Multiplying Modular Forms

IntroductionCompact Groups

Discrete SeriesFuture Directions

Multiplying Modular Formsassociated to discrete series

M. Weissman

July 24, 2007 / Hausdorff Institute of Mathematics

M. Weissman Multiplying Modular Forms

Outline

1 IntroductionA remark of DeligneHolomorphic Modular FormsMultiplication Datum

2 Compact GroupsBorel-Weil TheoremMultiplying Algebraic Modular Forms

3 Discrete SeriesConstruction via lowest K -typeMultiplication

4 Future Directions

A remark of DeligneHolomorphicMultiplication Datum

Outline

4 Future Directions

Remarque 2.1.4.Formes Modulaires et Representations de GL(2)

Remarque 2.1.4. L’espace F (G, GL(2, Z)) ci-dessus est stablepar produit. D’autre part, Dk−1 ⊗ D`−1 contient les Dk+`+2m−1(m ≥ 0). Pour m = 0, ceci correspond au fait que le produit fgd’une forme modulaire holomorphe de poids k par une depoids ` en est une de poids k + `. Pour m = 1, en coordonnées(1.5.2), on trouve que ` ∂f

∂z · g − kf · ∂g∂z est modulaire

holomorphe de poids k + ` + 2, et ainsi de suite. De mêmedans le cadre adélique.

Remark 2.1.4.Modular forms and representations of GL(2)

Remark 2.1.4. The space F (G, GL(2, Z)) [smooth functions ofmoderate growth on GL(2, R), invariant on the right byGL(2, Z)] is stable under products. On the other hand,Dk−1 ⊗ D`−1 [the tensor product of holomorphic discrete seriesrepresentations] contains Dk+`+2m−1 (m ≥ 0). For m = 0, thiscorresponds to the fact that the product fg of a modular form ofweight k and one of weight ` is one of weight k + `. Whenm = 1, in the coordinates of (1.5.2) [a typo - should be 1.2.5],we find that ` ∂f

∂z · g − kf · ∂g∂z is a holomorphic modular form of

weight k + ` + 2, and so on. A similar phenomenon occurs inthe adelic case.

Outline

4 Future Directions

The Simplest Case of Multiplication.(of modular forms)

Let H be the upper half plane.Let M(k) be the space of holomorphic modular forms onH, of level 1.If f1 ∈ M(k1), and f2 ∈ M(k2), then f1 · f2 ∈ M(k1 + k2).

Holomorphic Discrete Series

A family of representations of SL2(R) . . .For every integer k ≥ 2 :

Dk = f : H → C . . .such that f is holomorphic . . .and

∫H f (z)yk−2dxdy < ∞.

The action of SL2(R) on such f :

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

Dk has a one-dimensional lowest K -type Ck . Fix a nonzerolowest K -type vector j . (Canonical normalization later...)

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

(a bc d

)⇒ [gf ](z) = (−bz + d)−k f

(az − c−bz + d

Automorphic Forms

Take G = SL2(R), Γ fixed arithmetic, K = SO(2), for example.Let A = A(Γ\G) denote the space of automorphic forms:

Functions f ∈ C∞(Γ\G).K -finite.Z(gC)-finite.Moderate growth.

A is a (gC, K )-module (by right-translation).

Automorphic Forms

Modular Forms as Intertwining Operators

Fix an integer k ≥ 2.Suppose that F : Dk → A is (gC, K )-intertwining.There is a unique function f : H → C satisfying:

[g−1f ](i) = F (j)(g),

for all g ∈ SL2(R).This function f is in M(k). This yields an isomorphism(classicalization):

cl : Hom(gC,K )(Dk ,A) ∼= M(k).

[g−1f ](i) = F (j)(g),

Tensor Products of Holomorphic Discrete Seriesafter J. Repka

Fix k1, k2 ≥ 2.As a representation of SO(2), we have:

Dk =⊕n≥0

Ck+2n.

Hence, the tensor product (gC, K )-module decomposes:

Dk1 ⊗ Dk2 =⊕n≥0

(n + 1)Ck1+k2+2n.

Lowest K -type Ck1+k2 occurs with multiplicity one. There existsan embedding:

ν1,2 : Dk1+k2 → Dk1 ⊗ Dk2 .

Dk =⊕n≥0

Ck+2n.

Dk1 ⊗ Dk2 =⊕n≥0

(n + 1)Ck1+k2+2n.

ν1,2 : Dk1+k2 → Dk1 ⊗ Dk2 .

Dk =⊕n≥0

Ck+2n.

Dk1 ⊗ Dk2 =⊕n≥0

(n + 1)Ck1+k2+2n.

ν1,2 : Dk1+k2 → Dk1 ⊗ Dk2 .

Dk =⊕n≥0

Ck+2n.

Dk1 ⊗ Dk2 =⊕n≥0

(n + 1)Ck1+k2+2n.

ν1,2 : Dk1+k2 → Dk1 ⊗ Dk2 .

Dk =⊕n≥0

Ck+2n.

Dk1 ⊗ Dk2 =⊕n≥0

(n + 1)Ck1+k2+2n.

ν1,2 : Dk1+k2 → Dk1 ⊗ Dk2 .

Multiplication via the Tensor Product

Multiplication of modular forms is a collection of bilinear maps:

M(k1)⊗M(k2) → M(k1 + k2).

We can identify:

M(k) ∼= Hom(gC,K )(Dk ,A).

Multiplication of automorphic forms yields a bilinear map:

m : A⊗A → C∞(Γ\G).

Multiplication arises from representation theory:

F1 ∈ Hom(gC,K )(Dk1 ,A), F2 ∈ Hom(gC,K )(Dk2 ,A),

Define F ∈ Hom(gC,K )(Dk ,A) by:

F (v) = m((F1 ⊗ F2)(ν1,2(v))).

M(k1)⊗M(k2) → M(k1 + k2).

We can identify:

m : A⊗A → C∞(Γ\G).

F (v) = m((F1 ⊗ F2)(ν1,2(v))).

M(k1)⊗M(k2) → M(k1 + k2).

We can identify:

m : A⊗A → C∞(Γ\G).

F (v) = m((F1 ⊗ F2)(ν1,2(v))).

M(k1)⊗M(k2) → M(k1 + k2).

We can identify:

m : A⊗A → C∞(Γ\G).

F (v) = m((F1 ⊗ F2)(ν1,2(v))).

M(k1)⊗M(k2) → M(k1 + k2).

We can identify:

m : A⊗A → C∞(Γ\G).

F (v) = m((F1 ⊗ F2)(ν1,2(v))).

The diagram commutes

Extend the map (F1, F2) 7→ F linearly to a “multiplication”:

m1,2 : Hom(Dk1 ,A)⊗ Hom(Dk2 ,A) → Hom(Dk1+k2 ,A).

Then the following diagram commutes:

Hom(Dk1 ,A)⊗ Hom(Dk2 ,A)m1,2 //

clk1⊗clk2

Hom(Dk1+k2 ,A)

clk1+k2

M(k1)⊗M(k2)mult // M(k1 + k2)

The diagram commutes

Extend the map (F1, F2) 7→ F linearly to a “multiplication”:

m1,2 : Hom(Dk1 ,A)⊗ Hom(Dk2 ,A) → Hom(Dk1+k2 ,A).

Then the following diagram commutes:

Hom(Dk1 ,A)⊗ Hom(Dk2 ,A)m1,2 //

clk1⊗clk2

Hom(Dk1+k2 ,A)

clk1+k2

M(k1)⊗M(k2)mult // M(k1 + k2)

Outline

4 Future Directions

General Modular Forms

Suppose that G connected semisimple algebraic group over Q.Write G = G(R), Γ arithmetic subgroup, K maximal compact.Space of automorphic forms A = A(Γ\G), a (gC, K )-module.Suppose that J is an irreducible Harish-Chandra module (Aweight).

DefinitionThe space of modular forms of weight J is:

M(J ) = Hom(gC,K )(J ,A).

Multiplication Datum

Suppose that J , J1, J2 are three weights. Suppose that:

ν1,2 : J → J1 ⊗ J2

is an intertwining operator (of (gC, K )-modules).

TheoremSuppose that F1 ∈ M(J1), F2 ∈ M(J2). The composition ofmaps F = m (F1 ⊗ F2) ν1,2, described below, is in M(J ).

J → J1 ⊗ J2 → A⊗A → C∞(Γ\G).

This gives a bilinear map:

m1,2 : M(J1)⊗M(J2) → M(J ).

ν1,2 : J → J1 ⊗ J2

J → J1 ⊗ J2 → A⊗A → C∞(Γ\G).

m1,2 : M(J1)⊗M(J2) → M(J ).

ν1,2 : J → J1 ⊗ J2

J → J1 ⊗ J2 → A⊗A → C∞(Γ\G).

m1,2 : M(J1)⊗M(J2) → M(J ).

ν1,2 : J → J1 ⊗ J2

J → J1 ⊗ J2 → A⊗A → C∞(Γ\G).

m1,2 : M(J1)⊗M(J2) → M(J ).

ν1,2 : J → J1 ⊗ J2

J → J1 ⊗ J2 → A⊗A → C∞(Γ\G).

m1,2 : M(J1)⊗M(J2) → M(J ).

Problems

Multiplication of modular forms requires more than the previoustheorem:

We would like a collection of weights Jλ, indexed by acommutative monoid C.We would like a collection of intertwining operators:

νλ1,λ2 : Jλ1+λ2 → Jλ1 ⊗ Jλ2 ,

for all λ1, λ2 ∈ C.We require that the collection νλ1,λ2 satisfies axioms ofco-identity, co-commutativity, and co-associativity.

Fulfilling these conditions yields a ring of modular forms:

MC =⊕λ∈C

M(Jλ).

Problems

νλ1,λ2 : Jλ1+λ2 → Jλ1 ⊗ Jλ2 ,

MC =⊕λ∈C

M(Jλ).

Problems

νλ1,λ2 : Jλ1+λ2 → Jλ1 ⊗ Jλ2 ,

MC =⊕λ∈C

M(Jλ).

Problems

νλ1,λ2 : Jλ1+λ2 → Jλ1 ⊗ Jλ2 ,

MC =⊕λ∈C

M(Jλ).

Problems

νλ1,λ2 : Jλ1+λ2 → Jλ1 ⊗ Jλ2 ,

MC =⊕λ∈C

M(Jλ).

Co-commutativity

We leave most of the axioms to the audience imagination. Buthere’s a sample:For all λ1, λ2 ∈ C, the following diagram commutes:

Jλ1 ⊗ Jλ2

Jλ1+λ2

νλ1,λ288rrrrrrrrrr

νλ2,λ1 &&LLLLLLLLLL

Jλ2 ⊗ Jλ1

Co-commutativity

Jλ1 ⊗ Jλ2

Jλ1+λ2

Jλ2 ⊗ Jλ1

Co-commutativity

Jλ1 ⊗ Jλ2

Jλ1+λ2

Jλ2 ⊗ Jλ1

Borel-Weil TheoremMultiplying Algebraic Modular Forms

Outline

4 Future Directions

Algebraic Modular Forms

Suppose that G (a connected semisimple group over Q) hasthe property that G = G(R) is a compact Lie group.In this situation, “weights” are finite-dimensional irreduciblerepresentations of G.Modular forms have been studied by B. Gross, under the name“Algebraic Modular Forms”.Such modular forms can be multiplied. This has been carriedout by Khuri-Makdisi.The case when G = B×1 (norm one elements in a definitequaternion algebra over Q) has been studied extensively.

Realization of Representations

The Borel-Weil theorem yields a uniform realization of allirreducible representations of G.Irreducible representations of G are indexed by dominantweights λ ∈ Λ+.Fix a maximal torus T ⊂ G.Fix a Borel subgroup BC ⊂ GC, containing TC.Any λ ∈ Λ+ corresponds to an algebraic homomorphismTC → C×, and extends to BC.

The Borel-Weil Theorem

Consider the space Hλ of functions h : GC → C such that:h is holomorphic.h(gb) = λ(b)h(g), for all g ∈ GC, b ∈ BC.

In other words, Hλ = H0(Y ,O(−λ)), where Y is the complexflag variety, and O(−λ) is the line bundle associated to thecharacter −λ of T .

Theorem (Borel-Weil)The space Hλ, with the left-translation action τλ of G, is anirreducible representation of G with highest weight λ.

Additional structures

The Borel-Weil realization yields additional structures on therepresentations Hλ, defined uniformly in λ.We may define a Hilbert space structure on Hλ via:

〈h1, h2〉λ =

h1(g)h2(g)dg,

where Haar measure is normalized so that Vol(G) = 1.A highest weight vector hλ ∈ Hλ, unique up to scalar, can benormalized by requiring hλ(1) = 1.Another highest weight vector h∨λ is given by requiring that〈hλ, h∨λ 〉 = 1.

〈h1, h2〉λ =

h1(g)h2(g)dg,

〈h1, h2〉λ =

h1(g)h2(g)dg,

〈h1, h2〉λ =

h1(g)h2(g)dg,

Outline

4 Future Directions

Intertwining Operators

Suppose that λ1, λ2 ∈ Λ+. Let λ = λ1 + λ2.There is a natural G-intertwining operator:

µ∗1,2 : Hλ1 ⊗ Hλ2 → Hλ,

µ∗1,2(h1 ⊗ h2) = h1 · h2.

We may dualize, via the Hilbert space structure, to get aG-intertwining operator:

µ1,2 : Hλ → Hλ1 ⊗ Hλ2 .

One may compute:

µ1,2(h∨λ ) = h∨λ1⊗ h∨λ2

µ∗1,2 : Hλ1 ⊗ Hλ2 → Hλ,

µ∗1,2(h1 ⊗ h2) = h1 · h2.

µ1,2 : Hλ → Hλ1 ⊗ Hλ2 .

One may compute:

µ1,2(h∨λ ) = h∨λ1⊗ h∨λ2

µ∗1,2 : Hλ1 ⊗ Hλ2 → Hλ,

µ∗1,2(h1 ⊗ h2) = h1 · h2.

µ1,2 : Hλ → Hλ1 ⊗ Hλ2 .

One may compute:

µ1,2(h∨λ ) = h∨λ1⊗ h∨λ2

µ∗1,2 : Hλ1 ⊗ Hλ2 → Hλ,

µ∗1,2(h1 ⊗ h2) = h1 · h2.

µ1,2 : Hλ → Hλ1 ⊗ Hλ2 .

One may compute:

µ1,2(h∨λ ) = h∨λ1⊗ h∨λ2

µ∗1,2 : Hλ1 ⊗ Hλ2 → Hλ,

µ∗1,2(h1 ⊗ h2) = h1 · h2.

µ1,2 : Hλ → Hλ1 ⊗ Hλ2 .

One may compute:

µ1,2(h∨λ ) = h∨λ1⊗ h∨λ2

Properties of these Intertwining Operators

We have a monoid Λ+ of dominant weights.For each λ ∈ Λ+, an irreducible representation Hλ of G.For each pair λ1, λ2 ∈ Λ+, we have an intertwining operator:

µλ1,λ2 : Hλ1+λ2 → Hλ1 ⊗ Hλ2 .

TheoremThe collection of operators µλ1,λ2 satisfies axioms ofco-identity, co-commutativity, and co-associativity. Thus, for anyfixed arithmetic subgroup Γ ⊂ G, there is a Λ+-graded ring ofmodular forms:

M =⊕λ∈Λ+

M(Hλ).

µλ1,λ2 : Hλ1+λ2 → Hλ1 ⊗ Hλ2 .

M =⊕λ∈Λ+

M(Hλ).

µλ1,λ2 : Hλ1+λ2 → Hλ1 ⊗ Hλ2 .

M =⊕λ∈Λ+

M(Hλ).

µλ1,λ2 : Hλ1+λ2 → Hλ1 ⊗ Hλ2 .

M =⊕λ∈Λ+

M(Hλ).

The SU(2) case

The previous results have essentially been obtained byKhuri-Makdisi.Especially, when G = B×1, so G ∼= SU(2), the rings of modularforms have been studied.In this case, Λ+ = N, and we can define:

X (Γ) = Proj(M).

This is a “modular curve” parameterizing complex 2-tori, withendomorphims by an order in B.

The SU(2) case

X (Γ) = Proj(M).

The SU(2) case

X (Γ) = Proj(M).

The SU(2) case

X (Γ) = Proj(M).

Construction via lowest K -typeMultiplication

Outline

4 Future Directions

Discrete series representations

Now, we consider when G is not necessarily compact.Suppose that G is a connected semisimple linear algebraicgroup over Q.A discrete series representation of G is a pair (π, V ), with V aHilbert space, π a continuous action of G on V , and all matrixcoefficients in L2(G).G has discrete series representations iff rank(G) = rank(K ).Hereafter, suppose that rank(G) = rank(K ).

Construction of Discrete Series

Harish-Chandra proved that discrete series representations ofsuch groups G exist, and are parameterized by certain weightsλ ∈ Λ.His proof did not immediately lead to a construction. Manyconstructions were given shortly afterwards:

L2 or Dolbeaux cohomology of line bundles on G/T .(Langlands conjecture, proven by Schmid)D-module on GC/BC. (Beilinson-Bernstein)Cohomological induction. (Vogan)Harmonic spinors on G/K (Hotta-Parthasarathy).

The Hotta-Parthasarathy Realization

Suppose that λ ∈ Λ+k . Thus (τλ, Hλ) is a representation of K of

highest weight λ.Define Jλ = J (Y , Hλ), the set of functions j : G → Hλ suchthat:

j ∈ [L2 ∩ C∞](G, Hλ).f (gk) = τλ(k)−1f (g), for all g ∈ G, k ∈ K .Ωf = (‖λ‖2 −‖ρ‖2)f . (a second-order differential equation).

G acts on J (Y , Hλ) by left-translation.

Lowest K-types

Not every representation of K is the lowest K -type of a discreteseries representation.But... every discrete series representation is uniquelydetermined by its lowest K -type.If (τλ, Hλ) is isomorphic to the lowest K -type of a discreteseries Dλ, then Dλ is isomorphic to J (Y , Hλ), as unitaryrepresentations of G.(Proven by Hotta-Parthasarathy, for λ away from the walls, andextended fully by Wallach).

Lowest K-types

Outline

4 Future Directions

What is needed for multiplication

Suppose that C ⊂ Λ+k is a submonoid, such that: for all λ ∈ C,

Hλ is the lowest K -tye of a discrete series representation.Suppose that we can find intertwining operators, for allλ1, λ2 ∈ C:

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

which descends to Harish-Chandra modules (more about thislater).Suppose co-ring axioms are satisfied. Then we get a ring ofmodular forms:

MC =⊕λ∈C

M(Jλ).

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

MC =⊕λ∈C

M(Jλ).

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

MC =⊕λ∈C

M(Jλ).

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

MC =⊕λ∈C

M(Jλ).

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

MC =⊕λ∈C

M(Jλ).

νλ1,λ2 : Jλ1+λ2 → Jλ1⊗Jλ2 ,

MC =⊕λ∈C

M(Jλ).

A branching problemin the unitary category

Finding an intertwining operator Jλ1+λ2 → Jλ1⊗Jλ2 can bethought of as a branching problem.Jλ1Jλ2 is a discrete series representation of G ×G.We would like to find a specific irreducible constituent, whenrestricted to G ⊂ G ×G.Geometrically, Jλ1Jλ2 can be realized as:

J (Y × Y , Hλ1 Hλ2).

One may restrict such sections to the diagonally embeddedY ⊂ Y × Y .

J (Y × Y , Hλ1 Hλ2).

A result of Orsted and Vargas

Suppose that j ∈ J (Y × Y , Hλ1 Hλ2).Then ResY (j) is naturally in C∞(Y , Hλ1 ⊗ Hλ2).

Theorem (Orsted-Vargas)

ResY (j) is contained in [L2 ∩ C∞](Y , Hλ1 ⊗ Hλ2). Moreover,ResY is a continuous G-intertwining map from the Hilbert spaceJ (Y × Y , Hλ1 Hλ2) to the Hilbert space L2(Y , Hλ1 ⊗ Hλ2).

An intertwining operator

Given the continuous map ResY , we have the adjoint map:

Res∗Y : L2(Y , Hλ1 ⊗ Hλ2) → J (Y × Y , Hλ1 Hλ2).

The intertwining operator of Borel-Weil representations yields: