilde{P} and \tilde{P}^{*} of P and P^{*} by Q and Q^{*} respectively, satisfy a functional. dual. the. the. Then,. ,. equation of the form (0.1) and the. in term of those for P and P^{*}. explicit expression. an. will be. given. gamma factors for the. in Section 1. For the. In Section. 2,. apply. we. the. proof. .. refer to. we. new. functional equation have. A precise formulation of this result. [7].. result in Section 1 to the. general. where V=V^{*}=. case. P=P^{*}=(x_{1}^{2}+\cdots+x_{p}^{2})-(x_{p+1}^{2}+\cdots+x_{p+q}^{2}) Let C_{p} and C_{q} be the Clifford algebras of the positive definite quadratic forms x_{1}^{2}+\cdots+x_{p}^{2} and x_{p+1}^{2}+\cdots+x_{p+q}^{2},. \mathbb{R}^{n} , and. .. Then. respectively. Q. W\rightarrow V. :. we. correspond. can. to. from representations of. satisfying several. from. prove that. non‐degenerate. representations of the tensor. C_{p}\otimes C_{q}. we. ,. can. construct. quadratic mappings. of. C_{p}\otimes C_{q} and, starting. product. quartic polynomials \tilde{P}=Po Q. (0.1). Among. functional equations of the form. new. self‐dual. these. polynomials. examples of polynomials satisfying functional equations. prehomogenous. The. vector spaces.. we. that do not. find. come. with the. non‐prehomogeneous polynomials. (0.1) appearing in the work of Faraut, Koranyi and Clerc is a special case where signature of the quadratic forms P is (1, n-1) To prove that a given homogeneous. property the. .. polynomial \tilde{P}. does not. know about the group. give. the. quadratic form. a. G_{P^{-}}. conjecture of. We. come. from. a. prehomogeneous. of linear transformations that leave the. the structure of the Lie. P and. vector space, it is necessary to. explain. some. partial. algebras. polynomial. Lie (G_{P^{-}}) for the. invariant.. pullback \tilde{P} of. results.. global zeta functions with functional equations can be polynomials \tilde{P} and \tilde{P}^{*} given in [7]. For polynomials obtained from the. It is natural to ask whether. associated with. theory. of Faraut and. method works not. of. apply. to. only. our. Koranyi,. for the. case. this. was. where the fibers. general setting.. prehomogeneous. problem. vector spaces,. If the. solved. by Achab. in. Q^{-1}(v)(P(v)\neq 0). polynomials. P and P^{*}. then, by generalizing. [1]. are. are. and. [2].. But her. compact and. can. relative invariants. the method of Arakawa. [3]. [11], we can define global zeta functions for \tilde{P} and prove their analytic properties (analytic continuation and functional equation) (work with K. Tamura). We and Suzuki. shall discuss. global. zeta functions elsewhere..

(4) Representations. 0F. Clifford. algebras and local functional. Pullback of local functional. §1. In this. section,. a. as a. polynomial. V. on. .. Let. P_{1}. .. ,. .. \mathrm{V}^{*} ) defined. (resp.. .. ,. with real‐structure V and \mathrm{V}^{*} the. n. of the real vector space V. The dual vector space V^{:}. real‐structure of \mathrm{V}^{*}. functions. equations. vector space of dimension. complex. vector space dual to V.. regarded. [7].. Local functional. §1.1. Let V be. equations by quadratic mappings. recall the main result of. we. 55. equations. P_{r}. P_{1}^{*}. resp.. \mathbb{R}. over. ,. .. .. .. ,. P_{r}^{*} ). be. be. can. homogeneous. We put. .. $\Omega$=\{v\in \mathrm{V}|P_{1}(v)\cdots P_{r}(v)\neq 0\}, $\Omega$= $\Omega$\cap V,. $\Omega$^{*}=\{v*\in \mathrm{V}^{*}|P_{1}^{*}(v^{*})\cdots P_{r}^{*}(v^{*})\neq 0\}, $\Omega$^{*}=$\Omega$^{*}\cap V. We. that. assume. (A.1). there exists. a. biregular. rational. mapping $\phi$. $\Omega$\rightarrow$\Omega$^{*} defined. :. over. \mathbb{R}.. Let. $\Omega$=$\Omega$_{1}\cup\cdots\cup$\Omega$_{l $\nu$}, $\Omega$^{*}=$\Omega$_{1}^{*}\cup\cdots\cup$\Omega$_{l $\nu$}^{*} be the. decompositions. into connected components of $\Omega$ and $\Omega$^{*}. that the numbers of connected components of $\Omega$ and $\Omega$^{*} assume. (sl,. an s=. |P(v)|_{j}^{s}. are. Note that. (A.1) implies. the. and. same. we. may. that. $\Omega$_{j}^{*}= $\phi$($\Omega$_{j}) (j=1, \ldots, v) For. .. V. on. .. .. .. ,. s_{r}. ). \in \mathbb{C}^{r} with. \Re(s_{1}). ,. .. .. .. ,. \Re(s_{r})>0. ,. .. we. define. a. continuous function. by. |P(v)|_{j}^{s}. The function. |P(v)|_{j}^s}=\left\{ begin{ar y}{l \prod_{i=1}^{r}|P_{i}(v)|^{s_i},&v\in$\Omega$_{j},\ 0,&v\not\in$\Omega$_{j}. \end{ar y}\right. can. be extended to. meromorphically. Similarly We denote. by S(V). we. and. real vector spaces V and V^{*} , the local zeta functions. define. S(V^{*}). a. tempered. distribution. |P^{*}(v^{*})|_{j}^{s}(s\in \mathbb{C}^{r}). the spaces of. respectively.. For. depending. on s. in \mathbb{C}^{r}. .. rapidly decreasing. $\Phi$\in S(V). and. functions. $\Phi$^{*}\in S(V^{*}). ,. we. on. the. define. by setting. $\zeta$_{i}(s, $\Phi$)=\displaystyle \int_{V}|P(v)|_{i}^{s} $\Phi$(v)dv, $\zeta$_{i}^{*}(s, $\Phi$^{*})=\displaystyle \int_{V^{*} |P^{*}(v^{*})|_{i}^{s}$\Phi$^{*}(v^{*})dv^{*} (i=1, \ldots, v) It is well‐known that the local zeta functions. for. \Re(s_{1}). in \mathbb{C}^{r}. .. ,. We. .. .. .. ,. \Re(s_{r})>0. assume. the. and have. following:. analytic. $\zeta$_{i}(s, $\Phi$) $\zeta$_{i}^{*}(s, $\Phi$^{*}) ,. continuations to. are. .. absolutely convergent. meromorphic functions of. s.

(5) Fumihiro Sato. 56. (A.2). There exist. A\in GL_{r}(). an. and. and. a. Takeyoshi Kogiso. $\lambda$\in \mathbb{C}^{r} such that. a. functional equation of the. form. $\zeta$_{i}^{*}( s+$\lambda$)A,\displaystyle\hat{$\Phi$})=\sum_{j=1}^{l$\nu$} \Gam a$_{ij}(s)$\zeta$_{j}(s,$\Phi$) (i=1, \ldots, v). (1.1).$\Phi$\in S(V) where$\Gamma$_{ij}(s) with \det($\Gamma$_{ij}(s))\neq 0 and. holds for every. depending. on.$\Phi$. meromorphic functions. are. ,. on. \mathbb{C}^{r} not. \displaystyle \hat{$\Phi$}(v^{*})=\int_{V}$\Phi$(v)\exp(-2$\pi$\sqrt{-1}\langle v, v^{*}\rangle)dv, the Fourier transform of$\Phi$.. A lot of. examples of \{P_{1}, . . . , P_{r}\}. be obtained from relative invariants of. [10], [6]). However, relates the. §1,. in. polynomials. to. complex. a. do not. regular prehomogeneous. assume. prehomogeneous. vector. vector spaces.. vector space of dimension. m. equations. with real structure W and \mathrm{W}^{*} the. vector space dual to W. We consider the dual vector space W^{*} of W. of \mathrm{W}^{*}. .. defined. Suppose \mathbb{R}. over. that The. .. we are. given quadratic mappings Q. mappings B_{Q}. :. (A.2) can spaces (see [9], and. here the existence of group action that. Pullback of local functional. §1.2. Let \mathrm{W} be. we. \{P_{1}^{*}, . . . , P_{r}^{*}\} satisfying (A.1). and. \mathrm{W}\times \mathrm{W}\rightarrow \mathrm{V} and. as a. \mathrm{W}\rightarrow \mathrm{V} and. :. real structure. Q^{*}:\mathrm{W}^{*}\rightarrow \mathrm{V}^{*}. B_{Q^{*} :\mathrm{W}^{*}\times \mathrm{W}^{*}\rightarrow \mathrm{V}^{*}. defined. by. B_{Q}(w_{1}, w_{2}):=Q(w_{1}+w_{2})-Q(w_{1})-Q(w_{2}) are. bilinear.. For. \mathrm{W}^{*}\rightarrow \mathbb{C} defined. given v\in \mathrm{V}. ,. B_{Q^{*}}(w_{1}^{*}, w_{2}^{*}):=Q^{*}(w_{1}^{*}+w_{2}^{*})-Q^{*}(w_{1}^{*})-Q^{*}(w_{2}^{*}). and v^{*}\in \mathrm{V}^{*} , the. mappings Q_{v}*. :. \mathrm{W}\rightarrow \mathbb{C} and. Q_{v}^{*}. :. by. Q_{v^{*}}(w)=\langle Q(w) , v^{*}\rangle, Q_{v}^{*}(w^{*})=\langle v, Q^{*}(w^{*})\rangle are. quadratic forms. We. assume. the. biregular mapping$\phi$. that. Q. on. \mathrm{W} and \mathrm{W}^{*} , which take values in \mathbb{R}. and. Q^{*} in. (A.3) (i) (Nondegeneracy). are. This. means. The open set. that. n. and. Namely,. S_{v}^{*}. Then. and. Q^{*} satisfy. the. following:. at. is not. w\in\tilde{$\Omega$} (resp. w^{*}\in\tilde{$\Omega$}^{*} ). is. .. ,. other.. Q. (resp. Q^{*} ). (In particular, m\geq n. ) (ii) (Duality) For any v\in$\Omega$the quadratic to. respectively.. \tilde{$\Omega$}:=Q^{-1} () (resp. \tilde{$\Omega$}^{*}=Q^{*-1}($\Omega$^{*}) ). empty and the rank of the differential of Q. equal. W and W^{*} ,. and dual to each other with respect to. non‐degenerate. (A.1).. on. fix. a. basis of W W and the basis of W^{:} W. the matrices of the. S_{$\phi$(v)}. and. forms. S_{v}^{*}(v\in$\Omega$). Q_{$\phi$(v)}. dual to. quadratic forms Q_{Q_{v}*\\matmathrmhrm{{a}\mata}\hmatrm{nh}\rm{langlne}( Q_{v}^{*} are. non‐degenerate. and. and. Q_{v}^{*}. are. dual to each. it, and denote by S_{v}*. with respect to the bases.. S_{$\phi$(v)}=(S_{v}^{*})^{-1}.. (6) Representations. Now. collect. we. First note that. a.$\Omega$^{*}) (A.3) (ii) imply is. a. some. algebras and local functional. elementary. consequences of the. rational function defined. monomial of P_{1} , the. .. .. .. ,. following If. Lemma 1.1.. necessary), (1) there. Clifford. 0F. exists. a. P_{r} (resp. P_{1}^{*}. ,. .. .. .. ,. P_{r}^{*} ).. assumptions (A.1) and (A.3).. no zeros. Hence the. P_{j}^{*},$\phi$by. B=(b_{ij})\in GL(). There exist$\kappa$,$\kappa$^{*}\in \mathbb{Z}^{r} and. The. mapping$\phi$. a non‐zero. of degree. is. constant.$\alpha$. -1 and there exists. \displaystyle \det(\frac{\partial$\phi$(v)_{i} {\partial v_{j} )=\pm P^{$\mu$}(v) If P_{1} ,. .. .. .. ,. P_{r} and P_{1}^{*}. prehomogeneous the. Indeed, by is. a. ,. .. .. .. vector space. regularity,. G ‐equivariant. ,. P_{r}^{*}. we. (G,$\rho$, \mathrm{V}). there exists. have.$\Omega$. on. .. such that. a. ..$\mu$\in \mathbb{Z}^{r}. such that. .. a. and its dual. (G,$\rho$^{*}, \mathrm{V}^{*}). ,. then. relative invariant P for which. we. a. (see [6]). It is very likely assumption (A.1) and (A.2) and,. regular. have B=A^{-1}..$\phi$(v)= gradlog P. From the G ‐equivariance of the. B=A^{-1}. B=A^{-1} always holds under the. multiples (if. the fundamental relative invariants of. are. morphism satisfying (A.1)..$\phi$([8, §4, Prop. 9]),. poles. such that. \det S_{v}^{*}=$\alpha$^{-1}P^{$\kappa$}(v) , \det S_{v}*=$\alpha$P^{*$\kappa$^{*}}(v^{*}) (3). no. their suitable real constant. P_{i}^{*}($\phi$(v) =\displaystyle \prod_{j=1}^{r}P_{j}(v)^{b_{ij} (i=1, \ldots, r) (2). (resp. assumptions (A.1) and. and. lemma.. replace P_{i},. we. \mathbb{R} with. over. 57. equations. that the. for. mapping. identity. simplicity,. we. assume. (A.4) B=A^{-1}. Since. we. assumed that.$\Omega$_{i} (resp.$\Omega$_{i}^{*} ). are. connected components, the. signature of. quadratic form Q_{v}^{*}(w^{*}) (resp. Q_{v}*(w) ) on W^{*} (resp. W ) do not change when v (resp. v^{*} ) varies on$\Omega$_{i} (resp.$\Omega$_{i}^{*} ). Let p_{i} and q_{i} be the numbers of positive and negative. the. eigenvalues. of. Q_{v}^{*}. for. v\in$\Omega$_{i} and put.$\gamma$_{i}=\displaystyle \exp(\frac{(p_{i}-q_{i})$\pi$\sqrt{-1} {4}) (i=1, \ldots, v). (1.2). .. We put. \tilde{P}_{i}(w)=P_{i}(Q(w)) , \tilde{P}_{i}^{*}(w^{*})=P_{i}^{*}(Q^{*}(w^{*})) (i=1, \ldots, r) \tilde{$\Omega$}_{i}=Q^{-1}($\Omega$_{i}) , \tilde{$\Omega$}_{i}^{*}=Q^{*-1}($\Omega$_{i}^{*}) (i=1, \ldots, v) .. (7) Fumihiro Sato. 58. Some of. \tilde{$\Omega$}_{i} s in. manner as. \tilde{$\Omega$}_{i}^{*\prime}\mathrm{s}. and. §1.1.. may be. For. and. Takeyoshi Kogiso. empty. We define.$\Psi$\in S(W). and. |\tilde{P}^{*}(w^{*})|_{i}^{s}. in the. same. define the zeta functions associated with these. we. ,. |\tilde{P}(w)|_{i}^{s}. polynomials by. \displaystyle \tilde{$\zeta$}_{i}(s,$\Psi$)=\int_{W}|\tilde{P}(w)|_{i}^{s}$\Psi$(w)dw, \tilde{$\zeta$}_{i}^{*}(s,$\Psi$^{*})=\int_{W^{*} |\tilde{P}^{*}(w^{*})|_{i}^{s}$\Psi$^{*}(w^{*})dw^{*} by \hat{$\Psi$}. We denote. the Fourier transform of$\Psi$:. \displaystyle \hat{$\Psi$}(w^{*})=\int_{W}$\Psi$(w)\exp(2$\pi$\sqrt{-1}\langle w, w^{*}\rangle)dw. Then a. our. main result is that the functional. functional equation for. equation. can. be written. Theorem 1.2. functions. \tilde{$\zeta$}_{i}(s,$\Psi$). ([7],. and. \tilde{P}_{i} s. and. \tilde{P}_{j}^{*\prime}\mathrm{s}. and the gamma factors in the. explicitly. Namely, Theorem. \tilde{$\zeta$}_{i}^{*}(s,$\Psi$^{*}). equation (1.1) for P_{i} s and. 4).. we. have the. following. Under the assumptions. satisfy. the. factors. \tilde{$\Gamma$}_{ij}(s). are. new. (A.1)(A.4),. we. denote. by d(s)(s\in \mathbb{C}^{r}). the zeta. functional equation. ,. given by. \displaystyle \tilde{$\Gam a$}_{ij}(s)=2^{-2d(s)-m/2}|$\alpha$|^{1/2}\sum_{k=1}^{l$\nu$}$\gam a$_{k}$\Gam a$_{ik}(s+$\lambda$+$\kap a$/2+$\mu$)$\Gam a$_{kj}(s) Here. functional. theorem.. \displaystyle\tilde{$\zeta$}_{i}^{*}( s+2$\lambda$+$\kap a$/2+$\mu$)A,\hat{$\Psi$})=\sum_{j=1}^{l$\nu$}\tilde{$\Gam a$}_{ij}(s)\tilde{$\zeta$}_{j}(s,$\Psi$) where the gamma. P_{j}^{*\prime}\mathrm{s} implies. homogeneous degree of. the. P^{s} ,. .. namely, d(s)=. \displaystyle \sum_{i=1}^{r}s_{i}\deg P_{i}. In the. case. P=P_{1} and. of. single. P^{*}=P_{1}^{*}. Lemma 1.3.. ,. we. variable zeta have the. functions, namely,. following. Assume that r=1. .. in the. case. of r=1 ,. writing. lemma.. Then. we. have. A=B=-1, d:=\displaystyle \deg P=\deg P^{*},$\lambda$=\frac{n}{d},$\mu$=-\frac{2n}{d},$\kappa$=\frac{m}{d}. By. Lemma. 1.3, if. r=1 , then the functional. equation for local. zeta functions takes. the form. \displaystyle\tilde{$\zeta$}_{i}^{*}(-s\frac{m}{2d},\hat{$\Psi$})=\sum_{j=1}^{l$\nu$}\tilde{$\Gam a$}_{ij}(s)\tilde{$\zeta$}_{j}(s,$\Psi$) \displaystyle \tilde{$\Gamma$}_{ij}(s)=2^{-2ds-m/2}|$\alpha$|^{1/2}\sum_{k=1}^{l$\nu$}$\gamma$_{k}$\Gamma$_{ik}(s+\frac{m-2n}{2d})$\Gamma$_{kj}(s) ,. (1.3). (8) Representations. and the b ‐function is. Clifford. 0F. \displaystyle \tilde{b}(s)=b(s)b(s+\frac{m-2n}{2d}). where. b(s). \tilde{b}(s). and. \tilde{b}(s)\tilde{P}^{s-1}(v). 59. equations. given by. (1.4). defined. are. ,. P^{*}(\partial_{v})P^{s}(v)=b(s)P^{s-1}(v). by. and. \tilde{P}^{*}(\partial_{w})\tilde{P}^{s}(w)=. .. Example put. algebras and local functional. Let V be the vector space of real. 1.4.. P(v)=\det v. Take. .. non‐degenerate. a. real. symmetric. symmetric. matrices of size. n. and. matrix Y of size m>n with. arbitrary signature. Set W=M_{m,n}(\mathbb{R}) and define the quadratic mapping Q:W\rightarrow V by Q(w)=t_{wYw} Then \tilde{P}(w)=\det(^{t}wYw ). The polynomial \tilde{P} is the fundamental .. (SO(Y)\times GL(n), M_{m,n}) If we identify the dual space of V (resp. W ) with V (resp. W ) via the inner product \langle v, v^{*}\rangle= \mathrm{t}\mathrm{r}(^{t}vv^{*}) (resp. \langle w, w^{*}\rangle=\mathrm{t}\mathrm{r}(^{t}ww^{*} ), the dual of the mapping Q is given by Q^{*}(w^{*})= relative invariant of the. prehomogeneous. t_{w^{*}Y^{-1}w^{*}} and the theorem. Example a. to this. apply. [5, Chap. 16],. In. 1.5.. can. vector space. case.. Clerc. equations.. generalized. by. some. are. cases. dimensional real. for which the. quadratic. without. space of. quadratic form. is. by. (see [7, §2.2]).. to several variable zeta. Koranyi. obtained. prehomogeneous. (without specifying. non‐prehomogeneous example. polynomials satisfying. Theorem 1.2. polynomials \tilde{P}. not relative invariants of. low‐dimensional. construct. [4],. In. functions,. 1.2, and noted that, if the Euclidean Jordan alge‐. Theorem. bra V is of Lorentzian type, then the construction. one can. Their result is covered. the result of Faraut and. which is also covered. Koranyi proved that, starting from. Faraut and. representation of a Euclidean Jordan algebra,. local functional. .. referring. the. .. We fix. algebra. a. the. Faraut‐Koranyi. vector spaces. exceptions).. to Jordan. signature ( 1, q). by. basis. Let. except for. explain. us. Let V be the. this. q+1-. \{e_{0}, e_{1}, . . . , e_{q}\}. of. V,. given by. P ( x_{0} , xl,. .. .. .. ,. x_{q} ). =x_{0}^{2}-x_{1}^{2}-\cdots-x_{q}^{2}.. by C_{q} the Clifford algebra of the positive definite quadratic form x_{1}^{2}+\cdots+x_{q}^{2} and consider a representation S:C_{q}\rightarrow M_{m}() of C_{q} on an m ‐dimensional \mathbb{R}‐vector space. Denote. We may. assume. W=\mathbb{R}^{m} the. that. S_{i}:=S(e_{i})(i=1, \ldots, q). are. representation space of S and define. a. symmetric. matrices. We denote. quadratic mapping Q. :. W\rightarrow V. by by. Q(w)=({}^{t}ww)e_{0}+\displaystyle \sum_{i=1}^{q}({}^{t}wS_{1}w, \ldots,{}^{t}wS_{q}w)e_{i}. Then, if a. \displaystyle \tilde{P}(w)=P(Q(w))=(^{t}ww)^{2}-\sum_{i=1}^{q}(^{t}wS_{i}w)^{2}. self‐dual. non‐degenerate quadratic mapping and, by. functional equation. In the next section, the. prehomogeneity.. we. does not vanish Theorem. generalize. identically, Q. 1.2, \tilde{P} satisfies. a. is. local. this construction and examine. (9) Fumihiro Sato. 60. Remark. In harmonic. [4],. Clerc. proved. Takeyoshi Kogiso. local functional. This part is not covered. polynomials.. Quartic polynomials. §2.. and. equations also for. zeta functions with. Theorem 1.2.. by. obtained from. representations. of Clifford. algebras Let p, q be. \displaystyle \sum_{j=1}^{q}x_{p+j}^{2}. non‐negative integers. signature (p, q). of. on. and consider the. V=\mathbb{R}^{p+q}. .. quadratic form. identify. We. product (x, y)=x_{1}y_{1}+\cdots+x_{p+q}y_{p+q} Put$\Omega$=V\backslash \{P=0\}. quadratic mappings Q:W\rightarrow V that are self‐dual with respect to. via the standard inner. We determine the the. By. .. biregular mapping$\phi$.$\phi$(v).$\Omega$\rightarrow$\Omega$defined. :. by. :=\displayst le\frac{1}2 grad \displaystyle \log P(v)=\frac{1}{P(v)} (vl,. 1.2, for such. Theorem. a. .. .. .. v_{p}, -v_{p+1},. ,. quadratic mapping Q. ,. the. a. matrices. quadratic mapping Q of W=\mathbb{R}^{m}. S_{1}. ,. .. .. .. ,. S_{p+q}. of size. m. v=(x_{1}, \ldots, x_{p+q})\in \mathbb{R}^{p+q}. ,. with. V=\mathbb{R}^{p+q}. to. ,. -v_{p+q} ). power of the. explicit. gamma. there exist. quartic. factor,. symmetric. such that. Q(w)=({}^{t}wS_{1}w, \ldots,{}^{t}wS_{p+q}w) For. \ldots,. complex. polynomial \tilde{P}(w) :=P(Q(w)) satisfies a functional equation unless \tilde{P} vanishes identically (see [7, Lemma 6]). For. P(x)=\displaystyle \sum_{i=1}^{p}x_{i}^{2}-. V with its dual vector space. we. .. put. S(v)=\displaystyle \sum_{i=1}^{p+q}x_{i}S_{i}. Then the. mapping Q. is self‐dual with respect to.$\phi$. if and. S(v)S($\phi$(v))=I_{m} (v\in$\Omega$) If. we. the. define$\epsilon$_{i} to be 1. or. -1. according. as. i\leq p. or. only. if. .. i>p this condition is equivalent ,. polynomial identity. \displaystyle \sum_{i=1}^{p}x_{i}^{2}S_{i}^{2}-\sum_{j=1}^{q}x_{p+j}^{2}S_{p+j}^{2}+\sum_{1\leq i<j\underline{<}p+q}x_{i}x_{j}($\epsilon$_{j}S_{i}S_{j}+$\epsilon$_{i}S_{j}S_{i})=P(x)I_{m}. This. identity. holds if and. only. if. S_{i}^{2}=I_{m}(1\leq i\leq p+q). ,. S_{i}S_{j}=\left\{ begin{ar ay}{l} S_{j}S_{i}&(1\leqi\leqp<j\leqp+q\mathrm{o}\mathrm{r}1\leqj\leqp<i\leqp+q)\ -S_{j}S_{i}&(1\leqi,j\leqporp+1\leqi,j\leqp+q). \end{ar ay}\right.. to. (10) Representations. This. of the tensor. C_{q}. algebras and local functional. mapping S:V\rightarrow Sym(). that the. means. Clifford. 0F. product. of the Clifford. algebra C_{p}. of. x_{p+1}^{2}+\cdots+x_{p+q}^{2}.. of. Conversely, representation S tensor. if. we. is. a. are. direct. product of simple. representation. space. given. so. sum. that. x_{1}^{2}+\cdots+x_{p}^{2}. representation S. of. simple. S(\mathbb{R}^{p+q}). modules. and. C_{p}. C_{q}. and the Clifford. :. algebra. ,. Since. .. representation. a. C_{p}\otimes C_{q}\rightarrow M_{m}() then and a simple C_{p}\otimes C_{q} ‐module. a. modules of. be extended to. can. 61. equations. is contained in. one. can. Sym_{m}(\mathbb{R}). ,. choose. we. a. have. the is. a. basis of the. proved. that. Theorem 2.1. space. Self‐ dual quadratic mappings Q of W=\mathbb{R}^{m} to the quadratic (V, P) correspond to representations S of C_{p}\otimes C_{q} such that S(V)\subset Sym_{m}(\mathbb{R}) .. The construction above is. generalization. a. of. Example. 1.5 related to representa‐. (p, q)=(1, q) quadratic mappings quadratic signature (1, q) correspond to representations of C_{1}\otimes C_{q}\cong C_{q}\oplus C_{q} Representations of C_{1}\otimes C_{q} can be identified with the direct sum of 2 C_{q} ‐modules W+ and W_{-} On W+( resp. W_{-}) e_{0} acts as mul‐ tiplication by +1 (resp. -1 ). The Lorentzian case in the Faraut‐Koranyi construction is the one for which W_{-}=\{0\}. tions of. simple. Euclidean Jordan. the self‐dual. algebra. over. of Lorentzian type. In the. case. ,. space of. the .. .. The. quartic polynomials. variants of. prehomogeneous. interesting problem Theorem 2.2. variants. The. following. to. \tilde{P}(=\tilde{P}^{*}). above. vector spaces. classify. the. prehomogeneous. are. conjectured. not to be relative in‐. except for low‐dimensional. prehomogeneous. If p+q=\dim V\leq 4. of prehomogeneous. ,. ,. cases.. polynomials \tilde{P}. are. relative in‐. vector spaces.. vector spaces. appearing. in the. case. p+q\leq 4. are. given. table: vector space. (1, 0) (\mathrm{G}\mathrm{L}(1, ) \mathrm{S}\mathrm{O}(\mathrm{k} ,k), ) (2,0) (\mathrm{G}\mathrm{L}(1, ) \mathrm{S}\mathrm{O}(\mathrm{k}, ), ) \ma t h r m { G } \ ma t h r m { L } (1,1)( (1, ) \mathrm{S}\mathrm{O}(\mathrm{k} ,k), ) (\mathrm{G}\mathrm{L}(1, ) \mathrm{S}\mathrm{O}(\mathrm{k} ,k), (3, 0) (\mathrm{G}\mathrm{L}(1, ) SU(2) SO (2k), ) (2 1) (\mathrm{G}\mathrm{L}(2, ) \mathrm{S}\mathrm{O}(\mathrm{k} k), \mathrm{M}(2,\mathrm{k} + \mathrm{k} )) (4, 0) (\mathrm{G}\mathrm{L}(1, ) \mathrm{G}\mathrm{L}(1, ) \mathrm{G}\mathrm{L}(\mathrm{k}, ),\mathrm{M}(2,\mathrm{k}, )) (3 1) (\mathrm{G}\mathrm{L} () SU (\mathrm{k} k), \mathrm{M}(2, \mathrm{k} + \mathrm{k} ; )) (2, 2) (\mathrm{G}\mathrm{L} () \mathrm{G}\mathrm{L} () \mathrm{S}\mathrm{L}(\mathrm{k}, ), \mathrm{M}(2,\mathrm{k};) ,. ,. ,. ). ,. ,. (Table 1) k, k_{1}, k_{2}, k_{3}, k_{4}. an. case.. then the. prehomogeneous. Here. It is. denote the. multiplicities of simple C_{p}\otimes C_{q} ‐modules. in W.. in the. (11) Fumihiro Sato. 62. \tilde{P}. If p+q=5 , then. except the. spaces. However, if W. case. is not. of. is not. simple. relative invariant of any. a. module. If. simple. Takeyoshi Kogiso. and. but pure, then. p+q=6 and W. \tilde{P}. is. a. is. prehomogeneous vector simple, then \tilde{P} vanishes.. relative invariant of the. prehomogeous. vector space. (3, 3) (\mathrm{G}\mathrm{L} () \mathrm{S}\mathrm{p}(\mathrm{k}, ),\mathrm{M}(4,2\mathrm{k};)) (\mathrm{k} (5, 1) (\mathrm{G}\mathrm{L} () \mathrm{S}\mathrm{p}(\mathrm{k} k), \mathrm{M}(2, \mathrm{k} + \mathrm{k} ; )) (\mathrm{k} ,. 2, \mathrm{W} + \mathrm{k}. =. pure). 2, \mathrm{W}. =. pure). (Table 1) Here. a. C_{p}\otimes C_{q} ‐module. subalgebra. (6,0). and. of. even. W is called pure if. elements in. (C_{p}\otimes C_{q})\otimes_{\mathbb{R} \mathbb{C}. higher. dimensional cases,. G=\{g\in GL(W)|\tilde{P}(gw)\equiv\tilde{P}(w)\} \{h\in GL(W)|Q(hw)\equiv Q(w)\}. Our. .. as. module of the. a. (p, q)=. Pure modules do not appear for. (4, 2).. To examine. the Lie. W\otimes_{\mathbb{R} \mathbb{C} is isotypic. algebras. problem. \mathfrak{g} and. \mathfrak{h} depend. on. consider the Lie. \mathfrak{g} of the group. algebra. algebra \mathfrak{h}(=\mathfrak{h}_{p,q}) of the algebra \mathfrak{h} is a Lie subalgebra of \mathfrak{g}. and the Lie. The Lie. .. we. p, q and the choice of the. is to determine all the. cases. where. group H= .. Note that. representation of C_{p}\otimes C_{q}.. (GL_{1}\times G, W). is. a. prehomogeneous. vector space.. Conjecture. We have. 2.3.. \mathfrak{g}\cong \mathfrak{s}0(p, q)\oplus \mathfrak{h} except for. some. low dimensional. The structure of. \mathfrak{h}. can. be described. C_{p+8}\cong C_{p}\otimes M(16, \mathbb{R}) C_{p+8}\otimes C_{q} and representations ,. there exists. of. \mathfrak{h}. is the. same. for. cases.. of. a. explicitly. By the periodicity. natural. C_{p}\otimes C_{q}. correspondence. and it. corresponding representations.. (2.1). can. This. be. between. proved. implies. of Clifford. the. algebras. representations. that the structure of. isomorphisms. \mathfrak{h}_{p,q}\cong \mathfrak{h}_{q,p}\cong \mathfrak{h}_{p+8,q}\cong \mathfrak{h}_{p,q+8}.. Similarly, by C_{p+4}\cong C_{p}\otimes M(2, \mathbb{H}) isomorphism. (2.2). and. M_{2}(\mathbb{H})\otimes M_{2}()\cong M_{16}(\mathbb{R}). ,. we. have the. \mathfrak{h}_{p,q}\cong \mathfrak{h}_{p+4,q\pm 4}.. Hence it is sufficient to. Theorem 2.4.. give. the structure of. The Lie. algebra \mathfrak{h}. is. \mathfrak{h} only for 0\leq p\leq 7. isomorphic. and. 0\leq q\leq 4.. to the reductive Lie. algebra given. (12) Representations. in the. following. Clifford. 0F. algebras and local functional. 63. equations. table:. so(\mathrm{k}, so(\mathrm{k} k) ) \mathrm{g}\mathrm{l}(\mathrm{k}, ) \ m a t h r m { k } so(\mathrm{k},\mathrm{k}) \mathrm{s}\mathrm{o}(\mathrm{k},\mathrm{k}) so(\mathrm{k},\mathrm{k})\mathrm{s}\mathrm{o}( ,k) \mathrm{g}\mathrm{l}(\mathrm{k}, ) \mathrm{s}\mathrm{p}(\mathrm{k}, ) \mathrm{s}\mathrm{p}(\mathrm{k}, ) \mathrm{s}\mathrm{p}(\mathrm{k}, ) \mathrm{s}\mathrm{p}(\mathrm{k}, ) \mathrm{s}\mathrm{p}(\mathrm{k},\mathrm{k})\mathrm{s}\mathrm{p}(\mathrm{k},\mathrm{k}) \mathrm{s}\mathrm{p}(\mathrm{k},\mathrm{k})\mathrm{s}\mathrm{p}(\mathrm{k} ,k) \mathrm{g}\mathrm{l}(\mathrm{k}, ) ,. 1. 5. so. (Table 2) Here \mathrm{p}=p mod8 and \overline{q}=q mod8 and. k_{1}, k_{2}, k_{3}, k_{4},. k. the. are. multiplicities of simple. modules in W. Put. R_{p,q}^{+}. is. of the. R_{p,q}=C_{p}\otimes C_{q}. isomorphic. C_{p,q}^{+}. to. quadratic form. structure of the Lie. and let ,. the. P of. R_{p,q}^{+}. be the. subalgebra. of. signature (p, q). algebra \mathfrak{h}. is. subalgebra .. completely. even. As. of. even. elements in. elements of the Clifford. we can see. determined. by. R_{p,q} Then, algebra C_{p,q}. from the table. .. below,. the structure of. R_{p,q}. the and. R_{p,q}^{+}. \mathrm{p},\mathrm{q}. \mathrm{p},\mathrm{q}. (\mathrm{p}\overline{},\overline{\mathrm{q} ) (\overline{\mathrm{p} \overline{\mathrm{q} ) (0 2),(4, 6) (0 7),(2, 3),(3, 4),(6, 7) (0 3),(2, 7),(3, 6),(4, 7) (0 6),(2, 4) (0,0),(2, 2),(4, 4),(6, 6) (0 4),(2, 6) (0 1),(1, 2),(4, 5),(5, 6) (1, 3),(1, 7),(3, 5),(5, 7) (0 5),(1, 4),(1, 6),(2, 5) (3, 3),(7, 7) \mathrm{s}\mathrm{p}(\mathrm{k}, ) (2\mathrm{k}) (3, 7) \ m a t h r m { k } so( k) (1, 1),(5, 5) (1, 5) \mathrm{s}\mathrm{p}(\mathrm{k} k) ,. ,. ,. ,. ,. ,. \mathrm{T}, \mathrm{T}. (\mathrm{T}. ,. (\mathrm{T} (\mathrm{T}. \mathrm{T}. \mathrm{T}, \mathrm{T} \mathrm{T}. \mathrm{T}, \mathrm{T}. (, (, (, (,. ) \mathrm{s}\mathrm{p}(\mathrm{k}, ) ) so ) so(\mathrm{k},\mathrm{k}) ) \mathrm{s}\mathrm{p}(\mathrm{k},\mathrm{k}). ,. ,. (Table 2) Here T and T' denote the matrix size.. algebras. over. \mathrm{K} and \mathrm{K}' ,. respectively,. of appropriate. (13) Fumihiro Sato. 64. Sketch of the End (W) e_{1}. ,. .. .. .. For. .. R_{p,q}. \mathcal{A}^{r}=\{X\in \mathcal{A}|{}^{t}XS(r)+S(r)X=0\}. put. e_{p+q} the standard basis of V=\mathbb{R}^{p+q}. ,. coincides with. As. \mathfrak{h}. .. For. example,. an. an. 1. The action of e_{1}, e_{2}. are. e_{2}\cdot w_{1}=w_{1},. subalgebra row. Take r=e_{1}. r,. R_{1,1}^{+}. case. r=e_{i}r'. with. .. Denote. by. r'\in(R_{p,q}^{+})^{\times}. ,. then \mathcal{A}^{r}. it is not hard to calculate \mathcal{A}^{r}. .. given by e_{1}. .. \left\{ begin{ar ay}{l e_{1}.W3=w_{3},\ \{ \end{ar ay}\right.. w_{2}=-w_{2},. e_{2}. W3=-w_{3},. e_{2}\cdot w_{2}=-w_{2},. is. generated by. W=W_{1}^{k_{1}}\oplus W_{2}^{k_{2}}\oplus W_{3}^{k_{3}}\oplus W_{4}^{k_{4}}. dimensional. in. (p, q)=(1,1) Then R_{1,1} is isomorphic to \mathbb{R}\oplus \mathbb{R}\oplus inequivalent simple R_{1,1} ‐modules W_{1}, W_{2}, W_{3}, W_{4} of dimension. \left\{ begin{ar ay}{l e_{1}.w_{1}=w_{1},\ \{ \end{ar ay}\right. Since the. If. .. appropriate choice of. consider the. \mathbb{R}\oplus \mathbb{R} and there exist 4. Let. we. ,. R_{p,q}^{+}. Let \mathcal{A} be the commutant of. of Theorem 2.4.. proof. in. an r. Takeyoshi Kogiso. and. vectors. Then. we. and. e_{1}. .. w_{4}=-w_{4},. e_{2}\cdot w_{4}=. W_{1}\cong W_{2} and W_{3}\cong W_{4}. ee,. identify. W with the space of. as. w_{4}.. R_{1,1}^{+} ‐modules.. k_{1}+k_{2}+k_{3}+k_{4^{-}}. have. \mathcal{A}=\{\left(\begin{ar ay}{l} A0\ 0B \end{ar ay}\right)|A\in M_{k_{1}+k_{2} (\mathb {R}) , B\in M_{k_{3}+k_{4} (\mathb {R})\}. Then the action of. .. r. on. W is. given by. the. right multiplication. of the. matrix. and. we. \left(bgin{ary}l I_{k1}&0 \ 0&-I_{k2}&0 \ 0& I_{k3}&0\ &0 -I_{k4} \end{ary}\ight). obtain. \mathfrak{h}=\mathcal{A}^{r}=\{ left(\begin{ar ay}{l} A&0\ 0B& \end{ar ay}\right)|B\in\mathfrak{s}0(k_{3},k_{4}A\in\mathfrak{s}0(k_{1},k_{2}\} cong\mathfrak{s}0(k_{1},k_{2})\oplus\mathfrak{s}0(k_{3},k_{4}). If p+q is calculation the. relatively small,. engine. following. such. Maple. or. Conjecture. check. 2.3 with the aid of. Mathematica and the result. can. a. symbolic. be summarized in. table.. Theorem 2.5. m_{0}. as. we can. .. =. Under the convention. minimum of the dimensions of the. simple C_{p}\otimes C_{q} ‐modules,. m=\dim W,. 0\Leftrightarrow\tilde{P}\equiv 0. (degenerate case),. \mathrm{T}\Leftrightarrow \mathfrak{g}_{p,q}($\rho$)=\mathfrak{s}0(p, q)\oplus \mathfrak{h}_{p,q}($\rho$). (Conjecture. \mathrm{F}\Leftrightar ow \mathfrak{g}_{p,q}($\rho$)\neq\supset \mathfrak{s}0(p, q)\oplus \mathfrak{h}_{p,q}($\rho\$) (Conjecture \mathrm{p}\mathrm{v}\Leftrightar ow\tilde{P} is relative invariant of pv, a. pure \Leftrightarrow. mixed. \Leftrightarrow. (all (. the. 2.3 is 2.3. true.),. fails.),. a. R_{p,q}^{+}\otimes_{\mathb {R} \mathb {C} ‐simple. W is not. pure),. modules in. W\otimes_{\mathbb{R} \mathbb{C}. are. isomorphic),.

(14) Representations. we. 0F. Clifford. algebras and local functional. equations. 65. have. \mathrm{p} + \mathrm{q}. \mathrm{m}. \mathrm{m} = \mathrm{m}. \mathrm{m} =. \mathrm{T} pv. 2\mathrm{m}. \mathrm{T} pv. ,. \mathrm{T} pv. ,. ,. (mixed) \mathrm{T} pv (pure) 0. (mixed) \mathrm{T} pv (pure) 0. \mathrm{F} pv. \mathrm{T} pv. \mathrm{F} pv. \mathrm{T} pv. ,. ,. ,. ,. ,. ,. \mathrm{F} pv ,. (pure) pv (mixed) non‐pv. \mathrm{F} pv ,. 16. \mathrm{F} pv. 16. \mathrm{F} pv. 16. \mathrm{F} pv. 10. 16. 11. 32. ,. ,. ,. (pure) pv (mixed) non‐pv \mathrm{F} pv ,. (Table 3) Almost all the are. non‐degenerate. prehomogeneous \mathrm{p} + \mathrm{q}. cases. and. are. cases. given. in Theorem 2.5 for which in the. following. \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{W}. 16 16. (pure) (mixed) 16 16 16. 10. 32. 11. (pure) 32. table:. (so(\mathrm{p},q) \mathrm{s}\mathrm{l}(2) \mathrm{s}\mathrm{l}(2) \mathrm{s}\mathrm{l}(2) \mathrm{s}\mathrm{l}(2) \mathrm{s}\mathrm{l}(2) \mathrm{g}\mathrm{l}(2) so(8) \mathrm{s}\mathrm{l}(4) \mathrm{s}\mathrm{l}(4) \mathrm{s}\mathrm{l}(4) \mathrm{s}\mathrm{l}(2) \mathrm{s}\mathrm{l}(2) \mathrm{g}\mathrm{l}(1) so(8) \mathrm{s}\mathrm{l}(2) so(8) so(8) \mathrm{g}\mathrm{l}(1) so(16) so(10) \mathrm{s}\mathrm{l}(2) so(12). Conjecture. h). \mathrm{s}\mathrm{l}(2). \mathrm{s}\mathrm{l}(2). \mathrm{g}\mathrm{l}(2). \mathrm{s}\mathrm{l}(4). \mathrm{s}\mathrm{l}(2). \mathrm{s}\mathrm{l}(2). so(9) so(11). (Table 4) The. unique non‐prehomogeneous \mathrm{p} + \mathrm{q}. 10. case. is. \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{W} 32. (so(\mathrm{p},q) h) so(10). (mixed) (Table 5). 2.3 fail.

(15) Fumihiro Sato. 66. The. following. Conjecture. Conjecture and all the. is. a. refinement of. Conjecture. 2.6.. 2.6. implies. prehomogeneous. (Added. Remark.. in. that. and. Takeyoshi Kogiso. Conjecture. 2.3.. 2.3 is true for. prehomogeneous. cases are. p+q\geq 12.. cases. listed in Tables. proof) Conjectures. do not appear for. 1, 1 and. 2.3 and 2.6. p+q\geq 12. 4.. proof. theorems. The. are now. will appear elsewhere.. References. [1]. D.. [2]. D.. Achab, Représentations. Fourier. 45(1995),. Achab,. des. algèbres. de rang 2 et fonctions zêta. associées, Ann.. Inst.. 437‐451.. Zeta functions of Jordan. algebras representations, Ann.. Inst. Fourier. 45(1995),. 1283‐1303.. [3] T.Arakawa,. Dirichlet series related to the Eisenstein series. Comment. Math. Univ. St. Pauli. [4]. J.‐L. Z.. [5] [6]. Clerc,. Zeta distributions associated to. 239(2002), Sato,. Siegel. upper. half‐plane,. 29‐42. a. representation of. Koranyi, Analysis of symmetric. a. Jordan. cones, Oxford. Zeta functions in several variables associated with. I: Functional. the. algebra,. Math.. 263‐276.. J. Faraut and A. F.. 27(1978),. on. University Press, prehomogeneous vector. 1994. spaces. equations, Tôhoku Math. J. 34(1982), 437‐483.. [7]. F.. [8]. Kimura, A classication of irreducible prehomogeneous vector spaces and their invariants, Nagoya Math. J. 65(1977), 1‐155. M. Sato, Theory of prehomogeneous vector spaces (Notes taken by T.Shintani in Japanese), Sugaku no Ayumi 15(1970), 85‐157. M. Sato and T. Shintani, On zeta functions associated with prehomogenous vector spaces, Ann. of Math. 100(1974), 131‐170. T.Suzuki, Distributions with automorphy and Dirichlet series, Nagoya Math. J. 73(1979),. Sato, Quadratic. maps and. Math. Univ. St. Pauli. [9] [10] [11]. M. Sato and T.. 157‐169.. nonprehomogeneous. 56(2007),. local functional. equations, Comment.. 163‐184..

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