(December 25, 2014) Unwinding and integration on quotients Paul Garrett [email protected] http:=/www.math.umn.edu/egarrett/ [This document is http://www.math.umn.edu/~garrett/m/mfms/notes 2013-14/11a integration on quots.pdf] 1. Surjectivity of averaging maps 2. Invariant measures and integrals on quotients HnG 3. Uniqueness of invariant integrals 4. Preview of vector-valued integrals 5. Mapping property of Gelfand-Pettis integrals 6. Appendix: apocryphal lemma X ≈ G=Gx and other background o The simplest case of unwinding is for f 2 Cc (R): Z X Z f(x + n) dx = f(x) dx = R Z n2Z R [1] In fact, the integral on the quotient R=Z is unequivocally characterized by this relation, once we know P o o that the averaged functions n f(x + n) are at least dense in C (R=Z). As corollary, for F 2 C (R=Z), o since F · f 2 Cc (R), Z X Z X F (x) f(x + n) dx = F (x) f(x + n) dx = = R Z n2Z R Z n2Z Z X Z = F (x + n) f(x + n) dx = F (x) f(x) dx = R Z n2Z R We need analogous assertions with less elementary group actions and less transparent representatives for the quotients. For example, with Γ = SL2(Z) and H the upper half-plane, integration on ΓnH is characterized o by requiring, for all f 2 Cc (H), Z X dx dy Z dx dy f(γz) = f(z) y2 y2 ΓnH γ2Γ H P o once we know that the averages γ2Γ f(γz) are at least dense in Cc (ΓnH). In fact, such averaging maps are universally surjective on compactly-supported continuous functions, as demonstrated just below. [2] o An important variant uses f 2 Cc (Γ1nH) for a subgroup Γ1 of Γ. By the surjectivity of averaging maps, o take ' 2 Cc (H) such that X ' ◦ β = f β2Γ1 [1] The Riesz-Markov-Kakutani theorem asserts that every (continuous) functional on compactly-supported R continuous functions on a reasonable topological space X is f ! X f(x) dµ(x) for some measure µ. Relying o on this, specification of a functional (integration) on Cc (X) specifies a measure. In fact, we care more about the integral than about the measure. [2] R 2 This variant of unwinding arose most prominently in the Rankin-Selberg method, where ΓnH jfj ·Es for cuspform s f and Eisenstein series Es is unwound using the definition of Es as wound up from y . This theme is pervasive in the theory of automorphic forms. 1 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) so then Z X Z X X Z X f ◦ γ = (' ◦ β) ◦ γ = ' ◦ γ ΓnH ΓnH ΓnH γ2Γ1nΓ γ2Γ1nΓ β2Γ1 γ2Γ Z Z X Z = ' = ' ◦ β = f H Γ1nH Γ1nH β2Γ1 o o The corollary with F 2 C (ΓnH) and f 2 Cc (Γ1nH) is Z X Z X Z F · f ◦ γ = (F · f) ◦ γ = F · f ΓnH ΓnH Γ1nH γ2Γ1nΓ γ2Γ1nΓ 1 1 1 o o o Letting L (G), L (ΓnH), and L (Γ1nH) be the completions of Cc (H), Cc (ΓnH), and Cc (Γ1nH) with respect to the corresponding L1 norms Z Z Z 1 1 1 jfjL (H) = jfj jfjL (ΓnH) = jfj jfjL (Γ1nH) = jfj H ΓnH Γ1nH Extension by continuity gives the same unwinding property of integrals on L1 spaces. 1. Surjectivity of averaging maps By convention, a topological group is a locally compact, Hausdorff topological space G with a continuous group operation G × G ! G, and continuous inversion map g ! g−1. To avoid pathologies with regard to measures on products, we require that topological groups have a countable basis. [3] o Let dg be a right G-invariant measure on G, meaning that for f 2 Cc (G) Z Z Z Z f(g) dg = f(gh) dg = f(g) d(gh−1) = f(g) dg (for all h 2 G) G G G G Let δG : G ! (0; +1) be the modular function of G, gauging the discrepancy between left and right invariant −1 measures, in the sense that meas (gE) = δG(g) · meas (E) for a measurable set E ⊂ G. Then δG (g) dg is a left invariant measure. Let H be a closed subgroup of G, with right invariant measure dh, and modular function δH . o o [1.0.1] Lemma: The averaging map α : Cc (G) ! Cc (HnG) by Z (αf)(g) = f(hg) dh H is surjective. Proof: Let q : G ! HnG be the quotient map. Let U be a neighborhood of 1 2 G having compact closure U. For each g 2 G, gU is a neighborhood of g. The images q(gU) are open, by the characterization of the o quotient topology. Given F 2 Cc (HnG), the support spt(F ) of F is covered by the opens q(gU), and admits a finite subcover q(g1U); : : : ; q(gnU). The set −1 C = q (spt(F )) \ g1U [ ::: [ gnU ⊂ G [3] Right or left G-invariant positive regular Borel measures on G are (right or left) Haar measures on G. 2 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) o is compact, and q(C) = spt(F ) ⊂ HnG. Let f 2 Cc (G) be identically 1 on a compact neighborhood of o C, and non-negative real-valued everywhere. Then αf 2 Cc (HnG) is strictly positive on the compact set spt(F ), so has a strictly positive lower bound µ there. By continuity, V = fx 2 HnG : αf(x) > µ/2g is an open subset of HnG containing q(C) = spt(F ), and αf(x) ≥ µ/2 on the closure V . Then F/αf is continuous on V . Let 8 F (qx) < f(x) · αf(x) (for x 2 V ) Φ(x) = : 0 (for x 62 V ) By design, α(Φ) = F , and Φ is continuous in V . This is the main argument. Continuity of Φ in the interior of the complement of V is clear, but one might worry about continuity of Φ at points on the boundary @V of V : Since V is compact, @V is compact. Thus, there is a neighborhood N of @V on which αf > µ/4. Every neighborhood of every point of @V contains a point not in V , so not in spt(F ), so by continuity f is 0 on @V . Given " > 0, there is an open neighborhood N 0 of the compact set @V on which jF j < ". On the neighborhood N 00 = N \ N 0 of @V the continuous quotient F/αf is bounded by 4ε/µ. Thus, assigning values 0 to Φ on @V is compatible both with values 0 off V and values F/αf in V . === 2. Invariant measures and integrals on quotients HnG [2.0.1] Theorem: The quotient HnG has a right G-invariant measure if and only if δG H = δH . In that case, the integral is unique up to scalars, and is characterized as follows. For given right Haar measure dh on H and for given right Haar measure dg on G there is a unique invariant measure dg_ on HnG such that for f 2 Co(G) c Z Z Z o f(hg) dh dg = f(g) dg (for f 2 Cc (G)) HnG H G Proof: First, prove the necessity of the condition on the modular functions. Suppose that there is a right R o G-invariant measure on HnG. Let α be the averaging map f ! H f(hg) dh. For f 2 Cc (G) the map Z f ! αf(_g) dg_ HnG momentarily emphasizing the coordinateg _ on the quotient, is a right G-invariant functional (with the continuity property as above), so by uniqueness of right invariant measure on G must be a constant multiple of the Haar integral Z f −! f(g) dg G −1 The averaging map behaves in a straightforward manner under left translation Lhf(g) = f(h g) for h 2 H: o for f 2 Cc (G) and for h 2 H Z Z −1 α(Lhf)(g) = f(h xg) dx = δH (h) f(xg) dx H H by replacing x by hx. Then Z Z Z Z −1 −1 −1 f(g) dg = α(f)(g) dg_ = δ(h) α(Lhf)(g) dg_ = δ(h) f(h g) dg G HnG HnG G 3 Paul Garrett: Unwinding and integration on quotients (December 25, 2014) by comparing the iterated integral to the single integral. Replacing g by hg in the integral gives Z Z −1 f(g) dg = δ(h) δG(h) f(g) dg G G Choosing f such that the integral is not 0 implies the stated condition on the modular functions. Proof of sufficiency starts from existence of Haar measures on G and on H. For simplicity, first suppose that o both groups are unimodular. As expected, attempt to define an integral on Cc (HnG) by Z Z αf(_g) dg_ = f(g) dg HnG G o o invoking the fact that the averaging map α from Cc (G) to Cc (HnG) is surjective. The potential problem is R o well-definedness. It suffices to prove that G f(g) dg = 0 for αf = 0. Indeed, for αf = 0, for all F 2 Cc (G), the integral of F against αf is certainly 0. Rearrange Z Z Z Z Z 0 = F (g) αf(g) dg = F (g) f(hg) dh dg = F (h−1g) f(g) dg dh G G H H G by replacing g by h−1g.