## $$H$$-fixed distribution vectors for generalized principal series of reductive symmetric spaces and meromorphic continuation of Eisenstein integrals. (Vecteurs distributions $$H$$-invariants pour les séries principales généralisées d’espaces symétriques reductifs et prolongement méromorphe d’intégrales d’Eisenstein.)(French)Zbl 0785.22014

Let $$G$$ be the group of real points of a Zariski-connected linear algebraic group defined over $$\mathbb{R}$$, equipped with an involution $$\sigma$$ defined over $$\mathbb{R}$$. Let $$H$$ be the group of fixed points of $$\sigma$$ in $$G$$. Then $$G/H$$ is a reductive symmetric space. Let $$\theta$$ be a Cartan involution of $$G$$ commuting with $$\sigma$$, and let $$P\subset G$$ be a parabolic subgroup of $$G$$ which is $$\sigma\circ\theta$$-stable. Then $$P$$ has a $$\sigma$$-Langlands decomposition $$P=MAN$$. Here $$N$$ is the unipotent radical of $$P$$, and $$L=P\cap \theta P$$ the $$\theta$$-stable Levi component. Moreover, $$A$$ is the subgroup of the usual Langlands $$A$$ consisting of elements $$a$$ with $$\sigma a=a^{-1}$$. Suppose that the reductive symmetric space $$M/M\cap H$$ satisfies Flensted-Jensen’s rank condition so that it possesses a discrete series representation $$\delta$$. If $$\nu\in{\mathfrak a}_{\mathbb{C}}^*$$, then the representation $$\pi_{\delta,\nu}$$ of $$G$$ induced from the representation $$\delta\otimes (\nu+\rho)\otimes 1$$ of $$P$$ is called a generalized principal series representation of $$G/H$$.
The authors establish the existence of an $$H$$-fixed distribution vector in $$\pi_{\delta,\nu}$$ which depends meromorphically on the parameter $$\nu$$. The main interest in such a family stems from its expected contribution to the Plancherel decomposition of $$L^ 2(G/H)$$.
As a consequence of the above mentioned result the authors establish meromorphic continuation of Eisenstein integrals, i.e. matrix coefficients of $$K$$-finite vectors with $$H$$-fixed distribution vectors in $$\pi_{\delta,\nu}$$.
The authors also give an application of their result to standard intertwining operators: their distribution kernels are examples of the $$H$$-fixed distribution vectors under consideration, in the setting of the group $$G$$ viewed as the symmetric space $$G\times G/\text{diagonal }G$$.
The present work generalizes earlier work by T. Oshima and J. Sekiguchi [Invent. Math. 57, 1-81 (1980; Zbl 0434.58020)], G. ’Olafsson [ibid. 90, 605-629 (1987; Zbl 0665.43004)] and E. P. van den Ban [Ann. Sci. Ec. Norm. Supér., IV. Sér. 21, 359-412 (1988; Zbl 0714.22009)] for minimal $$\sigma\circ\theta$$-stable parabolic subgroups. The main tool of the first two papers is the use of a Bernstein-Soto functional equation. This tool does not directly apply to the present situation. Instead the authors use a generalization due to Kashiwara, involving the meromorphic continuation of a product of complex powers of polynomials with the solution of a holonomic system of differential equations.

### MSC:

 2.2e+47 Semisimple Lie groups and their representations 2.2e+46 Representations of Lie and linear algebraic groups over real fields: analytic methods

### Citations:

Zbl 0434.58020; Zbl 0665.43004; Zbl 0714.22009
Full Text:

### References:

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