Reducible principal series representations, and Langlands parameters for\n real groups

2017/05/03 by Dipendra Prasad, Prasad, Dipendra
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1705.01445

openalex publication_date 2017/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The work of Bernstein-Zelevinsky and Zelevinsky gives a good understanding of\nirreducible subquotients of a reducible principal series representation of\nGL_n(F), F a p-adic field (without specifying their multiplicities which\nis done by a Kazhdan-Lusztig type conjecture). In this paper we make a proposal\nof a similar kind for principal series representations of GL_n( mathbb R).\nOur investigation on principal series representations naturally led us to\nconsider the Steinberg representation for real groups, which has curiuosly not\nbeen paid much attention to in the subject (unlike the p-adic case). Our\nproposal for Steinberg is the simplest possible: for a real reductive group\nG, the Steinberg of G( mathbb R) is a discrete series representation if\nand only if G( mathbb R) has a discrete series, and makes up a full\nL-packet of representations of G( mathbb R) (of size W_G/W_K), so is\ntypically not irreducible.\n

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