Tamely ramified intertwining algebras
type: Item
instance of: scholarly article
title: Tamely ramified intertwining algebras
zbMATH Open document ID: 0854.22022
DOI: 10.1007/BF01232662
author: Lawrence Morris
published in: Inventiones Mathematicae
publication date: 1997-01-15T00:00:00Z
full work available at URL: https://eudml.org/doc/144141
review text: Let \(\mathbf G\) be a (not necessarily split) reductive group defined over a local non archimedean field \(F\); set \(G = {\mathbf G}(F)\). Suppose that \(P\) is a parahoric subgroup of \(G\) with pro-unipotent radical \(U\). Then \(M = U \setminus P\) is the group of points of a reductive group defined over the residue field \(\mathbb{F}_q\) of \(F\). Suppose that \((\sigma, V)\) is an irreducible cuspidal representation of \(M\); we can view \(\sigma\) as a representation of \(P\). We define \({\mathcal H} = {\mathcal H} (\sigma)\) to be the space of compactly supported functions from \(G\) to \(V\) which transform on the left and right under \(P\) by \(\sigma\). Then \({\mathcal H}(\sigma)\) is an algebra under convolution which can be identified with the algebra of intertwining operators of the (smooth) representation \(c\)-\(\text{Ind}^G_P (\sigma)\); it is an important problem to give an explicit description of \(\mathcal H\) by generators and relations. For example, if \(B = P\) is an Iwahori subgroup of \(G\) and \(\sigma\) is the trivial representation, the structure of \(\mathcal H\) is well understood [\textit{N. Iwahori} and \textit{H. Matsumoto}, Inst. Haut. Étud. Sci., Publ. Math. 25, 5-48 (1965; Zbl 0228.20015)]; it is also well understood when \(G = \text{GL}_n\) and \(P\), \(\sigma\) are arbitrary. (For more complete results in this case we refer the reader to [\textit{C. J. Bushnell} and \textit{P. C. Kutzko}, The admissible dual of \(\text{GL}_n\) over a local field via compact open subgroups. Ann. Math. Studies 129. (Princeton, NJ 1993; Zbl 0787.22016)].) Other cases that have been treated include \(B\) as above and \(\sigma\) non trivial [\textit{D. Goldstein}, Thesis, Univ. Chicago 1990], and some groups of rank of at most 2 (see \textit{A. Moy} [J. Reine Angew. Math. 372, 178-208 (1986; Zbl 0589.22015); Compos. Math. 66, 237-284 (1988; Zbl 0662.22012)]). In this paper, we prove that in general \(\mathcal H\) has a structure that is similar to…
Mathematics Subject Classification ID: 22E50
zbMATH DE Number: 549641
zbMATH Keywords: reductive group
zbMATH Keywords: local non archimedean field
zbMATH Keywords: parahoric subgroup
zbMATH Keywords: cuspidal representation
zbMATH Keywords: intertwining operators
zbMATH Keywords: generators
zbMATH Keywords: relations
zbMATH Keywords: Iwahori subgroup
zbMATH Keywords: Hecke algebra
zbMATH Keywords: parabolic subgroup
MaRDI profile type: Publication
cites work: Les sous-groupes fermes de rang maximum des groupes de Lie clos
cites work: Reductive groups over a local field
cites work: Reductive groups on a local field. II. Groups schemes. Existence of valuated root datum
cites work: The Admissible Dual of GL(N) via Compact Open Subgroups. (AM-129)
cites work: Q3682663
cites work: An Assortment of Results on Representations of GL2(k)
cites work: Q4150220
cites work: Q3049038
cites work: Q3722752
cites work: Normalizers of Parabolic Subgroups of Reflection Groups
cites work: Induced cuspidal representations and generalised Hecke rings
cites work: On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups
cites work: Q4067288
cites work: Automorphic forms on GL (2)
cites work: Mackey's Theorem for Nonunitary Representations
cites work: On the restriction of supercuspidal representations to compact, open subgroups
cites work: Some Examples of Square Integrable Representations of Semisimple p-Adic Groups
cites work: Characters of Reductive Groups over a Finite Field. (AM-107)
cites work: Q4011109
cites work: Affine root systems and Dedekind's \(\eta\)-function
cites work: P -Cuspidal Representations of Level One
cites work: Representations of U (2,1) over a p-adic field.
cites work: Q3810863
cites work: Q5679031
OpenAlex ID: W2006436225
description: scientific article; zbMATH DE number 549641