Subconvex bounds for Hecke-Maass forms on compact arithmetic quotients of semisimple Lie groups

Let $H$ be a semisimple algebraic group, $K$ a maximal compact subgroup of $G:=H(\mathbb{R})$, and $\Gamma\subset H(\mathbb{Q})$ a congruence arithmetic subgroup. In this paper, we generalize existing subconvex bounds for Hecke-Maass forms on the locally symmetric space $\Gamma \backslash G/K$ to corresponding bounds on the arithmetic quotient $\Gamma \backslash G$ for cocompact lattices using the spectral function of an elliptic operator. The bounds obtained extend known subconvex bounds for automorphic forms to non-trivial $K$-types, yielding subconvex bounds for new classes of automorphic representations, and constitute subconvex bounds for eigenfunctions on compact manifolds with both positive and negative sectional curvature. We also obtain new subconvex bounds for holomorphic modular forms in the weight aspect.


Introduction
Let M be a closed 1 Riemannian manifold M of dimension d and P 0 : C ∞ (M ) → L 2 (M ) an elliptic classical pseudodifferential operator on M of degree m, where C ∞ (M ) denotes the space of smooth functions on M and L 2 (M ) the space of square-integrable functions on M . Assume that P 0 is positive and symmetric. Denote its unique self-adjoint extension by P with the m-th Sobolev space as domain, and let {φ j } j≥0 be an orthonormal basis of L 2 (M ) consisting of eigenfunctions of P with eigenvalues {λ j } j≥0 repeated according to their multiplicity. By a classical result of Avacumovic, Levitan, and Hörmander [1,30,22] one has for any j ∈ N the convex bound 2 If the φ j are eigenfunctions of a larger family of commuting differential operators on M containing P 0 , this bound can be improved. Thus, assume that M carries an isometric action of a compact Lie group K such that all orbits have the same dimension κ ≤ d − 1. Denote by K the set of equivalence classes of irreducible unitary representations of K, which can be identified with the set of irreducible characters of K. Suppose further that P commutes with the family of differential operators generated Date: September 8, 2018. 1 By a closed manifold we shall understand a compact boundaryless manifold. 2 Here and in what follows we shall write a ≪γ b for two real numbers a and b and a variable γ, if there exists a constant Cγ > 0 depending only on γ such that |a| ≤ Cγ b. If there are no relevant variables involved, we shall simply write a ≪ b.
by the action of K, so that the eigenfunctions φ j can be chosen to be compatible with the Peter-Weyl decomposition of L 2 (M ) into σ-isotypic components L 2 σ (M ), where σ ∈ K. It was then shown in [40,41] that the equivariant convex bound holds, where d σ denotes the dimension of a representation of class σ, and D u is a differential operator of order u on K. If K = T is a torus, one actually has the almost sharp estimate where W λ denotes the subset of K-types occuring in the Peter-Weyl decomposition of L 2 (M ) that grow at most with rate λ 1/m / log λ. 3 The bounds (1.1) and (1.2) are known to be sharp in the eigenvalue aspect on the standard d-sphere, but if the considered eigenfunctions are joint eigenfunctions of an even larger family of commuting operators, they can be improved. Thus, let G be a semisimple real Lie group, K a maximal compact subgroup of G, Γ ⊂ G a lattice, and Y := Γ\G/K the corresponding locally symmetric space of dimension d and rank r. If {ψ j } j≥0 constitutes an orthonormal basis in L 2 (Y ) of simultaneous eigenfunctions of the full ring of invariant differential operators on Y , which is isomorphic to a finitely generated polynomial ring in r variables and contains the Beltrami-Laplace operator ∆, Sarnak [43] was able to show the spherical convex bound j for arbitrary compacta Ω ⊂ Y , λ j being the Beltrami-Laplace eigenvalue of ψ j . From an arithmetic point of view, there is still an additional family of commuting operators on Y given by the Hecke operators, and in the case G = SL(2, R) and K = SO(2), Iwaniec and Sarnak [27] were able to strengthen the bound (1.4) for certain compact locally symmetric spaces Y = Γ\H of rank r = 1, given as quotients of the complex upper half plane H ≃ G/K by suitable congruence arithmetic lattices Γ, and proved for any ε > 0 and j ∈ N the substantially stronger spherical subconvex bound (1.5) ψ j ∞ ≪ ε λ 5 24 +ε j , provided that the ψ j are also eigenfunctions of the ring of Hecke operators on L 2 (Γ\H). More generally, if H is a semisimple algebraic group over Q satisfying certain conditions, Γ ⊂ H(Q) an arithmetic congruence lattice, and G = H(R), Marshall [33] was able to strengthen the bound (1.4) and prove spherical subconvex bounds of the form for some δ > 0 and arbitrary compacta Ω ⊂ Y , if the ψ j are also eigenfunctions of the ring of Hecke operators on L 2 (Y ), generalizing previous work of Blomer-Maga [3,4] and Blomer-Pohl [7], among others. In fact, for negatively curved manifolds, much better bounds are expected to hold generically, the bound (1.5) being the strongest known bound up to now. The estimates (1.4)-(1.6) represent bounds for automorphic forms on G which are right K-invariant, and for this reason are called spherical.
In this paper, left Γ-invariant functions on G which are simultaneous eigenfunctions of an invariant elliptic differential operator and some module of Hecke operators will be called Hecke-Maass forms of rank 1. This class encompasses the usual concept of an automorphic form on G, and coincides with it in the rank 1 case, compare Section 5.3 and 7.4. Nevertheless, note that a Hecke-Maass form of rank 1 is not necessarily an eigenfunction of the full ring of invariant differential operators, since one can choose a very small submodule of the ring of Hecke operators, see Remark 7.3 for details. The goal of this paper is to extend the spherical subconvex bounds (1.5) and (1.6) to non-spherical situations, that 3 The estimate is almost sharp in the sense that as a consequence of the equivariant Weyl law T -types in L 2 (M ) can grow at most with rate λ 1/m j , see [41].
is, to non-trivial K-types in the Peter-Weyl decomposition of L 2 (Γ\G) for a large class of compact arithmetic quotients Γ\G, sharpening the bounds (1.1) and (1.2) in case that the eigenfunctions φ j are Hecke-Maass forms.
As our first main result, we extend the bound (1.5) to automorphic forms on G of arbitrary Ktype and Nebentypus character. Thus, let R be an Eichler order in an indefinite division quaternion algebra A over Q. Denote by N (x) the reduced norm of an element x ∈ A, and write R(m) := {α ∈ R | N (α) = m} for any m ∈ N * . Choose an embedding θ : ⊔ ∞ m=1 R(m) → G, and set Γ := θ(R(1)). Then Γ constitutes a congruence arithmetic subgroup, and Γ\H ≃ Γ\G/K becomes a compact hyperbolic surface. Now, let χ be a Nebentypus character on Γ, and denote by L 2 χ (Γ\G) the Hilbert space of measurable functions on G such that The space L 2 χ (Γ\G) can be regarded as a closed subspace in L 2 (Γ χ \G), where Γ χ := ker χ. Identifying R(n) with its image θ(R(n)) for each n prime to a fixed natural number which depends only on R, the finite cosets Γ\R(n) give rise to Hecke operators on L 2 χ (Γ\G). Now, with the identification K ≃ S 1 ≃ [0, 2π), any K-type σ l ∈ K can be realized as the character σ l (θ) = e ilθ , θ ∈ [0, 2π), l ∈ Z, and we denote by L 2 σ l ,χ (Γ\G) the σ l -isotypic component of L 2 χ (Γ\G). It is then shown in Theorem 5.5 that for any orthonormal basis {φ j } j≥0 of L 2 (Γ χ \G) consisting of Hecke-Maass forms (of rank 1) with Beltrami-Laplace eigenvalues 0 ≤ λ 0 ≤ λ 1 ≤ λ 2 ≤ · · · and compatible with the Peter-Weyl decomposition one has the hybrid subconvex bound , for arbitrary small ε > 0 in the eigenvalue and isotypic aspect. This bound is the first sharpening the bound (1.3) for arbitrary K-types. If σ l and χ are trivial, one recovers the spherical subconvex bound (1.5). Note that (1.7) is a subconvex bounds on a manifold which does have both positive and negative sectional curvature. It is stated from the perspective of elliptic operator theory, which is the natural one in our approach, while in the theory of automorphic forms it is more common to work within a representation-theoretic framework, and use the Casimir operator C of G instead of the Beltrami-Laplace operator ∆, the former being no longer elliptic. But since on L 2 σ l ,χ (Γ\G) the operators in question are related according to ∆ = −C + l 2 4 id , the bound (1.7) can be rephrased accordingly. Thus, for any Hecke eigenform φ ∈ L 2 σ l ,χ (Γ\G) satisfying φ L 2 = 1 and Cφ = s 2 −1 8 φ one has the hybrid subconvex bound φ ∞ ≪ ε (1 − s 2 + 2l 2 ) 5 24 +ε , see Theorem 5.8. In this way, we obtain subconvex bounds for new classes of automorphic representations, in particular for the discrete series D s and their limits D ±,0 , as well as the principal series H(1, s), compare Section 5.3. Let us note that for fixed s we obtain the bound φ j ∞ ≪ ε (1 + |l|) 5 12 +ε for any φ j ∈ L 2 σ l ,χ (Γ\G). This agrees with results of Venkatesh [50, p. 993], though by work of Reznikov [42,Theorem 1.5] one has in this case the much better bound φ j ∞ ≪ ε (1 + |l|) in the weight aspect. The best previously known subconvex bound, proved by Das and Sengupta [12], had the exponent 1 2 − 1 33 = 31 66 . In an analogous way, we are able to derive equivariant and non-equivariant subconvex bounds for G = SU(2), K = SO(2), and Γ := {±1} in the setting of [31,32] by identifying G with the group of units in the quaternion algebra over R, and defining corresponding Hecke operators T n on L 2 (Γ\G). Thus, we obtain again in Theorem 6.1 the equivariant subconvex bound (1.7) for any simultaneous eigenfunction φ j ∈ L 2 σ l (Γ\G) of the Beltrami-Laplace operator ∆ on G with eigenvalue λ j and the T n , where now Γ\G ∼ = SO(3). This generalizes a result of VanderKam [49,Theorem 1.1], where the case l = 0 with L 2 σ0 (Γ\G) ∼ = L 2 (Γ\G/K) ∼ = L 2 (S 2 ) is treated, S 2 being the 2-sphere. Our second main result concerns bounds of the form (1.6). As before, let H be a semisimple algebraic group over Q which is assumed to be connected in the sense of Zariski. Write A fin for the finite adele ring of Q and A := R × A fin for the adele ring. Choosing an open compact subgroup K 0 in H(A fin ), we obtain an arithmetic subgroup Γ := H(Q) ∩ (H(R)K 0 ) in the semisimple Lie group G = H(R). Assume that H(A) = H(Q)(H(R)K 0 ) and that H(Q)\H(A) is compact, so that Γ\G is also compact. 4 From the point of view of automorphic representations, one has a suitable family of Hecke operators on L 2 (Γ\G), which is given by unramified Hecke algebras over Q p for infinitely many primes p [33]. Now, let K be a maximal compact subgroup of G and {φ j } j≥0 an orthonormal basis of L 2 (Γ\G) consisting of Hecke-Maass forms of rank 1 with respect to an elliptic left-invariant differential operator P 0 on Γ\G of order m which commutes with the right regular representation of K. Assume that P 0 is positive and symmetric, and that the cosphere bundle defined by its principal symbol is strictly convex. Then, assuming the condition (WS) made in [33], we show in Theorem 7.4 that there exists a constant δ > 0 independent of σ such that one has the equivariant subconvex bound where λ j denotes the spectral eigenvalue of φ j with respect to P 0 ; if K = T is a torus, one has the stronger estimate  6). An example would be given by H = SL(1, D), where D is any central division algebra of index n over Q, and G = SL(n, R). Furthermore, we show in Theorem 7.9 for some δ > 0 the weaker non-equivariant subconvex bound for an orthonormal basis of L 2 (Γ\G) consisting of suitable Hecke-Maass forms, sharpening the bound (1.1), but without assuming the condition (WS) of [33]. An example is again H = SL(1, D), where now D is any central division algebra over Q, except when G = SL(1, H). As before, (1.8) and (1.9) constitute first arithmetic subconvex bounds on a large class of manifolds which are both positively and negatively curved, and if P 0 is the Beltrami-Laplace operator, the bounds can be rephrased in terms of the eigenvalues of the Casimir operator of G. Indeed, by Theorem 7.12 we have for each φ j ∈ L 2 σ (Γ\G) with Casimir eigenvalue µ j the bound µ σ being the eigenvalue of the Casimir operator of K on σ. If K = T is a torus, Let us briefly say a few words about the methods employed. While in the theory of automorphic forms representation-theoretic tools prevail, our analysis is mainly based on the spectral theory of elliptic operators, and uses Fourier integral operator methods. Thus, let P be an elliptic pseudodifferential operator on a closed Riemannian manifold M as above. Our main tool is the spectral function e(x, y, µ) of the m-th root Q := m √ P of P given by In the spherical situations [27,3,4,7,33] examined before, a crucial role is played by asymptotics for spherical functions, see [27,Eq. (1.3)] and [33,Eq. (8)]. Since we cannot rely on them in our setting 5 , we consider instead the spectral expansion of e(x, y, µ) itself and the asymptotic behaviour of s µ (x, y) := e(x, y, µ + 1) − e(x, y, µ), which represents the Schwartz kernel of the spectral projection s µ onto the sum of eigenspaces of Q with eigenvalues in the interval (µ, µ + 1]. More precisely, if M carries an effective and isometric action of a compact Lie group K and σ ∈ K, denote by Π σ the projector onto the σ-isotypic component in the Peter-Weyl decomposition of L 2 (M ). In order to show the L ∞ -bounds (1.2), and analogous equivariant convex L p -bounds, an asymptotic formula for the Schwartz kernel of s µ • Π σ , or rather of s µ • Π σ , where s µ represents certain smooth approximation to s µ , was derived in [40, Corollary 2.2. and Theorem 3.3] in a neighbourhood of the diagonal relying on the theory of Fourier integral operators. Now, let G be a semisimple Lie group with finite center, Γ a discrete cocompact subgroup, and K a maximal compact subgroup of G. Let Γ denote the set consisting of characters of Γ of finite order. For χ ∈ Γ, introduce on L 2 (Γ χ \G) the Hecke operators T χ where β belongs to a certain set containing the commensurator C(Γ) of Γ. Based on the asymptotics for the kernel of s µ • Π σ mentioned above, we deduce in Proposition 4.1 for any small δ > 0 and some constant C > 0 the equivariant bound uniformly in x ∈ Γ χ \G for the Schwartz kernel of T χ ΓβΓ •s µ • Π σ , where we introduced the lattice point counting function given in terms of the distance function on the Riemannian symmetric space G/K. In case that K = T is a torus, a corresponding better estimate holds. From this, we obtain the subconvex bounds (1.7) and (1.8) by using known uniform upper bounds [27,33] for M (x, β, δ) combined with arithmetic amplification. The bound (1.9) is inferred by analogous methods. In both cases, it is crucial to control the caustic behaviour of the kernels ofs µ • Π σ ands µ near the diagonal as µ → +∞, respectively.
Let us close this introduction with some comments. There exist several variants of the bounds (1.5), beginning with [27,Appendix], where the non compact hyperbolic surface SL(2, Z)\H is considered. On the other hand, bounds in the level aspect are shown in [47] for compact locally symmetric spaces of arithmetic type, while bounds in the eigenvalue and level aspect are derived for the modular surfaces Γ 0 (N )\H in [2,48] and other papers. It is likely that our work can be extended to these settings, and we plan to deal with these questions in a future paper. Also, we intend to widen our results to Hecke-Maass forms of rank r, that is, simultaneous eigenfunctions of the Hecke operators and the full ring of invariant differential operators associated to the center of the universal envelopping algebra of the complexification of the Lie algebra of G. For such forms, the exponent −1/2m in (1.8) and (1.9) should be improvable by a factor r. Finally, we expect the factor d σ sup This paper is structured as follows. In Section 2, we introduce Hecke operators with character on semisimple Lie groups with finite center, in Section 3 we give a description of the asymptotic behaviour of spectral function of an elliptic operator by means of Fourier integral operators, and explain how convex bounds can be deduced from this in equivariant and non-equivariant situations. Based on these results, we derive spectral asymptotics for kernels of Hecke operators in Section 4. Relying on the latter, we finally prove subconvex bounds for arithmetic congruence lattices in SL(2, R), SO(3), and a large class of semisimple algebraic groups in Sections 5, 6, and 7, respectively. Throughout the paper, N := {0, 1, 2, 3, . . .} will denote the set of natural numbers, while N * := {1, 2, 3, . . .}.

Hecke operators with character on semisimple Lie groups
To introduce our setting, let G be a real semisimple Lie group with finite center and Lie algebra g. Denote by X, Y := tr (ad X • ad Y ) the Cartan-Killing form on g and by θ a Cartan involution of g. Let g = k ⊕ p be the Cartan decomposition of g into the eigenspaces of θ, corresponding to the eigenvalues +1 and −1 , respectively, and denote the maximal compact subgroup of G with Lie algebra k by K. Put X, Y θ := − X, θY . Then ·, · θ defines a left-invariant Riemannian metric on G, which in general will possess some strictly positive sectional curvature, compare Milnor [35, p. 298 and p. 317]. Dividing by the K-action, the quotient G/K becomes a Riemannian symmetric space of non-positive sectional curvature. With respect to the left-invariant metric on G, a distance function dist (g, h) is defined on each connected component of G as the geodesic distance between two points g, h in that component. Note that dist (g 1 g, g 1 h) = dist (g, h) for all g 1 ∈ G. In contrast to the Killing form, ·, · θ is no longer Ad (G)-invariant, but still Ad (K)-invariant, so that dist (gk, hk) = dist (g, h) for all k ∈ K.
Next, let X 1 , . . . X dim p be an orthonormal basis of p and Y 1 , . . . , Y dim k an orthonormal basis for k with respect to ·, · θ . If Ω and Ω K denote the Casimir elements of G and K, one has and we put Θ := −Ω + 2Ω K . Then dR(Θ) is the Beltrami-Laplace operator ∆ on G with respect to the left invariant metric defined by ·, · θ , while C := dR(Ω) represents the Casimir operator, R being the right regular representation of G on C ∞ (G), see [37,Section 3] and [8, Section 2.10]. Thus, all three operators commuting with each other.
We consider now a discrete cocompact subgroup Γ of G, together with the set Γ of its characters of finite order, and let χ ∈ Γ. Then Γ χ := ker χ is a subgroup of finite index in Γ and the quotient Γ χ \G a compact manifold without boundary. By requiring that the projection G → Γ χ \G is a Riemannian submersion, we obtain a Riemannian structure on Γ χ \G which locally has the same curvature than G. Furthermore, the Riemannian structure on G/K induces a Riemannian metric on Γ χ \G/K, becoming a locally symmetric space of negative curvature. In both cases, dist induces corresponding distances on Γ χ \G and Γ χ \G/K. Before we proceed, note that due to the compactness of Γ χ \G, the right regular representation of G on L 2 (Γ χ \G) decomposes into an orthogonal direct sum of countably many irreducible unitary representations with finite multiplicities, that is, where G denotes the unitary dual of G and m(π, Γ χ ) is a non-negative integer, see [18]. Furthermore, both the spectra of ∆ and C in L 2 (Γ χ \G) are discrete. Note that the eigenvalues of ∆ are positive, while the ones of C can be both negative and positive.
In what follows, we introduce Hecke operators on Γ\X, where X := G or G/K, following [21, Section 2], and consider the commensurator of Γ, where we say that two subgroups Γ 1 and Γ 2 are commensurable iff the indices [Γ 1 : Γ 1 ∩ Γ 2 ] and [Γ 2 : Γ 1 ∩ Γ 2 ] are finite. Let β ∈ C(Γ). Since the mapping is bijective, the double coset ΓβΓ is a finite union of right cosets of Γ, that is, there exist representative elements β 1 , β 2 , . . . , β t in ΓβΓ such that One can then associate to each double coset a linear operator T ΓβΓ on L 2 (Γ\X) by setting where β j · x ≡ Γβ j · Γx := Γβ j x depends on the choice of the representative x, but the sum does not depend on the choice of the representatives x and β j . Summing up, one writes 6 and calls T ΓβΓ a Hecke operator.
If a subset U of C(Γ) is decomposed into a finite disjoint union of double cosets of Γ, a linear operator T U can be defined in the same manner according to (2.4) T Γβ k Γ, β k ∈ C(Γ).
More generally, one can introduce Hecke operators as follows. Write H(Γ, C(Γ)) for the space of left and right Γ-invariant C-valued functions h on C(Γ) such that the support of h is included in a finite union of double Γ-cosets. Endowed with the convolution product H(Γ, C(Γ)) becomes an associative algebra over C with the characteristic function 1 Γ of Γ as unit element. For each h ∈ H(Γ, C(Γ)), a linear operator T h on L 2 (Γ\X) can then be defined by and one has T h1 * h2 = T h1 • T h2 . If U is as above and h is the characteristic function of U , then it is obvious that T h equals T U . We call H(Γ, C(Γ)) the Hecke algebra and refer the reader to [21,6 Here and above α ≡ Γα (resp. β j ≡ Γβ j ) and x ≡ Γx are considered both as right cosets of Γ and representatives in G, and the products αx (resp. β j x) are taken in G.
Example 2.1. One of the main examples we are having in mind is G ′ := GL(n, R) with where ω is a Dirichlet character on (Z/N Z) × .

The spectral function of an elliptic operator and convex bounds for eigenfunctions
The main tool underlying our analysis is the spectral function of an elliptic operator on a smooth manifold, which contains essential information on the spectrum. For large spectral parameters, an asymptotic description of it can be derived within the theory of Fourier integral operators, yielding in particular convex bounds for eigenfunctions. In what follows, we shall briefly recall the main arguments in non-equivariant and equivariant situations, and provide the results that will be needed later.
3.1. The spectral function and convex bounds for eigenfunctions. Let M be a closed Riemannian manifold of dimension d and P 0 an elliptic classical pseudodifferential operator on M of degree m, which is assumed to be positive and symmetric. Denote its unique self-adjoint extension by P , and let {φ j } j≥0 be an orthonormal basis of L 2 (M ) consisting of eigenfunctions of P with eigenvalues {λ j } j≥0 repeated according to their multiplicity. Let p(x, ξ) be the principal symbol of P 0 , which is strictly positive and homogeneous in ξ of degree m as a function on T * M \ {0}, that is, the cotangent bundle of M without the zero section. Here and in what follows (x, ξ) denotes an element in T * Y ≃ Y × R d with respect to the canonical trivialization of the cotangent bundle over a chart domain Y ⊂ M . Consider further the m-th root Q := m √ P of P given by the spectral theorem. It is well known that Q is a classical pseudodifferential operator of order 1 with principal symbol q(x, ξ) := m p(x, ξ) and the first Sobolev space as domain. Again, Q has discrete spectrum, and its eigenvalues are given by µ j := m λ j . The spectral function e(x, y, λ) of P can then be described by studying the spectral function of Q, which in terms of the basis {φ j } is given by and belongs to C ∞ (M × M ) as a function of x and y for any µ ∈ R. Let s µ be the spectral projection onto the sum of eigenspaces of Q with eigenvalues in the interval (µ, µ + 1], and denote its Schwartz kernel by s µ (x, y) := e(x, y, µ + 1) − e(x, y, µ).
To obtain an asymptotic description of the spectral function of Q, one first derives a description of s µ (x, y) by approximating s µ by Fourier integral operators. To do so, let ̺ ∈ S(R, R + ) be such that ̺(0) = 1 and supp̺ ∈ (−δ/2, δ/2) for an arbitrarily small δ > 0, and define the approximate spectral projection operator where E j denotes the orthogonal projection onto the subspace spanned by φ j . Clearly, constitutes the kernel of s µ . Now, notice that for µ, τ ∈ R one has where̺(t) denotes the Fourier transform of ̺, so that for u ∈ L 2 (M ) we obtain where U (t) stands for the one-parameter group of unitary operators in L 2 (M ) given by the Fourier transform of the spectral measure, {E Q µ } being a spectral resolution of Q. The central result of Hörmander [22] then says that U (t) : L 2 (M ) → L 2 (M ) can be approximated by Fourier integral operators. More where a ι ∈ S 0 phg is a classical polyhomogeneous symbol satisfying a ι (0,x, η) = 1 and ψ ι the defining phase function given as the solution of the Hamilton-Jacobi equation see [24,Page 254]. Let us remark that ψ ι is homogeneous in η of degree 1, so that Taylor expansion for small t gives . In other words, there exists a smooth function ζ ι which is homogeneous in η of degree 1 and satisfies where F ι ,F ι denote the multiplication operators with f ι andf ι , respectively. Then Hörmander showed that for small |t| Approximating in (3.3) the operator U (t) byŪ (t), one obtains a description for the kernel of s µ as the double oscillatory integral up to terms of order O(µ −∞ ) which are uniform in x and y, where ι,x (ω) denotes the quotient of Lebesgue measure in R d by Lebesgue measure in R with respect to ζ ι (t,x, ω). Furthermore, for sufficiently small δ > 0 one can assume that the R-integration is over a compact set, and R and t are close to 1 and 0, respectively. From (3.5), an asymptotic description can be inferred as µ → +∞ by means of the stationary phase principle. In fact, one has the following are strictly convex. 8 Then, for any fixed x, y ∈ M , andÑ = 0, 1, 2, 3, . . . one has the expansion The coefficients in the expansion and the remainder RÑ (x, y, µ) = O x,y (µ −Ñ ) term can be computed explicitly; if y = x, they are uniformly bounded in x and y, while if y = x, they satisfy the bounds where dist (x, y) denotes the geodesic distance between two points belonging to the same connected component. Otherwise, dist (x, y) := ∞.
Consequently, the dependence of the amplitude on µ does not interfer with the asymptotics, compare [14,Proposition 1.2.4]. Applying the stationary phase principle [23,Theorem 7.7.5] to the (R, t)integral in (3.5) with t(1 − R) as phase function then yields the assertion for x = y, the unique critical point being (R 0 , t 0 ) = (1, 0) in this case. Let us now assume that x = y. By assumption, the cospheres S * x M are strictly convex, so that for small |t|, the hypersurfaces Σ R,t ι,x will be strictly convex, too.
Applying [40,Lemma 3.5] to the integrals I ι (µ, R, t, x, y) with κι(x)−κι(y) κι(x)−κι(y) , ω as phase function and ν := µ κ ι (x) − κ ι (y) as asymptotic parameter yields for anyÑ ∈ N the expansion where the coefficients and the remainder are smooth in R and t, and satisfy the bounds Regarding the value of Φ ι,x,y on its critical set, one computes for is homogeneous of degree 1 in η, the gradient grad η ζ(t, κ ι (x), ω 0 ) only depends on the direction of ω 0 , and is therefore independent of R. From this and (3.5) we deduce for K sµ (x, y) as µ → +∞ the expansion . Again, we apply the stationary phase principle to the (R, t)-integrals, where now the phase function reads t(1 − R) + Φ ι,x,y (ω 0 ). The determinant of the matrix of its second derivatives is given by By choosing the charts Y ι sufficiently small so that κ ι (x)−κ ι (y) ≪ 1, we can therefore achieve that in a sufficiently small neighborhood of (R, t) = (1, 0), which is where the amplitude of the (R, t)-integral is supported, the phase function t(1 − R) + Φ ι,x,y (ω 0 ) has, if at all, only non-degenerate, hence isolated, critical points. If we now apply the stationary phase theorem, the assertion follows in the case x = y as well.
Remark 3.2. By Cauchy-Schwarz and the positivity of the test function ̺ we infer from the previous proposition forÑ = 0 that uniformly in x, y ∈ M . Also, note that the asymptotics in the proposition off the diagonal are only meaningful if dist (x, y) −1 is small with respect to µ.
As the previous proposition shows, the kernel ofs µ exhibits a caustic behaviour 9 in a neighbourhood of the diagonal since for x = y the integrals I ι (µ, R, t, x, y) no longer oscillate and are of order O(µ 0 ). From Proposition 3.1, similar asymptotics for s µ can be deduced. By looking at asymptotics on the diagonal, one obtains Weyl's law for the spectral function of Q and convex L ∞ -bounds for eigenfunctions, since s µ 2 L 2 →L ∞ ≡ sup x∈M s µ (x, x), yielding for any eigenfunction the convex bound ]. Nevertheless, in order to prove subconvex bounds, we shall also need asymptotics off the diagonal, so that the full caustic behaviour of Ks µ (x, y) near the diagonal becomes relevant. 9 For the terminology, see Appendix A in [40].

The reduced spectral function and equivariant convex bounds for eigenfunctions.
Keeping the notation of Section 3.1, assume now that M carries an effective and isometric action of a compact Lie group K, and consider the right regular representation π of K on L 2 (M ) with corresponding Peter-Weyl decomposition where K denotes the unitary dual of K and the orthogonal projector onto the σ-isotypic component, dk being Haar measure and d σ the dimension of an irreducible representation of K in the class σ ∈ K. Further, suppose that P commutes with π, and that the orthonormal basis {φ j } j≥0 is compatible with the decomposition (3.8) in the sense that each φ j lies in some L 2 σ (M ). Then every eigenspace of P is invariant under π, and decomposes into irreducible K-modules spanned by eigenfunctions. In order to study eigenfunctions of P of a certain K-type, one is interested in the spectral function of the operator also called the reduced spectral function, given by For this, one considers the composition s µ • Π σ , or rathers µ • Π σ , whose kernel has the spectral expansion Similarly to (3.5), it was shown in [40, Eq. (2.8)] that by approximating U (t) in (3.3) by the Fourier integral operatorŪ (t) one obtains a description for the kernel of s µ • Π σ as the double oscillatory integral x, y) dR dt (3.11) up to terms of order O(µ −∞ ) which are uniform in x and y, where y) being a Jacobian. Write O x := x · K for the K-orbit through x ∈ M . We then have the following The coefficients in the expansion and the remainder term can be computed explicitly; if y ∈ O x , they satisfy the bounds uniformly in x and y, where D u denote differential operators on K of order u, and if y / ∈ O x , the bounds T is a torus, let T ′ ⊂ T be the subset of representations occuring in the decomposition (3.8), and identify T with the set of integral linear forms on t. Then the remainder estimates can be improved to Remark 3.4.
(1) Proposition 3.3 implies forÑ = 0 by Cauchy-Schwarz that uniformly in x, y ∈ M and σ ∈ K, while takingÑ = 1 would yield an estimate of order If K = T is a torus, better remainder estimates hold. . To obtain an asymptotic expansion off the diagonal from (3.11), we shall first apply the stationary phase theorem to the integrals I σ ι (µ, R, t, x, y), and then to the ( [40,Theorem 3.3] implies for sufficiently small Y ι , fixed R, t ∈ R, and anyÑ ∈ N the asymptotic expansion where Crit Φ ι,x,y denotes the critical set of Φ ι,x,y , and The coefficients and the remainder term are given by distributions depending smoothly on R, t with support in Crit Φ ι,x,y and Σ R,t ι,x × K, respectively. Furthermore, they and their derivatives with respect to R, t satisfy for y ∈ O x the bounds uniformly in x and y. If K = T is a torus and σ ∈ V µ , the remainder estimates can be improved [41, Theorem 3.2] to contain only derivatives of σ up to order 2Ñ . Finally, denotes the constant value(s) of Φ ι,x,y on (the components of) its critical set, where (ω 0 , k 0 ) is some point in Crit Φ ι,x,y . If y ∈ O x one has Φ 0 ι,x,y (R, t) = 0. As already noted in the proof of Proposition 3.1, a ι is a polyhomogeneous symbol of order 0, so that the dependence of the amplitude on µ does not interfer with the asymptotics. Putting (3.11) and (3.12) together we obtain x, y, µ) dR dt up to terms of order O(µ −∞ ) uniform in x and y. We now apply the stationary phase principle [23, Theorem 7.7.5] to the (R, t)-integral. If y ∈ O x , the phase function simply reads t(1 − R), and the only critical point is (R 0 , t 0 ) = (1, 0), which is non-degenerate, the determinant of the Hessian being −1. Therefore, the necessary conditions for an application of the principle are fulfilled, yielding the assertion of the proposition in this case. In case that y ∈ O x , the phase function is given by t(1 − R) + Φ 0 ι,x,y (R, t), and the determinant of the matrix of its second derivatives is given by By choosing the charts Y ι sufficiently small so that κ ι (x) − κ ι (y · k −1 0 ) ≪ 1, we can achieve that in a sufficiently small neighborhood of (R, t) = (1, 0) the phase function t(1 − R) + Φ 0 ι,x,y (R, t) has, if at all, only non-degenerate, hence isolated, critical points. If we now apply the stationary phase theorem, the proposition follows.
From Proposition 3.3 equivariant convex bounds for eigenfunctions can be easily inferred. Indeed, recall that the test function ̺ ∈ S(R, R + ) was chosen such that̺(0) = 1 and supp̺ ⊂ (−δ/2, δ/2) for some arbitrary δ > 0. By choosing δ sufficiently small, one can even achieve that ̺ > 0 on [ yielding a corresponding bound for K sµ•Πσ (x, x), compare [40,Remark 4.4 (2)]. In view of the equality , one finally obtains the equivariant convex bound for any φ j ∈ L 2 σ (M ) and σ ∈ K, see [40, Proposition 5.1 and Eq. (5.4)]. As in the non-equivariant case, the kernels Ks µ •Πσ (x, y) exhibit a caustic behaviour in their dependence on x, y, which will be crucial for the derivation of equivariant subconvex bounds. In case that K = T is a torus and σ ∈ V µ , the bounds above are independent of σ, see [41, Proposition 5.1].

Spectral asymptotics for kernels of Hecke operators
Keep the notation of Sections 2 and 3. The main goal of this paper consists in proving subconvex bounds for Hecke-Maass forms of rank 1 on the compact d-dimensional Riemannian manifold M = Γ χ \G. To this purpose, we shall first derive asymptotics for kernels of Hecke operators in the eigenvalue and isotypic aspect. Recall that K acts on G and M from the right in an isometric and effective way, the isotropy group of a point Γ χ g ∈ Γ χ \G being conjugate to the finite group gKg −1 ∩ Γ χ . Hence, all K-orbits in Γ χ \G are either principal or exceptional, and of dimension dim K. Since the maximal compact subgroups of G are precisely the conjugates of K, exceptional K-orbits arise from elements in Γ χ of finite order. Consider now the right regular representation π of K on L 2 (Γ χ \G) together with the corresponding Peter-Weyl decomposition (3.8), and suppose that P commutes with π and the Hecke operators T χ ΓβΓ , which commute with the right regular K-representation as well. To describe the growth of simultaneous eigenfunctions of P and T χ ΓβΓ in the σ-isotypic component of L 2 χ (Γ\G), we are interested in spectral asymptotics for the Schwartz kernel of the operator . Let {φ j } j≥0 be an orthonormal basis of L 2 (Γ χ \G) consisting of simultaneous eigenfunctions of P and T χ ΓβΓ compatible with the decompositions (2.9) and (3.8). Applying the Hecke operators T χ ΓβΓ to the spectral expansion (3.9) of the spectral function of Q • Π σ yields In order to get an asymptotic description of the right-hand side of (4.2), we consider the composition T χ ΓβΓ •s µ • Π σ with the approximate spectral projections µ . Clearly, its Schwartz kernel can be written as Ks µ•Πσ (x, y) being as in (3.10), and by Remark 3.4 (1) one immediately deduces Nevertheless, to obtain subconvex bounds, more subtle estimates are necessary that take into account the caustic behaviour of the kernels Ks µ•Πσ (x, y) near the diagonal.

Proof. By Proposition 3.3 one deduces as
Furthermore, by Remark 3.4 (1) one has the uniform bound In view of (4.3) we therefore obtain by definition of the Stieltjies integral. In case that K = T is a torus, corresponding better estimates hold, and the assertion follows.

Subconvex bounds on Γ χ \SL(2, R) for arithmetic congruence lattices
In this section, we shall use the kernel asymptotics derived in the previous section to prove subconvex bounds on the quotient Γ\SL(2, R), where Γ is an arithmetic congruence lattice as considered by Iwaniec and Sarnak [27].

Arithmetic congruence lattices.
To introduce the setting, let A be an indefinite quaternion division algebra over Q. Hence, there exist two square-free integers a and b such that a > 0 and where ω 2 = a, Ω 2 = b, and ωΩ = −Ωω. For each element x = x 0 + x 1 ω + x 2 Ω + x 3 ωΩ, its conjugate is defined as x := x 0 − x 1 ω − x 2 Ω − x 3 ωΩ, and its trace and norm as tr(x) := x + x and N (x) := xx, respectively. Let R be an order of A, that is, R is a finitely generated free Z-module, R is a subring of A cotaining 1, and R ⊗ Z Q = A. For each prime number p, set A p := A ⊗ Q p and R p := R ⊗ Z p . Let d A be the product of all primes p such that A p is a division algebra. Then d A is called the discriminant of A. d A is greater than 1 and square free, and A p is isomorphic to M (2, Q p ) if p does not divide q. Throughout this section, we assume that R is an Eichler order of level L, where L is a natural number such that (d A , L) = 1. Hence, R satisfies (1) R p is the maximal order of A p if p divides d A , or Note that any Eichler order is included in a maximal order. Particularly, R is maximal when L = 1. Now, choose an embedding θ : For each natural number n ∈ N * , we set Then Γ := θ(R(1)) becomes a cocompact lattice of G := SL(2, R). Note that tr(x) = tr(θ(x)) and N (x) = det (θ(x)) hold for any x in A. In what follows, we identify A with θ(A). Especially, we will often use Γ instead of R(1). Next, let χ be a Dirichlet character on (Z/LZ) × . In view of the product isomorphism (Z/LZ) × ∼ = p|L (Z p /LZ p ) × given by the diagonal embedding a → (a) p , a character χ p can be defined on (Z p /LZ p ) × by restriction of χ to each factor. Set Ξ R := {α ∈ R | N (α) > 0, (N (α), L) = 1} and R L := {(x p ) p|L | x p ∈ R p , N (x p ) ∈ pZ p }, and define a character χ L on the semigroup R L by Composing χ L and the diagonal embedding Ξ R ⊂ R L , we obtain a character χ on the sub-semigroup Ξ R of A × . By the inclusion Γ ⊂ Ξ R , χ becomes a character on Γ which is called a Nebentypus character.

Since the subset R(n) is left and right Γ-invariant, and it is known that Γ\R(n) is finite [36, Section 5.3], we can introduce the Hecke operators
Indeed, since ψ(R(n)) is given as a disjoint union of double cosets ⊔ j Γα j Γ, the operator T χ n coincides with the sum j T χ Γαj Γ of Hecke operators defined in (2.10). In particular, we have also the Hecke operators For natural numbers n such that (n, q) = 1, the T χ n are self-dual, commute with the Beltrami-Laplace operator ∆ on G, and satisfy the composition rule [36, Section 5.3] Next, recall that the group GL(2, R) + := {x ∈ M (2, R) | det (x) > 0} acts transitively on the upper half plane H := {z ∈ C | Im z > 0} by fractional transformations 11 by which H becomes isomorphic to the homogeneous space G/K, where K := SO(2), and we define In what follows, we shall identify Γ χ \G/K ≃ Γ χ \H with a subset in H, and endow it with the standard hyperbolic distance on H given by [25, Section 1.1] By this, Γ χ \H becomes a compact hyperbolic surface. Note that dist H agrees with the distance function dist introduced at the beginning of Section 2. Furthermore, one has the following important result of Iwaniec and Sarnak.

Equivariant subconvex bounds.
With the notation of the previous section, we shall first derive subconvex bounds for Hecke-Maass forms in L 2 χ (Γ\G) in the eigenvalue and isotypic aspect for the Beltrami-Laplace operator ∆. For this, let {φ j } j≥0 be an orthonormal basis of L 2 (Γ χ \G) consisting of simultaneous eigenfunctions of P 0 = ∆ and T χ n compatible with the decompositions (2.9) and (3.8), where X = G and M = Γ χ \G, respectively, so that with (n, q) = 1 . Further, let σ ∈ K be a fixed K-type, and L 2 σ,χ (Γ\G) be defined as in (4.1). When σ is trivial, the space L 2 σ,χ (Γ\G) can be identified with L 2 χ (Γ\G/K). In what follows, we shall make the identification SO(2) ≃ S 1 ⊂ C, so that the characters of K are given by the exponentials σ l (e iθ ) := e ilθ , θ ∈ [0, 2π), l ∈ Z. Since all irreducible representations of K are one-dimensional, Proposition 4.1 yields for any x ∈ Γ χ \G and σ l with |l| ≪ µ/ log µ the estimate 11 Note that the center a 0 0 a , a ∈ R * , acts trivially on H. regarding xK ∈ Γ χ \G/K ≃ Γ χ \H as an element in C, and took into account that for suitable constants c 1 , c 2 > 0 In order to derive a uniform bound for K T χ n • sµ•Πσ l (x, x), note that by Lemma 5.1 one has with N (s) := s −1/2 and δ = µ −1 Taking everything together we have shown Theorem 5.2. For any n ∈ N * , µ > 0, and σ l with |l| ≪ µ/ log µ the uniform bound , where x ∈ Γ χ \G, and χ ∈ Γ is a Nebentypus character.
Remark 5.3. The previous theorem is the non-spherical analogon of [27,Lemma 1.2]. Note that the bounds for the point pair invariants on H used by Iwaniec and Sarnak in order to show [27, Lemma 1.2] are better than ours by a factor (1 + u(α · z, z)) −5/4 in the Stieltjes integral, but the lattice point counting function considered by them is unbounded, while ours is a priori bounded.
Following the original approach of Iwaniec and Sarnak, we shall now make use of arithmetic amplification to deduce from Theorem 5.2 equivariant subconvex bounds. Since we will later choose n ≪ µ A for some A ∈ N, we can neglect the contributions of order O(n 1+ε µ −∞ ) in the following. Thus, let χ ∈ Γ, σ l ∈ K be arbitrary, and {φ j } j∈N as in (5.4). Writing η j (n) := λ j (n)/ √ n we deduce with (3.10), (4.3), (5.1), and (5.4) that 12 If one replaces µ by µ log µ in Theorem 5.2 one obtains for any σ l ∈ K K T χ n • sµ•Πσ l (x, x) ≪ ε log µ (µ + nµ 1/2 log µ) n ε , yielding for arbitrary N ∈ N * and σ l j≥0, φj ∈L 2 Since the T χ n are adjoint operators for all n with (n, q) = 1, one has η j (n) = η j (n), compare [36,Theorem 5.3.8]. More generally, (T χ since |χ(d)| = 1, where z n ∈ C are arbitrary complex numbers. A simple computation then gives We thus arrive at Proposition 5.4. For any µ > 0, σ l ∈ K, χ ∈ Γ, and N ∈ N * one has the estimate Proof. As explained at the end of Section 3, the test function ̺ ∈ S(R, R + ) can be chosen such that ̺ > 0 on [−1, 1]. The proposition now follows from (5.6) and the estimate Next, one proceeds as follows. Let j 0 ≥ 0 be fixed such that φ j0 ∈ L 2 σ l ,χ (Γ\G), and consider the amplifier where p is a prime not dividing q. Note that (2.13) and (5.1) imply by the Prime Number Theorem. Writing λ j = 1/4 + r 2 j and taking µ = r j0 Proposition 5.4 then gives As a next step, note that Jacquet-Langlands correspondence [29] and the study of Rankin-Selberg convolutions ( [26,Theorem 8.3] and [16,Proposition 19.6]) imply for any j ∈ N with φ j ∈ L 2 χ (Γ\G) the bound where n moves over natural numbers prime to q. Here we used the facts that the Strong Multiplicity One Theorem holds for GL (2) and each automorphic representation factors as a tensor product of local representations. Consequently, with (5.9) and Cauchy's inequality one deduces uniformly in x ∈ Γ χ \G. Thus, we have shown our first main result.
Proof. If σ l = id is trivial, L 2 σ l ,χ (Γ\G) ≃ L 2 χ (Γ\H), and the assertion follows form the previous theorem. Note that since all K-orbits in G have the same volume, each eigenfunction of the Beltrami-Laplace operator on H ≃ G/K lifts to a unique K-invariant eigenfunction of the Beltrami-Laplace operator on G.

Automorphic forms on SL(2, R) and representation-theoretic interpretation.
In what follows, we would like to discuss our results within the theory of automorphic forms and their representation-theoretic meaning. For this, let us first recall the concept of an automorphic form on G = SL(2, R) for a discrete co-compact subgroup Γ, compare [8, Section 5].
Definition 5.7. A smooth function f : G → C is called an automorphic form on G for Γ iff: (A1) f (γg) = f (g) for all γ ∈ Γ and g ∈ G, (A2) f is K-finite on the right, where K = SO (2), (A3) f is Z-finite, where Z denotes the center of the universal envelopping algebra U(g C ) of the complexification of g.
Note that (A2) means that f is a finite sum of functions f l belonging to a specific K-type σ l , while (A3) is equivalent to the existence of a polynomial p(C) in the Casimir operator C = dR(Ω) that annihilates f , the notation being as in Section 2. Now, g = {X ∈ M (2, R) | tr(X) = 0}, while a Cartan involution is given by − t X. With respect to the basis of g = p ⊕ k, the modified Killing form ·, · θ is represented by the matrix Consequently, a corresponding orthonormal basis of g is given by so that the Casimir element reads compare (2.1). Note that our normalization of C differs from the one in [8, p. 20], where Ω ≡ 1 2 id . Writing p(C) = i (C − µ i ) and µ σ l = l 2 /8 for the eigenvalue of dR(Ω K ) on the σ l -isotypic component one sees that where we took into account (2.2). Thus, p(C)f = 0 iff q l (∆)f l = 0 for all l, by orthogonality. Since q l (∆) is an elliptic differential operator of the same order than p(C), and any subspace defined by a K-type and a Casimir eigenvalue is finite dimensional by Harish-Chandra's theorem [9,Theorem 1.7], we see that f is essentially given by a finite sum of Hecke-Maass forms in the sense of this paper.
To interprete our results in terms of the representation theory of G, let us first notice that, since −I 2 belongs to Γ, one has the limits of discrete series D +,0 (resp. D −,0 ) with l > 0 (resp. l < 0) can appear. Note that in each of the above unitary representations the σ l -isotypic component is 1-dimensional. By the above list, one sees that representations occuring in L 2 σ l ,χ (Γ\G) are in general different from the spherical case L 2 (Γ\H). Hence, Theorem 5.5 implies subconvex bounds for new classes of automorphic representations, in particular for the discrete series D s and their limits D ±,0 , as well as the principal series H(1, s).
Here H ∞ denotes the subspace of differentiable vectors in a Hilbert representation H. Consequently, the subconvex bound in Theorem 5.5 can be restated as follows.
Classically, an automorphic form of weight l ∈ N and Nebentypus character χ was first introduced as a holomorphic function f : where j(g, z) is as in (5.2). Its liftf (g) := f (g · i)j(g, i) −l constitutes an automorphic form on G in the sense of Definition 5.7; it is of K-type σ l and satisfies Cf = 1 4 (l 2 /2 − l)f , see [8,Sections 5.14 and 5.15]. In particular, if l > 1,f belongs to the discrete series representation D l−1 in L 2 σ l ,χ (Γ\G). If, in addition,f is a Hecke eigenform with f 2 = 1, one deduces from Theorem 5.8 since f p ≡ f p for all p, compare [36, p. 219], yielding subconvex bounds for classical automorphic forms on H in the weight aspect. This is consistent with Godement's formula [19], by which one has the convex bound f ∞ ≪ l 1 2 , see [12,10]. Furthermore, a corresponding subconvex bound was proven in [12], the exponent there being 1 2 − 1 33 = 31 66 . Thus, our results do imply new results about holomorphic modular forms on H. Note that in the case Γ = SL(2, Z) one can even show [52] that l 1 4 −ε ≪ ε f ∞ ≪ ε l 1 4 +ε by using the Fourier expansion of f and Deligne's bound [13], though this method is not available for cocompact arithmetic subgroups. In the non-cocompact case, hybrid bounds in the eigenvalue and the level aspect were considered in [2].

Subconvex bounds on SO(3)
In this section, we shall derive equivariant and non-equivariant subconvex bounds on SO(3) in the setting of [31,32]. They are proven in an analogous way than the ones proven in Section 5 using results of [49]. To begin, consider the quaternion algebra H(R) := {a 0 + a 1 i + a 2 j + a 3 k | a 0 , a 1 , a 2 , a 3 ∈ R} over a given commutative ring R, where i 2 = j 2 = −1, ij = −ji = k, and recall that for an element a := a 0 + a 1 i + a 2 j + a 3 k ∈ H(R) its conjugate is given by a := a 0 − a 1 i − a 2 j − a 3 k ∈ H(R) while its norm reads N (a) := aa = a 2 0 + a 2 1 + a 2 2 + a 2 3 . Note that H(R) corresponds to the field of Hamilton's quaternions and H(Z) to the ring of Lipschitz integers [11]. Write H(R) 1 := {a ∈ H(R) | N (a) = 1} and put G := H(R) 1 . As a group G, can be identified with SU(2) via the mapping G is compact, while H(Z) 1 = {±1, ±i, ±j, ±k} is finite, so that by choosing the lattice Γ := {±1} in G we have Γ\G ∼ = SO (3) via the adjoint action of G on its Lie algebra. Next, we introduce Hecke operators on SO(3) following [31,32]. Thus, for each α ∈ H(R) \ {0} and x ∈ G set α · x := N (α) −1/2 αx ∈ G.
As in Section 2, one can associate to each double coset ΓαΓ, α ∈ H(Q), a Hecke operator T ΓαΓ . We then have T Γα1Γ • T Γα2Γ = T Γα2α1Γ . Further, setting 14 R(n) := {a ∈ H(Z) | N (a) = n, a ≡ 1 mod 2}, one can define the Hecke operator For natural numbers r, s ≡ 1 mod 4, one has T r T s = d|(r,s) d T rs/d 2 , see [32,Remark 1] and [49, p. 331]. Since Hecke operators commute with the right regular representation of G on L 2 (Γ\G), we may replace L 2 (Γ\G) by L 2 (Γ\G/K) for any subgroup K of G in the above argument on Hecke operators. Choose K = SO(2) ≃ C 1 , and denote the corresponding characters by σ l : e iθ → e ilθ , l ∈ Z. Let ∆ denote the Beltrami-Laplace operator on G. Since ∆ and T n commute, there exists an orthonormal basis {φ j } j≥0 of L 2 (Γ\G) consisting of simultaneous eigenfunctions compatible with the decompositions (2.9) and (3.8), where X = G and M = Γ\G, respectively. Further, note that the action of K on Γ\G is isometric and non-singular. We then can prove the following equivariant subconvex bounds. ♯{α ∈ R(n) | dist(x, α · x) < δ} ≪ ε δ 1 2 n 1+ε + n ε if δ < 1/n, n 1 2 +ε + δ 2 3 n 1+ε otherwise. Further, Proposition 4.1 also holds in the present case, since Proposition 3.3 is true for arbitrary compact manifolds and symmetry groups. By repeating the arguments given in Section 5.2 we therefore get for any |l| ≪ µ/ log µ, µ > n, and n ≡ 1 mod 4 the uniform bound K Tn• sµ•Πσ l (x, x) ≪ ε (µ + nµ 1/2 log µ) n ε up to neglegible terms. Now, by the Dirichlet prime number theorem on arithmetic progressions it is well-known that # {p < x | p is a prime, p ≡ 1 mod 4} ∼ 1 2 x log x .
Furthermore, the Ramanujan conjecture proved by Deligne [13] together with the Jacquet-Langlands correspondence for GL(2) [29] implies that the Hecke eigenvalues λ j (p) of T p are bounded from above by 2p 1 2 +ε for prime levels. Hence, the argument of Iwaniec-Sarnak already used in Section 5.2, but now applied to L 2 (SO(3)), yields 14 a 0 + a 1 i + a 2 j + a 3 k ≡ 1 (2) means that a 0 is odd and a 1 , a 2 , a 3 are even. Note that R(n) is empty unless n ≡ 1 mod 4.
The theorem now follows by taking N = µ 1/3 . Note that the right regular representation of Γ\G ∼ = SO(3) on L 2 (Γ\G) decomposes according to where π k denotes the irreducible representation of SO(3) of dimension 2k + 1. In particular, the Beltrami-Laplace eigenvalue corresponding to π k is k(k+1), and the restriction of π k to K is isomorphic to k l=−k σ l . Hence, if we choose an orthonormal sequence {ψ j } j≥0 in L 2 (Γ\G) consisting of Hecke-Maass forms with ψ j ∈ (σ lj ) ⊕2kj +1 ⊂ M kj , where |l j | ≤ k j , Theorem 6.1 yields In what follows, we assume that there exists a submodule H of H χ Ξ such that there is an orthonormal basis {φ j } j∈N of L 2 (Γ χ \G) compatible with the decomposition (2.9), and in case that P commutes with the right regular K-representation, also with the decomposition (3.8), consisting of simultaneous eigenfunctions of P and all T ∈ H with P φ j = λ j φ j . As before, such simultaneous eigenfunctions will be called Hecke-Maass forms of rank 1. We also suppose that T * belongs to H for each T ∈ H and that the cospheres S * x (Γ χ \G) := {(x, ξ) ∈ T * (Γ χ \G) | p(x, ξ) = 1} are strictly convex for all x ∈ Γ χ \G. Further, consider the lattice point counting functions M(x, β, δ) := M(x, β, δ) or M (x, β, δ) corresponding to L 2 , respectively; that is, where dist (α · x, x) ≡ dist (Γαx, Γx), and M (x, β, δ) is as in (4.4). We then have the following Lemma 7.1. Fix a character χ in Γ such that [Γ : Γ χ ] < ∞. Let φ j0 be a Hecke-Maass form in L 2 with corresponding spectral eigenvalue λ j0 . Let P ′ be an infinite set and N ′ : P ′ → N a mapping such that Assume that for each element v ∈ P ′ there exists a Hecke operator T ′ v ∈ H satisfying T ′ v φ j0 = φ j0 , and that for any N ∈ N and any x ∈ Γ χ \G we have a suitable finite subset Q ′ N,x ⊂ P ′ such that As a linear operator on L 2 , T ′ N,x • (T ′ N,x ) * can be represented as a u T χ ΓαuΓ for certain l ∈ N, a u ∈ C, and α u ∈ Ξ depending on x. Further, suppose that there exist numbers 0 < κ ≪ 1 and A 1 , A 2 > 2 such that for each N ≫ 1 and each x ∈ Γ χ \G one has Then, if L 2 = L 2 χ (Γ\G), there exists a constant δ > 0 such that σ,χ (Γ\G), there exists a constant δ > 0, which does not depend on σ, such that Finally, if K = T is a torus and L 2 = L 2 χ (Γ\G), there exists a constant δ > 0 such that Proof. Let us consider first the case L 2 = L 2 σ,χ (Γ\G). Set µ := m λ j0 and denote by λ ′ j,N the eigenvalue of T ′ N,x for φ j , so that With the same arguments than at the end of Section 3 and in the proof of Proposition 4.1 one now deduces with µ j := m λ j for any . Hence, the assertion follows from (7.1) by taking N ∼ µ B , B := dim G/K−1 2(A−2+2κ) . The case L 2 = L 2 χ (Γ\G) is seen in a similar way taking into account Proposition 3.1 and the toric case in Proposition 3.3.
Remark 7.2. The assumptions of the previous lemma are primarily motivated by the work of Marshall [33] in the case that χ is trivial. One can easily verify that they are fulfilled in the setup of Section 5 when P ′ is the totality of primes and N ′ is the inclusion mapping P ′ ⊂ N. Indeed, for each prime p with (p, q) = 1, there exists an element β p in R(p 2 ) such thatT χ p 2 := T χ ΓβpΓ = T χ p 2 − T χ Γ(pI2)Γ , see [36, p. 217]. Now, letλ j (p 2 ) be the eigenvalue ofT χ p 2 belonging to the eigenfunction φ j ∈ L 2 σ,χ (Γ\G).
Remark 7.3. When using Lemma 7.1, we shall take for P ′ a subset of the totality of primes. Note that it is unnecessary to suppose that φ j is an eigenfunction of the operators T p for all primes p to prove the subconvex bound in Section 5, as we did already see in Section 6. In fact, one can make the conditions on P ′ and H weaker. Namely, we can replace P ′ by a smaller subset satisfying (7.1) and replace H by the submodule T ′ v | v ∈ P ′ . This means that our concept of a Hecke-Maass form is much weaker than the usual one in Section 5. Such forms are not eigenfunctions of the center of the universal enveloping algebra in general, and can be obtained in abundance by functorial lifts of Hecke characters.

Equivariant subconvex bounds.
In what follows, we shall derive equivariant subconvex bounds on arithmetic quotients for a large class of semisimple algebraic groups, extending the work of Marshall [33] to non-spherical situations. Thus, let G be a connected semisimple algebraic group over a number field F . We write G(k) for the set of k-rational points in G for a field k ⊃ F and F v for the completion of F by a place v of F . Following [33], we assume that there exists a real place v 0 of F such that (WS) The group G(F v0 ) is quasi-split, and not isogeneous to a product of odd special unitary groups. is compact, so that Γ\G is compact as well. Further, assume that H satisfies the condition (WS). Now, let P 0 be an elliptic leftinvariant differential operator on G of degree m which gives rise to a positive and symmetric operator on Γ\G that commutes with the right regular K-representation and has strictly convex cospheres S * x (Γ\G). Then, there exist a submodule H of H χ=1 Ξ=H(Q) and a constant δ > 0 such that (1) there is an orthonormal basis {φ j } j∈N of L 2 (Γ\G) which consists of simultaneous eigenfunctions for the unique self-adjoint extension P of P 0 and all T ∈ H; (2) for each φ j ∈ L 2 σ (Γ\G) with spectral eigenvalue λ j one has If K = T is a torus, one has the stronger estimate Remark 7.5. The equivariant subconvex bound of the previous theorem can be rephrased using the Cartan-Weyl classification of unitary irreducible representations of compact groups. In fact, assume that K is a compact connected semisimple Lie group, k its Lie algebra, and T ⊂ K a maximal torus with Lie algebra t. Denote by k C and t C the complexifications of k and t, respectively. Then t C is a Cartan subalgebra of k C , and we write Σ(k C , t C ) for the corresponding system of roots and Σ + for a set of positive roots. Now, as a consequence of the Cartan-Weyl classification of irreducible finite-dimensional representations of reductive Lie algebras over C one has the identification K ≃ {Λ ∈ t * C : Λ is dominant integral and T -integral} , compare [51], and we write Λ σ ∈ t * C for the highest weight corresponding to σ ∈ K under this isomorphism. Weyl's dimension formula then implies that d σ = O |Λ σ | |Σ + | , while from Weyl's character formula one infers that if D u is a differential operator on K of order u, compare [39,Eq. (3.5)]. Consequently, the bound in Theorem 7.4 can be rewritten as Proof of Theorem 7.4. By translating the results in [33, Section 3] to our non-adelic setting, one verifies that the assumptions of Lemma 7.1 are fulfilled under the hypothesis of the theorem. Note that it is unnecessary to relate the subgroup K to the specific maximal connected compact subgroup considered in [33], because the assumptions in question are concerned only with the structure of the Hecke algebra and the lattice point counting function M (x, α, δ).
Let us explain this in a more detailed way. Since For each double coset K 0 αK 0 with α ∈ H(A fin ), a linear operator T K0αK0 on L 2 σ (H(Q)\H(A)/K 0 ) can then be defined by setting Moreover, there exist finitely many elements β 1 , . . . , β m in H(Q) such that Γβ i Γ, the intersection being non-empty due to the assumption H(A) = H(Q)(H(R)K 0 ). 15 This implies that for all ϕ ∈ L 2 σ (Γ\G) Hence, any adelic Hecke operator T K0αK0 can be regarded as a sum of non-adelic Hecke operators via the identification ϕ ≡ ϕ A . In order to apply Lemma 7.1 in the present context, we choose χ = 1 and Ξ = H(Q). Let P ′ be the set denoted by P in [33, Section 2.5], that is, an infinite subset of the totality of finite places of F . A map N ′ : P ′ → N is defined by the order of the residue field of F v . Then, G(F v ) is split for each v ∈ P ′ and (7.1) holds by the prime ideal theorem and the Chebotarev density theorem. Now, put  15 Without this assumption, the intersection in (7.5) might be empty, and the following arguments make no sense.
Along the same lines, recall that Hecke operators on SO(3) are only defined in the case p ≡ 1 mod 4.
for P and all T ∈ H. Now, let φ j0 ∈ L 2 σ (Γ\G) be fixed. Applying the results in [33] to the function ψ := φ j0,A , that is also denoted by ψ there, one can verify the assumptions of Lemma 7.1 for φ j0 . Indeed, by [33, Propositions 6.1], for each place v ∈ P ′ there exists a Hecke operator T v ∈ H such that T v ψ = ψ holds and T v is a linear combination of operators T K0αK0 with α ∈ G(F v ). In view of (7.6) we can identify T v with a non-adelic Hecke operator T ′ v on L 2 σ (Γ\G) such that Similarly, set T N,x := v∈Q ′ N,x T v , and denote the corresponding non-adelic Hecke operators by T ′ N,x , where Q ′ N,x is chosen as the set denoted by Q N in [33,Section 3.4]. By the convolution on H(A fin ), there exist n ∈ N, b k ∈ C, and ω k ∈ H(A fin ) such that The corresponding l ∈ N, a u ∈ C, and α u ∈ C(Γ) in the decomposition of T ′ N,x • (T ′ N,x ) * in Lemma 7.1 are then obtained from this equality via the identification (7.6). Finally, the upper bounds (7.2) in Lemma 7.1 can be verified using (7.5), (7.6) and the arguments in [33,Section 3], completing the proof of the theorem.   and that Γ\G is compact. Let P 0 be an elliptic left-invariant differential operator on G of degree m that gives rise to a positive and symmetric operator on Γ\G with strictly convex cospheres S * x (Γ\G). Then, there exist a submodule H of H χ=1 Ξ=H(Q) and a constant δ > 0 such that there is an orthonormal basis {φ j } j∈N of L 2 (Γ\G) consisting of simultaneous eigenfunctions for P and all T ∈ H, so that for each φ j with spectral eigenvalue λ j one has Proof. To prove this theorem, we need an explicit distance on G. We may assume that H is a closed subgroup of SL(m) over Q for some sufficiently large m ∈ N, so that G = H(R) becomes a closed subgroup of SL(m, R) with respect to the topology induced from the Euclidean topology on R m 2 , compare [38,Chapter 3]. Note that H(R) might consist of finitely many connected components with respect to the usual topology, even if H is connected in the sense of Zariski [38, Corollary 1]. One then defines on M (m, R) the Euclidean distance dist 1 (x, y) := x − y , x := Tr( t xx), x, y ∈ M (m, R), obtaining a distance on G by the inclusions G ⊂ SL(m, R) ⊂ M (m, R). In fact, the distance dist is locally equivalent to the distance dist 1 . Indeed, dist 1 is equivalent to dist in a small neighborhood U of the identity. Furthermore, for fixed g ∈ G one computes dist 1 (gx, gy) ≤ g dist 1 (x, y), dist 1 (gx, gy) ≥ g −1 −1 dist 1 (x, y), so that dist 1 (x, y) is equivalent to the distance (x, y) → dist 1 (gx, gy) on G. The assertion now follows by covering G by translates of U.
The first assertion follows from the corresponding argument in Theorem 7.4. It remains to show that the assumptions in Lemma 7.1 are satisfied for the module H given in (7.7), for which we shall follow the considerations in [33]. Let us choose the same norm * as in [33, Section 2.2] on the group of cocharacters of a maxial torus over Q, and regard * as a norm on the cocharacters of each Q p -torus by conjugation. Let P ′ be the set denoted by P in [33, Section 2.5] for F = Q and G = H. Then P ′ is an infinite set of prime numbers, (7.1) holds, and for each prime p ∈ P ′ the group H(Q p ) is split. Furthermore, a Hecke operator τ (p, µ) is defined by the product of p − µ * with the characteristic function of H(Z p )µ(p)H(Z p ), where µ is a cocharacter on a suitable maximal split torus T p in H(Q p ). In addition, several conditions are imposed on H, P ′ , and T p , and we refer the reader to [33, Section 2] for details. By [33, Proposition 6.1], there exists for each p ∈ P ′ a Hecke operator T p such that T p φ j,A = φ j,A , T p = µ * ≤R a(p, µ) τ (p, µ), T p T * p = µ * ≤R b(p, µ) τ (p, µ), a(p, µ) ≪ 1, a(p, 0) = 0, b(p, µ) ≪ 1 for some constant R ∈ N. Now, choose a compact subset Ω of G such that G = ΓΩ, let x be an element in Ω, and set T N,x := p∈P ′ , p≤N T p . 16 In order to verify the necessary conditions in Lemma 7.1 we proceed as in the proof of Theorem 7.4, and let T ′ p and T ′ N,x be non-adelic Hecke operators corresponding to T p and T N,x , respectively. For γ ∈ H(Q) ⊂ SL(m, Q), let γ f denote the least common multiple of denominators of components of γ. By [33,Corollary 3.6], one has γ f ≪ N A ′ 2 for some A ′ 2 > 0 if γ ∈ H(Q) ∩ supp (T N,x (T N,x ) * ), where supp (T N,x (T N,x ) * ) means the support of T N,x (T N,x ) * in H(A fin ), and the second bound in (7.2) follows. Furthermore, for the distance dist 1 , one can show that for some A 2 > A ′ 2 the inequality dist 1 (γx, x) < c 1 N −A2 does not hold for any non-trivial element γ ∈ H(Q) ∩ supp(T N,x (T N,x ) * ), where we choose c 1 > 0 such that c 1 dist (y 1 , y 2 ) < dist 1 (y 1 , y 2 ) H(Q)(H(R)K 0 ) and H(Q)\H(A) is compact. Then, there exist a submodule H of H χ=1 Ξ=H(Q) and a constant δ > 0, which are independent of σ ∈ K, such that (1) there exists an orthonormal basis {φ j } j∈N of L 2 σ (Γ\G) which consists of simultaneous eigenfunctions for the Casimir operator C and all Hecke operators T ∈ H; (2) for each φ j with Casimir eigenvalue µ j one has provided that H = Res F/Q G and (WS) is fulfilled, while in general −δ µ σ being the eigenvalue of dR(Ω K ) on σ. If K = T is a torus,