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**Zeta functions in several variables associated with prehomogeneous vector spaces. III: Eisenstein series for indefinite quadratic forms.**
*(English)*
Zbl 0497.10012

The reviewer feels that it is appropriate to pick up the statements of the results in the introduction of this paper. Let \(Y\) be an \(n+1\) by \(n+1\) rational nondegenerate symmetric matrix with signature \((p,q)\). Denote by \(d_i(A)\) the principal \(i\)-th minor determinant of a matrix \(A\). Let \(\Gamma_\infty\) be the group of upper triangular matrices of size \(n+1\) with diagonal entries \(1\). Then the Eisenstein series for \(Y\) are defined by
\[
E(Y, \varepsilon; s) = \sum_U \prod_{i=1}^n \vert d_i(Y[U])\vert^{-s_i},
\]
where \(\varepsilon = (\varepsilon_1, \ldots,\varepsilon_{n+1})\) is an \(n+1\)-tuple of \(\pm 1\), \(s_i\) are complex variables \(Y[U] = ^+UYU\) and \(U\) runs through a set of all representatives of the double cosets belonging to \(\mathrm{SO}(Y)_{\mathbb Z}\backslash \mathrm{SL}(n+1)_{\mathbb Z}/ \Gamma_\infty\) such that \(d_i(Y[U])/\vert d_i(Y[U])\vert =\varepsilon_1\cdots\varepsilon_i\) \((1\le i\le n+1)\). Let \(z = (z_1,\ldots, z_{n+1})\) be a sequence of variables which are connected to \(s_i\) by \(s_i = z_{i+1} - z_i + 1/2\) \((1\le i\le n)\). Set
\[
\Lambda(Y, \varepsilon; z) = \prod_{1\le j< i\le n+1} \eta(2z_i - 2z_j + 1) \vert \det Y\vert^{z_{n+1}} E(Y, \varepsilon; s),
\]
where \(\eta(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)\) and \(\zeta(s)\) is the Riemann zeta-function. Denote by \(\mathfrak S_{n+1}\) the symmetric group on \(n+1\) letters. It will be shown that the Eisenstein series \(E(Y,\varepsilon;s)\) have the following properties:

(1) The functions \(E(Y,\varepsilon;s)\) multiplied by \[ \prod_{1\le j< i\le n} (s_i + \cdots + s_j - \frac{j-1}{2} - 1)^2 \zeta(2(s_i + \cdots + s_j) - j + 1) \] have analytic continuations to entire functions of \(s_i\).

(2) For any \(\sigma\in\mathfrak S_{n+1}\) and any \(\varepsilon, \eta\in \{\pm 1\}^{n+1}\) such that \(\operatorname{sgn} \varepsilon = \operatorname{sgn} \eta = (p,q)\), there exists \(A^\sigma(\varepsilon,\eta;z)\), a rational function of trigonometric functions of \(z\), such that \[ \Lambda(Y, \varepsilon; \sigma, z) = \sum_{\operatorname{sgn} \eta = (p,q)} A^\sigma(\varepsilon,\eta;z)\Lambda(Y, \eta; z), \] where \(\sigma z = (z_{\sigma(1)},\ldots,z_{\sigma(n+1)})\).

The paper consists of four sections. In Section 1, the prehomogeneous vector spaces of descending chains are introduced, and their relative invariants and regular subspaces are studied. Section 2 is devoted to the calculation of the partial Fourier transforms of complex powers of relative invariants. In Section 3, zeta functions associated with the prehomogeneous vector spaces of descending chains are studied. In the final section, the author discusses another family of Dirichlet series related to \(E(Y, \varepsilon;s)\).

The reviewer would like to add some comments. As one may guess, the conception of the zeta function associated to the prehomogeneous space has the nature that it theoretically unifies many existing zeta-functions. However this conception has not much proved to have a penetrating power to investigate further arithmetical properties (e.g. Euler product formula) of many zeta-functions.

[Part I, cf. Tôhoku Math. J., II. Ser. 34, 437–483 (1982; Zbl. 497.14007); ibid. 35, 77–99 (1983; Zbl 0513.14011).]

(1) The functions \(E(Y,\varepsilon;s)\) multiplied by \[ \prod_{1\le j< i\le n} (s_i + \cdots + s_j - \frac{j-1}{2} - 1)^2 \zeta(2(s_i + \cdots + s_j) - j + 1) \] have analytic continuations to entire functions of \(s_i\).

(2) For any \(\sigma\in\mathfrak S_{n+1}\) and any \(\varepsilon, \eta\in \{\pm 1\}^{n+1}\) such that \(\operatorname{sgn} \varepsilon = \operatorname{sgn} \eta = (p,q)\), there exists \(A^\sigma(\varepsilon,\eta;z)\), a rational function of trigonometric functions of \(z\), such that \[ \Lambda(Y, \varepsilon; \sigma, z) = \sum_{\operatorname{sgn} \eta = (p,q)} A^\sigma(\varepsilon,\eta;z)\Lambda(Y, \eta; z), \] where \(\sigma z = (z_{\sigma(1)},\ldots,z_{\sigma(n+1)})\).

The paper consists of four sections. In Section 1, the prehomogeneous vector spaces of descending chains are introduced, and their relative invariants and regular subspaces are studied. Section 2 is devoted to the calculation of the partial Fourier transforms of complex powers of relative invariants. In Section 3, zeta functions associated with the prehomogeneous vector spaces of descending chains are studied. In the final section, the author discusses another family of Dirichlet series related to \(E(Y, \varepsilon;s)\).

The reviewer would like to add some comments. As one may guess, the conception of the zeta function associated to the prehomogeneous space has the nature that it theoretically unifies many existing zeta-functions. However this conception has not much proved to have a penetrating power to investigate further arithmetical properties (e.g. Euler product formula) of many zeta-functions.

[Part I, cf. Tôhoku Math. J., II. Ser. 34, 437–483 (1982; Zbl. 497.14007); ibid. 35, 77–99 (1983; Zbl 0513.14011).]

Reviewer: Michio Ozeki (Yamagata)

### MSC:

11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |

11M41 | Other Dirichlet series and zeta functions |

11S90 | Prehomogeneous vector spaces |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |