AUTOMORPHIC FORMS ON REDUCTIVE GROUPS ARMAND BOREL 1. Introduction The goal of these notes is the basic theory of automorphic forms and reductive groups, up to and including the analytic continuation of Eisenstein series. 2. Notation 2.1. Let X be a set and f, g strictly positive real functions on X. We write f ≺ g if there exists a constant c > 0 such that f(x) ≤ cg(x) for all x ∈ X; similarly, f g if g ≺ f, and f g if f ≺ g and g ≺ f. 2.2. Let G be a group. The left (resp. right) translation by g ∈ G is denoted lg (resp rg); these act on functions via −1 lg · f(x) = f(g x) , rg · f(x) = f(xg)(1) 2.3. Let G be a Lie group and g its Lie algebra. The latter may be viewed as the tangent space T1(G) at the identity, or as the space of left-invariant vector fields on G. If X1 ∈ T1(G), the associated vector field is x 7→ x · X1. The action of X on functions is given by d Xf = f(xetX )| dt t=0 The universal enveloping algebra U(g) is identified with the algebra of left-invariant differential operators; the element X1X2 ...Xn acts via dn t1X1 t2X2 tnXn X1X2 ...Xnf(x) = f(xe e . e )|ti = 0. dt1 . dtn Let Z(g) be the center of U(g). If G is connected, it is identified with the left and right invariant differential operators. If G is connected and reductive, it is a polynomial algebra of rank equal to the rank of G. 1 2 ARMAND BOREL 2.4. Let G be unimodular. The convolution u ? v of two functions is defined by Z Z (2) u ? v(x) = u(xy)v(y−1)dy = u(y)v(y−1x)dy G G whenever the integral converges. It is a smoothing operator: if u is ∞ continuous and v ∈ Cc (G), then (3) D(u ? v) = u ? Dv, D ∈ U(g) and, in particular, u ? v ∈ C∞(G). It extends to distributions and is associative. If g is identified to distributions with support {1}, then Xf = f ? (−X); see section 2.2 of [6]. 3. Notion of automorphic form Let G be a subgroup of finite index in the group of real points of a connected semisimple algebraic group G defined over R. Let K be a maximal compact subgroup of G. Then X = G/K is the Riemannian symmetric space of noncompact type of G. Let Γ ⊂ G be a discrete subgroup. 3.1. A continuous function f ∈ C(G; C) is an automorphic form for Γ if it satisfes the following conditions: (A1) f(γx) = f(x). (A2) f is K-finite on the right. (A3) f is Z(g)-finite. (A4) f is of moderate growth (or slowly increasing). Explanation 1. f is K-finite on the right right means that the set of right translates rkf, k ∈ K is contained in a finite dimensional space. f is Z(g)-finite means that there exists an ideal J of finite co-dimension in Z(g) which annihilates f. If f is not C∞, this is understood in the sense of distributions, but in any case f will be analytic, cf. below. By definition, G ⊂ SLN (R), and is closed. Let kgk be the Hilbert-Schmidt 2 t P 2 norm of g ∈ SLN (R). Thus kgk = tr ( g.g) = i,j gij. Then f is of moderate growth or slowly increasing if there exists m ∈ Z such that |f(x)| ≺ kxkm , (x ∈ G). −m Let νm be the semi-norm on C(G, C) defined by νm(f) = sup f(x).kxk . Then f is slowly increasing if and only if νm(f) < ∞ for some m. We note some elementary properties of k k: (n1) kx.yk ≤ kxk.kyk, and there is an m such that kx−1k ≺ kxkm. (n2) If C ⊂ G is compact, kx.yk kyk for x ∈ C, y ∈ G. AUTOMORPHIC FORMS ON REDUCTIVE GROUPS 3 Remark 3.1. The notion of moderate growth (but not the exponent m) is independent of the embedding. One can also define a canonical Hilbert-Schmidt norm as follows: On g, let K(x, y) = tr (ad x ◦ ad y) be the Killing form, and let θ be the Cartan involution of G with respect to K. Then the form (x, y) = −K(θx, y) is positive definite on g. Then the associated Hilbert-Schmidt norm on the adjoint group is kgk2 = tr (Ad θg−1.Ad g) (Exercise). 3.2. Relation with classical automorphic forms on the upper half plane. Here G = SL2(R), K = SO2, and X = {z ∈ C | =z > 0 }, the action of G being defined by a b az + b .z = , (z ∈ X). c d cz + d Let (cz + d)m = µ(g, z). It is an automorphy factor, i.e. (4) µ(g.g0, z) = µ(g, g0.z)µ(g0, z). K is the isotropy group of i ∈ X. Equation 4 gives for k, k0 ∈ K, and z = i (5) µ(kk0, i) = µ(k, i)µ(k0, i), i.e. k 7→ µ(k, i) is a character χm of K. Let Γ be a subgroup of finite index in SL2(Z). An automorphic form f on X of weight m is a function satisfying (A1’) f(γ.z) = µ(γ, z)f(z) (A2’) f is holomorphic (A3’) f is regular at the cusps. Let f˜ be the function on G defined by f˜(g) = µ(g, i)−1f(g.i). Then (A1’) for f implies (A1) and (A2) for f˜ by a simple computation using 5. Note in particular ˜ ˜ (6) f(g.k) = f(g)χ−m(k). The condition (A2’) implies that f˜ is an eigenfunction of the Casimir operator C. As C generates Z(g), this yields (A3). Consider the cusp at ∞. In the inverse image of the “Siegel set” |x| ≤ c and y > t, 1 where c and t are positive constants, it is easily seen that kgk y 2 , hence moderate growth means ≺ ym for some m. On the other hand, Γ contains a translation x 7→ x + p, for some non-zero integer p. Since f is invariant under this translation in the x direction, f admits a P 2πinz development in a Laurent series n an exp( p ). For f to be bounded 4 ARMAND BOREL m by y in the Siegel set, it is necessary and sufficient that an = 0 for n < 0. This is the regularity condition (A3’). (cf. [6], 5.14) 4. First properties of automorphic forms In this section, G, K, X, and Γ are as in 1.1, and f is an automorphic form for Γ. 4.1. f is analytic. For this it suffices, by a regularity theorem, to show that it is annihilated by an analytic elliptic operator. Let g = k ⊕ p be the Cartan decomposition of g, where p is the orthogonal complement of k with respect to the Killing form. The latter is negative (resp. positive) definite on k (resp. p). Let {xi} (resp. {yj}) be an orthonormal basis of k (resp. p). Then a Casimir operator P 2 P 2 P 2 can be written Cg = − xi + yj , and Ck = i xi is a Casimir operator for k. Let Ω = 2Ck + Cg. It is an analytic elliptic operator. We claim that f is annihilated by some non-constant polynomial in Ω, which will prove our assertion. The function f is annihilated by an ideal J of finite codimension in Z(g) (by (A3) ) and, since f is K-finite on the right, it is annihilated by an ideal l of finite codimension of U(k). Therefore f is annihilated by an idea of finite codimension of the subalgebra Z(g).U(k) of U(g). But then there exists a polynomial of strictly positive degree P (Ω) in Ω belonging to that ideal. 4.2. A function α on G is said to be K-invariant if α(k.x) = α(x.k) for all x ∈ G, k ∈ K. We have the following theorem: Theorem. Given a neighborhood U of 1 in G, there exists a K-invariant ∞ function α ∈ Cc (U) such that f = f ? α. This follows from the fact that f is Z-finite and K-finite on one side, by a theorem of Harish-Chandra. See ([12], theorem 1) or ([1], 3.1), and for SL2(R) ([6], 2.13). 4.3. A smooth function u on G is said to be of uniform moderate growth if there exists m ∈ Z such that νm(Df) < ∞ for all D ∈ U(g). An elementary computation shows that if νm(u) < ∞, then νm(u ? α) < ∞ ∞ for any α ∈ Cc (G). Since, D(f ? α) = f ? Dα, the previous theorem implies that Corollary. An automorphic form is of uniform moderate growth. More generally, if f has moderate growth, then f ? α is of uniform moderate growth. AUTOMORPHIC FORMS ON REDUCTIVE GROUPS 5 4.4. We intercalate some facts needed in the sequel. Since G is a closed subgroup of SLN (R), it is clear that k.k has a strictly positive minimum, say t0 on G. Fix a Haar measure on G. For t ≥ t0, let (7) Gt = {g ∈ G | kgk ≤ t}. There there exists m ∈ N such that m (8) vol (Gt) ≺ t , (t ≥ t0). (cf Lemma 37 of [13]). The proof will be sketched later, once we have reviewed some structure theory of reductive groups.