Automorphic Forms and Automorphic Vector Bundles Weixiao Lu June 2020 Contents 1 Automorphic Forms and Automorphic Representations 1 2 Automorphic Vector Bundles and Automorphic Representations 3 3 Relations to (g;K) Cohomology 5 References 8 Happy Dragon Boat Festival! 1 Automorphic Forms and Automorphic Representations Denote A = AQ and Af = AQ;f be the finite adele. Let G be a reductive group over Q, Z is the center of G and g is the complexification of the Lie algebra of G(R), U(g) and Z(g) stands for the universal enveloping algebra and the center of universal enveloping algebra respectively. Kmax;f ⊂ G(Af ) be a maximal compact subgroup, K1 ⊂ G(R) be a maximal compact subgroup of Lie group G(R). Denote Kmax = Kmax;f × K1. Definition 1.1 (Following [1]). A function ' : G(A) ! C is called an automorphic form if • ' is smooth. i.e. locally constant on the non-archimedean part and C1 on the archimedean part. • ' is left G(Q) invariant. 1 • ' is Kmax finite. i.e. The subspace generated by Kmax · ' is finite dimensional. Where the action is given by (k')(g) = '(gk). • ' is Z(g) finite. i.e. ' is annihilated by a finite codimensional ideal of Z(g). • ' is moderate growth. The space of all automorphic forms is denoted by A(G). Let ! : Z(Q)nZ(A) ! C× be a central character, define A(G; !) as follows A(G; !) = f' 2 A(G)j'(zg) = !(z)'(g); 8z 2 Z(A); x 2 G(A)g Remark. We will pretend that G(Q)nG(A)=Z(R) is compact, which is equivalent to say that there is no growth condition. Although, in our main example G = GL2 this will be false, but we pretend it is true(We will lost automorphic forms that are not cusp forms, for example, Eisenstein series) Example 1.2. For G = GL2;Z = Gm,in this case g = gl(2; C) and Z(g) = C[∆;Z], where 1 2 ∆ = EF + FE + 2 H is the Casimir operater and Z is the center of g. Let Γ = Γ0(N) . For a modular form f 2 Sk(Γ0(N); ), we can define a adelic lift −k 'f (g) = f(g1(i))j(g1; i) (k0) Which can be checked to be a automorphic form. Moreever, we have an isomorphism 1 Γ\0(N) S (Γ (N)) =∼ A h∆ − (k2 − 1)i; σ k 0 cusp 4 k (See [2]) Back to general discussion, We have a natural action of G(A) on A(G) or A(G; !). i.e. these space is natural a (g;K1) × G(Af ) module. The action is given by (X; k; g; ') 7! (g 7! X'(xkg)) Recall that we have assumed that all automorphic forms are cuspidal, then we have a spectrum decomposition(called the cuspidal spectrum) M 0 m(π) A(G; !) = (π1 ⊗ (⊗lπl)) π Where m(π) is a finite number, called the automorphic multiplicity. ⊗0 is the restricted tensor 0 product. And for all but finitely many finite prime l,πl is unramified principal series. and vl is the sperical vector for πl. 2 (For a proof, see [1] Chapter 9 or [3] Chapter 3, where A is replaced by Acusp) G(Ql) Here unramified principal series means πl = Ind χl. Where B is Borel subgroup and T is B(Ql) the Levi quotient. And × χl : T (Ql) ! T (Ql)=T (Zl) ! C G(Zl) G(Ql) is an unramified character. As G( l) = B( l)G( l) (Iwasawa decomposition), so if ' 2 Ind χl . Q Q Z B(Ql) Then '(g) = '(bk) = χl(b)'(1) G(Zl) 0 So dim πl = 1, we then choose vl = '(1) ! ! ∗ ∗ ∗ 0 Example 1.3. For G = GL2;B = , and T = . 0 ∗ 0 ∗ × ! Q 0 χ is of the form l ! × ! ; (x; y) 7! αvl(x)βvl(y) l × Z Z C 0 Ql Definition 1.4. An automorphic representation is a representation of (g;K1) × G(Af ) of the 0 form π1 ⊗ (⊗lπl) which occurs in the cuspidal spectrum decomposition(for some !). Remark. In general, an automorphic representation is an irreducible admissible subrepresen- tation of natural action of (g;K1) × G(Af ) on A(G). And automorphic representation defined above is called cuspidal automorphic representation. 2 Automorphic Vector Bundles and Automorphic Represen- tations Fix an algebraic representation W of K1. And fix an open compact subgroup Kf ⊂ G(Af ). We have an locally symmetric space(real manifolds) ShG(Kf ) = G(Q)n (G(Af )=Kf ) × (G(R)=K1) and an vector bundle on ShG(Kf ): K1 W = G(Q)n (G(Af )=Kf ) × G(R) × W So we have 1 fC sections of W g = f' : G(Q)nG(A) ! W; K1 equivariant, Kf invariantg Which is roughly equal to (A(G; !)Kf ⊗ W )K1 if we omit the cusp issue and choose ! compatible with W . 3 Example 2.1. For G = GL2 List of irreducible admissible (g;K) representation π1: ([4], Chapter 7 or [3] 2.3): • holomorphic discrete series: π1 = πk ⊕ πk+2 ⊕ · · · ! cos θ sin θ 2πilθ Where k > 0, πl stands for K = SO(2) acts as 7! e . And the g action − sin θ cos θ is like an \anti" Verma module with integer weights: k k Ev = ( + l)v; F v = ( − l)v l 2 l 2 l−2 • antiholomorphic discrete series: π1 = ··· π−k−4 ⊕ π−k−2 ⊕ π−k where k > 0 and g acts as a Verma module of integral weight. • principal series π1 = ··· πk−2 ⊕ πk ⊕ πk+2 ⊕ · · · g action looks like \Verma module with non-integer weight" • finite dimensional representation • limit of discrete series(corresponds to weight 1 form) Fix the representation × × −k χ−k : K1 = R · SO(2) ! C ; (r; z) 7! z From previous lecture, we learned that this corresponds to !k And we have 1 : Kf K1 M Kf ⊕m(π) K1 C (ShG(Kf );W −k) = (A(G) ⊗ W−k) = (πf ) ⊗ (π1 ⊗ W−k) π K1=χk The last term is just π1 It has a subspace M Kf ⊕m(π) K1=χk F =0 Hol(ShG(Kf );W −k) = (πf ) ⊗ (π1 ) π 4 K1 So k is the lowest weight, by the list of classification above, when π1 ⊗ W−k) non zero, π1 must be discrete series representation. We can use above discussion to explain old/new form theory. M Γ^ K1=χk Sk(Γ) = Hol(ShG(Γ)b ; χ−k) = (πf ) ⊗ π1 π (p) 0 0 Consider Γ = Γ1(N); p 6 jN; K = Γb = K GL2(Zp). And Γ ⊃ Γ = Γ1(N) \ Γ0(p);K = (p) K Iwp.Where × ! Iw = Zp Zp p × pZp Zp is the Iwasawa subgroup. We have two types of operators, which fits into the following diagram (p);K(p) L ⊕m(π) GL2(Zp) Sk(Γ) = π(πf ) ⊗ (πp) (p) 0 L (p);K ⊕m(π) Iwp Sk(Γ ) = π(πf ) ⊗ (πp) ( M (p);K p) ⊕m(π) Iwp GL2(Zp) New forms: = (πf ) ⊗ (πp) ,sum for those π such that πp = 0 but π Iwp Iwp πp 6= 0. Special for GL2 theory, for these π, dim πp = 1. Old Forms: If πp is an unramified principal series, we have an isomorphism GL2( p);⊕2 ∼ Iwp (πp) Z = (πp) 3 Relations to (g;K) Cohomology Now, we move to the general situation: Recall that we have gotten 1 M Kf ⊕m(π) K1 fC sections of W g = (πf ) ⊗ (π1 ⊗ W ) π Let's assume that we are in the situation of Shimura varieties, we want cohomology for holo- • morphic sections H (ShG(Kf );W ). Use Dolbeault resolution 1 2 d Ohol C1 @ Ω @ Ω @ ··· @ Ω 0 ShG(Kf ) 5 + 1 ∼ + ∗ 1 ∼ − ∗ Note that TCShG(Kf ) = g=q = p ,so Ω = (p ) and Ω = (p ) Tensoring with W , we get a resolution of W hol, which implies ! • hol • M Kf ⊕m(π) • − H (ShG(Kf );W ) = H (πf ) ⊗ (π1 ⊗ Hom(^ (p );W )) π And the latter term is exactly the (g;K) cohomology we now introduce. What is (g;K) cohomology?[5] (1) Lie algebra cohomology Let g be a Lie algebra and V be a g module. Define q q C (g; V ) = HomkΛ (g);V ) and d : Cq(g; V ) ! Cq+1(g;V ) is given by X i X i+j df(x0; ··· ; xq) = (−1) xif(x0; ··· ; xbi; ··· ; xq)+ (−1) f([xi; xj]; x0; ··· ; xbi; ··· ; xbj; ··· ; xq) i i<j The cohomology is denoted by H•(g;V ). Which is also the right derived functor of V 7! V g := fv 2 V jxv = 0; 8x 2 gg.i.e. p p H (g;V ) = ExtUg(k; V ) where k is the base field. (2) Relative Lie algebra cohomology Let k ⊂ g be a Lie subalgebra.Define q q C (g; k;V ) = Homk(Λ (g=k);V ) q X = ff 2 C (g;V )jf only depends on xi 2 g=k; f(x1; ··· ; [x; xi]; ··· ; xq) = xf(x1; ··· ; xq); for x 2 kg i It can be verified that C•(g; k;V ) is a sub-cochain complex of C•(g;V ), whose cohomology is denoted by H•(g; k;V ). (3)( g;K) cohomology and (q;K) cohomology. Now we work over C, let g be a Lie algebra, not necessrily reductive. Let G be a real Lie group with complexified Lie algebra g and K be the maximal compact with connected component K0. 6 Let V be a (g;K) module, define q q ∼ q K=K0 C (g; K; V ) := HomK (Λ (g=k);V ) = C (g; k;V ) Whose cohomology groups is denoted by H•(g;K; V ).