Sato-Tate Problem for GL(3) Fan Zhou

Transcription

1 Sato-Tate Problem for GL(3) Fan Zhou Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 203

2 c 203 Fan Zhou All Rights Reserved

3 ABSTRACT Sato-Tate Problem for GL(3) Fan Zhou Based upon the work of Goldfeld and Kontorovich on the Kuznetsov trace formula of Maass forms for SL(3, Z), we prove a weighted vertical equidistribution theorem (with respect to the generalized Sato-Tate measure) for the p th Hecke eigenvalue of Maass forms, with the rate of convergence. With a conectured orthogonality relation between the Fourier coefficients of Maass forms for SL(N, Z) for N 4, we generalize the above equidistribution theorem to N 4.

4 Table of Contents Introduction 2 Historical Equidistribution Results on Hecke-Maass Forms for SL(2, Z) 6 3 Hecke-Maass Froms for SL(3, Z) 4 Sato-Tate Problem and Equidistribution Theorems 6 4. Sato-Tate Measure for GL(N) Equidistribution Main Theorem 9 6 Hecke Relations and Random Walks in the First Quadrant 2 7 Lie Theory and Random Walks in a Weyl Chamber Basic Lie Algebra Reflection Method Special Linear Lie Algebra sl(n, C) More about sl(3, C) Proof of the Main Theorem 38 9 Rate of Convergence 4 0 Case of GL(N) Two Types of Random Walks i

5 0.2 Completion of the Proof Bibliography 48 ii

6 Acknowledgments The author is grateful to his thesis advisor, Dorian Goldfeld, who brought the author to this topic of research, gave the author much guidance, and painstakingly read the manuscript. The author also wants to thank Gautam Chinta, Johan de Jong, Daniel Disegni, Patrick Gallagher, Hervé Jacquet, Min Lee, Chiu-Chu (Melissa) Liu, Yifeng Liu, Qing Lu, Xuanyu Pan, You Qi, Peter Sarnak, Chenyan Wu, Zhengyu Xiang, Hang Xue, Shou-Wu Zhang, and Wei Zhang. iii

7 To My Parents iv

8 CHAPTER. INTRODUCTION Chapter Introduction Let N be a positive integer and k a non-negative integer. We consider S(N, k) the space of holomorphic modular forms of level N and weight k for a congruence subgroup of SL(2, Z). For each positive integer n with (n, N ), the Hecke operator T n acts on S(N, k). Let φ S(N, k) be an eigenfunction of all the Hecke operators T n ((n, N ) ) and a φ (n) the eigenvalue of φ under the action of T n. The Ramanuan conecture states that for a prime number p N a φ (p) 2. We define a measure on R dµ (2) (x) p k 2 π x2 4 dx, when x 2, 0, otherwise, called the Sato-Tate measure for GL(2), or the semi-circle measure. The Sato-Tate conecture is a more refined statement about the statistics of the Hecke eigenvalues of non-cm holomorphic modular form φ of weight k 2. It states that a φ(p) is an equidistributed sequence as p with p k 2 respect to µ (2). More precisely the Sato-Tate conecture predicts that ( ) lim T p T f p T a φ(p) p k 2 R f dµ (2) for any continuous test function f : R R. In recent years, there has been enormous progress toward proving this conecture and its generalizations, most notably in [Barnet-Lamb-Gee-Geraghty, 20] and [Barnet-Lamb-Geraghty-Harris-Taylor, 20].

9 CHAPTER. INTRODUCTION 2 Considering this problem from another perspective, we can fix the prime number p and investigate the distribution of aφ(p) as φ runs over different modular forms. Most notably in [Serre, 997], p k 2 it is proved that a φ(p) is equidistributed with respect to the Plancherel measure p k 2 dµ (2) p p + (p /2 + p /2 ) 2 x 2 dµ(2) as φ runs over all Hecke eigenforms in S(N, k) and N + k (p N ). Almost at the same time [Conrey-Duke-Farmer, 997] obtained the same result independently for N. Recently, [Murty- Sinha, 2009] investigated the effective version of [Serre, 997], which gives an explicit estimate on the rate of convergence. From the same perspective of fixing p and varying φ, two earlier papers [Sarnak, 987] and [Bruggeman, 978] proved that a φ (p) is an equidistributed sequence with respect to the Plancherel measure µ (2) p by varying φ over all Hecke-Maass forms for SL(2, Z). As in [Murty-Sinha, 2009] an effective version of [Sarnak, 987] appeared in [Lau-Wang, 20]. Very recently [Shin-Templier, 202] gave a highbrow generalization of [Serre, 997] et al. It is understandable that by fixing a prime number p instead of a Hecke eigenform φ we get the Plancherel measure instead of the Sato-Tate measure. Strikingly, if we give each Hecke eigenvalue a φ (p) a weight Res s L(s, φ φ) ( or ) L(, φ, Ad) and do the same statistics of fixing p and varying φ, the same Sato-Tate measure appears once again, instead of the Plancherel measure. In [Bruggeman, 978] it is essentially proved that f(a φ (p)) Res L(s,φ φ) λ φ T s f dµ (2) lim T Res λ φ T L(s,φ φ) s for any continuous test function f : R R, where φ runs over all Hecke-Maass forms for SL(2, Z) and λ φ is the eigenvalue of φ under the action of the Laplace operator on the upper half plane. Later [Li, 2004] and [Gun-Murty-Rath, 2008] proved similar theorems for modular forms. weight /Res L(s, φ φ) appears naturally in the Petersson and Kuznetsov trace formulae. s The theory of Maass forms for SL(3, Z) has been studied since the 980s. The definitions and results are summarized in [Goldfeld, 2006]. Let ϕ be a Maass form for SL(3, Z) which is an eigenfunction of all the Hecke operators T n, where T n corresponds to the union of double cosets R The

10 CHAPTER. INTRODUCTION 3 ( m0 m m 2 ) SL(3, Z) m0 m SL(3, Z). Let a m 3 0 m2 m m ϕ (p) be the Hecke eigenvalue of ϕ under the 2n 0 action of the Hecke operator T p. In this context the Ramanuan conecture predicts { 3 a ϕ (p) e iθ i : i } 3 θ i 0, θ i R, i, 2, 3. i Note that a ϕ (p) is not necessarily a real number. We define µ (3), the Sato-Tate measure for GL(3), as the pushforward measure of the Haar measure of SU(3) by the trace map Tr : SU(3) C. It is unlikely that this measure can be expressed explicitly by elementary functions. The Sato-Tate conecture should be modified correspondingly, predicting that a ϕ (p) is an equidistributed sequence as p with respect to the measure µ (3). Every Hecke-Maass form ϕ has spectral parameter ν ϕ (ν ϕ, νϕ 2 ) C2, and λ ν ϕ( 2 ) 3(ν ϕ 2 + ν ϕ νϕ 2 + νϕ 2 2 ) is the Laplace eigenvalue of ϕ. Theorem. (Main theorem). For T, let {h T : C 2 R} be a family of test functions where h T is essentially supported on {ν (ν, ν 2 ) : λ ν ( 2 ) T 2 } as in Definition 5.. Let a ϕ (p) be the p th Hecke eigenvalue of a Hecke-Maass form ϕ. For any continuous test function f : C C, we have where ϕ lim T f(a ϕ (p)) ϕ ϕ Res s h T (ν ϕ ) L(s,ϕ ϕ) h T (ν ϕ ) Res L(s,ϕ ϕ) s sums over all Hecke-Maass forms for SL(3, Z). C f dµ (3), Our proof of Theorem. is based upon the orthogonality relation in [Goldfeld-Kontorovich, 203] for Fourier coefficients of Maass forms for SL(3, Z). Let A ϕ (m, m 2 ) be the (m, m 2 ) th Fourier coefficient of a Maass form ϕ. If we assume that ϕ is normalized so that A ϕ (, ), then we have A ϕ (p, ) a ϕ (p) for all prime numbers p. Goldfeld and Kontorovich s orthogonality relation states A ϕ (m, m 2 )A ϕ (n, n 2 ) ϕ ϕ h T (ν ϕ ) Res L(s,ϕ ϕ) s Res s h T (ν ϕ ) L(s,ϕ ϕ) δ m,n δ m2,n 2 + O {ht },ϵ((m m 2 n n 2 ) 2 T ϵ 2 ) (.) as T. A similar result was also obtained by [Blomer] and is equally usable to us. Combining the orthogonality relation with the Hecke relations we can compute the average of a ϕ (p) k a ϕ (p) k 2

11 CHAPTER. INTRODUCTION 4 for non-negative integers k and k 2. More precisely, we find that a ϕ (p) k a ϕ (p) k 2 h T (ν ϕ ) lim T ϕ ϕ h T (ν ϕ ) Res L(s,ϕ ϕ) s Res L(s,ϕ ϕ) s equals the number of random walks of some special type in the integer points of the first quadrant of the coordinate plane, which will be defined in later chapters. This type of random walk comes from the Hecke relations for SL(3, Z). In the proof of Theorem. in the case of f(z) z k z k 2, we show that z k z k 2 dµ (3) C equals the multiplicity of the trivial representation in the tensor product representation H k H k 2 where H is the defining representation SU(3) GL(3, C). Relying on the Weyl character formula and two papers [Gessel-Zeilberger, 992] and [Grabiner-Magyar, 993] on the reflection method, we show that the multiplicity equals the number of random walks of certain type in a Weyl chamber of the root system A 2. By matching the random walk in the first quadrant and the random walk in a Weyl chamber, we obtain Theorem. when f(z) z k z k 2. Since all continuous functions on a compact set can be approximated by polynomials this completes the proof. As in [Murty-Sinha, 2009], [Lau-Wang, 20] and [Shin-Templier, 202], we obtain an effective version of Theorem., which gives the rate of convergence, but only for a large family of special test functions. Theorem.2 (Rate of Convergence). For T, let {h T : C 2 R} be a family of test functions where h T is essentially supported on {ν (ν, ν 2 ) : λ ν ( 2 ) T 2 } as in Definition 5.. Fix ϵ > 0. For a power series f(z) a k,k 2 z k z k 2 which is absolutely convergent on {z : z p 2 + p 2 + }, we have f(a ϕ (p)) ϕ where ϕ ϕ Res s Res s k,k 2 0 h T (ν ϕ ) L(s,ϕ ϕ) h T (ν ϕ ) L(s,ϕ ϕ) C sums over all Hecke-Maass forms for SL(3, Z). f dµ (3) ϵ,f T ϵ 2, Our proof of Theorem.2 is based on the error terms O h,ϵ ((m m 2 n n 2 ) 2 T ϵ 2 ) in the orthogonality relation (Equation.). By exploiting more equivalences between the random walk in the

12 CHAPTER. INTRODUCTION 5 first quadrant and the random walk in a Weyl chamber, we get the error term accumulated by each monomial in the power series a k,k 2 z k z k 2, with the help of its large convergence domain. k,k 2 0 The mechanism we use in the proof of Theorem. can be generalized to GL(N) (N 2). In general we expect that the Kuznetsov trace formula would give an orthogonality relation for Fourier coefficients with the weight /Res s L(s, ϕ ϕ), as in [Bruggeman, 978], [Goldfeld-Kontorovich, 203], [Blomer] et al. For GL(2) automorphic forms, both the Hecke relations and the Sato-Tate measure are simple enough to allow elementary techniques to connect them, as in [Bruggeman, 978], [Gun-Murty-Rath, 2008] and [Li, 2004]. For GL(N) (N 4) the orthogonality relation is conectured but not proved yet. Hopefully, a version of the orthogonality relation will appear in the near future for GL(N) (N 4). Let µ (N) be the Sato-Tate measure for GL(N) which is the pushforward measure of the Haar measure of SU(N) by the trace map Tr : SU(N) C. If we assume the orthogonality relation of type lim T A ϕ (m,..., m N )A ϕ (n,..., n N ) ϕ ϕ h T (ν ϕ ) Res L(s,ϕ ϕ) s Res s h T (ν ϕ ) L(s,ϕ ϕ) N i δ mi,n i for some family of test functions h T, we prove lim T f(a ϕ (p)) ϕ ϕ Res s h T (ν ϕ ) L(s,ϕ ϕ) h T (ν ϕ ) Res L(s,ϕ ϕ) s C f dµ (N) for any continuous test function f : C C. The idea of our proof is the same as the proof for GL(3). The set of integer points in the first quadrant is replaced with a lattice in higher dimensions. The root system A 2 is replaced by A N. By matching the two types of random walk and approximation by polynomials we complete the proof.

13 CHAPTER 2. HISTORICAL EQUIDISTRIBUTION RESULTS ON HECKE-MAASS FORMS FOR SL(2, Z) 6 Chapter 2 Historical Equidistribution Results on Hecke-Maass Forms for SL(2, Z) In this chapter we will recall the definitions of Maass forms for SL(2, Z) and review the historical results of Sarnak and of Bruggeman. This chapter is independent from the rest chapters. The main references are [Goldfeld, 2006], [Sarnak, 987] and [Bruggeman, 978]. Definition 2.. Let H 2 be the upper half plane {z x + iy : x, y R, y > 0}. The group SL(2, Z) acts on H 2 by where a c b SL(2, Z) and z H 2. d a c b z az + b d cz + d, Definition 2.2. Let φ and φ be two complex-valued functions on SL(2, Z) \ H 2. We define their inner product We define the L 2 -norm < φ, φ > φ(z)φ (z) dxdy SL(2,Z)\H 2 y 2. φ 2 < φ, φ >. Definition 2.3. We define L 2 (SL(2, Z)\H 2 ) to be {φ : SL(2, Z)\H 2 C : φ 2 < }. We define

14 CHAPTER 2. HISTORICAL EQUIDISTRIBUTION RESULTS ON HECKE-MAASS FORMS FOR SL(2, Z) 7 L 2 cusp(sl(2, Z)\H 2 ) to be the space of all φ L 2 (SL(2, Z)\H 2 ) such that φ is cuspidal, i.e., 0 φ(z) dx 0. Definition 2.4 (Maass form for SL(2, Z)). A Maass form of type ν C for SL(2, Z) is a non-zero smooth function φ L 2 (SL(2, Z)\H 2 ) which satisfies φ(γz) φ(z) for any z H 2 and any γ SL(2, Z), φ is an eigenfunction of the Laplace operator, i.e., φ is cuspidal or 0 φ(z) dx 0. ( ) y 2 2 x y 2 φ(z) ( 4 ν2 )φ(z), Definition 2.5 (Hecke operator). For n > 0, we define the Hecke operator T n L 2 cusp(sl(2, Z)\H 2 ) by T n φ(z) n 0 b<d adn ( az + b φ d ). acting on Remark 2.6 (Hecke-Maass form). The Hecke operators T n (n, 2, 3,...) commute with the Laplace operator y 2 ( 2 x y 2 ). All Hecke operators commute. We can simultaneously diagonalize the space L 2 cusp(sl(2, Z)\H 2 ) by these operators. The space L 2 cusp(sl(2, Z)\H 2 ) has an orthogonal basis φ, φ 2, φ 3,... Each φ is a Maass form of type ν and an eigenfunction of all Hecke operators T n. We arrange the order of φ s by their Laplace eigenvalues so that These φ s are called Hecke-Maass forms. 4 ν2 4 ν2 2 4 ν Definition 2.7 (Hecke eigenvalue). Let a (n) be the eigenvalue of of φ under the Hecke operator T n, i.e., T n φ a (n)φ.

15 CHAPTER 2. HISTORICAL EQUIDISTRIBUTION RESULTS ON HECKE-MAASS FORMS FOR SL(2, Z) 8 Definition 2.8. We define the convolution L-function for a Hecke-Maass form φ to be where ζ(s) n n s Maass form φ and it is given by L(s, φ φ ) ζ(2s) n a (n) 2 n s, is the Riemann-Zeta function. Let L be a number attached to each Hecke- L Res s L(s, φ φ ). Definition 2.9 (the Sato-Tate measure and the Plancherel measure). We define the Sato- Tate measure for GL(2) as a measure on R dµ (2) (x) π x2 4 dx, if x 2, 0, otherwise. For a prime number p, we define the p-adic Plancherel measure for GL(2) as a measure on R given by dµ (2) p (x) (p+) 4 x 2 2π((p /2 +p /2 ) 2 x 2 ) dx, if x 2, 0, otherwise. Both measures are probability measure and supported on [ 2, 2]. Obviously we observe that lim p µ(2) p µ (2). Theorem 2.0 (Sarnak). Let p be a prime number. For, 2, 3,..., let a (p) be the Hecke eigenvalue of the Hecke-Maass form φ as in Definition 2.7. The sequence a (p), a 2 (p), a 3 (p),... is equidistributed with respect to the p-adic Plancherel measure µ (2) p. In another word, for any continuous test function f : R R we have lim T f(a (p)) 4 ν2 T 4 ν2 T R f dµ (2) p. Proof. See [Sarnak, 987]. It is an application of the Selberg trace formula.

16 CHAPTER 2. HISTORICAL EQUIDISTRIBUTION RESULTS ON HECKE-MAASS FORMS FOR SL(2, Z) 9 Theorem 2. (Bruggeman). Let p be a prime number. For, 2, 3,..., let a (p) be the Hecke eigenvalue of the Hecke-Maass form φ as in Definition 2.7. Define a family of test functions h T (ν) e 4 ν 2 T for ν C and T. For any continuous function f : R R, we have f(a (p))h T (ν ) lim f dµ (2) p. T h T (ν ) Proof. See [Bruggeman, 978]. It is an application of the Selberg trace formula. Remark 2.2. The test function h T (ν) e 4 ν 2 T is essentially supported on {ν : 4 ν2 T }. It acts as a characteristic function for the set {ν : 4 ν2 T } and it essentially counts the Hecke-Maass forms with eigenvalues no greater than T. Additionally we have the Weyl law h T (ν ) T 2. Theorem 2.3 (Bruggeman). Let p be a prime number. For, 2, 3,..., let a (p) be the Hecke eigenvalue of the Hecke-Maass form φ as in Definition 2.7. Let L be as in Definiton 2.8. For a continuous function f : R R, we have lim T f(a (p)) h T (ν () ) L h T (ν () ) L R R f dµ (2), where h T is the same as in Theorem 2.. Additionally we have the Weyl law h T (ν ) L T 4π. Proof. See [Bruggeman, 978]. It is an application of the Kuznetsov trace formula. Remark 2.4. Theorem 2.0 and Theorem 2. are essentially the same. involve the notion of equidistributed sequences. equidistributed sequences. The later does not We refer to Definition 4.5 for the definition of

17 CHAPTER 2. HISTORICAL EQUIDISTRIBUTION RESULTS ON HECKE-MAASS FORMS FOR SL(2, Z) 0 Remark 2.5. The contrasting difference between the Sato-Tate measure µ (2) in Theorem 2.3 and the p-adic Plancherel measure µ (2) p in Theorem 2.0 and Theorem 2. indicates that the number L Res s L(s, φ φ ) which is attached to each Hecke-Maass form φ plays a key role. We can view Theorem 2.3 as a weighted version of the equidistribution problem. The sequence {a (p)} is equidistributed with respect to the Sato-Tate measure µ (2), if each a (p) is given a weight L. On the other hand, as in Theorem 2.0 and Theorem 2., the sequence a (p) is equidistributed with respect to µ (2) p, if each a (p) given a uniform weight. The same thing happens in the papers [Serre, 997], [Li, 2004], [Gun- Murty-Rath, 2008] and [Conrey-Duke-Farmer, 997] in the case of holomorphic modular forms.

18 CHAPTER 3. HECKE-MAASS FROMS FOR SL(3, Z) Chapter 3 Hecke-Maass Froms for SL(3, Z) In this chapter, we will summarize the definitions and some of the most important theorems of Maass forms for SL(3, Z). The main reference for this chapter is [Goldfeld, 2006]. Definition 3. (Generalized upper half plane). The generalized upper half plane H 3 is the set of all matrices z x y with x 2 x 3 x 0 x 0 0 y y and y 0 y 0, 0 0 where x, x 2, x 3 R and y, y 2 > 0. Remark 3.2. By the Iwasawa decomposition, we have GL(3, R) H 3 O(3, R) R and H 3 GL(3, R)/(O(3, R) R ). Definition 3.3. Let gl(3, R) be the Lie algebra of GL(3, R). It consists of 3 3 real matrices with the Lie bracket give by [α, β] α β β α for all α, β gl(3, R). Definition 3.4. For α gl(3, R) and a smooth function ϕ : GL(3, R) C, we define D α ϕ(g) t ϕ(g exp(tα)) t0.

19 CHAPTER 3. HECKE-MAASS FROMS FOR SL(3, Z) 2 Definition 3.5 (Casimir operator). Let E i, gl(3, C) be the matrix with at the (i, ) th entry and 0 elsewhere, for i, 3. We define three operators on smooth functions on GL(3, R) D Ei,i, 2 D Ei,i 2 D Ei2,i, 3 D Ei,i 2 D Ei2,i 3 D Ei3,i. i i i 2 i i 2 i 2 These three operators generate a commutative ring D 3 C[, 2, 3 ], and any element in D 3 is called a Casimir operator. Explicitly, we have 2 y 2 y 2 + y 2 2 y 2 2 y y 2 2 y y 2 + y 2 (x y 2 2) 2 x y 2 x 2 + y 2 2 x y 2 x 2 2 x x 3 and we call 2 the Laplace operator. Lemma 3.6. Any D 3 is well-defined for smooth functions on SL(3, Z) \ H 3, i.e., for ϕ : SL(3, Z) \ GL(3, R)/(O(3, R) R ) C, we have ( ϕ)(γ g k δ) ϕ(g) for all g GL(3, R), γ SL(3, Z), δ R and k O(3, R). Proof. This is Proposition 2.3. of [Goldfeld, 2006]. Definition 3.7. For ν (ν, ν 2 ) C 2, we define a function on H 3, I ν x 2 x 3 y y x 0 y 0 y+ν +2ν y +ν 2+2ν 2. Lemma 3.8. The function I ν is an eigenfunction of for any D 3. Proof. See Equation 6.. of [Goldfeld, 2006]. Definition 3.9. For any D 3, we define λ ν ( ) to be the eigenvalue in Lemma 3.8, i.e., I ν λ ν ( )I ν.

20 CHAPTER 3. HECKE-MAASS FROMS FOR SL(3, Z) 3 Definition 3.0 (Upper triangular subgroups). We define three upper triangular subgroups of GL(3, R): U,, 0, U, and U 2, Definition 3. (Inner product and L 2 -norm). Let ϕ and ϕ be two functions on SL(3, Z)\H 3. We define their inner product < ϕ, ϕ > ϕ(z)ϕ (z) d z, SL(3,Z)\H 3 where d z dx dx 2 dx 3 dy dy 2 (y y 2 ) 3. We define the L2 -norm ϕ 2 SL(3,Z)\H 3 ϕ(z) 2 d z. Definition 3.2. We define L 2 (SL(3, Z)\H 3 ) to be { ϕ : SL(3, Z)\H 3 C : ϕ 2 < }. We define L 2 cusp(sl(3, Z)\H 3 ) to be all ϕ L 2 (SL(3, Z)\H 3 ) such that ϕ is cuspidal, i.e., SL(3,Z) U\U ϕ(uz) du 0, for U U,,, U,2, U 2,. Definition 3.3 (Maass form). Let ν (ν, ν 2 ) C 2. A Maass form of type ν for SL(3, Z) is a smooth function ϕ L 2 (SL(3, Z)\H 3 ) which satisfies ϕ(γz) ϕ(z) for any z H 3 and any γ SL(3, Z), for any Casimir operator D 3, we have ϕ(z) λ ν ( )ϕ(z), ϕ is cuspidal or ϕ L 2 cusp(sl(3, Z)\H 3 ). Definition 3.4 (Jacquet s Whittaker function). Let m and m 2 be two integers. We define Jacquet s Whittaker function ( where ω 3 W Jacquet (z; ν, ψ m,m 2 ) ), d u du du 2 du 3 for u U 3 (R) ( u2 u 3 ) 0 u 0 0 I ν (ω 3 u z)ψ m,m 2 (u) d u, (( u2 u 3 )) and ψ m,m 2 0 u e 2πi(m u +m 2 u 2 ). 0 0

21 CHAPTER 3. HECKE-MAASS FROMS FOR SL(3, Z) 4 Theorem 3.5 (Fourier-Whittaker expansion). A Maass form ϕ of type ν for SL(3, Z) has Fourier-Whittaker expansion ϕ(z) γ U 2 (Z)\SL(2,Z) m m 2 0 A(m, m 2 ) (( m ) ) m 2 W Jacquet m ( γ m m 2 ) z; ν, ψ, m 2, m 2 where A(m, m 2 ) C is the (m, m 2 ) th Fourier coefficient and U 2 {( 0 )}. Proof. This is Theorem and Equation 6.2. of [Goldfeld, 2006]. Definition 3.6 (Hecke operators). For n, 2, 3,..., we define the Hecke operator T n acting on L 2 cusp(sl(3, Z)\H 3 ) by T n ϕ(z) n abcn 0 c,c 2 <c 0 b <b a b c ϕ 0 b c 2 z. 0 0 c Theorem 3.7 (Hecke relations). Let ϕ be a Maass form for SL(3, Z) as in Definition 3.3. Assume that ϕ is an eigenfunction of every Hecke operator T n. If A(, ) 0, then ϕ vanishes identically. Assume ϕ 0 and it is normalized so that A(, ). Then T n ϕ A(n, )ϕ. Furthermore, we have the following multiplicativity relations and A(n, )A(m, m 2 ) A(, n)a(m, m 2 ) b m,a m 2 abcn a m,b m 2 abcn A(m, m 2 ) A(m 2, m ). A( m c b, m 2b a ), A( m b a, m 2c b ), Proof. This is Theorem 6.4. of [Goldfeld, 2006]. Remark 3.8 (Hecke-Maass forms). The Hecke operators commute with any Casimir operator D 3. All Hecke operators commute. We can simultaneously diagonalize the space L 2 cusp(sl(3, Z)\H 3 ) by these operators. The space L 2 cusp(sl(3, Z)\H 3 ) has an orthogonal basis ϕ, ϕ 2, ϕ 3,...

22 CHAPTER 3. HECKE-MAASS FROMS FOR SL(3, Z) 5 Each ϕ is a Maass form of type ν () (ν (), ν() 2 ) C2 and an eigenfunction of all Hecke operators T n. They are normalized so that the (, ) th Fourier coefficient is. We arrange the order of ϕ s by their eigenvalue under the Laplace operator 2 (Definition 3.5) so that 0 < λ ( 2 ) λ 2 ( 2 ) λ 3 ( 2 )..., where 2 ϕ λ ( 2 )ϕ. These ϕ s are called Hecke-Maass forms. Definition 3.9. We define A (m, m 2 ) to be the (m, m 2 ) th Fourier coefficient of ϕ. Since ϕ is normalized we have A (, ) for all. Theorem 3.20 (Weyl law). We have the following asymptotic formula #{ : λ ( 2 ) < T } ct 5 2, as T for some constant c > 0. Proof. See [Miller, 200]. Definition 3.2. We define the convolution L-function for a Hecke-Maass form ϕ to be where ζ(s) n n s Maass form ϕ which is given by L(s, ϕ ϕ ) ζ(3s) m m 2 A (m, m 2 ) 2, m 2s ms 2 is the Riemann-Zeta function. Let L be a number attached to each Hecke- L Res s L(s, ϕ ϕ ).

23 CHAPTER 4. SATO-TATE PROBLEM AND EQUIDISTRIBUTION THEOREMS 6 Chapter 4 Sato-Tate Problem and Equidistribution Theorems 4. Sato-Tate Measure for GL(N) Definition 4. (Pushforward measure). Let (X, Σ ) and (X 2, Σ 2 ) be measurable spaces in which Σ i is a σ-algebra for X i. Let F : X X 2 by a measurable map and a measure µ : Σ [0, + ). The pushforward measure of µ by F is defined to be the measure F (µ) : Σ 2 [0, + ) given by The change of variables formula is given by F (µ)(b) µ(f (B)) for any B Σ 2. g df (µ) X 2 (g F ) dµ X (4.) for any integrable function g on (X 2, Σ 2 ) with respect to the measure F (µ). Definition 4.2 (Sato-Tate measure for GL(N)). The Sato-Tate measure for GL(2) is a measure on R given by dµ (2) (x) π x2 4 dx, when x 2, 0, otherwise. When N 3, the Sato-Tate measure for GL(N) is a measure on C given by

24 CHAPTER 4. SATO-TATE PROBLEM AND EQUIDISTRIBUTION THEOREMS 7 dµ (N) (z) d(tr (ω))(z), when z N, 0, otherwise, where ω is the unique normalized Haar measure on SU(N) and the map Tr : SU(N) {z C : z N} takes the trace of each element of SU(N) and Tr (ω) is the pushforward measure of ω by Tr. Remark 4.3. The Sato-Tate measure for GL(N) is supported inside {z C : z N}, but we shall note that the support of this measure does not fill {z C : z N}. Its support is the set { N } N e iθ i : θ i 0, θ i R, i, 2,..., N. It is the area inside the curve i i { } (N )e iθ e inθ : θ R. In particular, for N 3, the measure µ (3) is supported on the area inside the delta-shaped curve {2e iθ + e 2iθ : θ R}. Remark 4.4. We shall note that the Sato-Tate measure for GL(2), or, the semi-circle measure dµ (2) π x2 4 dx, when x 2, (x) 0, otherwise, is also a pushforward measure from a special unitary group. The unitary group cos θeiα sin θe SU(2) iβ : θ [0, π sin θe iβ cos θe iα 2 ], α [0, 2π), β [0, 2π) has the unique normalized Haar measure cos θ sin θ dθ dα dβ. 2π2 ( ) The trace map Tr : SU(2) [ 2, 2] maps cos θe iα sin θe iβ to 2 cos θ cos α. The limit sin θe iβ cos θe iα lim x 0 x 2π 2π π cos θ cos α [x,x+ x] cos θ sin θ dθ dα dβ (4.2) 2π2

25 CHAPTER 4. SATO-TATE PROBLEM AND EQUIDISTRIBUTION THEOREMS 8 gives us the pushforward measure of the Haar measure by the trace map. Computing the limit in Equation 4.2, we get lim x 0 x 2π 2π π cos θ cos α [x,x+ x] 2π 2 cos θ sin θ dθ dα dβ 2π 2π 2 dβ 0 arccos x 2 π 0 π x2 4, arccos x 2 0 sin θdθ ( d 2 sin θ cos θ dx arccos x 2 cos θ dθ x 2 cos θ )2 which is the same as in Definition Equidistribution There are multiple definitions for equidistributed sequence in different settings. definition is taken from [Gun-Murty-Rath, 2008]. The following Definition 4.5 (Equidistributed sequence). Let X be a Hausdorff space with a regular normalized Borel measure µ. We denote by R(X) the space of continuous functions of compact support on X. A sequence {x n } of elements in X is called equidistributed with respect to the measure µ if for all f R(X). lim N N N f(x n ) n Remark 4.6. In the cases we are interested in, the space X is C, or R. The Borel measure µ is the Sato-Tate measure given in Definition 4.2. X f dµ

26 CHAPTER 5. MAIN THEOREM 9 Chapter 5 Main Theorem The recent paper [Goldfeld-Kontorovich, 203] establishes an orthogonality relation for Fourier coefficients of Hecke-Maass forms for SL(3, Z), via the Kuznetsov trace formula. Definition 5. (Family of test functions). Let ν 3 ν + ν 2. For T and fixed R 0, we define a test function on C 2 h T,R (ν) e 6(ν 2 +ν2 2 +ν ν 2 ) T 2 ( Γ( 2+R+3ν i 4 )Γ( 2+R 3ν i i 3 Γ( +3ν i 2 )Γ( 3ν i 2 ) i 3 4 ) Remark 5.2. The family of test functions {h T,R : T } appears in the Kuznetsov trace formula and should not be confused with the test function f in the definition of equidistributed sequences. Theorem 5.3 (Goldfeld-Kontorovich orthogonality relation). Assume the Ramanuan conecture at the infinite place, i.e., ν () (ir) 2 for all. For four positive integers m, m 2, n, n 2, and T, we have A (m, m 2 )A (n, n 2 ) h T,R(ν () ) L O R,ϵ (T 3+3R+ϵ (m m 2 n n 2 ) 2 ), ) 2 h T,R (ν () ) L + O R,ϵ (T 3+3R+ϵ (m m 2 n n 2 ) 2 ), if m n m 2 n 2, For this family of test functions {h T,R : T } we have the Weyl law for some constant c > 0. h T,R (ν () ) L ct 5+3R. otherwise.

27 CHAPTER 5. MAIN THEOREM 20 Proof. See [Goldfeld-Kontorovich, 203]. Remark 5.4. A recent paper [Blomer] also establishes a similar orthogonality relation between Fourier coefficients of Hecke-Maass forms for SL(3, Z). It may also be used to prove a similar version of our main theorem. Theorem 5.5 (Main theorem). For any continuous test function f : C C, we have lim T f(a (p, )) h T,R(ν () ) L h T,R (ν () ) L C f dµ (3). Remark 5.6. We can interpret our main theorem in the context of the Sato-Tate conecture and equidistribution. The Sato-Tate conecture for GL(3) states that if ϕ is not a symmetric square lift from GL(2), the sequence A (2, ), A (3, ), A (5, ),..., A (p, ),..., is equidistributed with respect to the Sato-Tate measure µ (3) for a fixed Hecke-Maass form ϕ. Considering the same problem in vertical perspective, we can fix the prime number p and vary ϕ. One can investigate the distribution of the sequence A (p, ), A 2 (p, ), A 3 (p, ),... (5.) as in Theorem 2.0 and Theorem 2.. But no such theorem has been proved yet. We conecture that this sequence should be equidistributed with respect to a properly defined measure on C which depends on p. Our main theorem is a weighted version of this conecture and essentially claims that Sequence 5. is equidistributed with respect to the Sato-Tate measure µ (3), if each A (p, ) is given a weight L Res s L(s, ϕ ϕ ) Remark 5.7. We shall note that the Sato-Tate measure µ (3) curve {2e iθ + e 2iθ : θ R} is supported inside the delta-shaped and this is equivalent to the Ramanuan conecture at a finite prime p since A (p, ) A (, p).

28 CHAPTER 6. HECKE RELATIONS AND RANDOM WALKS IN THE FIRST QUADRANT2 Chapter 6 Hecke Relations and Random Walks in the First Quadrant Let p be a fixed prime number. Let A(m, m 2 ) be the (m, m 2 ) th Fourier coefficient of a Maass form ϕ as in Theorem 3.5. This chapter will heavily rely on the following three Hecke relations from Theorem 3.7: and A(n, )A(m, m 2 ) A(, n)a(m, m 2 ) b m,a m 2 abcn a m,b m 2 abcn ( m c A b, m ) 2b, (6.) a ( m b A a, m ) 2c b (6.2) A(m, m 2 ) A(m 2, m ). (6.3) We intend to use the above three relations to express A(p, ) k A(p, ) k 2 as a linear combination of the A(p i, p i 2 ) s. For example, by an easy application of the above relations, we obtain the following identities: A(p, ) 2 A(p 2, ) + A(, p), A(p, )A(p, ) A(p, )A(, p) A(, ) + A(p, p), A(p, ) 2 A(, p) 2 A(, p 2 ) + A(p, ), A(p, ) 3 A(p, )(A(p 2, ) + A(, p)) A(, ) + 2A(p, p) + A(p 3, ),

29 CHAPTER 6. HECKE RELATIONS AND RANDOM WALKS IN THE FIRST QUADRANT22 A(p, ) 2 A(p, ) A(p, )(A(, ) + A(p, p)) 2A(p, ) + A(, p 2 ) + A(p 2, p), A(p, )A(p, ) 2 2A(, p) + A(p 2, ) + A(p, p 2 ), A(p, ) 3 A(, ) + 2A(p, p) + A(, p 3 ), A(p, ) 4 A(p, )(A(, ) + 2A(p, p) + A(p 3, )) 3A(p, ) + 2A(, p 2 ) + 3A(p 2, p) + A(p 4, ). The algorithm to express A(p, ) k A(p, ) k 2 as a linear combination of A(p i, p i 2 ) s is inductive. Assume that we have expressed A(p, ) k A(p, ) k 2 as a linear combination of A(p i, p i 2 ) s, i.e., A(p, ) k A(p, ) k 2, 2 b, 2 A(p, p 2 ) for some b, 2 C. Multiply A(p, ) on both sides and we get A(p, ) k A(p, ) k 2, 2 b, 2 A(p, )A(p, p 2 ). For each A(p, )A(p, p 2 ) on the right hand side, we apply Equation 6. and it can be expressed as a sum of A(p i, p i 2 ) s. Similarly assume that we have expressed A(p, ) k A(p, ) k2 as a linear combination of A(p i, p i 2 ) s, i.e., A(p, ) k A(p, ) k2 b, 2 A(p, p 2 ), 2 for some b, 2 C. Multiply A(, p) A(p, ) (Equation 6.3) on both sides and we get A(p, ) k A(p, ) k 2, 2 b, 2 A(, p)a(p, p 2 ). For each A(, p)a(p, p 2 ) on the right hand side, we apply Equation 6.2 and it can be expressed as a sum of A(p i, p i 2 ) s. This inductive algorithm allows us to express the coefficient before A(p i, p i 2 ) as the number of random walks of certain type in the integer points of the first quadrant. Definition 6.. We define a random walk in the set C (3) {(i, i 2 ) Z 2 : i 0, i 2 0}, which is the first quadrant of Z 2 and the allowable steps are from

30 CHAPTER 6. HECKE RELATIONS AND RANDOM WALKS IN THE FIRST QUADRANT23 where e <, 0 > and e 2 < 0, >. S (3) {e, e 2 e, e 2 }, Remark 6.2. We consider random walks in C (3) with allowable steps of S (3). Because of the restriction of C (3), any walk at the the boundary of C (3) can take only one or two allowable steps from S (3) as its next step. More specifically, a walk at the origin (0, 0) can only take e as its next step; a walk at a point of {(i, i 2 ) Z 2 : i > 0, i 2 0} can only take e or e 2 e as its next step; a walk at a point of {(i, i 2 ) Z 2 : i 0, i 2 > 0} can only take e 2 or e 2 e as its next step. Definition 6.3. Let λ C (3). Let q (k) λ steps which stay in C (3). be the number of walks from the origin (0, 0) to λ of k Lemma 6.4. We have q (k+) λ q (k) λ+e 2 + q (k) λ e + q (k) λ+e e 2, if λ {(i, i 2 ) C (3) : i > 0, i 2 > 0}, q (k) λ+e 2 + q (k) λ e, if λ {(i, i 2 ) C (3) : i > 0, i 2 0}, q (k) λ+e 2 + q (k) λ+e e 2, if λ {(i, i 2 ) C (3) : i 0, i 2 > 0}, q (k) λ+e 2, if λ (0, 0). Proof. Any walk of (k + ) steps from (0, 0) to λ must be at either λ + e 2, λ e or λ + e e 2 after completing the k th steps, if they are in C (3). Thus, the number of walks of (k + ) steps from (0, 0) to λ is the sum of the numbers of walks of k steps from (0, 0) to λ + e 2, λ e and λ + e e 2, if any of them is in C (3). Theorem 6.5. We can express A(p, ) k as a sum of A(p i, p i 2 ) s and the coefficient before A(p i, p i 2 ) is q (k) (i,i 2 ), i.e., A(p, ) k i,i 2 0 q (k) (i,i 2 ) A(pi, p i 2 ). Proof. We shall note that the right hand side is a finite sum since for all except a finite number of (i, i 2 ) s such that q (k) (i,i 2 ) 0. For k 0 and, this theorem is obvious. Assume this theorem is true for some k > 0, i.e., A(p, ) k i,i 2 0 q (k) (i,i 2 ) A(pi, p i 2 ).

31 CHAPTER 6. HECKE RELATIONS AND RANDOM WALKS IN THE FIRST QUADRANT24 Multiply A(p, ) on both sides and we have A(p, ) k+ i,i 2 0 i,i 2 >0 + q (k) (i,i 2 ) A(p, )A(pi, p i 2 ) q (k) (i,i 2 ) (A(pi +, p i 2 ) + A(p i, p i 2+ ) + A(p i, p i 2 )) i 0,i 2 >0 + i >0,i 2 0 i,i 2 0 q (k) (i,i 2 ) (A(pi +, p i 2 ) + A(p i, p i 2 )) q (k) (i,i 2 ) (A(pi +, p i 2 ) + A(p i, p i2+ )) + q (k) (0,0) A(p, ) q (k+) (i,i 2 ) A(pi, p i 2 ). The last equality follows from Lemma 6.4. By induction, we complete the proof. Definition 6.6. We define another set of allowable steps S (3) 2 {e 2, e e 2, e }. Remark 6.7. We shall now consider a random walk of (k + k 2 ) steps, where the first k steps are from S (3) and the remaining k 2 steps are from S (3) 2. Definition 6.8. Let λ C (3) and let q (k,k 2 ) λ be the number of walks from the origin (0, 0) to λ confined in C (3) of (k + k 2 ) steps in which the first k steps are from S (3) and the remaining k 2 steps are from S (3) 2. Lemma 6.9. We have q (k,k 2 +) λ q (k,k 2 ) λ+e + q (k,k 2 ) λ e 2 + q (k,k 2 ) λ+e 2 e, if λ {(i, i 2 ) C (3) : i > 0, i 2 > 0}, q (k,k 2 ) λ+e + q (k,k 2 ) λ+e 2 e, if λ {(i, i 2 ) C (3) : i > 0, i 2 0}, q (k,k 2 ) λ+e + q (k,k 2 ) λ e 2, if λ {(i, i 2 ) C (3) : i 0, i 2 > 0}, q (k,k 2 ) λ+e, if λ (0, 0). Proof. This proof is essentially the same of that of Lemma 6.4. Theorem 6.0. We can express A(p, ) k A(p, ) k 2 before A(p i, p i 2 ) is q (k,k 2 ) (i,i 2 ), i.e., A(p, ) k A(p, ) k 2 i,i 2 0 as a sum of A(p i, p i 2 ) s and the coefficient q (k,k 2 ) (i,i 2 ) A(pi, p i 2 ).

32 CHAPTER 6. HECKE RELATIONS AND RANDOM WALKS IN THE FIRST QUADRANT25 Proof. In a manner similar to Theorem 6.5, this proof is based on induction. If k 2 0, this theorem is reduced to Theorem 6.5. Assume this theorem is valid for k 2 0, i.e., A(p, ) k A(p, ) k 2 i,i 2 0 q (k,k 2 ) (i,i 2 ) A(pi, p i 2 ). Multiply A(, p) A(p, ) (see Equation 6.3) on both sides of the above identity to obtain A(p, ) k A(p, ) k 2+ i,i 2 0 i,i 2 >0 + i0,i 2 >0 + i>0,i 2 0 i,i 2 0 q (k,k 2 ) (i,i 2 ) A(, p)a(pi, p i 2 ) q (k,k 2 ) (i,i 2 ) (A(pi, p i 2+ ) + A(p i +, p i 2 ) + A(p i, p i 2 )) q (k,k 2 ) (i,i 2 ) (A(pi, p i 2+ ) + A(p i +, p i 2 )) q (k,k 2 ) (i,i 2 ) (A(pi, p i 2+ ) + A(p i, p i 2 )) + q (k,k 2 ) (0,0) A(, p) q (k,k 2 +) (i,i 2 ) A(p i, p i 2 ). The last equality follows from Lemma 6.9. By induction, we complete the proof.

33 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 26 Chapter 7 Lie Theory and Random Walks in a Weyl Chamber In this chapter we will summarize some basic concepts and results of Lie theory. Our main goal is to relate the following integral C z k z k 2 dµ (N) to some random walks in a Weyl chamber, via the decomposition of tensor power representations. A number of text books have treated the theory of finite-dimensional complex linear representations of Lie groups and Lie algebras, such as [Bump, 2004] and [Hall, 2003]. For random walks in a Weyl chamber, our main references are [Gessel-Zeilberger, 992] and [Grabiner-Magyar, 993]. Definition 7.. Let (π, V ) be a representation of a Lie group G, where V is finite-dimensional complex vector space and π : G GL(V ) is a group homomorphism. By abuse of language, we will also refer to (π, V ) as V, on many occasions. In our context a representation is always finite-dimensional and linear over C. Definition 7.2. We define the character χ : G C of a representation (π, V ) by taking χ(g) to be the trace of the linear map π(g) : V V. Proposition 7.3. Let G be a compact Lie group and dg its unique normalized Haar measure such that G dg. If (π, V ) is a representation of G and χ its character, then we have ( χ(g) dg dim C V G ) G

34 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 27 where V G {v V : g v v for any g G}. Proof. This is Proposition 2.8 of [Bump, 2004]. Remark 7.4. We have χ(g) dg G is the multiplicity of the trivial representation in the decomposition of π into a direct sum of irreducible representations. Definition 7.5. Let ι : SU(N) GL(N, C) be the standard inclusion map. This inclusion map defines a representation (ι, H), where H C N is an N-dimensional complex vector space. The Lie group SU(N) acts on H by matrix multiplication on the left if we assume H C N as column vectors. Let (ι, H) be the contragredient representation of (ι, H). Theorem 7.6. We have C z k z k 2 dµ (N) dim C ((H k H k 2 ) G), which is the multiplicity of the trivial representation in H k H k 2. Proof. Obviously the trace of the representation (ι, H) is Tr in Definition 4.2. For two representations V and V 2 with characters χ and χ 2, the character of the tensor product representation V V 2 is χ χ 2. For a representation V with character χ, the character of its contragredient representation is χ. Inductively, we prove that the character of the representation H k H k 2 Tr k Tr k 2. Let ω be the unique normalized Haar measure on SU(N). Recalling Definition 4.2, we have C z k z k 2 dµ (N) Tr k Tr k 2 dω, SU(N) which is the change of variable formula in Definition 4.. By Proposition 7.3, we complete the proof. Theorem 7.7. There is a one-to-one correspondence between the finite-dimensional complex representations of SU(N) and the finite-dimensional complex representations of the Lie algebra sl(n, C). This correspondence keeps irreducibility. is

35 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 28 Proof. Since SU(N) is simply connected, the finite-dimensional representations of SU(N) are in one-to-one correspondence with the finite-dimensional representations of the Lie algebra su(n). The complex representations of su(n) are in one-to-one correspondence with the complex representations of the complexified Lie algebra su(n) C sl(n, C). (This proof is modified from p.27 of [Hall, 2003].) Definition 7.8. Let (π, V ) be a representation of SU(N). We use the same symbol V or (π, V ) for the corresponding sl(n, C) representation in Theorem 7.7. Therefore we will study representations of the Lie algebra sl(n, C) instead of representations of the Lie group SU(N). 7. Basic Lie Algebra Definition 7.9. Let g be a semi-simple complex Lie algebra. Let [, ] : g g g be its Lie bracket. Let h be a Cartan subalgebra of g. A weight of g is a complex linear map from h to C. Let h be the set of all weights of g. Let 0 be the zero weight of h. Definition 7.0. For λ h, we define g λ {a g : [h, a] λ(h)a for all h h}. Definition 7.. We call a non-zero λ h a root if g λ is not empty. One can show that g λ is one-dimensional for λ is a root. Let Φ be the set of all roots. Definition 7.2. Let (, ) : h h C be the inner product on h which is inherited from the Killing form on g. Definition 7.3. Let h 0 be the real subspace of h spanned by all the roots in Φ. The Euclidean space h 0, along with its inner product (, ) and the roots Φ, forms the root system of the Lie algebra g. Definition 7.4. Let Λ be the set of integral weights Λ { ( ) } µ h 2λ 0 : µ, Z for all λ Φ. (λ, λ)

36 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 29 Definition 7.5. Let σ λ be the reflection through the hyperplane orthogonal to a root λ, i.e., for µ h 0. (µ, λ) σ λ (µ) µ 2 (λ, λ) λ Definition 7.6. Let W be the Weyl group of this root system and W is generated by σ λ (λ Φ). Definition 7.7. Assume we have picked up a set of positive roots Φ + Φ. Let ρ 2 λ Φ + λ be the half sum of all positive roots. Assume we have picked up a set of simple positive roots with respect to the chosen set of positive roots Φ +. Every positive root can be written as a sum of simple roots. Definition 7.8. Let E be the free Z-module on the set of symbols {e λ : λ Λ}. It consists of all the formal sums n λ e λ λ Λ with n λ Z such that n λ 0 for all but finitely many λ. It is a ring with the multiplication ( ) n λ e λ m µ e µ n λ m µ e ν. µ Λ ν Λ λ Λ λ+µν Definition 7.9. The Weyl group W acts on Λ and E. Let E W be the invariant elements of E under the action of W. Definition The length of a Weyl group element is the length of the shortest word representing that element in terms of the standard generators σ λ for λ Φ. Let sgn(w) ( ) length of w be the sign of w W. Definition 7.2. For each finite-dimensional representation V of the Lie algebra g, we define ch(v ) λ Λ dim(v λ )e λ E, in which V λ {v V : h v λ(h)v, for all h h}.

37 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 30 Lemma Let V and V 2 be two representations of g. We have ch(v V 2 ) ch(v )ch(v 2 ) and ch(v V 2 ) ch(v ) + ch(v 2 ). Proof. See [Bump, 2004] or [Hall, 2003]. Definition If x E and λ Λ, we define to be the coefficient of e λ in x. Namely, if x ν x e λ n ν e ν, we have x n λ. e λ Definition Let C {µ h 0 : (µ, λ) 0 for all λ Φ+ } be the Weyl chamber defined by the positive roots of Φ +. Let C {µ h 0 : (µ, λ) > 0 for all λ Φ+ } be the interior of that Weyl chamber. Definition The hyperplane {λ h 0 : (λ, α) 0} orthogonal to a root α Φ is called a Weyl chamber wall if {λ : (λ, α) 0} C. Theorem 7.26 (Weyl character formula). Let λ be a weight in Λ C. Let V λ be the (irreducible) highest weight representation of λ. We have the following formula sgn(w)e w(ρ+λ) ch(v λ ) Proof. This is Chapter 25 of [Bump, 2004]. w W w W sgn(w)e w(ρ). Theorem Every finite-dimensional representation V is a direct sum of finitely many V µ s, where V µ is the highest weight representation of the weight µ Λ C. Namely, for a finitedimensional representation V, we have V V m i µ i µ i Λ C where m i is a non-negative integer and m i 0 for all but finitely i. The number m i is the multiplicity of V µi in V.

38 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 3 Proof. This follows from the assumption that g is semi-simple. The following lemma is taken from [Grabiner-Magyar, 993]. Lemma The multiplicity m i of V µi in Theorem 7.27 can be obtained by m i sgn(w)ch(v ). w W e µ i +ρ w(ρ) Proof. By Lemma 7.22, we have Multiply w W ch(v ) sgn(w)e w(ρ) on both sides, we get sgn(w)e w(ρ) ch(v ) w W m ch(v µ ). m sgn(w)e w(ρ) ch(v µ ). (7.) w W By Theorem 7.26, we have sgn(w)e w(ρ) ch(v µ ) sgn(w)e w(ρ+µ ). w W w W And we know that for µ i Λ C w W sgn(w)e w(ρ+µ ), if µ i µ, e ρ+µ i 0, if µ i µ. Thus, taking e ρ+µ i on both sides of Equation 7., we get m i sgn(w)ch(v ). e µ i +ρ w(ρ) w W 7.2 Reflection Method w W sgn(w) ( e w(ρ) ch(v ) ) e ρ+µ i D. André developed the reflection method (a.k.a. the reflection principle) to solve Bertrand s ballot problem. The same method is applied to random walk (of discrete time) and Brownian motion (of continuous time). The reflection method is generalized to random walks in a Weyl chamber in [Gessel-Zeilberger, 992]. André s reflection on the straight line is replaced with the Weyl group action. This is an

39 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 32 elegant method, which covers both classical ballot problem and the hook-length formula for Young tableau. We are going to present the theorem of Gessel and Zeilberger in detail in our context. We are not going to apply their theorem directly but we will slightly modify their theorem and its proof for our purpose. Definition For i and 2, let Σ i be a finite subset of Λ which is invariant under the action of the Weyl group W. We will consider random walks in the lattices Λ and Λ C. We shall note that the latter is the set of integral weights that lie strictly inside the positive Weyl chamber and it excludes all weights on the Weyl chamber walls. Each allowable step is taken either in Σ or in Σ 2. Definition If λ, ν Λ, let WALK (k,k 2 )(λ ν) be the number of walks of (k +k 2 ) steps from λ to ν with the first k steps in Σ and the remaining k 2 steps in Σ 2. Definition 7.3. If λ, ν Λ C, let WALK W (k,k 2 )(λ ν) be the number of walks of (k + k 2 ) steps from λ to ν that always stay strictly in Λ C, with the first k steps in Σ and the remaining k 2 steps in Σ 2. Lemma We have WALK (k,k 2 )(λ ν) σ Σ e σ k σ Σ e σ k 2 e λ+ν. Proof. Obviously we have WALK (k,k 2 )(λ ν) WALK (k,k 2 )(0 ν λ). The polynomials ( ) ( ) e σ and e σ are the generating functions for the random walks in WALK (k,k 2 )(0 σ Σ σ Σ ν λ). Each term in σ Σ e σ σ Σ e σ } {{ } k σ Σ 2 e σ σ Σ 2 e σ } {{ } k 2

40 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 33 corresponds to a walk with (k + k 2 ) steps with the first k steps in Σ and the remaining k 2 steps in Σ 2. The following definition of reflectable random walks is key to the success of the reflection method. It is due to [Gessel-Zeilberger, 992]. Definition We say a random walk starting from λ is reflectable if for each simple root α i, there is a real number k i, such that (α i, σ) ±k i or 0 for all allowable steps σ and (α i, λ) is an integer multiple of k i. Remark The previous definition guarantees that a reflectable random walk cannot leave the Weyl chamber C from its interior C before landing on a Weyl chamber wall at some step. We will only consider reflectable random walks, i.e., the set of allowable steps and the start point are properly chosen to satisfy the conditions in the previous definition. Because we have two sets of allowable steps, we need to specify the meaning of being reflectable. We require that for each simple root α i, there is a real number k i, such that (α i, σ) ±k i or 0 for all steps σ Σ Σ 2 and (α i, λ) is an integer multiple of k i, where the random walk starts from λ. Theorem 7.35 (Gessel-Zeilberger). Let λ, ν Λ C and λ, Σ and Σ 2 are properly chosen so that Remark 7.34 is satisfied. We have the equality WALK W (k,k 2 ) (λ ν) w W sgn(w)walk (k,k 2 )(w(λ) ν). Proof. The following proof is due to [Gessel-Zeilberger, 992] but modified to our case. Let WALK (k,k 2 ) ( ) be the number of walks which appear in WALK (k,k 2 )( ) and touch at least one wall of the Weyl chamber C. Obviously we have WALK (k,k 2 ) (λ ν) + WALKW (k,k 2 ) (λ ν) WALK (k,k 2 )(λ ν), and for w not the identity in W. We claim that WALK (k,k 2 ) (w(λ) ν) WALK (k,k 2 )(w(λ) ν) w W sgn(w)walk (k,k 2 )(w(λ) ν) 0. (7.2)

41 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 34 We will prove that all these walks which appears on the left side of Equation 7.2 can be divided into cancelling pairs. We put the positive roots in Φ + into a total order. This total order can be chosen arbitrarily but it must be fixed. Let a walk from w(λ) to ν on the left side of Equation 7.2 be δ, δ 2, δ 3,..., δ k +k 2, where δ, δ 2,..., δ k Σ and δ k +, δ k +2,..., δ k +k 2 Σ 2. After the th step δ, the walk is at w(λ) + δ. Because ν is in Λ C, we can find r the largest number such that after the r th step i δ r, the walk is on a Weyl chamber wall. After the (r + ) th step, it stays strictly in Λ C and will never leave Λ C again. After the r th step, the walk is on a Weyl chamber wall but it may be at the intersection of several Weyl chamber walls. Among the positive roots corresponding to these walls, we pick up the largest positive root α i according to the total order we have assigned for Φ +. We pair the above walk with the random walk from σ αi w(λ) to ν with steps σ αi (δ ), σ αi (δ 2 ), σ αi (δ 3 ),..., σ αi (δ r ), δ r+, δ r+2,..., δ k +k 2, where σ αi is the reflection through the wall orthogonal to α i. The signs of these two walks are sgn(w) and sgn(σ αi w) respectively and they cancel each other. 7.3 Special Linear Lie Algebra sl(n, C) We need to apply the results of the previous sections to the special case of sl(n, C). Let N be an integer greater than. The special linear Lie algebra of order N, or sl(n, C), is the Lie algebra of N N matrices of zero trace, with the Lie bracket [X, Y ] : XY Y X. It is a semi-simple complex Lie algebra. A Cartan subalgebra h of sl(n, C) is that of diagonal matrices of zero trace. It is spanned by E i,i E i+,i+ for i, 2,..., N, where E i, is the matrix with at the (i, ) th entry and 0 elsewhere. The Lie algebra sl(n, C) has the root system of type A N. We can identify h 0 with { } N (x, x 2,..., x N ) R N : x i 0. Let ϵ i be the vector in R N with at the i th entry and 0 elsewhere. We have the set of roots Φ {ϵ i ϵ : i } with the root ϵ i ϵ corresponding to E i, sl(n, C). Let the set of positive roots be i

42 CHAPTER 7. LIE THEORY AND RANDOM WALKS IN A WEYL CHAMBER 35 Φ + {ϵ i ϵ : i < }. We pick up ϵ i ϵ i+ for i, 2,..., N as the simple roots for Φ +. The half N (N 2i+)ϵ i i sum of positive roots is ρ 2. The inner product for this root system is the restriction { } N of the standard scalar product of R N to the subspace (x, x 2,..., x N ) R N : x i 0. Again } i we let 0 be the zero weight of h 0 {(x N, x 2,..., x N ) R N : x i 0. The representation (ι, H) of SU(N) in Definition 7.5 corresponds to the highest weight representation of the weight ϵ N N ϵ and hence we have i ch(h) N i e ϵ N i N ϵ. Since H is the contragredient of H, we have ch(h) Let Σ { ϵ i N N e i N ϵ N ϵ i. } { N ϵ : i, 2,..., N and Σ 2 } N ϵ ϵ i : i, 2,..., N. With this N pair of allowable steps Σ and Σ 2, we apply Theorem 7.35 to the case λ ν ρ. Hence we have WALK W (k,k 2 ) (ρ ρ) w W sgn(w)walk (k,k 2 )(w(ρ) ρ) ( ) sgn(w) ch(h) k ch(h) k 2 w W sgn(w)ch(h k H k 2 ) w W e ρ w(ρ) e 0+ρ w(ρ) the multiplicity of trivial representation in H k H k 2 z k z k 2 dµ (N). C The second to the last equality follows from Lemma 7.28 and we note that the highest weight representation of the zero weight 0 is the trivial representation. The main result that we will use from this chapter is the equality C z k z k 2 dµ (N) WALK W (k,k 2 )(ρ ρ). (7.3)