Survey of Basic Functions

Transcription

1 Automorphic Forms and Related Topics Hạ Long, August 21-24, 2017 Survey of Basic Functions Wen-Wei Li Chinese Academy of Sciences

2 The cover picture of Euclid (nguồn: Internet) is from the book Ai và Ky ở xứ sở những con số tàng hình by Nguyễn Phương Văn and Ngô Bảo Châu. Published by Nhã Nam & NXB Thế Giới, 2012.

3 The ideas of Braverman-Kazhdan Main references R. Godement and H. Jacquet. Zeta functions of simple algebras. Springer LNM 260, A. Braverman and D. Kazhdan. γ-functions of representations and lifting. In: Geom. Funct. Anal. Special Volume, Part I (2000). With an appendix by V. Vologodsky, GAFA 2000 (Tel Aviv, 1999) Ngô Bảo Châu, On geometry of arc spaces, the Hankel transform and function equation of L-functions, the 18th Takagi Lectures (2016). Sakellaridis talks, for a broader perspective. Goal: generalize Godement Jacquet theory to more general L-functions. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

4 Review of Godement Jacquet F: local field, and GL(n) M n (more generally: A A where A: central simple F-algebra); S (M n ): the space of Schwartz Bruhat functions on M n (F); for any φ S (M n ) and s C, let φ s = det s φ; for any admissible irreducible representation (π, V) of GL(n, F), let π s = det s π. Zeta integral Let v v V π V π and ξ S (M n ). Z GJ (s, v v, ξ) = v, π(x)v GL(n,F) matrix coefficient Here d x is a Haar measure and R(s) π 0. The case n = 1 is Tate s thesis (1950). det x n s+ ξ(x) d x. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

5 1 Meromorphic continuation to all s C, rational in q s for non-archimedean F. 2 Functional equation Z GJ (1 s, v v, F ξ) = γ GJ (s, π) Z GJ (s, v v, ξ); γ factor F S (M n ) S (M n ) is the Fourier transform. 3 In the unramified case with ξ = 1 Mn (o F ), vol(gl(n, o F )) = 1 and v, v unramified vectors with v, v = 1, we have Z GJ (s, v v, ξ ) = L(s, π). 4 In general (for non-archimedean F at least), define L(s, π) to be the gcd of zeta integrals, as ξ and v v vary. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

6 Global, adélic counterpart: use S (M n ) = v place of F S (M n,v ) w.r.t. ξ v and Poisson summation formula on M n. For archimedean F: take Casselman Wallach representations π. One can also improve the original Godement Jacquet to obtain continuity properties of zeta integrals in ξ, v v. A more canonical formalism: take S = Schwartz Bruhat half-densities on M n. Bonus: Z GJ (s, ) L s, π. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

7 Braverman Kazhdan [2], local case G: split connected reductive F-group with Langlands dual ρ G GL(N, C) an irreducible representation. Generalize (GL(n) M n, det) into (G X ρ, det ρ ), where X ρ : normal reductive monoid with unit group G, constructed in a canonical fashion from ρ, so that G X ρ is a G G-equivariant open immersion; det ρ G G m comes from the abelianization map X ρ G a of monoid, and is dual to C G such that ρ(z) = z id for z C. Z BK (s, v v, ξ) = ξ(x) v, π(x)v det G(F) ρ (x) s dx Now ξ is taken from some Schwartz space S ρ, and R(s) π 0, with π, v v are as before. G; W.-W. Li (AMSS-CAS) Survey of basic functions / 27

8 We seek to construct S ρ, and a basic function ξ S ρ in the unramified case. Desiderata. Cf. Ngô s Takagi talk Meromorphic/rational continuation of Z BK (s, ). Functional equation via some Hankel transform S ρ S ρ preserving ξ. As in the Godement Jacquet case, we require that Z BK (s, v v, ξ ) = L(s +?, π, ρ), v, v = 1. in the unramified case, with some explicit shift in s... The basic function ξ is closely related to the singularities of the monoid X ρ. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

9 Inverse Satake transform Let G be a split group over F: non-archimedean local field, q = #o F /p F. Let K = G(o F ). The spherical Hecke algebra (precisely: of measures) is H = C c (K\G(F)/K; C) +, vol(k) = 1. Classical Satake isomorphism Take T: Cartan torus, W: Weyl group. The constant term map yields Sat H C[ T] W = C[ T W]. The last term is also C[ G G] or Rep( G) C. One can replace C by Z[q ± / ], or work in the l-adic setting. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

10 Basic functions for monoids The K-unramified irreducible representations π of G(F) are in bijection with elements of T W. For every s, the π L(s, π, ρ) defines an element in the formal completion of C[ T W] relative to some explicit cone. The property Z BK (s, v v, ξ ) = L(s, π, ρ) turns out to be equivalent to ξ = ξ n, Supp(ψ n ) cpt G(F) v detρ=n, n such that ξ n Sat Tr Sym n ρ Rep( G). Not so easy to describe ξ by inverting Sat explicitly: some Kazhdan Lusztig polynomials for affine Weyl groups will appear. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

11 Approach 1 for basic functions The structure of Sat being combinatorial, one can assume char(f) = p > 0 and work l-adically, l p. Let L X be the formal arc space of (X: scheme of finite type over F q ). It is the functor L X(R) = X(R t ), F q algebra R. Thus L X(F q ) = X(F q t ) = X(o F ). Function-sheaf dictionary. Interesting constructible sheaves on Z /Fq (finite type) interesting functions on Z(F q ). Eg. the IC-complex IC Z IC-function. Normalize IC Z to be Q l on the smooth stratum. 1 Philosophy IC L Xρ ξ. Obstacle: L X ρ is really infinite-dimensional. 1 Sakellaridis remarked that it is not the most reasonable choice. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

12 When X ρ is smooth (eg. Godement Jacquet), it is reasonable to expect that IC L Xρ = Q l ; the IC-function is then 1 Xρ (o F ). Grinberg Kazhdan Drinfeld: finite-dimensional models of the singularities of L Z (for Z /Fq of finite type). This leads to a general definition of IC-functions and even IC-sheaves on L Z. By [1], ξ = IC L Xρ points to L(s η G, λ, π, ρ) instead of L(s, π, ρ). Here η G is the half-sum of positive roots, and λ is the highest weight of ρ. The local-global argument will be reviewed later. A. Bouthier, B. C. Ngo, Y. Sakellaridis. On the formal arc space of a reductive monoid. Amer. J. Math. 138 (2016), no. 1, With erratum. A. Bouthier and D. Kazhdan. Faisceaux pervers sur les espaces d arcs I: le cas d égales caractéristiques. Preprint, 2015, W.-W. Li (AMSS-CAS) Survey of basic functions / 27

13 Approach 2 for basic functions Let (V, ρ) be the representation of G in question. Write ξ Y = μ c μ (q)δ B (μ(π)) 1 Kμ(π)K Y det ρ(μ) where Y is a variable, μ ranges over the anti-dominant part of X (T), and det ρ X (T) Z. Substitution Y q s yields ξ det ρ s. We want to describe c μ. b = t u is the dual Borel with adjoint action of B. Ψ: the multi-set of T-weights on V u (as a B-representation). ρ : the half-sum of negative coroots. m μ λ, (q) = w W( 1) l(w) P w(λ + ρ ) (μ + ρ ); q W.-W. Li (AMSS-CAS) Survey of basic functions / 27

14 Here P is the q-analogue of Kostant s partition function (1 qe α ) = P (ν; q)e ν. α ν X (T) Theorem For all anti-dominant μ we have c μ (q ) = q det ρ(μ) m μ, (q). This is done by elementary invariant-theoretic arguments on G, based on prior works of Broer, R. Brylinski,... W.-W. Li (AMSS-CAS) Survey of basic functions / 27

15 Note: m μ λ, is a special case of generalized Kostka Foulkes polynomials (Panyushev). Many of its properties can be deduced from the geometry of the G-equivariant vector bundle G B (V u) G/ B. L. Basic functions and unramified local L-factors for split groups, Science China: Mathematics, Vol 60, No.5, 2017, pp Nevertheless... The formula in the Theorem is not convenient for computation when rank(g) is large. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

16 Approach 3 for basic functions (sketch) In [1], Sakellaridis gave another recipe for inverting the Satake transform, in the more general setting that Z: affine spherical homogeneous G-space, moreover, assume A Z = A Z,Gaitsgory Nadler. Sat C δ (Z) Z A Z,G N W C c (Z(F)) K Main techniques: boundary degeneration + theory of spherical functions on Z. The group case Z = G with G G-action: one can use this to invert L-factors, and recovers the formula in terms of m μ, (q). Y. Sakellaridis. Inverse Satake transforms arxiv: W.-W. Li (AMSS-CAS) Survey of basic functions / 27

17 Digression: The works of Finkelberg--Ionov and Hu Known (Achar Hendeson, Finkelberg Ginzburg Travkon): The IC-stalks of L GL(N)-orbit closures in the affine mirabolic Grassmannian yields the Kostka Shoji polynomials. Multi-variable case: Finkelberg Ionov arxiv: K λ, μ (t,, t r ) = σ S r N ( 1) σ L σ( λ+ ρ) ( μ+ ρ) r (t,, t r ) λ = (λ ( ),, λ (r) ), with λ (s) = [λ (s) μ; ρ = (ρ,, ρ) where ρ = (N,, 2, 1); λ(s) N ] (integers), same for L α r (t,, t r ): partition function of the pseudo-roots α mn = n l=m δ l where 1 m < n rn, n m 1 (mod r), and {δ l } l is the natural basis of Z rn ; each occurrence of α mn contributes a t s if m s (mod r), where s {1,, r}. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

18 When r = 2 and t = t = t, we recover the Kostka Shoji polynomials. Expectation: K λ, μ (t,, t r ) Z [t,, t d ]. This will follow from a vanishing property of H > of certain line bundles O ( μ) on T r B r N = GLr N Br N n r for some B r N -representation n r. When r = 1 we get T B N. Here B N GL N is the Borel subgroup; both are taken over C or Q l. This method is due to Brylinski and Broer. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

19 Recently, the required vanishing is proven by Yue Hu arxiv: In characteristic zero, it can be deduced from Panyushev s work. The scenario is also similar to that in the study of basic functions in the Approach 2. It would require some innovations to relate K λ, μ (t,, t r ) to IC-stalks. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

20 More general local zeta integrals Generalize Braverman Kazhdan by integrating a product of 1 suitable Schwartz functions on an affine normal X with a Zariski open G-orbit X +, such that X X + = {f = 0} for some G-eigenfunction f; 2 coefficients of an admissible representation π of G(F) on X + (F), in the sense of relative harmonic analysis. Assume X + is spherical and wavefront. Such integrals can be twisted by f s and we expect meromorphic continuation in s. Local functional equations are hopefully reflected by equivariant isomorphisms between Schwartz spaces. L. Zeta integrals, Schwartz spaces and local functional equations. arxiv: Needed: accessible examples! W.-W. Li (AMSS-CAS) Survey of basic functions / 27

21 Case study: prehomogeneous vector spaces 1 As in Braverman Kazhdan program, singularity creates difficulties. The easiest case: X is a vector space, on which G acts linearly the definition of Schwartz functions is then standard (for monoids, the only possibility is Godement Jacquet). 2 Then X is a prehomogeneous vector space. Zeta integrals of this type with π = 1 have a long history (Shintani, Igusa...) and are closely related to geometry. Eg. the poles are related to the log-canonical threshold of f at 0. Several results in the prehomogeneous case have been obtained for non-archimedean F. When F = R and X + is essentially a symmetric G-space the meromorphic continuation of the zeta integrals with coefficients. L. Towards generalized prehomogeneous zeta integrals. arxiv: W.-W. Li (AMSS-CAS) Survey of basic functions / 27

22 Case study: the doubling method Let G: a classical group, and π: irreducible admissible representation of G(F). Consider matrix coefficients c of π, against good sections f of a degenerate parabolic induction IP G (χ), where G is a bigger group and P is a parabolic, and that G G G, G = G {1} P\G G G-equivariant, open embedding Here P M is the Levi quotient, and χ is a character of M ab (F). 2 Piatetski Shapiro-Rallis [1] The integration of c against f (pulled back to G = G {1}) over G(F) yields the standard local L-factor for χ π, upon some sufficiently positive twist on χ. 2 We omit the other requirements on these data. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

23 What is more important is the quasi-affine G -space X P = P der \G. We have a natural right M ab G -action on X P P der x (m,g) P der m xg. The doubling method is rephrased in [2] using Schwartz functions on the affine closure X of X P, as universal good sections. See also [Getz Liu]. [1] I. Piatetski-Shapiro and S. Rallis. ε-factor of representations of classical groups. Proc. Nat. Acad. Sci. U.S.A., 83(13), [2] A. Braverman and D. Kazhdan. Normalized intertwining operators and nilpotent elements in the Langlands dual group. In: Mosc. Math. J. 2.3 (2002). W.-W. Li (AMSS-CAS) Survey of basic functions / 27

24 To simplify matters, take G = Sp(W), G = Sp(W W) where W is W with,. Let P G be the stabilizer of the Lagrangian diag(w). Doubling vs. monoid Let X + be the open M ab G G-orbit in X. There is a natural M ab G G-equivariant embedding M ab G X + X = X P aff. Fact (Rittatore). Such an equivariant normal affine embedding automatically makes X into a monoid with unit group M ab G. It is flat in Vinberg s sense, whose abelianzation map restricts to c M ab G M ab = GL(W) ab det G m. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

25 Identify M ab with G m by det. Relation to the L-monoids of Ngô et al. The monoid X is the L-monoid associated to the (2n + 1)-dimensional representation id std of (G m G). 1 We obtain another accessible case of Braverman Kazhdan program of L-monoids: the Schwartz spaces and Fourier/Hankel transforms are already available. 2 Shahidi and Getz Liu (2017) pursued these ideas further. 3 In particular, Braverman and Kazhdan defined a basic function c P = c P, from the geometry of X P X. Combinatorial formulas for c P exist, but the ultimate motivation seems to come from IC BunP (here: pass to equal-characteristics). W.-W. Li (AMSS-CAS) Survey of basic functions / 27

26 Idea: Over F q, Drinfeld s compactification Bun P Bun P serves as a global model of the singularities of L X. Let C be a complete, smooth, geometrically connected curve over F q, and [ ] = the quotient stacks; X X P X + are all defined over F q. Bun P = C φ X M ab G X P image M ab G generically, X as an open substack of Map C, M ab G over F q. More precisely, we should prescribe a groupoid Bun P (S) for test schemes S /Fq. As explained in [Bouthier Kazhdan] 3, one can relate c P or IC BunP to IC L X. 3 We only need the last section thereof; the idea is similar to the case of monoids in [Bouthier Ngô Sakellaridis], cf. the next slide. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

27 1 Viewing X as a monoid of unit group X + G = M ab G, there is another basic function L associated to X. 2 Bouthier Ngô Sakellaridis: L as the IC-function of L X, up to a shift s s + n. This is obtained by relating IC L X to IC M of the global model M = C φ X G G image G G generically. X + By relating both c P and L to the IC-function of L X in the equal-characteristic case, we obtain A comparison of basic functions, appendix to Shahidi s preprint As functions on X + M ab G, we have L = c P c n, where c is the abelianization map X + G m. W.-W. Li (AMSS-CAS) Survey of basic functions / 27

28 The shift in L = c P c n explained Key: The Schwartz spaces proposed by Braverman Kazhdan are always L (X). We are thus led to consider Schwartz half-densities (= square-roots of measures): what is basic about X P X (resp. X + X) is c P Ξ (resp. L Ω ), where 1 Ξ is a G -invariant measure on X P (F) = P der (F)\G (F) (note: Ξ is not M ab (F)-invariant); 2 Ω is a M ab G G-invariant measure on (M ab G)(F) X + (F), suitably normalized. We may restrict half-densities from X P (F) to its open subset X + (F). W.-W. Li (AMSS-CAS) Survey of basic functions / 27

29 If you prefer central L-values, L = L c is even more basic than L. Fact: one can actually take Ω = c n Ξ. Hence L = c P c n is equivalent to the equality of half-densities c P Ξ = c P c n+ Ω = L 1 2 Ω. In other words, basic = basic. For details, see the appendix to arxiv: W.-W. Li (AMSS-CAS) Survey of basic functions / 27