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Modular Forms on G2: Lecture 1

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Modular Forms on G2: Lecture 1 MODULAR FORMS ON G2: LECTURE 1 AARON POLLACK 1. Classical modular forms The purpose of this course is to (eventually) define and discuss modular forms on the exceptional group G2. I will not assume that you know what G2 is, and certainly I will not assume that you know what a modular form is on G2. Instead, the prerequisite for this course is basic graduate representation theory, complex analysis, algebra etc, and a familiarity with classical modular forms on GL2 or the upper half plane. Today I'll give an overview of what we will discuss over the course of the term, and the key results we will prove, time-permitting. Let me recall the definition of classical modular forms: SL2(R) acts on the upper half-plane h = fz 2 C : Im(z) > 0g −1 a b by fractional linear transformations γz = (az + b)(cz + d) for γ = c d 2 SL2(R). Suppose n ≥ 1, and f : h ! C is a holomorphic function. One says that f is a modular form of weight n if n • f(γz) = (cz + d) f(z) for all γ in a congruence subgroup of SL2(Z) • f(x + iy) is O(yN ) for some N as y ! 1 Recall that such an f has a Fourier expansion: if f is level 1 then f(z) = f(z + n) for all n 2 Z P n and this plus growth condition plus holomorphy implies f(z) = n≥0 af (n)q for some complex numbers af (n). Here q = exp(2πi), as usual. Examples Q n 24 2 3 4 (1) Ramanujan's function: ∆(z) = q n≥1 (1 − q ) = q − 24q + 252q − 1472q + ··· P 1 (2) Eisenstein series: E2k(z) = (m;n)6=0 (mz+n)2k , k ≥ 2. Modular forms as above are associated to SL2 (actually, better to use GL2, but for simplicity we will use SL2 for a few lectures) as follows: • SL2(R) acts on h transitively • The stabilizer of i is SO(2) ) SL2(R)= SO(2) ' h • Key point: SO(2) ⊆ SL2(R) is a compact subgroup, and is maximal among compact subgroups. (General theory: all maximal compact subgroups are conjugate) • Thus: modular forms give f : SL2(R)= SO(2) ! C satisfying f(γg) ≈ f(g) for all γ 2 Γ, some congruence subgroup. Better Given g 2 SL2(R), f a modular form on of weight n, define varphif : SL2(R) ! C −n 'f (g) = j(g; i) f(g · i) a b where j(g; z) = cz + d, g = c d is the usual factor of automorphy. Recall: j(g1g2; z) = j(g1; g2z)j(g2; z). The function φf has nice properties, just like f did: cos(θ) sin(θ) Lemma 1. Let the notation be as above. Suppose γ 2 Γ, and kθ = − cos(θ) sin(θ) 2 SO(2). Then (1) φf (γg) = φf (g). So, φf is actually left-invariant under Γ. 1 niθ (2) φf (gkθ) = e φf (g). So, φf now translates on the right by SO(2) under this nice 1- dimensional representation. Proof. This are immediate checks, from the definition: −n φf (γg) = j(γg; i) f(γg · i) = j(γ; gi)−nj(g; i)−nj(γ; gi)nf(g) = φf (g): Similarly: −n φf (gk) = j(gk; i) f(gk · i) = j(g; ki)−nj(k; i)−nf(g · i) −n = (cos(θ) − i sin(θ)) φf (g) inθ = e φf (g): Finally, the fact that f was holomorphic means that there is a linear differential operator (the Cauchy-Riemann equations) that kills f. This translates into the fact that Dnφf ≡ 0, for some specific linear differential operator Dn. 2. Automorphic functions Via the above lemma, it is clear that we can replace SL2 with other semisimple Lie groups G, and again ask for functions that are left-invariant under a discrete subgroup Γ of G. For example, we can take (1) G = SLn(R), and we can consider functions f : SLn(Z)n SLn(R) ! C (2) G = SO(p; q) = fg 2 SL : g 1p gt = 1p g p+1 −1q −1q (3) G = Sp = fg 2 GL : g 1n gt = 1n g 2n 2n −1n −1n All three of the above G can be considered generalizations of G = SL2: (1) SLn obviously generalizes SL2 (2) PGL2 = SO(2; 1), so this generalizes GL2 or SL2 (3) Sp2 = SL2, so this is a generalization as well One nice thing about SL2 was that SL2(R)=K was a complex manifold, where K was a maximal compact subgroup of SL2(R). Which of the above groups G have the property that G=K has (G-equivariant) complex structure? (1) K for SLn(R) is SO(n). SLn(R)=SO(n) is complex manifold only for n = 2 (2) K for SO(p; q) is (essentially) SO(p)×SO(q). G=K only complex manifold only when p = 2 or q = 2 (3) For G = Sp2n(R), K = U(n). Here G=K is a complex manifold for all n. 3. Siegel modular forms When we replace SL2 by Sp2n, the classical holomorphic modular forms get replaced by what are called Siegel modular forms. We now define these. Set t hn = fZ 2 Mn(C): Z = Z; Im(Z) > 0g 2 the so-called Siegel upper half-space of degree n. Thus Z 2 hn means Z = X + iY with X; Y n × n real symmetric matrices, and Y is positive definite. The group Sp2n(R) acts on hn as a b −1 g = c d in n× block form;Z 2 hn; g · Z = (aZ + b)(cZ + d) : More carefully, if g 2 Sp2n(R) and Z 2 hn, then it is a fact (we will discuss later) • cZ + d is invertible, as an n × n complex matrix • (aZ + b)(cZ + d)−1 is symmetric with positive-definite imaginary part • Sp2n(R) × hn ! hn as (g; Z) 7! g · Z defines an action Thus, when n = 1, this reduces to the usual action of SL2(R) on the upper half plane. Suppose k ≥ 1 is an integer. A Siegel modular form of weight k is a holomorphic function f : hn ! C satisfying k (1) f(γZ) = det(cZ + d) f(Z) for all γ 2 Γ ⊆ Sp2n(Z) some congruence subgroup (2) when n = 1, some slow growth condition (which turns out to be automatic for n ≥ 1). 1 V Siegel modular forms have a nice Fourier expansion: One has 1 2 Sp2n if and only if the n × n matrix V is symmetric. Assume (for simplicity) that Γ = Sp2n(Z), so that we are discussing Siegel modular forms of level 1. Then t f(Z) = f(Z + V ) for all V 2 Mn(Z);V = V which implies the Fourier expansion X 2πi tr(TZ) f(Z) = af (T )e _ T 2Sn(Z) t for complex numbers (the Fourier coefficients) af (T ). Here Sn(Z) = fV 2 Mn(Z);V = V g and _ Sn(Z) is the dual lattice in Sn(Q)(n × n symmetric matrices with rational coefficients) for the trace pairing (X; V ) = tr(XV ). In other words, the T appearing in the Fourier expansion are the half-integral symmetric matrices. Fact: af (T ) 6= 0 implies T ≥ 0, i.e., T is positive semi-definite. This is the analogue to Siegel modular forms of the Fourier expansion for regular modular forms only having non-negative terms. Summary: Suppose G a semisimple group over Q. Can consider (1) The automorphic forms, which are functions ' :ΓnG(R) ! C satisfying some conditions having to do with slow growth and being annihilated by some differential operator. (2) When G(R)=K has the structure of a complex manifold, there are special automorphic forms: (3) Namely, one can consider those ' that are related to holomorphic functions on G(R)=K (like we did for holomorphic modular forms φf $ f on SL2) In part 1 of course: we will take G = SL2 and G = Sp4, and consider the holomorphic modular forms, respectively the Siegel modular forms. In particular, we will discuss (1) Eisenstein series and L-functions for GL2 (2) Basic properties and examples of Siegel modular forms on Sp4, such as Eisenstein series and Θ functions (3) L-functions for Siegel modular forms on Sp4 (technically, GSp4 or PGSp4) 4. Modular forms on G2 In the second half of this course, we will consider G = G2. In particular • There is an algebraic group G2 over Q; we will define this group. • G2(R) is a 14-dimensional non-compact Lie group 3 • K ⊆ G2(R) a maximal compact subgroup is (SU(2) × SU(2))/µ2; so, we will discuss how this K sits inside G2(R) • There is a lattice G2(Z) ⊆ G2(R) and G2(Z)nG2(R) is non-compact, just like we saw with SL2 • But now, G2(R)=K has no invariant C-structure. • This implies that there is no a priori or obvious notion of modular forms or Fourier expan- sion However, it turns out that there is a notion of modular forms on G2. Their study was initiated by Gan, Gross, Savin, and Wallach. Let us now define modular forms on G2, at least in the way I like to define them. We have SU(2) acts on C2, and then the symmetric powers Sym`(C2). Definition 2. Let n ≥ 1 be an integer. A modular form on G2 of weight n (and level 1) is a function 2n 2 • ' : G2(R) ! Sym (C ) satisfying • '(γg) = '(g) for all γ 2 G2(Z) and g 2 G2(R) −1 1 • '(gk) = k · '(g) for all k 2 K, g 2 G2(R).
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