UPPSALA DISSERTATIONS IN MATHEMATICS

107

Department of MathematicsUppsala University

UPPSALA 2018

Some applications of representation theory in homogeneous dynamics and automorphic functions

Samuel Charles Edwards

Dissertation presented at Uppsala University to be publicly examined in Häggsalen,Ångströmlaboratoriet, Lägerhyddsvägen 1, Uppsala, Friday, 8 June 2018 at 13:15 for thedegree of Doctor of Philosophy. The examination will be conducted in English. Facultyexaminer: Professor Henrik Schlichtkrull (University of Copenhagen).

AbstractEdwards, S. C. 2018. Some applications of representation theory in homogeneous dynamicsand automorphic functions. Uppsala Dissertations in Mathematics 107. 25 pp. Uppsala:Department of Mathematics. ISBN 978-91-506-2698-8.

This thesis consists of an introduction and five papers in the general area of dynamics andfunctions on homogeneous spaces. A common feature is that representation theory plays a keyrole in all articles.

Papers I-IV are concerned with the effective equidistribution of translates of pieces ofsubgroup orbits in quotient spaces of semisimple Lie groups by discrete subgroups. In PaperI we focus on finite-volume quotients of SL(2,C) and study the speed of equdistribution forexpanding translates orbits of horospherical subgroups. Paper II also studies the effectiveequidistribution of translates of horospherical orbits, though now in the setting of a quotient of ageneral semisimple Lie group by a lattice subgroup. Like Paper II, Paper III considers effectiveequidistribution in quotients of general semisimple Lie groups, but now studies translates oforbits of symmetric subgroups. In all these papers we show that the translates equidistributewith the same exponential rate as for the decay of the corresponding matrix coefficients ofthe translating subgroup. In Paper IV we consider the effective equidistribution of translatesof pieces of horospheres in infinite-volume quotients of groups SO(n,1) by geometricallyfinite subgroups, and improve the dependency on the spectral gap for certain known effectiveequidistribution results.

In Paper V we study the Fourier coefficients of Eisenstein series for generic non-cocompactcofinite Fuchsian groups. We use Zagier's renormalization of certain divergent integrals toenable use of the so-called triple product method, and then combine this with the analyticcontinuation of irreducible representations of SL(2,R) due to Bernstein and Reznikov.

Keywords: automorphic functions, effective equidistribution, Eisenstein series, homogeneousdynamics, horospheres, lattices, Lie groups, representation theory

Samuel Charles Edwards, Department of Mathematics, Analysis and Probability Theory, Box480, Uppsala University, SE-75106 Uppsala, Sweden.

© Samuel Charles Edwards 2018

ISSN 1401-2049ISBN 978-91-506-2698-8urn:nbn:se:uu:diva-347976 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-347976)

To Gunta

List of papers

This thesis is based on the following papers, which are referred to in the textby their Roman numerals.

I S. C. EDWARDS, On the rate of equidistribution of expandinghorospheres in finite-volume quotients of SL(2,C).Journal of Modern Dynamics, Volume 11, 2017, pp. 155-188.

II S. C. EDWARDS, On the rate of equidistribution of translates ofhorospheres in Γ\G.Manuscript, 2017.

III S. C. EDWARDS, On the equidistribution of translates of orbits ofsymmetric subgroups in Γ\G.Manuscript, 2018.

IV S. C. EDWARDS, Effective equidistribution of horospheres ininfinite-volume quotients of SO(n,1) by geometrically finite groups.Manuscript, 2018.

V S. C. EDWARDS, Renormalization of integrals of products ofEisenstein series and analytic continuation of representations.Manuscript, 2018.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.1 Homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2 Homogeneous dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.3 Automorphic functions on the hyperbolic plane . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Group representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Summary of papers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Paper V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Summary in Swedish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1. Introduction

The central parts of this thesis are the following introduction, in which wesurvey some of the key concepts and objects related to the thesis, a short sum-mary of the articles, and finally the five articles which make up the majority ofthe thesis.

1.1 Homogeneous spacesThe central objects of study in this thesis are homogeneous spaces of Liegroups. A homogeneous space of a Lie group G is a non-empty set X equippedwith a transitive right G-action, which we denote by “·”. This means that for agiven element x ∈ X , x ·g is also an element of X for every g ∈ G, every otherelement y ∈ X may be written as y = x ·h for some h ∈ G, and

(x ·g) ·h = x · (gh) ∀g,h ∈ G, x ∈ X .

For a point x ∈ X , we let StabG(x) denote the stabiliser subgroup of x in G,i.e. StabG(x) = {g ∈ G : x · g = x}. By the orbit-stabiliser theorem the mapι : StabG(x)\G→ X given by ι(StabG(x0)g) = x · g is a bijection (recall thatfor a subgroup H < G, the quotient space H\G is the set of right H-cosets inG: H\G = {Hg : g ∈ G}). Moreover, StabG(x)\G is itself a homogeneousspace, with a natural transitive right G-action given by right multiplication:StabG(x)h ·g = StabG(x)hg, and ι(StabG(x)h ·g) = ι(StabG(x)h) ·g. Thus, inorder to study many properties of the G-action on X , one can instead considerthe corresponding action on the quotient space StabG(x)\G.

The homogeneous spaces that are studied in this thesis are, for the mostpart, quotient spaces X = Γ\G, where Γ is a discrete subgroup of G and theG-action is the natural right multiplication. We adopt the common practiceof suppressing the “·” when writing the result of the action of an elementof G on an element of X , and instead simply write xg for x · g. Since, forany x = Γg0 ∈ Γ\G, StabG(x) = g−1

0 Γg0 is a closed subgroup of G, by [19,Theorem 21.20] there exists a unique smooth manifold structure on X underwhich the group action is smooth. Furthermore, we will see that the groupaction preserves a natural measure on X . As such, the homogeneous spacesconsidered here are special cases of a more general construction. They are,however, ubiquitous throughout many fields of mathematics; for example, then-torus Tn may be realised as Tn = Zn\Rn. Other important examples of

9

homogeneous spaces of this type include the space of unimodular lattices inRn, which may be realised as the quotient SL(n,Z)\SL(n,R), and quotientsΓ\SO(n,1), which may be viewed as the orthonormal frame bundle F0(MΓ)of a hyperbolic n-orbifold MΓ.

As noted above, homogeneous spaces X of the form X = Γ\G possess sev-eral different structures under which the G-action is well-behaved. The studyof how these structures interact with the group action thus leads, in a natu-ral way, to an interplay of ideas in algebra, analysis, geometry, and numbertheory.

1.2 Homogeneous dynamicsHomogeneous dynamics refers to the study of dynamical systems on homoge-neous spaces. Over the last 30 years, homogeneous dynamics has developedinto a vibrant and highly active field of mathematical research. This is in partdue to the fact that several important problems in number theory have beensuccessfully resolved (or had significant progress made towards) by reinter-preting them in terms of homogeneous dynamics. Perhaps the most famousexample of this is Margulis’ proof of the Oppenheim conjecture [21, 22]. Werefer to [15] for a more comprehensive survey of the field of homogeneousdynamics and its applications to number theory.

Let G be a semisimple Lie group and Γ a discrete subgroup of G. Given aone-parameter subgroup {gt}t∈R of G, one defines a function Φ : R×Γ\G→Γ\G by

Φ(t,x) := xgt ∀t ∈ R, x ∈ Γ\G.

The triple (Γ\G,R,Φ) is then a dynamical system (cf. [12]); indeed, we haveΦ(0,x) = x and Φ(s+t,x) =Φ

(s,Φ(t,x)

)for all s, t ∈R and x∈ Γ\G. For the

sake of notational simplicity, we write gR in place of the triple (Γ\G,R,Φ),and xgt for Φ(t,x). The long-term behaviour of orbits of dynamical systemsof this type is one of the central questions in homogeneous dynamics; that isto say, the goal is to understand the topological and “statistical” (see below)properties of {Φ(t,x)}t∈[0,T ] ⊂ Γ\G as T →∞ for various choices of x ∈ Γ\G.

Since G is semisimple, it is unimodular, see [17, Chapter 8]; there existsa unique one-dimensional vector space of left Haar measures µl such thatµl(gA) = µl(A) for all Borel sets A ⊂ G and all g ∈ G, and this vector spacecoincides with the space of right Haar measures µr with the property thatµr(Ag) = µr(A) for all A, g as before (here gAh = {gah : a ∈ A} for all g, h ∈G, A ⊂ G). Thus every left Haar measure µ is also a right Haar measure,allowing us to (unambiguously) simply refer to µ as a Haar measure on G.The measure µ induces a measure µΓ\G on Γ\G as follows: firstly, let F ⊂ Gbe a Borel fundamental domain for Γ in G (i.e. #(Γg∩F) = 1 for all g ∈ G),and π : G→ Γ\G is the natural projection map given by π(g) := Γg for all

10

g ∈ G. Then for a Borel subset B⊂ Γ\G, define µΓ\G(B) := µ(F ∩π−1(B)

).

Note that µΓ\G is independent of the choice of F. Moreover, since µ is a rightHaar measure, µΓ\G is G-invariant: µΓ\G(Bg) = µΓ\G(B) for all g ∈ G andBorel subsets B⊂ Γ\G.

The dynamical system gR is thus in fact a measure-preserving system;µΓ\G(Bgt) = µΓ\G(B) for all Borel subsets B ⊂ Γ\G. If µ(F) < ∞, we callΓ a lattice, and assume that µ has been chosen so that µΓ\G is a probabil-ity measure. This allows one to use the methods of ergodic theory (cf. [12])when studying problems related to the behaviour of the dynamical system.For lattices Γ < G, recall that the orbit of a point x ∈ Γ\G is said to becomeequidistributed with respect to the probability measure µΓ\G if

limT→∞

1T

∫ T

0f (xgt)dt =

∫Γ\G

f (y)dµΓ\G(y) ∀ f ∈Cc(Γ\G).

Note that if the orbit {xgt}t∈R≥0 becomes equidistributed with respect to µΓ\G,then

{xgt}t∈R≥0 = Γ\G. (1.1)In a similar fashion, given any probability measure υ on Γ\G (again in thecase of general discrete subgroups Γ < G) we say that xgR≥0 becomes equidis-tributed with respect to υ if limT→∞

1T∫ T

0 f (xgt)dt =∫

Γ\G f dυ for all f ∈Cc(Γ\G). As in (1.1) above, if xgR≥0 becomes equidistributed with respect toa probability measure υ , then xgR≥0 = suppυ . Establishing results regardingthe equidistribution of xgR≥0 for various choices of x, subgroups gR, and mea-sures υ is a major direction of study in homogeneous dynamics, cf. [15]. In thecase that Γ is a lattice and gR is unipotent, important results of Ratner [27, 28](proving conjectures due to Raghunathan; a special case having been provedby Margulis in his proof of the Oppenheim conjecture) from the early 1990sgive a complete classification of the possible measures that orbits xgR≥0 canbecome equidistributed with respect to. Ratner’s theorems have since becomea centrepiece of homogeneous dynamics, with much research being focussedon applying, generalising, and strengthening them in various situations.

1.3 Automorphic functions on the hyperbolic planeThe Poincaré half-plane model of two-dimensional hyperbolic geometry con-sists of the following subset of the complex plane:

H= {z ∈ C : Im(z)> 0},

equipped with the hyperbolic metric ds2 = dx2+dy2

y2 , i.e. the length |γ| of a C1

path γ : [0,1]→H is given by

|γ| :=∫ 1

0

|γ ′(t)|Im(γ(t))

dt.

11

We denote the corresponding distance on H by dist:

dist(z,w) = arcosh(

1+|z−w|2

2Im(z)Im(w)

)∀z, w ∈H.

The group PSL(2,R) ={±(

a bc d

): a, b, c, d ∈ R, ad−bc = 1

}acts on H by

Möbius transformations:

g · z = az+bcz+d

∀g =±(

a bc d

)∈ PSL(2,R), z ∈H,

and this action is isometric;

dist(g · z,g ·w) = dist(z,w) ∀g ∈ PSL(2,R), z,w ∈H.

Given a discrete subgroup Γ < PSL(2,R) and k ∈ Z, a function f : H→ C issaid to be automorphic of weight k (with respect to Γ) if

f (γ · z) =(

cz+d|cz+d|

)2k

f (z) ∀γ =±(

a bc d

)∈ Γ, z ∈H.

Number theory is a particularly rich (and the primary) source of automorphicfunctions, cf., e.g., [1, 33]. However, they also occur, and have applications,in other areas of mathematics, for example in relation to Ramanujan graphsin network theory cf., e.g., [30]. In the most classical case, one considers thelattice Γ = PSL(2,Z) and functions f that are automorphic with respect toPSL(2,Z) (for some k ∈ Z), holomorphic, and have a Fourier expansion

f (z) =∞

∑n=0

ane2πinz (1.2)

(observe that since ±(

1 10 1

)∈ PSL(2,Z), f (z+ 1) = f (z) for all functions f

that are automorphic with respect to PSL(2,Z)). Examples of important au-tomorphic functions for PSL(2,Z) from number theory include powers of Ja-cobi theta functions, and Ramanujan’s tau function. Another class of impor-tant automorphic functions for PSL(2,Z) are Maass wave forms; these areeigenfunctions of the hyperbolic Laplacian ∆ = y−2( ∂ 2

∂x2 +∂ 2

∂y2 ) that are alsoautomorphic functions f of weight zero with respect to PSL(2,Z), and satisfya growth condition | f (x+ iy)| � yA for all x+ iy ∈H for some A > 0. Assum-ing f is a Maass wave form such that −∆ f = s(1− s) f for some s ∈ C, f hasa Fourier decomposition similar to (1.2):

f (x+ iy) = c0ys +d0y1−s + ∑n∈Zn6=0

an√

yKs− 12(2π|n|y) (1.3)

(here Kµ(z) denotes the K-Bessel function, cf. [11, Chapter 8.4]).

12

In a similar manner, one may also define holomorphic automorphic func-tions and Maass wave forms for other lattices Γ < PSL(2,R). If Γ containselements that are conjugate (in PSL(2,R)) to±

(1 10 1

), then such functions will

have Fourier decompositions of a similar nature to (1.2) and (1.3).The study of properties of automorphic functions, their Fourier expansions,

and the automorphic spectrum of ∆ (as well as their natural generalisations;see below) is a major field of research. We note that due to their connectionsto number theory, automorphic functions for Γ = PSL(2,Z), as well as forother arithmetic lattices, are particularly well-studied. For non-arithmetic Γ,the situation is more complicated, and many results that have been establishedfor arithmetic lattices remain wide open for generic lattices. A famous exam-ple of this is the Ramanujan-Petersson conjecture (proved by Deligne [4, 5]for PSL(2,Z), and subsequently for congruence subgroups of PSL(2,Z) byDeligne and Serre [6]) regarding the growth of the Fourier coefficients an in(1.2).

Using the Iwasawa decomposition (see [17, Chapter 6]) of SL(2,R), letting

N ={

nx =(

1 x0 1

): x ∈ R

}, A =

{ay =

(√y 0

0√

y−1

): y ∈ R>0

},

K ={

kθ =(

cosθ −sinθ

sinθ cosθ

): θ ∈ R/2πZ

}= SO(2),

we have SL(2,R) = NAK and N ∩ A = N ∩ K = A ∩ K = {e}; hence ev-ery element g may be written as g = nxayk for some uniquely determined(x,y,θ) ∈ R×R>0× (R/2πZ). Letting Γ < PSL(2,R) be a lattice and f anautomorphic function of weight j ∈ Z on H for Γ, we define a function fG onG by

fG(nxaykθ ) := f (x+ iy)e−2i jθ .

Letting Γ̃ < SL(2,R) denote the inverse image of Γ under the map fromSL(2,R) to PSL(2,R) given by g 7→ ±g, we have(

± (nxaykθ ))· i = x+ iy ∀x+ iy ∈H, k ∈ K,

and

γnxay =nRe((±γ)·z)aIm((±γ)·z)

(cx+d|cz+d| −

cy|cz+d|

cy|cz+d|

cx+d|cz+d|

)∀γ =

(a bc d

)∈ Γ̃, z = x+ iy ∈H.

This gives

fG(γnxaykθ ) = f((±γ) · z

)( cz+d|cz+d|

)−2 j

e−2i jθ = f (z)e−2i jθ = fG(nxaykθ )

∀γ =(

a bc d

)∈ Γ̃, z = x+ iy ∈H, θ ∈ R/2πZ,

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i.e. fG(γg) = fG(g) for all g ∈ SL(2,R), γ ∈ Γ̃. We may then define a functionf̃ on Γ̃\SL(2,R) by f̃ (Γg) := fG(g). Thus: the map f 7→ f̃ allows us toidentify automorphic functions for Γ on H with functions on the homogeneousspace Γ̃\SL(2,R). One can therefore generalise (in a natural way) the notionof an automorphic function to mean any function on a homogeneous spaceΓ\G for some Lie group G and a discrete subgroup Γ < G.

1.4 Group representationsA representation of a group G is a pair (π,V ), where V is a vector space (overa field K ) and π is a group homomorphism from G to GL(V ), hence

π(gh)vvv = π(g)(π(h)vvv

), π(g)

(auuu+bvvv

)= aπ(g)uuu+bπ(g)vvv

∀g,h ∈ G, uuu,vvv ∈V, a,b ∈K .

The study of group representations has grown to be one of the major areas ofmathematical research, both because it allows one to study the group structureof an abstract group by representing as a group of linear operators, and alsobecause many objects occurring in a wide range of topics in mathematics canbe interpreted as group representations.

The vector spaces for the group representations encountered in this thesisall have either C or R as the underlying field of scalars. Furthermore, therepresentations we consider carry the additional structure of being represen-tations of topological groups on topological vector spaces: for all sequencesvvvn → vvv (in V ) and gn → g (in G), we require that π(gn)vvvn → π(g)vvv (in V ).Most of these representation will carry even more structure: they are unitaryrepresentations of semisimple Lie groups G, i.e. V is a Hilbert space withinner product 〈·, ·,〉 and

〈π(g)uuu,π(g)vvv〉= 〈uuu,vvv〉 ∀g ∈ G, uuu,vvv ∈V.

Unitary representations are particularly well-behaved: given an arbitrary uni-tary representation, one can “decompose” it into so-called irreducible rep-resentations, which are (reasonably) well-understood (see, for example, [16,Chapter 1] for an introduction to the representation theory of semisimple Liegroups).

Given a semisimple Lie group G and a lattice Γ < G, a particular unitaryrepresentation that occurs naturally in connection with Sections 1.2 and 1.3 is(ρ,L2(Γ\G)

), where L2(Γ\G) = L2(Γ\G,µΓ\G) and[

ρ(g) f](x) := f (xg) ∀g ∈ G, f ∈ L2(Γ\G), µΓ\G-almost every x ∈ Γ\G.

This allows many problems in homogeneous dynamics and automorphic func-tions to be reinterpreted in terms of (and then attacked using methods from)representation theory.

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2. Summary of papers

2.1 Paper IIn Paper I, we study the effective equidistribution of expanding translates ofpieces of horospherical orbits in finite-volume quotients of SL(2,C). Thesetup is as follows: we denote G = SL(2,C) and let Γ be a lattice in G. Fort ∈ R, we define

gt =(

e t/2 00 e−t/2

)∈ G,

and let A = {gt}t∈R; note that this is a one-parameter subgroup of G. By theHowe-Moore vanishing theorem [14], for all f1, f2 ∈ L2(Γ\G), we have

limt→±∞

∫Γ\G

[ρ(gt) f1

](x) f2(x)dµΓ\G(x) =

∫Γ\G

f1 dµΓ\G

∫Γ\G

f2 dµΓ\G

(2.1)

Furthermore, by placing certain restrictions on f1, f2 one can obtain a quan-titative version of (2.1); for these functions, the integral in the left-hand sideof (2.1) equals the product of the integrals in the right-hand side plus an errorterm that decays exponentially in t.

By (2.1), the action of A on the probability space (Γ\G,µΓ\G) is mixing,and hence ergodic (cf., e.g., [8, Chapter 2]). By Birkhoff’s ergodic theorem,we then have

limT→∞

1T

∫ T

0f (xg±t)dt =

∫Γ\G

f dµΓ\G

for all f ∈ L1(Γ\G) and µΓ\G-almost every x ∈ Γ\G. While this shows thatthe subset {xg−t}t∈[0,T ] becomes equidistributed in Γ\G for almost all x, weconsider another type of subset related to gt that becomes equidistributed inΓ\G for all x.

Define N < G by

N := {h ∈ G : limt→−∞

gthg−t = e}. (2.2)

N is called the expanding horospherical subgroup with respect to gt . Note that

N ={

nz =(

1 z0 1

): z ∈ C

}.

For a set B ⊂ C, x ∈ Γ\G, and t ∈ R≤0 we consider the subset {xnzgt}z∈B ofΓ\G. This set is thus a piece of the orbit of x under the horospherical subgroup

15

N, which has been translated by gt . Again using the Howe-Moore theorem,combined with Margulis’ thickening technique [23], one can show that for anyrelatively compact subset of positive Lebesgue measure B⊂ C,

limt→−∞

1m(B)

∫B

f (xnzgt)dm(z) =∫

Γ\Gf dµΓ\G (2.3)

∀x ∈ Γ\G, f ∈ L2(Γ\G)∩C(Γ\G),

where m denotes the Lebesgue measure on C.In Paper I, we prove an effective version of (2.3); we show that for all

x,∈ Γ\G, t ≤ 0, f in an appropriate Sobolev space, and relatively compactsubsets B⊂C whose boundary satisfies a certain Lipschitz condition, we havethat the integral 1

m(B)

∫B f (xnzgt)dm(z) equals the right-hand side of (2.3) plus

an error term that decays exponentially with respect to t. Moreover, the rateof exponential decay we obtain matches that given in the corresponding quan-titative version of (2.1).

Similar results have previously been established for translates of closedhorocycles on finite-volume hyperbolic surfaces; these have been studied us-ing spectral theory by (amongst others) Hejhal [13], Sarnak [29], Selberg (un-published), Strömbergsson [35], and Zagier [37]. In [36], Södergren gener-alised these results to translates of closed horospheres in higher-dimensionalhyperbolic manifolds.

In order to prove our results, we use the method of Burger. This is arepresentation-theoretic method, first used in [3] to study the equidistribu-tion of horocycles in quotients Γ\SL(2,R), where Γ is a geometrically fi-nite convex cocompact subgroup of SL(2,R). Strömbergsson later used thesame method to study the equidistribution of horocycles in Γ\SL(2,R), forall lattices Γ < SL(2,R); cf. [34]. The idea behind the method is as follows:for f ∈ L2(Γ\G)∩C(Γ\G), let ft = 1

m(B)

∫B ρ(nzgt) f dm(z) ∈ L2(Γ\G). The

quantity from (2.3) that we are interested in is then given by ft(x) (x ∈ Γ\G).The decomposition of the unitary representation

(ρ,L2(Γ\G)

)into irreducible

representations and the fact that (by Schur’s lemma) the centre of the universalenveloping algebra of the Lie algebra of G acts by scalars on smooth vectorsof irreducible representations are then used to express an identity that capturesthe asymptotic behaviour of ft (as a vector in L2(Γ\G)). In order to study thepointwise behaviour of ft , we then use an automorphic Sobolev inequality dueto Bernstein and Reznikov [2].

2.2 Paper IIIn Paper II, we generalise Burger’s method to study the equidistribution oftranslates of pieces of horospherical orbits in homogeneous spaces Γ\G forarbitrary semisimple Lie groups G and lattices Γ < G that satisfy a certain

16

technical condition. We let G be such a group, and choose a one-parametersubgroup {gt}t∈R < G such that each operator Adgt (acting on the Lie algebraof G) is diagonalizable over R. This ensures that N, the expanding horosper-ical subgroup with respect to gt (defined as in (2.2)), is the unipotent radicalof some parabolic subgroup of G. Consequently, this enables us to use theHarish-Chandra isomorphism to construct differential equations related to gtfrom the centre of the universal enveloping algebra of the Lie algebra of G;this is key to the method.

The main difference from Paper I is that (for Lie algebraic reasons) in orderto get the relevant translates to equidistribute with the same rate of exponentialdecay as the matrix coefficients of gt , instead of integrating over subsets of N,we must consider averages against smooth, compactly supported test functionson N, i.e. we study integrals of the form∫

Nf (xngt)χ(n)dµN(n),

where x ∈ Γ\G, f ∈C(Γ\G)∩L2(Γ\G) , χ ∈C∞c (N) and µN denotes a Haar

measure on N (note that N is unimodular). Furthermore, we must also combinethe underlying method of [3] (and Paper I) with an iterative procedure so as toobtain the desired rate.

2.3 Paper IIILike Papers I and II, Paper III is concerned with the effective equidistributionin homogeneous spaces Γ\G (as before, G is a semisimple Lie group and Γ isa lattice in G) of translates of pieces of orbits of subgroups of G. However,Paper III studies the translates of orbits of symmetric subgroups of G, insteadof horospherical subgroups. A symmetric subgroup S < G is defined to bethe identity component of the subgroup of fixed points for some Lie groupinvolution σ ; i.e. S is the identity component of Gσ , where

Gσ := {g ∈ G : σ(g) = g}

(since σ is an involution, σ(gh) = σ(g)σ(h) and σ2(g) = g for all g,h ∈ G).We once again use the method of Burger to show that translates of orbits ofpoints x ∈ Γ\G under S by certain one-parameter subgroups {gt}t∈R equidis-tribute with the same exponential rate as the matrix coefficients decay. Themain arguments are quite similar to those of Paper II; the structure theory ofsymmetric subgroups permits us to associate a parabolic subgroup of G (andhence also a horospherical subgroup) to the translate being considered. Thispermits relatively straightforward modifications to be made to the proofs ofthe central results of Paper II that allow them to be carried over to the case ofsymmetric subgroups.

17

Figure 2.1. A bounded Apollonian circle packing.

The equidistribution of translates of symmetric subgroups is a much-studiedtopic in homogeneous dynamics. One reason for this is due to applications tocounting problems on affine symmetric spaces. An affine symmetric space isa homogeneous space G/S, where S is a symmetric subgroup of a semisimpleLie group G. Duke, Rudnick, and Sarnak [7], and Eskin and McMullen [9],showed how to relate counting problems for Γ-orbits on G/S (for the naturalleft G-action on G/S) to the equidistribution of translates of S-orbits in Γ\G.While many results regarding the equidistribution of symmetric translates areproved with the goal of obtaining similar counting results, our motivation hasinstead been on understanding the precise asymptotic behaviour of the trans-lates.

2.4 Paper IVAs in Papers I through III, in Paper IV, we use the method of Burger to studythe the effective equidistribution of translates of pieces of orbits in homoge-neous spaces Γ\G. We once again consider the translates of horosphericalorbits, though now consider G = SO0(n,1) (i.e. G is the identity componentof SO(n,1)), and discrete subgroups Γ < G which need no longer be lattices,but instead are geometrically finite (hence the measures µΓ\G are not neces-sarily finite). The study of dynamics on infinite-volume quotients Γ\G hasgrown in recent years; this is due to recent interest in counting problems forthin groups, to which such dynamics are connected, cf. [25].

One particularly beautiful example of this in Apollonian circle packings(see Figure 2.1). In [18], Kontorovich and Oh proved that translates of closedhorospheres equidistribute in Γ\SO0(3,1) with respect to what is now calledthe Burger-Roblin measure, and used this to obtain leading-term asymptotics

18

for various counting problems related to Apollonian packings, for examplecounting the number of circles in a given packing with curvature less than orequal to T > 0 (as T → ∞). Lee and Oh [20] subsequently proved effectiveversions of the equidistribution results of [18]; this enabled them to obtain anerror term with a power savings for counting problems related to Apolloniancircle packings. Using slightly different (but still related) methods, Moham-madi and Oh [24] proved effective equidistribution results for translates oforbits of horospherical and symmetric subgroups in Γ\SO0(n,1) for all n≥ 2and geometrically finite Γ that satisfy certain spectral gap conditions.

The main goal of Paper IV is to study the dependency of the error term inthe equidistribution results of [24] on the spectral gap. Using the method ofBurger instead of the thickening techniques of [24], we are able to improve thedependency on said spectral gap to match the correponding rate of decay ofmatrix coefficients. However, unlike in the case that Γ < SO0(n,1) is a lattice,the subrepresentation of

(ρ,L2(Γ\SO0(n,1))

)whose matrix coefficients have

the slowest rate of decay also contributes to the error term. For this reason,we are not able to improve the effective equidistribution results of [24] for allfunctions in C∞

c (Γ\SO0(n,1)) (since the quotients Γ\SO0(n,1) can have infi-nite volume, it is convenient to consider only functions with compact support);our results yield genuinely new equidistribution statements only for functionsin the orthogonal complement to the subrepresentations mentioned above.

In the special case G = SO0(3,1), and considering only translates of closedhorospheres, we combine the method of Burger with results of Lee and Oh[20] regarding functions in the subrepresentation whose matrix coefficientshave the slowest rate of exponential decay, and do in fact obtain an improve-ment in the effective equidistribution results of [20] for all SO(2)-invariantfunctions in C∞

c (Γ\G). This enables us to improve the error terms in somecounting problems of a similar nature to (and including) those for Apolloniancircle packings mentioned above.

2.5 Paper VPaper V is concerned with the growth properties of Fourier coefficients ofEisenstein series for the hyperbolic plane. We recall that for a non-cocompactlattice Γ⊂G = PSL(2,R), to each parabolic fixed point η ∈ ∂∞H=R∪{∞},we define the Eisenstein series Eη(z,s), where (z,s) ∈H×{ζ ∈ C : Im(ζ )>1}, by

Eη(z,s) := ∑γ∈Γη\Γ

Im(hηγ · z)s,

where Γη = {γ ∈ Γ : γ · η = η}, and hη ∈ G is such that hη · η = ∞ andhηΓηh−1

η = {nx : x ∈ Z} (that η is a parabolic fixed point for Γ ensures thatsuch an hη exists). For each z ∈ H, s 7→ Eη(z,s) has a meromorphic con-tinuation in s to the entire complex plane, and the location of the poles of

19

s 7→ Eη(z,s) are independent of z. On the other hand, for each fixed s ∈ Cthat is not a pole, z 7→ Eη(z,s) is an automorphic function of weight zerowith respect to Γ. Furthermore, each such Eη(z,s) is a Maass wave form:−∆Eη(x+ iy,s) = s(1− s)Eη(x+ iy,s). As such, each Eη(z,s) has a Fourierdecomposition similar to (1.3) (in fact, Eη(z,s) has one such decompositionfor each cusp of Γ\H). In Paper V, we prove formulas for sums of the form∑0<|n|≤N |an|2 (the numbers an being the Fourier coefficients for the Eisensteinseries, cf. (1.3)) as N→ ∞ for Eisenstein series Eη(z,1/2+ it) (where t ∈ R)for generic (in particular, non-arithmetic) lattices Γ < G. These formulas leadto the bound |an| �ε n

512+ε for the individual Fourier coefficients.

The Eisenstein series with Re(s) = 12 span the continuous spectrum of ∆ in

L2(Γ\H) (L2(Γ\H) is identified with L2(F ,µ), where F ⊂H is a reasonablefundamental domain for the action of Γ on H, and dµ(x+ iy) = dxdy

y2 ); each

individual Eisenstein series Eη(·, 12 + it) is not in L2(Γ\H), but appropriate

continuous superpositions of them are. For the most part, previous resultsregarding the Fourier coefficients of Maass wave forms for generic Γ havefocussed on Maass cusp forms; these are Maass wave forms that are of rapiddecay in all of the cusps of Γ\H. The “standard”, or Hecke, bound on theFourier coefficients for Maass cusp forms is |an| � |n|1/2.

The first improvement over the standard bound for Maass cusp forms forgeneric Γ is due to Sarnak; in [31], he combined the Rankin-Selberg method[26, 32] with a clever use of analytic continuation to obtain the bound |an| �ε

|n|5/12+ε . The analytic continuation in [31] was reinterpreted in terms of rep-resentation theory by Bernstein and Reznikov in [2], enabling them to developrepresentation-theoretic methods (in particular, G-invariant Sobolev norms) tofurther reduce the bound on the Fourier coefficients for Maass cusp forms to|an| �ε |n|1/3+ε . This bound matches the best available bound for the Fouriercoefficients of holomorphic cusp forms for generic lattices Γ, due to Good[10].

The aim of Paper V is to develop the Rankin-Selberg method and a the-ory of analytic continuation of representations as in [2] to go beyond theHecke bound for the Fourier coefficients for Eisenstein series for generic lat-tice Γ<G. Since the Rankin-Selberg method involves studying triple productsof Maass wave forms, and Eisenstein series are not in L2(Γ\H), the integralsthat a naïve attempt to use the Rankin-Selberg method suggests one shouldconsider are not finite. For this reason, we follow Zagier [38], and instead userenormalized integrals; these allow one to give meaning to certain integralsthat ordinarily would not have a finite value. Viewing Eisenstein series as theimages in C∞(Γ\G) of certain vectors in irreducible unitary representationsof G allows us to then make use of the results of [2] regarding the analyticcontinuation of representations of SL(2,R). We are, however, unable to makeuse of G-invariant Sobolev norms as in [2]; consequently, our bounds do notmatch those available for (Maass and holomorphic) cusp forms.

20

3. Summary in Swedish

Denna avhandling består av fem artiklar. De fyra första artiklarna handlar omproblem relaterade till effektiv likafördelning i homogena rum och den femtestuderar Fourierkofficienterna för Eisensteinserier på det hyperboliska övrehalvplanet.

De homogena rum som studeras i avhandlingen är alla kvotrum på formenΓ\G, där G är en halvenkel Liegrupp och Γ är en diskret delgrupp till G. Dessarum kommer utrustade med en naturlig höger G-verkan: givet Γh ∈ Γ\G ochg ∈ G, definierar vi Γh · g = Γhg. I homogen dynamik studeras asymptotiskaegenskaper hos gruppverkan på Γ\G för olika delgrupper till G. Ofta är mansärskilt intresserad av statistiska egenskaper (med avseende på ett naturligtmått på Γ\G; i många fall är detta ett G-invariant mått som induceras från ettHaarmått på G) hos dessa banor.

I artiklarna I-IV låter vi H < G vara antingen en horosfärisk eller sym-metrisk delgrupp till G och {gt}t∈R < G en 1-parameterdelgrupp till G som (iviss mening) “expanderar” H. Målet i artiklarna är att ge en så exakt kvan-titativ beskrivning som möjligt på hur jämnt utspridda, eller likafördelade,delmängder av Γ\G på formen xBgt , där x ∈ Γ\G och B är en delmängd i H,är i Γ\G då t → ±∞. Delmängderna på formen xBgt kallas (gt-)translat avdelar av H-banan av x.

Artikel I studerar likafördelningen av translat av delar av horosfäriska banori fallet då G = SL(2,C) och Γ är ett gitter i G, d.v.s. Γ har ändlig kovolym iG. Eftersom Γ < G är ett gitter finns det ett unikt G-invariant mått µΓ\G påΓ\G (som är inducerat från ett Haarmått på G). Vi väljer delgrupperna H och{gt}t∈R som ovan genom att göra följande definitioner:

H = N ={

nz =(

1 z0 1

): z ∈ C

}, gt =

(e t/2 0

0 e−t/2

).

För att mäta hur jämnt utspridd i Γ\G ett translat x{nz : z ∈ B}gt är, där B⊂ C är en delmängd med positivt Lebesguemått (som vi betecknar dm(z)),undersöker vi skillnaden mellan värdet av integraler över den med värdet avintegraler över hela rummet Γ\G för olika val av funktioner f på Γ\G d.v.s viundersöker hur stort följande uttryck:∣∣∣∣ 1

m(B)

∫B

f (xnzgt)dm(z)−∫

Γ\Gf dµΓ\G

∣∣∣∣ . (3.1)

Huvudresultatet i artikel I säger att (3.1) avtar exponentiellt med avseendepå t då t →−∞ för delmängder B ⊂ C och funktioner f som uppfyller vissa

21

tekniska krav. Dessutom gäller det att hastigheten hos detta exponentiella av-tagande är densamma som för avtagandet hos matriskoefficienterna för denunitära representationen

(ρ,L2(Γ\G)0

). Här betecknar L2(Γ\G)0 det ortogo-

nala komplementet till konstantfunktionerna i L2(Γ\G) = L2(Γ\G,µΓ\G), ochρ betecknar högertranslation: (ρ(g) f )(x) := f (xg) för alla g ∈ G, alla funk-tioner f ∈ L2(Γ\G) och µΓ\G-nästan alla x ∈ Γ\G.

För att visa resultaten i artikel I generaliserar vi en metod som först an-vändes av Burger i [3] för att studera liknande frågor i fallet G = SL(2,R).Denna metod är av en representationsteoretisk karaktär och är baserad påidentiteter för element i centret av den universella envelopperande algebrantill en halvenkel Liealgebra. I artiklarna II-IV vidareutvecklar vi metoden iolika situationer. I artikel II studeras återigen den effektiva likafördelningeni Γ\G av expanderande translat av banor för horosfäriska delgrupper, docknu för generella halvenkla Liegrupper G och gitter Γ < G som uppfyller ettvisst tekniskt krav. Även i artikel III betrakas likafördelningsproblem i Γ\Gför G och Γ som i artikel II, men nu för translat av symmetriska delgruppertill G. Artikel IV handlar om translat av banor av horosfäriska delgrupperi kvotrum Γ\SO0(n,1), där SO0(n,1) är identitetskomponenten av SO(n,1),och Γ < SO0(n,1) inte längre (nödvändigtvis) är ett gitter, utan geometrisktändligt. Faktumet att Γ\SO0(n,1) kan ha oändlig volym med avseende påde naturliga måtten inducerade från Haarmåtten på SO0(n,1) gör likafördel-ningspåståendena mer komplicerade; ofta involverar de flera mått utöver desom är relaterade till Haarmått. På senare tid har det funnits stort intresse förlikafördelningsresultat i rum Γ\SO0(n,1) med oändlig volym. Detta på grundav att de har tillämpningar i talteori, t.ex i räkneproblem för Apolloniska cirkel-packningar (se figur 2.1).

Artikel V handlar om Eisensteinserier på det hyperboliska övre halvplanetH. Gruppen G = PSL(2,R) verkar på H genom Möbiusavbildningar. Givetett gitter Γ < G med egenskapen att den hyperboliska ytan Γ\H har κ ≥1 stycken spetsar, så finns det en uppsättning Eisensteinserier E j(z,s), därz ∈ H, s ∈ C, och j ∈ {1, . . . ,κ}. För varje z ∈ H är s 7→ E j(z,s) en mero-morf funktion på C, och för s som är inte en pol så är E j(z,s) Γ-invariant:E j(γ · z,s) = E j(z,s) för alla z ∈ H, γ ∈ Γ. Dessutom är EisensteinseriernaMaass vågformer (d.v.s. egenfunktioner till den hyperboliska Laplaceoper-torn ∆ = y−2(∂ 2

x +∂ 2y )): −∆E j(x+ iy,s) = s(1− s)E j(x+ iy,s). Som följd av

detta har alla E j(z,s) en Fourierutveckling (1.3) i varje spets av Γ\H. Måletmed artikel V är att hitta nya begränsningar för Fourierkoefficienterna i dessautvecklingar för godtyckliga gitter Γ. För att göra detta använder vi Rankin-Selberg-metoden. Denna metod utvecklades ursprungligen för att studera lik-nande frågor för cuspidala funktioner, men tack vare en modifikation av Zagier[38] går den även att använda för Eisensteinserier. Genom att sedan omtolkaEisensteinserierna i termer av representationsteori kan vi använda en analytiskfortsättning av representationer som Bernstein och Reznikov gör i [2].

22

Acknowledgements

I am extremely grateful to Andreas Strömbergsson for the effort and dedica-tion he has put into being my supervisor over the last five years. It is a realprivilege to have been his student, and I thank him for the fun and interestingproblems, all his ideas and suggestions, his careful reading of my papers, andfor our many discussions about, among other things, maths, work, running,and music.

Thanks are also due to Anders Karlsson for being my second supervisor andteaching me analytic number theory, ergodic theory, and spectral graph theory,as well as for all the chocolates you brought that made your fun lectures evenmore enjoyable!

I thank Jens Marklof and Anders Södergren for their interest in, and inter-esting discussions about, my research. It has been a pleasure to attend manyof Jens’ beautiful lectures over the last few years. These include the inauguralEssén lectures, which served as a very inspiring introduction to the field ofhomogeneous dynamics. Thanks also to Anders for inviting me to give a talkat the number theory seminar in Copenhagen, as well as for hosting me duringmy brief (but productive) visit.

The completion of this thesis brings my time as a student of Uppsala Uni-versity to an end. I thank my teachers at the maths department for all their timeand effort spent teaching me. Special thanks to Professors Ernst Dieterich andWalter Mazorchuk for their inspiring lectures in algebraic structures and rep-resentation theory of finite groups.

Thanks to my fellow PhD students for keeping me company at work overthe last five years. In particular, I thank Andrea, Andreas, Filipe, Hannah,Jakob, and Linnéa for being good friends, and wish you all the best of luck forthe future.

I thank Mum, Dad, Jamie, Buddy, Lucia, Toby, and Mary for being suchan amazing family and always providing fun (and much-needed) distractionsfrom work.

Last but not least, I thank my darling Gunta for her endless love and supportover these years and for putting up with all the maths.

23

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