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**Eisenstein series on reductive symmetric spaces and representations of Hecke algebras.**
*(English)*
Zbl 0789.11031

For a reductive symmetric space \(X=G/H\) defined over \(\mathbb{Q}\) and an arithmetic subgroup \(\Gamma\) of \(G\), Eisenstein series are defined by taking an average over \(\Gamma\) of the complex powers of relative invariants on \(X\) of a \(\mathbb{Q}\)-parabolic subgroup. The generalized Eisenstein series are not real analytic functions on \(\Gamma \backslash X (\mathbb{R})\), but continuous functions on the completion \(\tilde X\) of \(X(\mathbb{Q})\) with respect to the congruence subgroup topology.

Then the authors introduce an integral transform of functions on \(\tilde X\) (the Fourier-Eisenstein transform) whose kernel function is the Eisenstein series. Under the assumption on the convergence of the Eisenstein series, the Fourier-Eisenstein transform defines a Hecke algebra-homomorphism of the space of Schwartz-Bruhat functions on \(\tilde X\) into a certain space of Dirichlet series (Theorems 1 and 2). This is the main result of the first half of the paper.

On the basis of the relation between \(\tilde X\) and the adelization \(X (\mathbb{A}_ f) \), a sufficient condition is given under which Eisenstein series have Euler product expansion (Theorem 3). If the condition is satisfied, the study of the Fourier-Eisenstein transform can be reduced to the study of the spherical Fourier transforms on the \(p\)-adic symmetric spaces \(X(\mathbb{Q}_ p)\). Typical examples are the following: \[ (GL(n) \times GL(n))/GL(n),\;GL(2m) /Sp(m),\;GL(n)/(GL(r) \times GL(n- r)). \] A detailed examination of the Fourier-Eisenstein transforms for these symmetric spaces are made in the latter half of the paper. It is expected that the generalized Eisenstein series have good analytic properties (such as analytic continuations and functional equations) similar to those of the Langlands Eisenstein series, though no general results have been obtained in this direction. Some examples have been examined by the second author [Ann. Math., II. Ser. 116, 177-212 (1982; Zbl 0497.10012); Adv. Stud. Pure Math. 7, 295-332 (1985; Zbl 0608.43007); Comment. Math. Univ. St. Pauli 37, 99-125 (1988; Zbl 0657.10024)].

Then the authors introduce an integral transform of functions on \(\tilde X\) (the Fourier-Eisenstein transform) whose kernel function is the Eisenstein series. Under the assumption on the convergence of the Eisenstein series, the Fourier-Eisenstein transform defines a Hecke algebra-homomorphism of the space of Schwartz-Bruhat functions on \(\tilde X\) into a certain space of Dirichlet series (Theorems 1 and 2). This is the main result of the first half of the paper.

On the basis of the relation between \(\tilde X\) and the adelization \(X (\mathbb{A}_ f) \), a sufficient condition is given under which Eisenstein series have Euler product expansion (Theorem 3). If the condition is satisfied, the study of the Fourier-Eisenstein transform can be reduced to the study of the spherical Fourier transforms on the \(p\)-adic symmetric spaces \(X(\mathbb{Q}_ p)\). Typical examples are the following: \[ (GL(n) \times GL(n))/GL(n),\;GL(2m) /Sp(m),\;GL(n)/(GL(r) \times GL(n- r)). \] A detailed examination of the Fourier-Eisenstein transforms for these symmetric spaces are made in the latter half of the paper. It is expected that the generalized Eisenstein series have good analytic properties (such as analytic continuations and functional equations) similar to those of the Langlands Eisenstein series, though no general results have been obtained in this direction. Some examples have been examined by the second author [Ann. Math., II. Ser. 116, 177-212 (1982; Zbl 0497.10012); Adv. Stud. Pure Math. 7, 295-332 (1985; Zbl 0608.43007); Comment. Math. Univ. St. Pauli 37, 99-125 (1988; Zbl 0657.10024)].

Reviewer: F.Sato (Tokyo)

### MSC:

11F55 | Other groups and their modular and automorphic forms (several variables) |

11F66 | Langlands \(L\)-functions; one variable Dirichlet series and functional equations |

43A85 | Harmonic analysis on homogeneous spaces |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

14M17 | Homogeneous spaces and generalizations |