Generalizing the MVW involution, and the contragredient

2017/05/09 by Dipendra Prasad, Prasad, Dipendra · 2 citations
Mathematics · #11F70 #22E55 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1705.03262

openalex publication_date 2017/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For certain quasi-split reductive groups G over a general field F, we construct an automorphism ι_G of G over F, well-defined as an element of \rm Aut(G)(F)/jG(F) where j:G(F) → \rm Aut(G)(F) is the inner-conjugation action of G(F) on G. The automorphism ι_G generalizes (although only for quasi-split groups) an involution due to Moeglin-Vigneras-Waldspurger in [MVW] for classical groups which takes any irreducible admissible representation π of G(F) for G a classical group and F a local field, to its contragredient π^\vee. The paper also formulates a conjecture on the contragredient of an irreducible admissible representation of G(F) for G a reductive algebraic group over a local field F in terms of the (enhanced) Langlands parameter of the representation.

Citations

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• A `relative' local Langlands correspondence

Cited by

• Rankin-Selberg duality via gluing
• The local Langlands conjecture

Related

• Generic representations for symmetric spaces
• The MVW involution of the metaplectic group
• Distinguished representations, Shintani base change and a finite field analogue of a conjecture of Prasad
• The twining character formula for reductive groups
• Semilinear automorphisms of reductive algebraic groups