The degenerate principal series representations of exceptional groups of type E 6 over p -adic fields
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THE DEGENERATE PRINCIPAL SERIES REPRESENTATIONS OF EXCEPTIONAL GROUPS OF TYPE E6 OVER p-ADIC FIELDS
BY
Hezi Halawi School of Mathematics, Ben Gurion University of the Negev POB 653, Be’er Sheva 84105, Israel e-mail: [email protected]
AND
Avner Segal Department of Mathematics, University of British Columbia Vancouver, BC V6T 1Z2, Canada and Department of Mathematics, Bar Ilan University, Ramat Gan 5290002, Israel e-mail: [email protected], [email protected]
ABSTRACT
In this paper, we study the reducibility of degenerate principal series of the simple, simply-connected exceptional group of type E6 . Furthermore, we calculate the maximal semi-simple subrepresentation and quotient of these representations.
Received December 20, 2018 and in revised form July 1, 2019
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H. HALAWI AND A. SEGAL
Isr. J. Math.
1. Introduction One of the main problems in the representation theory of p-adic groups is the question of reducibility and structure of parabolic induction. More precisely, let G be a p-adic group and let P be a parabolic subgroup of G. Let M denote the Levi subgroup of P and let σ be a smooth irreducible representation of M . One can ask: • Is the parabolic induction π = IndG P σ reducible? • If the answer to the previous question is positive, what is Jordan–H¨older series of π? In this paper we completely answer the first question for degenerate principal series of the exceptional group of type E6 . We further calculate the maximal semi-simple subrepresentation and quotient of such representations which partially answers the second question. More precisely, let G be a simple, split, simply-connected p-adic group of type E6 and let P be a maximal parabolic subgroup of G. For a one-dimensional representation Ω of P , we consider the normalized parabolic induction π = IndG P Ω. We determine for which Ω, π is reducible and its maximal semisimple subrepresentation and maximal semi-simple quotient. Our main result, Theorem 4.3, is summarized by the following corollary. Corollary 1.1: With the exception of one case, for any maximal Levi subgroup M of G (of type E6 ) and any one-dimensional representation Ω of M , if the degenerate principal series representation π = IndG P Ω is reducible, then π admits a unique irreducible subrepresentation and a unique irreducible quotient. In the one exception, up to contragredience, Ω = (χ ◦ ω4 )|ω4 |1/2 , where ω4 is the 4th fundamental weight and χ is a cubic character. In this case, π admits a maximal semi-simple quotient of length 3 and a unique irreducible subrepresentation. Similar studies were performed for both classical and smaller exceptional groups. For example, see: • [23, 4] for general linear groups. (It should be noted that the scope of these works goes beyond degenerate principal series.) • [11] for symplectic groups. • [2, 13] for orthogonal groups. • [15] for type G2 . • [7] for type F4 .
Vol. TBD, 2020
DEGENERATE PRINCIPAL SERIES OF E6
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The reason that such a study was not performed for groups of type En before, is that Weyl groups of these types
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