53C65 — 1 articles found.

Composition Series for Degenerate Principal Series of ${GL}(n)$

C. R. Math. Rep. Acad. Sci. Canada Vol. 39 (1) 2017, pp. 1-12
Vol.39 (1) 2017
Dmitry Gourevitch Details
(Received: 2015-11-01 , Revised: 2016-01-31 )
(Received: 2015-11-01 , Revised: 2016-01-31 )

Dmitry Gourevitch,The Incumbent of Dr. A. Edward Friedmann Career Development Chair in Mathematics, Department of Mathematics, Weizmann Institute of Science, POB 26, Rehovot 76100, Israel; e-mail: dimagur@weizmann.ac.il

Abstract/Résumé:

In this note we consider representations of the group \(GL(n,F)\), where \(F\) is the field of real or complex numbers or, more generally, an arbitrary local field, in the space of equivariant line bundles over Grassmannians over the same field \(F\). We study reducibility and composition series of such representations.

Similar results were obtained already in [4,20,31], but we give a short uniform proof in the general case, using the tools from [7]. We also indicate some applications to cosine transforms in integral geometry.

Dans cette note on considère des représentations du groupe \(GL(n,F)\), où \(F\) est le corps des nombres réels ou complexes ou plus généralement, un corps local arbitraire, dans l’espace de fibres en droites équivariants sur des Grassmanniennes sur le même corps \(F\). On étudie la réductibilité et la suite de composition de telles représentations.

Des résultats similaires ont déjà été obtenus dans [4,20,31], mais nous présentons une courte preuve dans le cas général en utilisant les outils de [7]. On donne aussi quelques applications aux transformées en cosinus en géométrie intégrale.

Keywords: Bernstein-Zelevinsky derivative, Degenerate principal series, Line bundle over Grassmannian, alpha-cosine transform

AMS Subject Classification: Analysis on real and complex Lie groups, Representations of Lie and linear algebraic groups over local fields, Homogeneous spaces, Integral transforms in distribution spaces, Integral geometry; differential forms; currents; etc. 22E30, 22E50, 22F30, 46F12, 53C65

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