Arthur packets for unipotent representations of the p -adic exceptional group G 2 Clifton Cunningham, Arthur packets for unipotent representations Andrew Fiori and Qing Zhang of the p -adic exceptional group G 2 Objective Strategy G2 Subregular Clifton Cunningham unipotent representations joint work with Andrew Fiori and Qing Zhang Artist’s impression of the main result University of Calgary Main result Proof by Fields Institute Geometry 2020 September 9 Forthcoming work A triple of Number Theory Seminar “superunipotent” representations References
Arthur packets for unipotent Abstract representations of the p -adic exceptional group G 2 This talk concerns work in progress on a generalization of the Clifton notion of local Arthur packets from Arthur-type representations Cunningham, of classical groups over p -adic fields to all admissible Andrew Fiori and Qing Zhang representations of all connected reductive algebraic groups over p -adic fields. The main ideas in this direction were described in Objective [CFM + 21], building on [Vog93]. In this talk our goal is much Strategy more modest: to report on this project for unipotent G2 representations of the exceptional group G 2 ( F ) for a p -adic field Subregular unipotent F . We will explain how to use the microlocal geometry of the representations moduli space of unramified Langlands parameters to compute Artist’s impression of the what we call Adams-Barbasch-Vogan packets, or ABV-packets main result for short, for all unipotent representations of G 2 ( F ) and how to Main result find the packet coefficients that are required to build stable Proof by Geometry distributions from ABV-packets. This talk will focus on the case Forthcoming work that is the most interesting geometrically and will include a A triple of discussion of unipotent representations that are not of Arthur “superunipotent” representations type. We will argue that ABV-packets provide the correct References extension of the notion of Arthur packets by explaining that the packet coefficients satisfy the expected endoscopic character identities.
Arthur packets for unipotent representations Motivation of the p -adic exceptional group G 2 We wish to adapt the main local result from Arthur’s book [Art13] to Clifton unipotent representations of the the p-adic group G 2 ( F ). Cunningham, Andrew Fiori and Qing Zhang Objective Strategy G2 Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References I reviewed the fundamental properties of A-packets in the introductory talk for this seminar.
Arthur packets for unipotent representations Strategy of the p -adic exceptional group G 2 Clifton Cunningham, Andrew Fiori and Qing Zhang 1 Use [CFM + 21] to define candidate packets – what we call ABV-packets – using a microlocal analysis of a moduli space of Objective Strategy unramified Langlands parameters. G2 2 Show that ABV-packets satisfy endoscopic character identities – Subregular this involves the groups SO 4 , PGL 3 , GL 2 and, of course, unipotent representations GL 1 × GL 1 . Artist’s impression of the 3 Attach stable distributions to these candidate packets. main result Main result Only then should the candidate packets be called Arthur packets! Proof by Geometry However, even then it should be noted that they correspond to cases Forthcoming work not treated in Arthur’s book, both because the group G 2 ( F ) is A triple of “superunipotent” classical and also because not all unipotent representations are of representations Arthur type. References
Arthur packets for unipotent representations The group G 2 of the p -adic exceptional group We view G 2 as a Chevalley group scheme with Dynkin diagram G 2 Clifton Cunningham, Andrew Fiori and Qing Zhang with short root γ 1 and long root γ 2 . There are 6 positive roots Objective Strategy { γ 1 , γ 2 , γ 1 + γ 1 , 2 γ 1 + γ 2 , 3 γ 1 + γ 2 , 3 γ 1 + 2 γ 2 } . G2 The dual root system is given by Subregular unipotent representations Artist’s impression of the main result Main result Proof by Geometry Forthcoming work A triple of “superunipotent” representations References
Arthur packets for unipotent representations Subregular Langlands parameter of the p -adic exceptional group G 2 • There are five unipotent conjugacy classes in � Clifton G 2 : Cunningham, Andrew Fiori and Qing Zhang O 0 < O long < O short < O sub < O reg . Objective Strategy • Each unipotent conjugacy classes determines a group G2 homomorphism SL 2 ( C ) → � G 2 . In particular, we have Subregular unipotent representations ϕ sub : SL 2 ( C ) → � G 2 . Artist’s impression of the main result • Consider the Langlands parameter φ 3 : W F × SL 2 ( C ) → � G 2 Main result defined by Proof by Geometry φ 3 ( w , x ) := ϕ sub ( x ) . Forthcoming work (Notation will be justified later.) A triple of “superunipotent” • The component group of φ 3 is non-Abelian: representations References G 2 ( φ 3 )) ∼ A φ 3 := π 0 ( Z � = S 3 .
Arthur packets for unipotent representations The “superunipotent” of the p -adic exceptional group G 2 representation of G 2 Clifton Cunningham, Andrew Fiori and • The L-packet for φ 3 is Qing Zhang Objective Π φ 3 ( G 2 ( F )) = { π 3 , π ̺ 3 , π ε 3 } , Strategy G2 where Subregular unipotent • π 3 corresponds to the trivial representation of S 3 , representations • π ̺ 3 corresponds to the unique 2-dimensional irreducible Artist’s impression of the representation ̺ of S 3 , and main result • π ε 3 corresponds to the sign character ε of S 3 . Main result • π ε 3 is supercuspidal: Proof by Geometry Forthcoming work 3 := cInd G 2 ( F ) π ε G 2 ( O F ) G 2 [1]; A triple of “superunipotent” representations where G 2 [1] is a cuspidal unipotent representation of G 2 ( F q ). References • All three unipotent representations are tempered. Both π 3 and π ̺ 3 have nonzero Iwahori-fixed vectors.
Arthur packets for unipotent representations The category of subregular of the p -adic exceptional group G 2 unipotent representations Clifton Cunningham, • This L-packet straddles two blocks in the Bernstein Andrew Fiori and Qing Zhang decomposition of the category of smooth representations of Objective G 2 ( F ): Strategy G2 Rep( G 2 ( F )) = Rep( G 2 ( F )) [ T ( F ) , ✶ ] ⊕ · · · ⊕ Rep( G 2 ( F )) [ G 2 ( F ) ,π ε 3 ] Subregular unipotent • We can refine this by specifying the relevant cuspidal support, representations not just the inertial classes. There is a unique unramified Artist’s impression of the character χ sub : T ( F ) → C × such that main result Main result Rep( G 2 ( F )) sub := Rep( G 2 ( F )) ( T ( F ) ,χ sub ) ⊕ Rep( G 2 ( F )) ( G 2 ( F ) ,π ε Proof by 3 ) Geometry Forthcoming work contains π 3 and π ̺ 3 in the first block and π ε 3 in the second. A triple of • We can identify all six irreducible representations in this “superunipotent” representations category: References � � irred / equiv = { π 0 , π 1 , π 2 , π 3 , π ̺ 3 , π ε Rep( G 2 ( F )) sub 3 } .
Arthur packets for unipotent representations Rep( G 2 ( F )) sub partitioned into of the p -adic exceptional group G 2 L-packets Clifton Cunningham, Andrew Fiori and Qing Zhang Objective φ 2 Strategy π 2 G2 Subregular unipotent representations Artist’s φ 3 φ 1 impression of the main result π ̺ π ε π 1 3 3 Main result Proof by Geometry Forthcoming work π 0 π 3 A triple of “superunipotent” representations φ 0 References
Arthur packets for unipotent representations Stabilizing representations and of the p -adic exceptional group G 2 stable distributions Clifton Cunningham, Andrew Fiori and Qing Zhang Objective Strategy π 2 G2 Subregular unipotent representations Artist’s impression of the main result π ̺ π ε π 1 3 3 Main result Proof by Geometry Forthcoming work π 0 π 3 A triple of “superunipotent” representations References
Arthur packets for unipotent representations Stabilizing representations of the p -adic exceptional group G 2 Clifton Cunningham, Andrew Fiori and Qing Zhang ABV-packet Objective Strategy L-packet G2 π 2 Subregular unipotent representations Artist’s impression of the main result Main result π ε π ̺ π 1 3 3 Proof by Geometry Forthcoming work A triple of “superunipotent” π 0 π 3 representations References
Arthur packets for unipotent representations Rep( G 2 ( F )) sub decomposed into of the p -adic exceptional group G 2 ABV-packets Clifton Cunningham, Andrew Fiori and Qing Zhang ψ 1 Objective ψ 2 Strategy π 2 G2 Subregular unipotent representations Artist’s impression of the main result π ̺ π 1 π ε 3 Main result 3 Proof by Geometry Forthcoming work A triple of π 0 π 3 “superunipotent” representations References ψ 0 ψ 3
Arthur packets for unipotent representations ABV-packets and Aubert duality of the p -adic exceptional group G 2 on K Rep( G 2 ( F )) sub Clifton Cunningham, Andrew Fiori and Qing Zhang ψ 1 Objective ψ 2 Strategy π 2 G2 Subregular unipotent representations Artist’s impression of the main result π ̺ π 1 π ε 3 Main result 3 Proof by Geometry Forthcoming work A triple of π 0 π 3 “superunipotent” representations References ψ 0 ψ 3
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