Inertial support of distinguished and inertial support - PowerPoint PPT Presentation

Introduction Inertial support Distinction Inertial support of distinguished and inertial support representations Examples G -data Distinction of tame repre- Fiona Murnaghan sentations Relatively supercuspidal University of Toronto representa- tions September 2014

Distinguished representations Introduction Inertial support Let H be a subgroup of a group G and let π be a Distinction representation of G . We say that π is H - distinguished if and inertial support there exists a nonzero H -invariant linear functional on the Examples space of π . G -data In this talk, all representation spaces are complex vector Distinction of tame repre- spaces. sentations We assume that G is a connected reductive p -adic group: Relatively supercuspidal G = G ( F ) , where G is a connected reductive F -group and F representa- tions is a local nonarchimedean field. (For technical reasons, we assume that the residual characteristic of F is odd.) When studying K -types contained in distinguished representations of G , we will be working with distinction of smooth representations of profinite groups.

Introduction We say that θ is an involution of G if θ is an Inertial F -automorphism of G of order two. Let H be the group of support fixed points of θ . We are interested in understanding Distinction and inertial (parametrizing, whenever possible) the H -distinguished support irreducible admissible representations of G . These are the Examples representations which play a role in harmonic analysis on G -data the p -adic symmetric variety G / H . Distinction of tame repre- sentations In some situations, we consider a slight generalization of the Relatively notion of distinction: If χ is a quasicharacter of H , let supercuspidal representa- tions Hom H ( π, χ ) = { λ ∈ V ∗ | λ ◦ π ( h ) = χ ( h ) λ ∀ h ∈ H } . If Hom H ( π, χ ) is nonzero, we say that π is ( H , χ ) - distinguished .

Introduction Various examples of H -distinguished representations of G Inertial occur as irreducible subquotients of representations of the support form Ind G P τ , where Distinction and inertial • M is a θ -stable Levi factor of a (not necessarily θ -stable) support Examples parabolic subgroup P of G . G -data • τ is an irreducible supercuspidal representation of M Distinction of such that some unramified twist of τ is tame repre- sentations M θ -distinguished. Relatively supercuspidal representa- tions In a recent paper(J. No. Theory, 2014), we obtain information about distinction of types contained in the (inertial) supports of distinguished depth-zero irreducible smooth representations.

Introduction Theorem Inertial The support of a depth-zero irreducible smooth support H-distinguished representation of G contains a pair ( M , τ ) Distinction and inertial where M is a θ -stable Levi subgroup of G and τ is an support irreducible (depth-zero) supercuspidal representation of M Examples containing a (depth-zero) unrefined minimal K-type ( K M , ρ M ) G -data such that K M is θ -stable and ρ M is K θ Distinction of M -distinguished. tame repre- sentations This suggests that the inertial supports of distinguished Relatively irreducible smooth representations may contain supercuspidal representa- distinguished representations of θ -stable Levi subgroups. tions We study the properties of K -types contained in distinguished “tame” representations and their inertial supports and use these properties to show that the inertial supports do have this property.

Introduction Suppose that ( K , ρ ) is a K -type contained in an irreducible Inertial support smooth distinguished representation π . Then, because π is Distinction distinguished, there exists g ∈ G such that and inertial support Hom K ∩ g H ( ρ, 1 ) � = 0. When ( K , ρ ) satisfies: Examples • ( K , ρ ) is a G -cover of a “sufficiently large” type G -data contained in the inertial support of π Distinction of tame repre- sentations • The inertial support of π is “tame” Relatively • Certain hypotheses concerning quasicharacters are supercuspidal representa- satisfied, tions then we can show that the inertial support of π is “distinguished”. More precise statements will be made later.

Let τ and τ ′ be irreducible supercuspidal representations of Levi subgroups M and M ′ of G , respectively. The pairs Introduction ( M , τ ) and ( M ′ , τ ′ ) are said to be inertially equivalent if Inertial support there exist g ∈ G and χ ∈ X ( M ′ ) such that g M = M ′ and Distinction g τ ≃ τ ′ χ . The inertial equivalence class of a pair ( M , τ ) will and inertial support be denoted by [ M , τ ] G . Examples G -data Recall that if π is an irreducible smooth representation of G , Distinction of there exists a pair ( M , τ ) , which is unique up to conjugacy, tame repre- sentations consising of a Levi subgroup M of G and an (equivalence Relatively class of an) irreducible supercuspidal representation τ of M supercuspidal representa- such that for any parabolic subgroup P ∈ P ( M ) , π occurs as tions an subquotient of Ind G P τ . (Here, P ( M ) is the set of parabolic subgroups of G having Levi factor M .) The conjugacy class of the pair ( M , τ ) is called the (cuspidal) support of π . The inertial equivalence class I ( π ) := [ M , τ ] G is called the inertial support of π .

We will say that an inertial equivalence class of G is θ - distinguished (or just distinguished ) if it contains a pair Introduction ( M , τ ) such that θ ( M ) = M and Hom M θ ( τ, 1 ) � = 0. Inertial support Remark Distinction and inertial The group G acts on the set of involutions of G . If g ∈ G , support the involution g · θ is defined by Examples G -data ( g · θ )( x ) = g θ ( g − 1 xg ) g − 1 , x ∈ G . Distinction of tame repre- sentations Note that G g · θ = gG θ g − 1 . It is clear that an inertial Relatively supercuspidal equivalence class is θ -distinguished if and only if it is representa- tions g · θ -distinguished for every g ∈ G . Question Let π be an irreducible smooth H-distinguished representation of G. Is the inertial support I ( π ) of π θ -distinguished?

Introduction Inertial support Distinction Example and inertial support Let G be a split group and let π be an (irreducible) Examples unramified representation of G . As shown by Helminck and G -data Wang, there exists a θ -stable maximal F -split torus A in G . Distinction of tame repre- The pair ( A , 1 ) belongs to the inertial support I ( π ) of π . sentations Relatively Hence I ( π ) is θ -distinguished (even when π is not supercuspidal distinguished). representa- tions

Example: Introduction Inertial support H = G θ = Sp 2 n ( F ) , • G = GL 2 n ( F ) , Distinction and inertial • M = GL n ( F ) × GL n ( F ) such that support θ ( g 1 , g 2 ) = ( t g − 1 2 , t g − 1 1 ) , g j ∈ GL n ( F ) . Examples Note that M θ = { ( g , t g − 1 ) | g ∈ GL n ( F ) } . An irreducible G -data Distinction of smooth representation τ 1 ⊗ τ 2 of M is M θ -distinguished if tame repre- sentations and only if τ 2 ≃ τ 1 . Relatively supercuspidal (Fix an irreducible smooth representation τ ′ of GL n ( F ) . Let representa- tions A be a nonzero intertwining operator between the representation g �→ τ ′ ( t g − 1 ) and � τ ′ . Define a linear form λ on V τ ′ ⊗ V τ ′ by λ ( v 1 ⊗ v 2 ) = �A v 2 , v 1 � , v 1 , v 2 ∈ V τ ′ . It is easy to see that λ is M θ -invariant.)

Assume that τ ′ is supercuspidal. If P is a parabolic subgroup of G with Levi factor M , the representation P ( τ ′ ⊗ τ ′ ) is irreducible and hence has a Whittaker model Introduction Ind G Inertial (since τ ′ has a Whittaker model). According to a result of support P ( τ ′ ⊗ τ ′ ) is not H -distinguished. So Heumos and Rallis, Ind G Distinction and inertial we cannot construct H -distinguished representations of G support by inducing from M θ -distinguished supercuspidal Examples G -data representations of M . Distinction of F , g ∈ GL n ( F ) and τ ν = τ ′ ⊗ ντ ′ . For P a Let ν ( g ) = | det g | − 1 tame repre- sentations particular parabolic subgroup of G with Levi factor M , the Relatively unique irreducible quotient π τ ν of the reducible supercuspidal representa- representation Ind G P τ ν is H -distinguished. (This is due to tions Heumos and Rallis.) The pair ( M , τ ν ) belongs to the support of π τ ν and τ ν is not M θ -distinguished. However, the representation τ ′ ⊗ τ ′ belongs to I ( π τ ν ) and τ ′ ⊗ τ ′ is M θ -distinguished. In this example, none of the distinguished pairs in the inertial support I ( π τ ν ) belong to the support of π τ ν .