Galois to Automorphic in Geometric Langlands
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1 Galois to Automorphic in Geometric Langlands Notes by Tony Feng for a talk by Tsao-Hsien Chen April 5, The classical case, G = GL n 1.1 Setup Let X/F q be a proper, smooth, geometrically irreducible curve. For each x X we denote O x = ÔX,x, F x its field of fractions, k x its residue field, and F = F q (X) its function field. Finally, we write A F := F x O F := O x. x X x X 1.2 Automorphic side Fix a prime l p. The main player is the space of unramified automorphic functions A := Funct(GL n (F)\ GL n (A F )/ GL n (O F ), Q l ). This admits an action of a Hecke algebra for each x X : H x := Funct(GL n (O x )\ GL n (F x )/ GL n (O x ), Q l ). The action is by convolution for T H x and f A, we have (T f )(g) = f (gh 1 x )T(h x ) dh x h x GL n (F x ) with measure normalized so that GL(O x ) dh x = 1. 1
2 Proposition 1.1. We have H x Q l [Tx 1, Tx 2,..., Tx n, (Tx n ) 1 ] where Tx i is the characteristic function of the double coset x GL n (O x ) ϖ... GL n(o x ) 1.3 Galois side Let σ: π 1 (X) GL n (Q l ) be an l-adic representation; we can think of this equivalently as a local system E σ Loc n (X). We define G := {σ: π 1 (X) GL n (Q l ) geometrically irreducible}/. Recall that for each x X we have a Frob x π 1 (X). Theorem 1.2 (Drinfeld, Lafforgue, Frenkel-Gaitsgory-Vilonen). To every σ G there corresponds a non-zero f σ A such that T i x f σ = Tr( i σ(frob x )) f σ. Moreover, f σ is unique up to scalar and cuspidal. Here i σ is the ith exterior power of σ and cuspidal means that f σ (ug) du = 0 U(F)\U(A F ) for all g GL n (A F ) and U the unipotent radical of proper standard parabolic subgroup of GL n. 2 Geometric reformulation 2.1 Geometrization of adeles Let Bun n be the stack of rank n vector bundles on X. Theorem 2.1 (Weil s uniformization theorem). We have Bun n (F q ) = GL n (F)\ GL n (A F )/ GL n (O F ). This allows us to interpret A = Funct(Bun n (F q ), Q l ). An obvious categorification of this is sheaves on Bun n. 2
3 2.2 Geometrization of Hecke operators A geometric version of Hecke algebra is the moduli stack of modifications x X, Hecke i = (x, M, M M, M, β): Bun n β: M M M /M k i x This admits maps where Hecke i h i h i π i Bun n Bun n X h i (x, M, M, β) = M, h i (x, M, M, β) = M, and π i (x, M, M, β) = x. To relate these Hecke stacks to the classical Hecke algebra, we define a local Hecke stack Hecke i x by the cartesian diagram Hecke i x x Hecke i X On rational points, we have a diagram Hecke i x(f q ) Bun n (F q ) Bun n (F q ) Then the classical Hecke operators can be interpreted as Tx i f := (h i )! (h i ) f for T H x and f A. 3
4 2.3 Hecke eigensheaves Based on this we make the following definition. Definition 2.2. (Geometric Hecke operators) We define T i : D b c(bun n, Q l ) D b c(bun n X, Q l ). by F (h i π i )! (h i ) F [i(n i)]. Let σ G. An object F D b c(bun n, Q l ) is called a Hecke eigensheaf with respect to σ if for all i = 1,..., n we have T i (F ) F i E σ. Theorem 2.3 (Frenkel-Gaitsgory-Vilonen). For every σ G, there exists a non-zero Hecke eigensheaf Aut σ which is cuspidal. What is the meaning of cuspidality? You can consider the moduli spaces of flags Fl n1,n 2 for n 1 + n 2 = n, which parametrize with rank E i = n i. This admits maps {0 E 1 E E 2 0} Bun n p Fl n1,n 2 q Bun n1 Bun n2 where p = E and q = (E 1, E 2 ). A F D b c(bun n, Q l ) is cuspidal if q! p F = 0 for all n 1, n 2. Remark 2.4. Frenkel-Gaitsgory-Vilonen show that Aut σ is a perverse sheaf and is irreducible on each connected component Bun d n. If we demand that Aut σ be irreducible, then it is unique. 3 Geometric Satake Let k = k. Let G be a connected reductive group over k and Ĝ/Q l be the dual group over Q l. Definition 3.1. The affine Grassmannian is Gr G := LG/L + G where the loop group LG is defined by LG(R) := G(R((t))) and L + G(R) := G(R[[t]]). Definition 3.2. We define the category Sat := P L + G(Gr G ), perverse sheaves equivariant for the arc group L + G. Properties. 4
5 1. There is a bijection L + G\ Gr G (k) X (T) + with L + G t λ λ for t T(k((t))). Here for λ: G m T we get k((t)) T(k((t))) sending t t λ. 2. Denoting L + G t λ by O λ, we have O λ = O µ. µ λ 3. We have Gr G (R) = Here we are using the notation D = Spec k[[t]], D 0 = Spec k((t)), D R = Spec R[[t]], D 0 R = Spec R((t)). 4. Consider the diagram { (E, β): E = G bundle on D } R β: E D 0 G D 0. R R Gr G Gr G p LG Gr G q LG L+G Gr G m Gr G For A, A Sat, we define A A P(LG L+ Gr G ) by the condition Define the fusion product p (A A ) = q (A A ) A A = m! (A A ) P L + G(Gr G ). Theorem 3.3 (Geometric Satake). We have an equivalence of categories (Sat, ) (Rep Ĝ, ) such that for λ X ( T) + X (T) + the highest weight representation V λ corresponds to IC λ = IC(O λ, Q l ). Under the fiber functor to vector spaces Sat Rep(Ĝ) Vec k this equivalence corresponds to taking cohomology. 5
6 4 Statement of global geometric Langlands for general G 4.1 The Hecke stacks Consider the stack Hecke = (x, M, M, β): x X, M, M Bun n β: M X x M X x This has maps h Hecke h π Bun G Bun G X where h (x, M, M, β) = M and h (x, M, M, β) = (M, x). There is an evaluation map [ L + G\LG/L + ] G ev: Hecke Aut(D) which is described as follows. After choosing an isomorphism D x D and a trivialization of M Dx and M Dx the map β describes some transition function g β LG, which is ev(x, M, M, β). Using this we can define an operator sending Hk : Rep(Ĝ) D b c(bun G,Ql ) ev D b c(bun G X, Q l ) (V, F ) (h π)! (h ) (F IC Hk V )[shift] where ICV Hk := ev (IC V ) is the pullback of the IC sheaf corresponding to the local system V under the Geometric Satake equivalence. Similarly, for V 1,..., V d Rep(Ĝ) we define in an analogous manner. These satisfy 1. Hk V1 V 2 (F ) BunG (X) Hk V1 V 2 (F ), Hk V1... V d (F ) D b c(bun G X d, Q l ) 2. For s: X X X X the swap function, we have s (Hk V1 V 2 (F )) Hk V2 V 1 (F ). Let E be a Ĝ-local system on X, viewed as a tensor functor denoted V E V. E : Rep(Ĝ) Loc(X) 6
7 Definition 4.1. A Hecke eigensheaf with eigenvalue E is a perverse sheaf F P(Bun G, Q l ) together with isomorphisms α V : Hk V (F ) F E V for all V Rep(Ĝ) that are compatible with the symmetric tensor structure on Rep(Ĝ) and composition of Hecke operators. Conjecture 4.2. To every irreducible Ĝ-local system E, there exists a non-zero Hecke eigensheaf Aut E with eigenvalue E. 7
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