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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

##### MSC:
 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.)