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Degenerate series representations for GL(2n,\({\mathbb{R}})\) and Fourier analysis. (English) Zbl 0715.22009
Indecomposable representations of Lie groups and their physical applications, Proc. Conf., Rome/Italy 1988, Symp. Math. 31, 45-69 (1989).
[For the entire collection see Zbl 0694.00013.]
Let \(G=GL(2n,{\mathbb{R}})\) and \(P_{\ell}\) (\(\ell \leq n)\) the parabolic subgroups of G with Levi factor GL(\(\ell)\times GL(2n-\ell)\). Then, \(Ind^ G_{P_{\ell}}(\chi)\), \(\chi\) is a one-dimensional character of \(P_{\ell}\), may be realized in a certain space \(V_{\ell}(\chi)\) of smooth functions on \(M_{(\ell,2n-\ell)}({\mathbb{R}})\) of \(\ell \times (2n- \ell)\) real matrices. In this paper the composition factors of the degenerate series \(Ind^ G_{P_ n}(\chi)\) are determined by constructing a rank filtration on \(V_ n(\chi):\) \(V_ n(\chi)\supseteq...\supseteq V_{\ell}\supseteq V_{\ell - 1}\supseteq..\). Each \(V_{\ell}\) is given by translating \(V_{\ell}(1)\) according to the Radon transform from \(M_{(\ell,2n-\ell)}({\mathbb{R}})\) to \(M_{(n,n)}({\mathbb{R}})\) and a Zuckerman translation functor. To find the possible candidates for the composition factors, the authors use a variant of some general results on the number of special unipotent representations of a real reductive group obtained D. Vogan and one of the authors. Then the condition of \(\chi\) for which all quotients of the filtration are irreducible is obtained.
Reviewer: T.Kawazoe

22E30 Analysis on real and complex Lie groups
43A80 Analysis on other specific Lie groups
22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)