On representations of \(\mathrm{GL}(n)\) distinguished by \(\mathrm{GL}(n - 1)\) over a \(p\)-adic field
type: Item
instance of: scholarly article
title: On representations of \(\mathrm{GL}(n)\) distinguished by \(\mathrm{GL}(n - 1)\) over a \(p\)-adic field
zbMATH Open document ID: 1276.22009
author: C. G. Venketasubramanian
published in: Israel Journal of Mathematics
publication date: 2013-06-18T00:00:00Z
review text: The present article is concerned with admissible representations of the linear group \(\mathrm{GL}_{n}(F)\) over a non-Archimedean local field \(F\) (\(n\geq 3\)) which are distinguished by \(\mathrm{GL}_{n-1}(F)\), i.e., which admit a nonzero \(\mathrm{GL}_{n-1}(F)\)-invariant linear form. More precisely, it contains a proof of the characterization of these representations in terms of their Langlands parameters (Theorem 1.1), conjectured and proved in the case \(n=3\) by \textit{D. Prasad} [Duke Math. J. 69, No. 1, 167--177 (1993; Zbl 0827.22006)], as well as the calculation of the dimension of the space of \(\mathrm{GL}_{n-1}(F)\)-invariant forms for all parabolically induced representations from irreducible representation of a Levi subgroup. The author uses the same methods as D. Prasad: a precise study of (normalized) induced representations from maximal parabolic subgroups via Mackey theory and segments theory of A. Zelevinsky which gives a list of \(\mathrm{GL}_{n-1}(F)\)-distinguished representations, then an argument by induction on \(n\) to prove that this list is complete (Corollary 6.15). These methods allow the author to show that the dimension of the space of \(\mathrm{GL}_{n-1}(F)\)-invariant forms for a parabolically induced representation from irreducible representation of a Levi subgroup is always smaller than 2 and equal to 2 for a precise list of representations which appears in Theorem 7.17.
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22D30
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 6176535
zbMATH Keywords: non-Archimedean field
zbMATH Keywords: linear group
zbMATH Keywords: distinguished representations
zbMATH Keywords: invariant linear forms
zbMATH Keywords: Langlands parameters
MaRDI profile type: Publication
full work available at URL: s11856-012-0152-7
OpenAlex ID: W1987978572
cites work: Multiplicity one theorems
cites work: (GLn+1(F), GLn(F)) is a Gelfand pair for any local field F
cites work: REPRESENTATIONS OF THE GROUPGL(n,F) WHEREFIS A NON-ARCHIMEDEAN LOCAL FIELD
cites work: Induced representations of reductive ${\germ p}$-adic groups. I
cites work: Q3139246
cites work: A note on trilinear forms for reducible representations and Beilinson's conjectures.
cites work: Q4297086
cites work: Q3356552
cites work: On the decomposition of a representation of \(GL(3)\) restricted to \(GL(2)\) over a \(p\)-adic field
cites work: Q5200139
cites work: Q4743014
cites work: Q3682555
cites work: Q5200141
cites work: Induced representations of reductive ${\germ p}$-adic groups. II. On irreducible representations of ${\rm GL}(n)$
DOI: 10.1007/S11856-012-0152-7
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description: scientific article; zbMATH DE number 6176535