Local L -values and geometric harmonic analysis on spherical - PowerPoint PPT Presentation

Local L -values and geometric harmonic analysis on spherical varieties Jonathan Wang (joint w/ Yiannis Sakellaridis) MIT Columbia Automorphic Forms and Arithmetic Seminar October 9, 2020 Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 1 / 21

Outline Integral representations of L -functions 1 What is a spherical variety? 2 Function-theoretic results 3 Geometry 4 G connected split reductive group / F q Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 2 / 21

Integral representations of L -functions C smooth, projective, geometrically connected curve over F q k = F q ( C ) global function field [ G ] = G ( k ) \ G ( A ) Automorphic period integral For a “nice” reductive subgroup H ⊂ G , the period integral � P H ( f ) := f ( h ) dh [ H ] for f a cusp form on [ G ] is related to a special value of an L -function. In these cases, X = H \ G is a homogeneous affine spherical variety. Theorem (Luna, Richardson) H \ G is affine if and only if H is reductive. Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 3 / 21

By formal manipulation � � � f ( h ) dh = f ( g ) · 1 X ( O ) ( γ g ) dg [ H ] [ G ] γ ∈ ( H \ G )( k ) where Definition � ΣΦ( g ) := Φ( γ g ) γ ∈ H \ G ( k ) is the X-Poincar´ e series (alias X-Theta series) on [ G ] G ( k ) X ( A ) ← ( H \ G )( k ) × G ( A ) → G ( k ) \ G ( A ) = [ G ] Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 4 / 21

General expectation (Sakellaridis) Start with X • = H \ G “nice” ( H not necessarily reductive) Choose an affine embedding X • ֒ → X (e.g., X = X • aff ) Let Φ 0 = IC X ( O ) denote the “IC function” of X ( O ) Define the X -Poincar´ e series � ΣΦ 0 ( g ) = Φ 0 ( γ g ) γ ∈ X • ( k ) Define the “ X -period” by � P X ( f ) = f · ΣΦ 0 , f cusp form on [ G ] [ G ] Conjecture (Sakellaridis, 2009) If f is unramified, then |P X ( f ) | 2 is “equal” to special value of L-function. Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 5 / 21

Example (Rankin–Selberg convolution) For π 1 , π 2 cuspidal GL 2 ( A )-representations, � L ( 1 f 1 ( g ) f 2 ( g ) E ∗ ( g , 1 2 + s , π 1 × π 2 , std ⊗ std) = 2 + s ) dg Z ( A ) \ [GL 2 ] for unramified f 1 ∈ π 1 , f 2 ∈ π 2 , Whittaker normalized. Think of the normalized Eisenstein series E ∗ ( g , s ) = ζ (2 s ) E ( g , s ) as a distribution on [GL 2 × GL 2 ] via diagonal embedding. RHS is obtained by Mellin transform from � P X ( f 1 × f 2 ) = ( f 1 × f 2 ) · Σ( 1 X ( O ) ) [ G ] G = GL 2 × GL 2 � X = A 2 × GL 2 open G -orbit X • = ( A 2 − 0) × GL 2 = H \ G H = ( ∗ ∗ 0 1 ) mirabolic subgroup, diagonally embedded Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 6 / 21

In all the previous examples, X was smooth. Example (Sakellaridis) G = GL × n × G m , H = 2 �� � � � � � � � a x 1 a x 2 a x n � × × · · · × × a � x 1 + · · · + x n = 0 � 1 1 1 � aff (usually singular). Let X = H \ G n = 2: Rankin–Selberg n = 3: P X is equivalent to the construction of Garrett This is a case where the integral P X “unfolds” and our local results imply: Theorem (Sakellaridis-W) Over a global function field, the Mellin transform of P X | π gives an integral representation of L ( s , π, std ⊗ n ⊗ std 1 ) for Re( s ) ≫ 0 on cuspidal representations π under Whittaker normalization. Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 7 / 21

What is a spherical variety? k = F q Definition A G -variety X / F q is called spherical if X k is normal and has an open dense orbit of B k ⊂ G k after base change to k Think of this as a finiteness condition (good combinatorics) Examples: Toric varieties G = T Symmetric spaces K \ G Group X = G ′ � G ′ × G ′ = G Reductive monoid X � X • = G ′ � G ′ × G ′ Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 8 / 21

Why are they relevant? Conjecture (Sakellaridis–Venkatesh) For any affine spherical G-variety X (*), and a cuspidal G ( A ) -representation π ֒ → A 0 ( G ) , 1 P X | π � = 0 implies that π lifts to σ ֒ → A 0 ( G X ) by functoriality along a map ˇ G X ( C ) → ˇ G ( C ) , 2 there should exist a ˇ G X -representation ρ X : ˇ G X ( C ) → GL( V X ) π = ( ∗ ) L ( s 0 ,σ,ρ X ) such that |P X | 2 L (0 ,σ, Ad) for a special value s 0 . Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 9 / 21

Some history on ˇ G X Goal: a map ˇ G X → ˇ G with finite kernel ˇ T X is easy to define Little Weyl group W X and spherical root system Symmetric variety: Cartan ’27 Spherical variety: Brion ’90, Knop ’90, ’93, ’94 Gaitsgory–Nadler ’06: define subgroup ˇ ⊂ ˇ G GN G using Tannakian X formalism Sakellaridis–Venkatesh ’12: normalized root system, define ˇ G X → ˇ G combinatorially with image ˇ G GN under assumptions about GN X Knop–Schalke ’17: define ˇ G X → ˇ G combinatorially unconditionally Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 10 / 21

ˇ X � G G X V X G ′ � G ′ × G ′ ˇ Usual Langlands G ′ ˇ g ′ Whittaker normal- ˇ ( N , ψ ) \ G G 0 ization ˇ T ∗ std Hecke G m \ PGL 2 G = SL 2 Rankin–Selberg, H \ GL n × GL n = ˇ T ∗ (std ⊗ std) Jacquet–Piatetski- G GL n × A n Shapiro–Shalika ˇ G = SO 2 n × Gan–Gross–Prasad SO 2 n \ SO 2 n +1 × SO 2 n std ⊗ std Sp 2 n Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 11 / 21

G X = ˇ ˇ G For this talk, assume ˇ G X = ˇ G (and X has no type N roots). [‘N’ is for normalizer] Equivalent to: (Base change to k ) X has open B -orbit X ◦ ∼ = B X ◦ P α / R ( P α ) ∼ = G m \ PGL 2 for every simple α , P α ⊃ B Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 12 / 21

Sakellaridis–Venkatesh ´ a la Bernstein Sakellaridis–Venkatesh: give generalized Ichino–Ikeda conjecture relating |P X | 2 to local harmonic analysis: |P X ( f ) | 2 I-I conjecture � = (local computation) v F = F q ( ( t ) ), O = F q [ [ t ] ] spherical functions (unramified Hecke eigenfunction) on X ( F ) unramified Plancherel measure on X ( F ) Fix x 0 ∈ X ◦ ( F q ) in open B -orbit. For Φ ∈ C ∞ c ( X ( F )) G ( O ) , define the X -Radon transform � π ! Φ( g ) := Φ( x 0 ng ) dn , g ∈ G ( F ) N ( F ) π ! Φ is a function on N ( F ) \ G ( F ) / G ( O ) = T ( F ) / T ( O ) = ˇ Λ. Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 13 / 21

Conjecture 1 (Sakellaridis–Venkatesh) Assume ˇ G X = ˇ G and X has no type N roots. There exists a symplectic V X ∈ Rep(ˇ G ) with a ˇ T polarization X ) ∗ such that V X = V + X ⊕ ( V + G (1 − q − 1 e ˇ α ) � α ∈ ˇ Φ + ˇ ∈ Fn(ˇ π ! IC X ( O ) = Λ) X ) (1 − q − 1 2 e ˇ λ ) � λ ∈ wt( V + ˇ where e ˇ λ is the indicator function of ˇ λ , e ˇ µ = e ˇ λ e ˇ λ +ˇ µ Mellin transform of right hand side gives T ( C ) �→ L ( 1 2 , χ, V + n ) , this is “half” of L ( 1 X ) 2 , χ, V X ) χ ∈ ˇ g / ˇ L (1 , χ, ˇ L (1 , χ, ˇ t ) Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 14 / 21

Previous work Conjecture 1 (possibly with ˇ G X � = ˇ G ) was proved in the following cases: Sakellaridis (’08, ’13): X = H \ G and H is reductive (iff H \ G is affine), no assumption on ˇ G X doesn’t consider X � H \ G Braverman–Finkelberg–Gaitsgory–Mirkovi´ c [BFGM] ’02: X = N − \ G , ˇ G X = ˇ T , V X = ˇ n Bouthier–Ngˆ o–Sakellaridis [BNS] ’16: X toric variety, G = T , ˇ G X = ˇ T , weights of V X correspond to lattice generators of a cone X ⊃ G ′ is L -monoid, G = G ′ × G ′ , ˇ g ′ ⊕ T ∗ V ˇ G X = ˇ λ G ′ , V X = ˇ Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 15 / 21

Theorem (Sakellaridis–W) Assume X affine spherical, ˇ G X = ˇ G and X has no type N roots. Then G (1 − q − 1 e ˇ α ) � α ∈ ˇ Φ + ˇ π ! IC X ( O ) = X ) (1 − q − 1 2 e ˇ λ ) � λ ∈ wt( V + ˇ X ∈ Rep( ˇ for some V + T ) such that: X ) ∗ has action of (SL 2 ) α for 1 (Functional equation) V X := V + X ⊕ ( V + every simple root α We do not check Serre relations 2 Assuming V X satisfies Serre relations (so it is a ˇ G-representation), we determine its highest weights with multiplicities (in terms of X) (2) gives recipe for conjectural ( ρ X , V X ) in terms of only data from X If V X is minuscule, then Serre relations hold Proposition If X = H \ G with H reductive, then V X is minuscule. Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 16 / 21

Enter geometry Base change to k = F q (or k = C ) X O ( k ) = X ( k [ [ t ] ]) Problem: X O is an infinite type scheme Bouthier–Ngˆ o–Sakellaridis: IC function still makes sense by Grinberg–Kazhdan theorem Jonathan Wang (MIT) Spherical varieties and L -functions October 9, 2020 17 / 21