\(H\)-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

Brylinski, Jean-Luc; Delorme, Patrick

Invent. Math. 109, No. 3, 619-664 (1992).

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.

Reviewer: E.P.van den Ban (Utrecht)

MSC:

22E46	Semisimple Lie groups and their representations
22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods

Keywords:

stable parabolic subgroups; reductive symmetric space; generalized principal series representation; \(H\)-fixed distribution vector; Plancherel decomposition; meromorphic continuation of Eisenstein integrals; matrix coefficients; intertwining operators

Citations:

Zbl 0434.58020; Zbl 0665.43004; Zbl 0714.22009

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