ASYMPTOTIC BEHAVIOR OF SUPERCUSPIDAL REPRESENTATIONS AND SATO-TATE EQUIDISTRIBUTION FOR FAMILIES

We establish properties of families of automorphic representations as we vary prescribed supercuspidal representations at a given finite set of primes. For the tame supercuspidals constructed by J.-K. Yu we prove the limit multiplicity property with error terms. Thereby we obtain a Sato-Tate equidistribution for the Hecke eigenvalues of these families. The main new ingredient is to show that the orbital integrals of matrix coefficients of tame supercuspidal representations with increasing formal degree on a connected reductive p-adic group tend to zero uniformly for every noncentral semisimple element.

1. Introduction 1.1. Limit multiplicity. We begin this introduction by explaining our results on counting discrete automorphic representations. Let G be a semisimple group over a totally real field F . Write F ∞ := F ⊗ Q R. Consider a sequence of lattices {Γ j } j≥1 in G(F ∞ ) whose covolumes tend to infinity as j → ∞. For an irreducible unitary representation π ∞ denote by m(π ∞ , Γ j ) the multiplicity of π ∞ occurring in the discrete spectrum of L 2 (Γ j \G(F ∞ )). DeGeorge-Wallach [dGW78,DW79] in the compact case and Rohlfs-Speh [RS87] and Savin [Sav89] in the arithmetic non-compact case proved that if {Γ j } is a normal series whose intersection is the identity, then (1.1) lim j→∞ m(π ∞ , Γ j ) vol(Γ j \G) = deg(π ∞ ).
Here deg(π ∞ ) is the formal degree which by convention is non-zero if and only if π ∞ is square-integrable. By different methods it is shown in [ is Benjamini-Schramm convergent and uniformly discrete (which recovers the compact case but not the non-compact case). Our goal is to investigate refinements where instead of the lattice Γ j we impose a prescribed supercuspidal representation σ j . Let u be a finite place, and consider a sequence {σ j } j≥1 of tame supercuspidal representations of G(F u ) whose formal degrees tend to infinity as j → ∞. For an irreducible algebraic representation ξ of G(F ∞ ) with regular highest weight, let Π ∞ (ξ) denote the L-packet of square-integrable representations of G(F ∞ ) whose infinitesimal and central characters are dual to those of ξ. Write m(ξ, σ j ) for the number of discrete automorphic representations π in L 2 (G(F )\G(A F )), counted with automorphic multiplicity m(π), such that π ∞ ∈ Π ∞ (ξ), π u σ j , and π is unramified at all finite places away from u. By results of Harish-Chandra the cardinality |Π ∞ (ξ)| is equal to the order of the Weyl group of G(F ∞ ) divided by the order of the Weyl group of a maximal compact subgroup of G(F ∞ ). Informally one of our main results (Corollary 5.8, cf. (1.4) below) states, provided that the residue characteristic of F u is sufficiently large, that for a positive constant c independent of ξ. We refer to §5.2 for the determination of c which is related to the Tamagawa number of G.
Note that the integer |Π ∞ (ξ)| dim ξ is equal to π∞∈Π∞(ξ) deg(π ∞ ) up to a multiplicative constant depending only on the Haar measure on G(F ∞ ). Compared with (1.1), we are averaging over π ∞ ranging in the L-packet Π ∞ (ξ) for technical simplicity in the trace formula; this simplification does not interfere with the new phenomena at the finite prime u that we are concentrating on.
Example 1.1. For the group PGL(2) consider discrete automorphic representations that ramify above a single prime q and are unramified elsewhere. Let D k be the discrete series representation of PGL(2, R) of weight k ≥ 2 (necessarily even). For each simple supercuspidal representation σ of PGL(2, Q q ) with q > 3, there is an exact multiplicity formula where σ is the other representation with the same affine generic character as σ, see [Gro11].
(The assumptions in [Gro11] differ slightly, but one can verify that the same argument applies, the key point being that PGL(2, Q) has no q-torsion.) Simple supercuspidal representations for PGL 2 (Q q ) coincide with the representations of GL 2 (Q q ) of conductor q 3 and trivial central character. There are 2(q − 1) distinct simple supercuspidal σ's, partitioned into q − 1 pairs {σ, σ }, thus (1.3) leads to the observation [Gro11] that for any even integer k ≥ 2 and any prime q > 3, the dimension of the space S k (q 3 ) new of newforms of weight k and level Γ 0 (q 3 ) is 1 m(D k , Γ 0 (q 3 )) new = dim S k (q 3 ) new = k − 1 12 (q + 1)(q − 1) 2 .
Since the formal degree of D k is k−1 12 , this is a strong form of the limit multiplicity property (1.1) as q → ∞.
Depth aspect. Recall the notion of depth of an admissible representation [MP94]. As a special case of Conjecture 3.15 below, we conjecture that for a sequence (σ j ) j≥1 of supercuspidal representations of a p-adic group, the condition deg(σ j ) → ∞ is equivalent to depth(σ j ) → ∞. In our present context of tame supercuspidals, this is easy to verify (see, e.g., the proof of Proposition 3.13).
We may refer to the asymptotic (1.2) as a limit multiplicity result in the depth aspect. Indeed we view it as an analogue at finite places of the limit multiplicity in the "weight aspect" [Shi12,ST16]. Indeed in the weight aspect, the roles of σ j and ξ are interchanged, namely σ j remains fixed and ξ → ∞ (in the sense that the highest weight for ξ gets arbitrarily far from the walls), whereas the above families have fixed ξ and deg(σ j ) → ∞. In fact, we also establish hybrid results in this paper, where both ξ and σ j tend to infinity. For example our results below allow us to obtain an error bound for (1.2), saving powers for both dim(ξ) and deg(σ j ) (Corollary 5.10 below).
1.2. Quantitative equidistribution for a family. In the same context as before, for simplicity, let G be a split semisimple group over a totally real field F with trivial center. (In the main text G need not be either split or semisimple with trivial center.) Let each of S 0 and S be a finite set of finite places of F such that S = ∅ (but S 0 could be empty) and S 0 ∩ S = ∅. Denote by S ∞ the set of infinite places of F and put S := S ∞ ∪ S 0 ∪ S. Set F S 0 := v∈S 0 F v and A S F := v / ∈S F v . Let K S 0 be an open compact subgroup of G(F S 0 ), and K S an open compact subgroup of G(A S F ) which is the product of hyperspecial subgroups over v / ∈ S (which arise from a global Chevalley group for G over Z). We will consider • irreducible algebraic representations ξ of G(F ∞ ) with regular highest weight, • irreducible supercuspidal representations σ of G(F S ). For a technical reason we will impose the condition that ξ ∈ Irr reg C (G(F ∞ )) for a fixed constant C ≥ 1 (see §5.4 for details; the error bound in the theorem depends on C).
Our main result in a simpler form is the following Sato-Tate equidistribution for the family F. See Theorem 5.4 and Corollaries 5.8 and 5.10 for precise statements. In the special case where σ is fixed, our result generalizes [ST16,Thm. 9.19].
The multiplicative constant depends on G, C, S, K S 0 but is independent of ξ, σ and φ.
Example 1.3. Suppose G = PGL(2) and F = Q. We take S 0 = ∅ and S any non-empty finite set of sufficiently large finite primes. We are counting, for even integer weights k ≥ 2, and irreducible supercuspidal representations σ of PGL(2, Q S ), the number m(D k , σ) of cusp forms f ∈ F(D k , σ) of weight k unramified outside of S and with local component σ at S. The limit multiplicity asymptotic (1.4) recovers [Wei09] as in Example 1.1: The second assertion (1.5) on Sato-Tate equidistribution is new already in this case of PGL(2). For example, if the function φ is a Hecke operator T n for some integer n ≥ 1 not divisible by any prime in S, then: where δ n= is one if n is a perfect square and zero otherwise. Note that we normalize a n (f ) in such a way that Deligne's bound reads |a p (f )| ≤ 2. The above precise error term (corresponding to ν = 1 and A = 0) is derived from the Sally-Shalika character formula [KST16,App. A]. Similarly as in §1.1, the equidistribution (1.6) is a refinement of an earlier result. Namely, Serre [Ser97] considered the trace of the Hecke operator T n on the space S k (N ) of cusp forms f of weight k and level Γ 0 (N ), and established for n ≥ 1 coprime with N , f ∈F (D k ,Γ 0 (N )) a n (f ) √ n = m(D k , Γ 0 (N ))δ n= + O(n).
Remark. Interestingly, the depth aspect families studied in this paper are rather thin compared to the families formed by varying a lattice subgroup. In favorable situations, and assuming that S 0 = ∅, the global root number of π ∈ F(ξ, σ, ∅) depends only on ξ and σ. This almost never happens for thicker families arising from limit multiplicity problems where the whole lattice subgroup Γ j varies.
Conductor vs depth. We view the sets F(ξ, σ, K S 0 ) with varying tame supercuspidal representations σ of G(F S ) as forming a harmonic family in the sense of [SST16]. Then Theorem 1.2 essentially gives us the Sato-Tate equidistribution for families stated as Conjecture 1 in [SST16]. One difference is that the formulation of [SST16, Conj. 1] involves analytic conductors whereas our results are expressed in terms of formal degrees. The relation between formal degree and conductor is not yet established in general, this is a problem closely related to that of the depth preservation in the local Langlands correspondence [Yu09].
1.3. Bounds towards Ramanujan. We can deduce from Theorem 1.2 an average bound towards Ramanujan. For every place v ∈ S and every θ > 0, there is > 0 such that where α(π v ) is the Satake parameter of the unramified representation π v . It is unitarily normalized so that |α(π v )| = 1 if and only if π v is tempered. The multiplicative constant and the exponent are independent of ξ and σ (they depend only on G, C, S, K S 0 , θ, v). The proof proceeds in the same way as for [MT,Cor. 1.8]. Namely we first construct a function φ 1 which is a bi-K v -invariant function on G(F v ) such that tr π v (φ 1 ) is uniformly small for |α(π v )| ≤ 1 and uniformly large for |α(π v )| ≥ q θ v . Then we apply Theorem 1.2 for φ := (φ 1 * φ ∨ 1 ) * k with the integer k ≥ 1 chosen proportional to log m(ξ, σ, K S 0 ), see [MT,§3].
The estimate (1.7) shows that exceptions to the Ramanujan bound are sparse. For quasisplit classical groups the Ramanujan bound may be reduced to the self-dual or conjugate self-dual case of general linear groups via work of Arthur [Art13] and Mok [Mok15]. The latter case is settled when cuspidal automorphic representations are cohomological (over totally real fields in the self-dual case; over CM fields in the conjugate self-dual case) by [Clo91,Kot92,Shi11,Clo13,Car12]. In particular the Ramanujan conjecture is known for the representations π ∈ F(ξ, σ, K S 0 ) if G is a split classical group. 2 For exceptional groups G very little is known and even a formulation of the Ramanujan conjecture is delicate, see [Sar05] and [Sha11] for recent treatments.
1.4. Trace formula and tame supercuspidal coefficients. We find that the limit multiplicity and quantitative equidistribution described above are related to asymptotic properties of orbital integrals. The first step in the proof of Theorem 1.2 is to express the left-hand side of (1.5) as the spectral side of the trace formula for a suitably chosen test function. Since the weight ξ is regular and σ is supercuspidal we can use the simple trace formula.
There exist test functions f σ that single out the given supercuspidal representation σ in the trace formula, obtained by forming matrix coefficients. 3 In our situation J.-K. Yu's construction gives σ as compactly induced from a finite dimensional representation on a compact open modulo center subgroup of G(F S ). (Every σ arises in this way if the residue characteristics of places in S are sufficiently large by the exhaustion theorem [Kim07].) This provides an explicit f σ which is essential for our purpose.
We can now explain in more details the geometric side in the application of the trace formula. The geometric side is a sum over conjugacy classes of semisimple elements γ ∈ G(F ) of a volume term times a global orbital integral. The global orbital integral is a product of orbital integrals at ramified places in S, for which the main contribution is O γ (f σ ), and orbital integrals at unramified places.
Here we are varying the supercuspidal coefficient f σ which is unlike the usual applications of the trace formula where it is fixed. A general approach to this situation appears in [ST16] in the weight aspect and we can use the results of [ST16] to estimate most of the terms in the geometric side of the trace formula, except for O γ (f σ ) which is new.
For the proof of (1.2) we establish that |O γ (f σ )| = o(deg(σ)) as deg(σ) → ∞, and for any fixed γ. The proof of Theorem 1.2 is much more difficult due to the uniformity in φ. As in [ST16] the number of terms in the geometric side is unbounded, and uniform estimates for orbital integrals are needed. Moreover the estimate for O γ (f σ ) has to be made quantitative and uniform in γ which we discuss in the next subsection.
1.5. Asymptotic behavior of orbital integrals. We have seen in the previous subsection that our approach leads to the problem of establishing uniform bounds for orbital integrals of supercuspidal coefficients. In general it would be desirable to develop a quantitative theory of orbital integrals. This is for example advocated in the introduction of [DS00]. Our present problem of establishing uniform bounds for O γ (f σ ) goes in this direction.
Theorem 3.11 below states that there exists a constant η < 1 depending only on the group G(F S ) such that for all noncentral elements γ and all tame supercuspidal representation σ of G(F S ), we have where D(γ) is the discriminant of γ. This result is the technical heart of the paper. The properties of O γ (f σ ) are related to the trace character Θ σ (γ). In fact the two quantities agree if γ is regular elliptic, and we derive some consequences in §3.5. However it should also be noted that for our application it is essential to include the case where γ is non-regular elliptic (in which case Θ σ (γ) is undefined). For explicit computations of Θ σ (γ) for regular semisimple γ, we refer to [AS09,DS,Kal].
In some special cases O γ (f σ ) can be computed exactly, especially if one allows an additional average of σ (over an L-packet). In fact one could allow σ to be not only supercuspidal but also discrete series representations. Notably if σ is the Steinberg representation, then Kottwitz [Kot88] constructed an Euler-Poincaré function f EP which is a pseudo-coefficient for σ. In this case (1.8) holds with f σ = f EP in the horizontal aspect as the residue characteristics of places in S grow to infinity, see Section 6.
Though exact formulas for orbital integrals and for trace characters are extremely difficult to obtain beyond some special cases, we manage to prove the desired asymptotic (1.8).
We indicate an outline of our proof. It follows from Yu's construction that the function f σ can be chosen as a matrix coefficient and is supported on an explicit open compact subgroup J ⊂ G(F S ). We recall in Section 2 how J is constructed from a generic G-datum. From this we reduce the estimate to the orbital integral of the characteristic function of a larger compact open subgroup L s which is generated by a principal congruence subgroup and a parahoric subgroup of a proper twisted Levi subgroup. We conclude the proof in Section 4 based on a detailed analysis of Moy-Prasad subgroups.
1.6. Prescribed Steinberg representations. In a direction somewhat orthogonal to our main results described above, we have developed the case of families with prescribed Steinberg representations. We let the group G and the finite sets S, S 0 of finite places be as before ( §1.2). Let St S be the Steinberg representation of G(F S ). We consider the multi-set F(ξ, St S , K S 0 ) of discrete automorphic representations π, counted with multiplicity m(π) dim(π S 0 ) K S 0 such that π ∞ ∈ Π ∞ (ξ), π S St S , and (π S ) K S = 0. We let S vary and refer to F(ξ, St S , K S 0 ) as an horizontal family. This is to be compared with the previous vertical families F(ξ, σ, K S 0 ) where S was fixed and σ was a varying supercuspidal representation of G(F S ). In this case the needed estimates for orbital integrals can be deduced from results of Kottwitz on Euler-Poincaré functions [Kot88]. We establish the Sato-Tate equidistribution for these horizontal families. The main point is that our method [ST16] described above for vertical families applies almost without change to these horizontal families, but with the simplification that the rather subtle bound on orbital integrals from Sections 3 and 4 is replaced with easier bounds such as (6.5) below. In the following example we explain the significance of the result for classical modular forms and refer to Theorem 6.4 for the precise statement in general.
Example 1.4. Consider again the group PGL(2). The Steinberg representation St q and the quadratic twist of the Steinberg representation St q ⊗ χ q are the two representations of PGL(2, Q q ) of level Γ 0 (q), thus We note that F(D k , St q ) (resp. F(D k , St q ⊗ χ q )) is the set of cuspidal modular forms of weight k, level Γ 0 (q) with global root number 1 (resp. −1), see e.g. [CM04]. Iwaniec-Luo-Sarnak [ILS00, Cor. 2.14] proved that the following asymptotic holds: and similarly for m(D k , St q ⊗ χ q ). We shall discuss this in a more general context in Section 6 and revisit the PGL(2) case in Example 6.6. 1.7. Notation. Let F be a number field, S (any) finite set of places of F , and S ∞ the set of all infinite places of F . Then set F S : where denotes the restricted product over all places v under the given constraint. Now let G be a connected reductive group over F . The center of G is denoted Z(G), the maximal Q-split torus in the center of Res F/Q G is A G , and Its Plancherel measure is written as µ pl S . We typically write φ S for an element of H(G(F S )) and φ S for the associated function π S → tr π S (φ S ) on G(F S ) ∧ .
When π is an admissible representation of a p-adic group G, write Θ π for its Harish-Chandra character. We write [g, h] := ghg −1 h −1 for g, h ∈ G.
1.8. Acknowledgment. We thank late Paul Sally for helpful discussions. We are grateful to the referees for their numerous corrections and suggestions. S.W.S. is partially supported by NSF grant DMS-1449558/1501882 and a Sloan Fellowship. N.T. is partially supported by NSF grant DMS-1200684/1454893.

Yu's construction of supercuspidal representations
In this section we review the construction of supercuspidal representations of a padic reductive group from the so-called generic data due to Jiu-Kang Yu and recall from [Kim07] that his construction exhausts all supercuspidal representations provided the residue characteristic of the base field is sufficiently large. The construction yields a supercuspidal representation concretely as a compactly induced representation, and this will be an important input in the next section.
2.1. Notation and definitions. The following local notation will be in use until Section 4. Let p be a prime. Let k be a finite extension of Q p . Denote by q the cardinality of the residue field of k. Let G be a connected reductive group over k, whose Lie algebra is denoted g. Denote the center of G by Z G . Write G and g for G(k) and g(k), respectively. For a tamely ramified extension E of k, denote by B(G, E) the extended building of G over E. If T is a maximal E-split k-torus, let A(T, G, E) denote the apartment associated to T in B(G, E). It is known that for any tamely ramified Galois extension E of E, A(T, G, E) can be identified with the set of all Gal(E /E)-fixed points in A(T, G, E ). Likewise, B(G, E) can be embedded into B(G, E ) and its image is equal to the set of the Galois fixed points in B(G, E ) [Rou77,Pra01].
For (x, r) ∈ B(G, E) × R, there is a filtration lattice g(E) x,r and a subgroup G(E) x,r if r ≥ 0 defined by Moy-Prasad [MP94]. We shall normalize the valuation of E to extend the valuation of k, which is a different convention than in [MP94] (where the valued group is normalized to be Z). Our convention is so that for a tamely ramified Galois extension . For simplicity, we put g x,r := g(k) x,r , etc, and B(G) := B(G, k). For r ∈ R and x ∈ B(G) we will also use the following notation: • g x,r + := ∪ s>r g x,s , and if r ≥ 0, G x,r + := ∪ s>r G x,s .
• g * x,r := χ ∈ g * | χ(g x,(−r) + ) ⊂ p k , where p k is the maximal ideal of the ring of integers of k. • g r := ∪ y∈B(G) g y,r and g r + := ∪ s>r g s • G r := ∪ y∈B(G) G y,r and G r + := ∪ s>r G s for r ≥ 0. Lastly, for x ∈ B(G), we denote the stabilizer of x in G by G [x] .
2.2. Generic G-datum. Yu's construction of supercuspidal representations starts with a generic G-datum, which we recall. The reader is referred to [Yu01] for further details and any notions undefined here.
Definition 2.1. A generic G-datum is a quintuple Σ = ( G, x, r, φ, ρ) satisfying the following: D3. r = (r 0 , r 1 , · · · , r d−1 , r d ) is a sequence of positive real numbers with 0 < r 0 < · · · < r d−2 < r D4. φ = (φ 0 , · · · , φ d ) is a sequence of quasi-characters, where φ i is a generic quasi-character of G i (see [Yu01,§9] for the definition of generic quasi-characters). When d = 0, φ 0 is trivial on G x,r + 0 , but, nontrivial on G x,r 0 . When d ≥ 1, φ i is trivial on G i x,r + i , but non-trivial , the stabilizer in G 0 of the image [x] of x in the reduced building of G 0 , such that ρ|G 0 x,0 + is isotrivial and c- ρ is irreducible and supercuspidal.
In D5, note that G 0 x is compact while G 0 [x] is only compact mod center. Recall from [Yu01,p.585] that there is a canonical sequence of embeddings Hence, x can be regarded as a point of each of B(G i ) = B(G i , k).
Also D5 implies that x is rational as a building point of B(G) because it is a vertex of B(G 0 ). This will become important in Hypothesis (E ).(ii) below.
Given a generic G-datum Σ = ( G, x, r, φ, ρ), we introduce an open compact-mod-center subgroup of G where we set s i := r i /2 for each i. Yu constructs a finite dimensional representation ρ Σ of J Σ from the datum. His key result is that Theorem 2.2 (Yu). π Σ = c-Ind G J Σ ρ Σ is irreducible and thus supercuspidal. Fix a positive Haar measure vol G on G and denote the formal degree of π Σ by deg(π Σ ). It is well-known (see for example [CMS90] or [KST16, Lem 2.9.(i)]) that we have The construction of ρ Σ is complicated, but in what follows we shall only need the inequality dim(ρ Σ ) ≤ q dim G , see [KST16, Lem 2.9.(ii)] for a proof. For later reference, we write s Σ := s d−1 .
2.3. Supercuspidal representations via compact induction. Denote by Irr(G) the set of (isomorphism classes of) irreducible smooth representations of G. Write Irr 2 (G) (resp. Irr sc (G)) for the subset of square-integrable (resp. supercuspidal) members. Define Irr Yu (G) ⊂ Irr sc (G) to be the subset of all supercuspidal representations which are of the form π Σ as above. Write Irr c-ind (G) ⊂ Irr sc (G) for the set of π compactly induced from a representation on an open compact-mod-center subgroup of G. We have that where the first inclusion comes from Theorem 2.2. The second inclusion is expected to be an equality but not known in general; see [KST16,§2.6] for references to partial results by Bushnell, Kutzko, Stevens and others in this direction. The main result of [Kim07] says that the above inclusions are equalities under a rather explicit set of four hypotheses (namely (Hk), (HB), (HGT), and (HN ) in [Kim07, §3.4]); in particular the equalities hold when p is greater than some lower bound depending only on the absolute root datum of G and the absolute ramification index of k. 4

Orbital integrals of pseudo-coefficients
We keep the notation from the last section and assume that G = G(k) has compact center throughout this section and the next section. (We will briefly explain how to carry over the results of the current section to the non compact center case in Remark 3.14 below.) For π ∈ Irr Yu (G) attached to a generic G-datum we will construct an explicit coefficient f π of π and study the asymptotic behavior of the orbital integral of f π on noncentral semisimple elements as deg(π) → ∞ (note that we use π instead of σ to denote a representation of G). The result admits an interpretation as an asymptotic formula for character values, cf. §3.5 below, and will be applied in §5 to obtain an equidistribution theorem for families of automorphic representations.
3.1. Pseudo-coefficients. As before we have G = G(k) and write Z for the center of G. Let us recall the definition of pseudo-coefficients, cf. [DKV84,A.4].
Although f π is not unique, the orbital integrals of f π and the trace values of f π against irreducible admissible representations of G are uniquely determined by the condition of Definition 3.1. So, by [Kaz86, Thm. 0], its orbital integrals are uniquely determined. Note that f π is a cuspidal function in the sense that the trace of every induced representation from a proper parabolic subgroup is zero against f π . (This fact is built into the construction of [Clo86, Prop 1].) Moreover the orbital integrals of f π are well known to encode the elliptic character values of π (recall that γ ∈ G is said to be elliptic if the centralizer Z G (γ) is compact). When γ is regular elliptic, we will use the Haar measure on the compact group G γ assigning total volume 1 in the definition of the orbital integral below.
Proof. The first assertion can be derived from [Art93, Thm 5.1] specialized to the M = G case. The last assertion is Lemma III.4.19 of [SS97] (noting that the Euler-Poincaré function in that lemma is a pseudo-coefficient in view of Proposition III.4.1 and Theorem III.4.6 of loc. cit.).

3.2.
Explicit supercuspidal coefficients. In the following lemma, we construct an explicit matrix coefficient (which is also a pseudo-coefficient) associated to a compactly induced supercuspidal representation.
We will need the following hypotheses on the group G for Theorem 3.11. Hypothesis (T ) will be used in Lemma 4.1. Hypothesis (E ) will be used in the proof of Proposition 4.4 and Lemma 4.7. Observe that the hypotheses are inherited by tame twisted Levi subgroups.

Hypotheses.
(T ) For any tame maximal torus T, and r > 0, every nontrivial coset in T r /T r + contains a G-good element. (E ) There is a tamely ramified extension E of k such that (i) every k-torus in G splits over E; (ii) the k-order, in the sense of [RY14, §3.3], of every building point x ∈ B(G), which is a vertex in the building of some tamely ramified twisted Levi subgroup of G divides the ramification index of E over k; (iii) for every r ≥ 1, the exponential map induces a homeomorphism exp : g(E) x,r → G(E) x,r , and an abelian group homomorphism G(E) x,r /G(E) x,r + g(E) x,r /g(E) x,r + . We recall the notion [AS08] of G-good elements (we shall simply write good when there is no risk of confusion). Define the depth d T (γ) of a compact element γ ∈ T 0 as the unique Hypothesis (E ) may appear complicated at first. In fact, it achieves different purposes simultaneously: • If the tame extension E of k satisfies (E ).(i), (resp. (E ).(ii)), and E /E is a larger tame extension, then E also satisfies (E ).(i), (resp. (E ).(ii)), because the ramification index of E over k is divisible by the ramification index of E over k. • On the other hand, the radius of convergence of the exponential map in (E ).(iii) decreases with the ramification index. In particular, Hypothesis (E ).(iii) may be satisfied for the extension E of k, and fail for larger extensions E /E. To show that Hypotheses (T ) and (E ) hold true for sufficiently large p, we begin with a few general group-theoretic lemmas.
Lemma 3.4. Fix a root datum R. Then there exists a constant d R > 0 such that for every field K 0 and every connected reductive group G over K 0 with absolute root datum R, the group G splits over an extension of K 0 with degree at most d R . In fact every maximal torus of G defined over K 0 splits over an extension with degree at most d R .
Proof. Let T be a maximal torus of G defined over K 0 . Then T splits over a finite separable extension K/K 0 such that Gal(K/K 0 ) acts faithfully on X * (T). Choose a Z-basis for X * (T) to identify X * (T) Z r so as to obtain a group embedding By Minkowski's lemma (see e.g. [Ser07]), GL(r, Z) has only finitely many finite subgroups up to conjugacy (for a fixed r), hence [K : K 0 ] admits a bound d r ∈ Z >0 only in terms of r. The lemma is proved with d R := d r , as r depends only on the root datum R.
When K 0 is a local field, we have an explicit and relatively tight bound on d R assuming that the residual characteristic is not too small.
Lemma 3.5. Fix a root datum R. Let M R be the largest integer m such that ϕ(m) ≤ rank(R), and p R := rank(R). Then for every local field K 0 of residual characteristic p > p R , every K 0 -torus T in a connected reductive group G over K 0 with absolute root datum R, splits over a tamely ramified extension of K 0 of degree ≤ M 2 R . Proof. We will deduce this from the argument for Lemma 3.4. In the proof of that lemma, retaining the same notation, it suffices that the order of the finite subgroup Gal(K/K 0 ) → GL(r, Z) be coprime to p for K/K 0 to be tame. By Minkowski's lemma [Ser07], this holds if p > r + 1. Moreover, the exponent m of every finite subgroup of GL(r, Z) satisfies ϕ(m) ≤ r, because the eigenvalues of an element of order m inside GL(r, Z) consist of all m-th roots of unity (see [Wald08, App. B, claim (1) p.256] for a similar argument). So K/K 0 is a finite tamely ramified extension of exponent m with m ≤ M R . If K denotes the maximal unramified subextension of K/K 0 then K/K and K /K 0 are cyclic so [K : K ] and [K : Lemma 3.6. Fix a root datum R. Then there exists a constant N R > 0 such that for every field K 0 of characteristic zero and every connected reductive group G over K 0 with absolute root datum R, there exists a K 0 -embedding G → GL(n) with n ≤ N R .
Proof. Denote by G 0 the Chevalley group over k 0 := Q determined by R. Fix a k 0embedding G 0 → GL k 0 (V 0 ) with V 0 a finite dimensional space over k 0 once and for all. We will prove the lemma with Let G and K 0 be as in the current lemma. By Lemma 3.4, there exists K/K 0 with [K :

The last embedding follows from viewing
Remark 3.7. For certain kinds of groups G, one can proceed more directly as follows: • If G is semisimple, then we have Aut(G) Inn(G) Aut(R), and the action of Gal(K 0 /K 0 ) on T factors through W Aut(R). Thus it is sufficient that p R be larger than every prime factor of the Weyl group W and the automorphism group Aut(R). • If G is already split over a tamely ramified extension of K 0 , then pick one tame K 0 -torus T. To ensure that all the other K 0 -tori are tame, it suffices that the image of H 1 (K 0 , N G (T)) → Hom(Gal(K 0 /K 0 ), W ) consists of elements of order coprime to p, which happens if p R is larger than every prime factor of W .
Proposition 3.8. Hypotheses (E ) and (T ) are satisfied if p is sufficiently large, depending only on the absolute root datum of G and the absolute ramification index of k over Q p .
Proof. Let R be the absolute root datum of G. Assuming p > p R , choose a tamely ramified extension K/k with [K : k] ≤ M 2 R as in Lemma 3.5. We shall construct E as an extension of K, thus (E ).(i) will be automatically satisfied.
Concerning Hypothesis (E ).(ii), the key is to observe that every x ∈ B(G, k) contained in a generic datum is a vertex of the sub-building B(G 0 , k) attached to a tamely ramified Similarly as in [RY14, §2.6], if we fix an alcove and consider the finitely many affine roots ψ i whose zero loci bound an alcove and which are positive valued on the alcove, then there is a linear relation We view Ψ(G 0 , S 0 , k), resp. Ψ(G, T 0 , K), as affine linear functions on X * (S 0 , k) ⊗ Z R, resp. X * (T 0 , K) ⊗ Z R. We need to recall the details of how B(G 0 , k) injects in B(G, k), see p.585, Rem.2.11, Rem.3.4 of [Yu01]. The affine isomorphism A(G 0 , T 0 , K) A(G, T 0 , K) is canonically defined only up to translation by X * (Z G 0 , K)⊗ Z R. The choice in [Yu01] is made essentially in such a way that the isomorphism be Gal(K/k)-equivariant. Recall that Z G 0 is anisotropic because Z G is anisotropic. Hence the origin is the only Gal(K/k)-fixed point in X * (Z G 0 , K)⊗ Z R, and the isomorphism is uniquely determined. To rigidify the situation, we identify the apartments Since the integers b G and e depend only on the relative root system attached to (G 0 , S 0 , k), the absolute root system R attached to (G, T 0 ), and the action of Gal(K/k), and since there are finitely many root systems of given rank (possibly non-reduced, and possibly reducible), we have b G e|e R for some positive integer constant e R depending only on R. If p e R , then a totally tamely ramified extension E/K of degree e R will satisfy (E ).(ii).
For (E ).(iii), one needs to show that for X, Y ∈ g(E) x,r , The Campbell-Hausdorff formula in the form given by Dynkin is By Legendre formula where σ is the sum of the base-p expansion digits of s i , we have implies r > r, and that ( * ) is satisfied.
Consider again a totally tamely ramified extension E/K of degree e R . Then ord E (p) = e R ord K (p) ≤ e R M R ord k (p). The conclusion is that if then the extension E satisfies the second statement of Hypothesis (E ).(iii). We observe that the above calculation shows that exp(g(E) x,r ) is a group when p satisfies ( * * ). We finally need to verify the first statement of Hypothesis (E ).(iii). Under the condition that p > dim(G)ord E (p) + 1, the map exp : g(E) 0 + → G(E) 0 + is a homeomorphism by [Wald08, App.B] (An alternative approach would be to consider a faithful representation G → GL n for some n, see also [DR, App.B]. Note that g(E) 0 + (resp. G(E) 0 + ) are the set of topologically nilpotent (resp. unipotent) elements since p > dim(G).) Hence, exp r = exp | g(E)x,r is injective. We also need to show that exp r maps onto G(E) x,r . To see , and the proof of assertion (4) on p.258 of [Wald08, App.B] shows that t 0 (E) r is exponentiated onto T 0 (E) r . Now, if p satisfies ( * * ), then exp(g(E) x,r ) is a group and hence equals G(E) x,r . In sum, Hypothesis (E ) holds when p satisfies ( * * ) and p > dim(G)e R M R ord k (p) + 1. The assertion that (T ) holds for p sufficiently large goes back to [AR00,§5], which treated the Lie algebra version. A recent treatment of the existence of good elements for the group case is [Fin18a, Thm 3.6], with the sharp result that Hypothesis (T ) holds when G splits over a tamely ramified extension and p does not divide the order of the Weyl group of G.
• By definition, a finite extension E of k is tame if the residue characteristic p is coprime to the ramification index of E over k. Thus, Hypothesis (E ).(ii) implies p is coprime to the order of every x contained in a generic datum. As we have seen in the proof, this is satisfied if p > h, the Coxeter number of G, and G splits over a tamely ramified extension of k. For convenience, we shall work under Hypotheses (T ) and (E ) in Section 4 below, which is devoted to prove Proposition 3.13. However, inspecting the recursive argument in Proposition 4.4, it is sufficient to have Hypothesis (E ).(iii) for large enough r, and this is always satisfied by the same argument. Also we shall need in Lemma 4.8 below that the isomorphisms in (E ).(iii) be G-equivariant, which is also satisfied for large enough r. Hypothesis (E ).(ii) is automatically satisfied for an extension of large enough degree. Hence a natural assumption for our setting is: (T ) Every k-torus T in G is tame, and for every large enough r, every nontrivial coset in T r /T r+ contains a G-good element.
More precisely, the preceding paragraph explains that the arguments in Section 4 prove Proposition 3.13 under Hypothesis (T ), instead of Hypotheses (T ) and (E ). This in turn implies Theorems 3.11 and 3.16 under Hypothesis (T ).
Remark 3.10. To reach an optimal assumption, one could also factor out the center Z G , which shouldn't play a role in estimating orbital integrals, and therefore it shouldn't be necessary to assume that Z G is tame.
3.4. A uniform bound on orbital integrals of supercuspidal coefficients. For a semisimple element γ ∈ G ss let g γ denote the Lie algebra of the connected centralizer of γ in G. Define Note that it is unnecessary to assume γ to be regular. Given a generic G-datum Σ, Lemma 3.3 provides us with the pseudocoefficient f π Σ ∈ H(G) coming from J Σ and ρ Σ .
The following is a key local result of this paper.
Theorem 3.11. Assume Hypothesis (T ). There exist constants C, ν > 0 depending only on G such that for every generic G-datum Σ, In fact we can take any ν < (dim G) −1 .
In particular the theorem implies that if γ / ∈ Z then where ω π is the central character of π. So the above limit is never zero. Remark 3.12. An interesting question is whether the above Theorem 3.11 remains valid if π is allowed more generally to run over Irr 2 (G). We are inclined to believe that it is at least true for every sequence in Irr sc (G), possibly with a different value of ν ∈ R >0 .
Proof of Theorem 3.11. The orbital integral vanishes unless γ is elliptic, so we assume that γ is noncentral and elliptic semisimple from now on. Let G γ := Z(γ) and O(γ) the G-orbit of γ in G.
For Σ = ( G, x, r, φ, ρ) as in Definition 2.1, we let Our strategy is to study a power-saving upper bound for O γ (1 Ls ) as γ runs over semisimple elements of G and as s → ∞. Indeed as Σ moves along a sequence of generic G-data such that deg(π Σ ) → ∞, up to conjugacy, there are only a finite number of choices for (G , x) appearing in Σ with x ∈ B(G 0 ). The crucial estimate is the following, whose proof is postponed to Section 4. (Regarding the hypothesis, see the discussion above Remark 3.10.) Proposition 3.13. Assume Hypothesis (T ). There exists a constant C 1 > 0 depending only on G such that for all generic G-datum Σ, s ∈ R >0 and all noncentral semisimple To relate deg(π Σ ) and s Σ , we deduce the following from (2.1) and the fact that Since G x,r /G x,r+1 is in bijection with g x,r /g x,r+1 for r > 0 by hypothesis (E ) and g We deduce from this, the inequality (3.2), and Proposition 3.13 above that The proof of Theorem 3.11 is complete.
Remark 3.14. Let us explain what can be done when the center Z is not compact. Theorem 3.11 and Proposition 3.13 remain valid as stated. The only modification to the proof in Section 4 is that we need to use the subgroups L s : Lemma 3.3 is still true in this case, if vol(J) is replaced with the volume of J/Z in G/Z, and verified by the same argument. It may be troubling at first that the support of f π is compact only modulo center but equality (3.1) remains valid. 5 Conjecture 3.15 below remains the same, even when the center Z is not compact. One only needs to remember that in our convention Irr sc (G) consist of unitary supercuspidal representations, so the central character of π is also unitary.
Finally, in the global application we appeal to the simple trace formula with fixed character on a closed central subgroup so as to allow f π as the local component of a test function. Alternatively one could work with a truncated (pseudo-)coefficient (cf. [HL04, 1.9]), which is compactly supported, in place of f π .
3.5. Asymptotic behavior of supercuspidal characters. The main result, Theorem 3.16, of this section may be rephrased as a uniform upper bound for the characters of supercuspidal representations constructed by Yu on elliptic regular elements via Proposition 3.2.
Let us recall the context of the problem. Even though the precise character formulas for supercuspidal (and discrete series) representations of a p-adic group remain largely mysterious, we conjectured in [KST16,Conjecture 4.1] that the character values behave in a controlled manner as the formal degree tends to infinity. Recall that Irr sc (G) denote the set of isomorphism classes of supercuspidal representations. Write G ell (resp. G rs ) for the set of elliptic (resp. regular semisimple) elements in G.
In other words, for every γ ∈ G rs and every > 0 there exists d γ, > 0 such that (ii) Let B ⊂ G be a bounded subset. Then there exist constants ν > 0 depending only on G and C B > 0 depending only on G and B such that (iii) There exist constants ν > 0 and C ell > 0 depending only on G such that Note that (ii) implies (i). Since G has finitely many (elliptic) maximal tori up to conjugacy ([PR94, p.320, Cor 3]), (ii) also implies (iii). In [KST16,Thm 4.2], we have proved a result for (ii), provided that the residue characteristic of k is sufficiently large and that γ runs over the set G 0 + of topologically unipotent elements (which are not necessarily elliptic). The argument of that paper is based on an estimate of the number of fixed points of γ acting on certain coset spaces (without estimating orbital integrals).
Theorem 3.11 restricted to regular elements may be interpreted in terms of character values to establish (iii) of the conjecture provided that the residue characteristic of k is large enough. Note that the proof of Theorem 3.11 is independent of what is done in [KST16] and relies on quite a different method. Proof. The first assertion follows from equality (3.1) and Theorem 3.11. The second is deduced from the first assertion and the exhaustion theorem of [Kim07].
It is natural to wonder whether the method of this paper may be pushed further to cover non-elliptic elements. Proposition 3.2 is only a special case of Arthur's formula [Art87] relating supercuspidal character values on non-elliptic regular elements to weighted orbital integrals of supercuspidal coefficients. (This extends to cover general discrete series via the local trace formula [Art93].) So the problem is to bound such weighted orbital integrals.
A different question concerning trace characters is whether the constant ν > 0 can be found independent of the field k and the residue characteristic p. In this direction we observe that there exist analogues for finite groups, notably a general estimate by Gluck [Glu95].
It is natural to ask for a common generalization of the families in the depth aspect and in the "level aspect". Here, level aspect means that q S → ∞. We would consider the multi-set F(ξ, σ, K S 0 ) where both the finite set of places S and the discrete series representation σ of G(F S ) are varying. In any non-trivial sequence, the formal degree deg(σ) with respect to the canonical measure (see [Gro97] or [ST16, §6.6] for definition) goes to infinity, either because the depth of σ goes to infinity or because the residue characteristic q S goes to infinity. Theorem 1.2 above corresponds to families in the depth aspect for which S is fixed. Theorem 6.4 below corresponds to a refinement of the families in the level aspect where σ = St S . To establish such a common generalization one would need to address the above question of uniformity of the constant ν > 0, and one also would need to keep track of the polynomial dependence in the constants C ell in Conjecture 3.15.

Proof of Proposition 3.13
In this section we work with a pair (G, G ) where G is a tamely ramified twisted Levi subgroup with G = G. We shall also fix x ∈ B(G ) = B(G , k) and recall that L s = G x,s G x .
The following is a special case of the decomposition theorem in [AS08]. Compared to [AS08], our situation is simpler and hypothesis (C) in [AS08] is not needed because we do not keep track of centralizers of good elements here. We include a proof for completeness.
Repeating the process, one can write Now suppose r = 0. By [Sp08, Prop 2.36], there is an absolutely semisimple element γ as ∈ T 0 such that γ = γ as γ tu where γ tu ∈ T 0 + . If γ as ∈ Z, take z = 1, γ = γ as and γ + = γ tu . If γ as ∈ Z, one proceeds as in the first case for γ tu to reach the conclusion.
We will prove Proposition 3.13 at the end of this section by induction based on the following two propositions. The first, Proposition 4.3, which is at the base of the induction, is concerned with orbital integrals of a fixed test function and proved by means of Shalika germ expansions. The second, Proposition 4.4, allows us to proceed inductively in the parameter s ∈ R ≥0 . Recall that L s = G x,s G x . The following power-saving bounds are essential ingredients in the proof of Proposition 3.13. Proposition 4.4 can be understood in comparison with its analogue Corollary 4.9 below, which is easier to grasp and gives a bound for the function 1 Gx,s in place of 1 Ls .
(i) Let X = X ss + X n be the Jordan decomposition of X with [X ss , X n ] = 0 and X ss (resp. X n ) semisimple (resp. X n nilpotent). Since the lemma may be proved after taking a finite extension of F, we may assume that M (thus also G) and X ss split over F and that M is a maximal proper Levi subgroup of G. Thus m is a maximal proper Levi subalgebra of g. Without loss of generality, we may assume that the Dynkin diagram of G is connected. Let T be a maximal F-split torus in M whose Lie algebra contains X ss . Let Φ be the set of T-roots. Let ∆ (resp. Φ + ) be the set of simple roots (resp. the set of positive roots) associated with T such that X n ∈ α∈Φ + g α , where g α is the root space of α in g. Let ∆ M (resp. Φ + M ) be the subset of ∆ (resp. Φ + ) associated with M. Let U and U − be the unipotent and the opposite unipotent subgroup respectively. Let β ∈ ∆ such that ∆ = ∆ M ∪ {β}.
We will prove assertion (i) by showing its contrapositive that g = m + g X implies that X ∈ Z g . Assuming g = m + g X , we have that g β ⊂ u ⊂ g X . We proceed in two steps, first showing that X ss is central, and then that X n = 0.
Without loss of generality, we may assume that X ss = 0, thus X = X n . Write X = α∈Φ + X α with X α ∈ g α . Fix α ∈ Φ. Let ∆ be a simple root system with α ∈ ∆. Let ∆ M , β, U and U − be as in the previous case.
Let α 1 , · · · , α j = α be distinct simple roots in ∆ M such that β and α 1 (resp. α i and α i+1 ) are adjacent to each other in the Dynkin diagram. Then, we have the following: This follows from β ∈ ∆ M and by inductively applying [Hum78,Prop 8.4] to the root string.
By (a), (b) and (c), X α = 0 since the Dynkin diagram is connected. As α runs over Φ, we have X = 0, which is what we wanted, completing the proof of (i).
Assertion (ii) can be proved similarly as in (i) using the Jordan decomposition δ = δ ss δ n in M.
We recall the notation [g, h] := ghg −1 h −1 for g, h ∈ G. We will frequently use the fact [MP94, (2.6)] that for a, b ∈ R ≥0 , It is also useful for us to introduce functions d x : G x,0 \{1} → R ≥0 and d Z x : G x,0 \Z 0 → R ≥0 as follows: , which is the first inequality. By Hypothesis (E ), there exists a tame maximal k-torus T containing δ. It follows from Corollary 4.2 that max{r ∈ R ≥0 | γ ∈ ZT r } = md(δ). The second inequality then follows from the definitions since G x,t G x,s ⊂ G x,t , and G x,t ∩T ⊂ T t .
The last assertion is a corollary of the first inequality because δ ∈ Z 0 G x,s + is equivalent to d Z x (δ) ≤ s. Lemma 4.7. Suppose Hypotheses (T ) and (E ) hold. Let s ∈ Z ≥2 and δ ∈ L s be such that d := d x,s (δ) < s − 1. Define the map Then the cardinality of the image of C δ is at least q.
Proof. The last assertion of Lemma 4.6 implies that d Z x (δ) = d. Thus the map C δ is well-defined in view of (4.1). Since C δ is unchanged if δ is multiplied by a central element, we may assume without loss of generality that δ ∈ G x,d .
Let E be a tamely ramified finite extension of k that satisfies Hypothesis (E ), and fix a uniformizer E of E. Let m be the ramification index of E over k. Hence ord( E ) = 1/m, in view of our normalization of valuation and filtration index in §2.1. Since the order of x divides m by (E ).(ii), we have g(E) x,t/m = t E g(E) x,0 for t ∈ Z and there is no break for non-integral values of t.
We now divide the proof into the following four steps (1-4): (1) Write (2) g(E) x,i/m and g (E) x,i/m are vector spaces over the residue field E of the maximal unramified extension E u of k in E. Let F = F q be the residue field of k. Let G (resp. G ) be a reductive group defined over F which is the reductive quotient at the building point x of G (resp. G ) given by Bruhat-Tits theory. Write g and g for the Lie algebras of G(F) and G (F) respectively. We have g(E) x,0 g(E). We claim dim E g(E) = dim E g(E). This equality can be established from [DeB, §4.2]. More precisely, we have the isomorphisms of E-vector spaces g(E) x := ⊕ r∈R/ 1 m Z g(E) x,r /g(E) x,r + = g(E) x,0 , where the first equality is a definition given in loc. cit. Note that the range r ∈ R/ 1 m Z comes from our normalization which is such that ord( E ) = 1/m, and hence E g(E) x,r = g(E) x,r+ 1 m as recalled above, so the second equality is clear since all breaks occur when r ∈ 1 m Z. Then, by the result in loc.cit., we have dim E g(E) = dim E g(E) x , which establishes the claim.
(3) If d = 0, then i m = s − 1, and the map g → [δ −1 , g] induces G(E) x,i/m → G(E) x,i/m thus a linear map g(E) → g(E). Moreover, it maps G x,i/m ⊂ G(E) x,i/m into itself since δ −1 lies in G x (rather than G(E) x ). Its composition with the projection g(E) → g(E) /g (E) is seen to be equal to the map is the image of δ.
(4) If d > 0, then i m < s − 1. Write X ∈ g x,d for the element whose exponential is δ. Then the map g → δ −1 gδg −1 induces a map G(E) x,i/m → G(E) x,s−1 and thus a linear map g(E) → g(E) via (2). Composed with the projection g(E) → g(E) /g (E), this map is equal to the map x,i/m to g(E) x,s−1 (a proof of this assertion is similar to that of Proposition 3.8). Let X denote the image of X under the isomorphism g(E) x,d g(E) in (2). Then X ∈ g(E). We see that C δ is given by C δ (Y ) = [Y , X] modulo g (E) for each Y ∈ g(E).
In summary, we have a series of maps such that the following diagram commutes: Here the three horizontal isomorphisms on the right are chosen to make the diagram commute.
Consider the map C δ : g(E) → g(E)/g (E). Since δ / ∈ ZG x,d + (because d Z x (δ) = d), we see that δ is not in the center of G(E) when d = 0 and similarly X is not in the center of g(E) when d > 0. (Indeed δ ∈ Z(E)G(E) x,d + , hence X ∈ z(E) + g(E) x,d + in view of Hypothesis (E ).(iii), therefore the image of X in g(E) x,d is noncentral). We apply Lemma 4.5 with M = G , which yields dim E Im(C δ ) ≥ 1.
In both cases (either d = 0 or d > 0), the map C δ sends the rational subspace g into the rational subspace g/g . Via identification given in the diagram ( †), after taking Galois invariants of E-vector spaces, we see that the dimension of Im(C δ )∩g (mod g (E)) over F is at least one. Indeed for each component in the above diagram, we have H(k) x,r → H(E) x,r where H = G, G and r = i m , s − 1, and observe also that the horizontal isomorphisms on the right can be also chosen such that H(k) x,r → H(E) x,r induces h → h(E) where h = g, g . Lifting this fact to C δ , we have (Im(C δ )) ≥ q.
Next we want to show that the sets G The same reasoning shows that . This is promoted to δ i ≡ δ i (mod G x,s−1 G x,s ) thanks to the fact that δ i , δ i ∈ L s , thus i = i , verifying the disjointness. Define a map C δ i as follows: given by C δ i (g) := [δ −1 i , g] (mod G x,s−1 G x,s ). By Lemma 4.7, for each i there exist q elements u i1 , u i2 , · · · , u iq ∈ G x,s−d i −1 such that C δ i (u ij ) are distinct, j = 1, · · · , q.
As consequence of the above claim we have To finish the proof of (i) it is enough to prove that the terms on the right hand side are mutually disjoint. Indeed, if for each open compact subset U ⊂ G we write vol G/Gγ (U ) to denote the volume of the image of U in G/G γ then we will have Since the sets G x,s−d i −1 V i,s are disjoint, it only remains to show that u ij V i,s and u ij V i,s are disjoint for j = j and 1 ≤ i ≤ n. Suppose u ij V i,s ∩ u ij V i,s = ∅ for some i, j, j . There are v, v ∈ V i,s such that u ij v = u ij v . As before, vγv −1 = δ i g for some g ∈ G x,s−1 G x,s . Hence, Since d i ≤ s − 2, we apply (4.1) to obtain [g −1 , u ij ] ∈ G x,2s−d i −2 ⊂ G x,s . Thus the term u ij vγ(u ij v) −1 above belongs to δ i C δ i (u ij )G x,s−1 G x,s . Since u ij v = u ij v , we deduce similarly that it also belongs to δ i C δ i (u ij )G x,s−1 G x,s . This implies C δ i (u ij ) = C δ i (u ij ), hence j = j .
Observe that O(γ) ∩ ZG x,s = ∅. This follows from combining Corollary 4.2 and the fact that for any torus T, there exists g ∈ G with ZT s+1 ⊂ ZG gx,s . Hence, one can assume γ ∈ ZG x,s . Without loss of generality, we assume that γ ∈ G x,s .
Write γ = zγ 1 with z ∈ Z and γ 1 ∈ T 0 \ Z 0 with d T (γ 1 ) = md(γ) as in Corollary 4.2. Then, since which is a contradiction. Hence, the claim follows. Now we can arrange the decomposition such that d i < s − 2 for 1 ≤ i ≤ n and δ i ∈ G x,s−2 G x,s for n + 1 ≤ i ≤ n: For 1 ≤ i ≤ n , choose u i and define V i,s as in Case (i). Then, ψ −1 γ (L s−1 ) contains n i=1 q j=1 u ij V i,s and the summands are mutually disjoint by a similar argument to Case (i). Arguing as in Case (i) but keeping in mind that G x,s−2 G x,s ∩ O(γ) accounts for O γ (1 G x,s−2 Gx,s ), we complete the proof of Case (ii) as follows.
In preparation for the end of the proof of Proposition 3.13 we shall need a final bound from Corollary 4.9 below.
Lemma 4.8. Assume that the isomorphisms of Hypothesis (E ).(iii) are G-equivariant. Let s ∈ R ≥1 , and γ ∈ G x,s . Write X := exp −1 (γ). Then We are going to follow the idea as in the proof of Theorem 3.2.3 of [Fer07]. Let G γ (resp. G X ) denote the connected centralizer of γ (resp. X) in G. Write K := G x,s and k := g x,s . By the definition of orbital integrals, Since G γ = G X and 1 K (g −1 γg) = 1 k (g −1 Xg), the equality in the lemma holds.
There is a constant C > 0 depending only on G such that the following holds: for every s ∈ R ≥0 , and every semisimple γ ∈ G \ Z, Proof. In view of Proposition 4.3, we may assume without loss of generality that s ∈ R ≥2 . It is enough to prove the bound for s ∈ Z ≥2 . Indeed, if the corollary is known for s ∈ Z ≥2 then the corollary holds for s ≤ t < s + 1 at the expense of increasing the constant: For the rest of this proof, s ∈ Z ≥2 and γ = exp(X) ∈ G x,s . Then Here the first and third equalities are from Lemma 4.8 and the second follows from a direct computation. Applying Proposition 4.3 to f = 1 G x,1 we obtain where the constant C = c(1 G x,1 ) ∈ R >0 depends only on G and x which had been fixed since the beginning of this section. We can arrange that c(1 G x,1 ) depend only on the G-orbit of the facet containing x: If x = gx with g ∈ G then G x ,1 = gG x,1 g −1 so 1 G x,1 and 1 G x ,1 have the same orbital integral on each conjugacy class. There are only finitely many G-orbits of facets, so the lemma holds true for a constant C > 0 depending only on G.
Proof of Proposition 3.13. As in the proof of Corollary 4.9, once we prove the proposition for a fixed x, the proposition also holds when x varies by a similar finiteness argument. So we may and will keep x fixed in this proof.
Let a s := D(γ) 1 2 O γ (1 Ls ), which is a decreasing function of s ∈ R ≥0 . We want to show that a s ≤ C 1 · q −s . It is sufficient to verify the inequality for s ∈ Z ≥0 at the expense of replacing C 1 by qC 1 . Applying Proposition 4.3 to f = 1 L 0 we have that a 1 ≤ a 0 ≤ c(1 L 0 ). Set m to be the largest integer such that m ≤ md(γ). Proposition 4.4.(ii) combined with Corollary 4.9 provides us with the recursive inequality (replacing C by q 2 C) This implies that for any 2 ≤ s ≤ m − 1, We have in particular Next, Proposition 4.4.(i) shows that for s ≥ m + 2, we have the inequality a s ≤ 1 q a s−1 . Hence Proposition 3.13 is verified with C 1 := (c(1 L 0 )q + C)q 2 , which is indeed a constant depending only on G.

Automorphic Plancherel equidistribution with error terms
In this section our asymptotic formula for supercuspidal characters and orbital integrals is applied to produce an equidistribution theorem for a family of automorphic representations. The theorem can be informally summarized as follows: Consider the set of L 2 -discrete automorphic representations with supercuspidals at a fixed finite place (suitably weighted). As the formal degree of the supercuspidal at the fixed place moves toward infinity, the local components (away from the fixed place) of the automorphic representations are equidistributed with respect to the Plancherel measure. We prove the theorem in the case where a suitable condition at infinite places simplifies the trace formula so that the technical difficulties with general terms in the trace formula do not blur the close relationship between the asymptotic formula for supercuspidal characters and the equidistribution.

Preliminaries.
In the rest of the article the following global setup will be in effect. Let G be a connected reductive group over a totally real number field F . (We used the symbol G differently in the preceding sections.) Put F ∞ := F ⊗ Q R. Write A disc (G) for the set of discrete automorphic representations of G(A F ) up to isomorphism (i.e. without multiplicity). The automorphic multiplicity of π ∈ A disc (G) is denoted m disc (π). Let S be a nonempty finite set of finite places of F . Fix a Haar measure µ S on G(F S ). Recall that the unitary dual G(F S ) ∧ is equipped with a positive Borel measure µ pl S , the Plancherel measure.
Write Ram(G) for the set of finite places v of F such that G is ramified over F v . For each finite place v / ∈ Ram(G) let K hs v be a hyperspecial subgroup of G(F v ). We choose K hs v such that at all but finitely many v / ∈ Ram(G), the group K hs v consists of the O Fv -points of some reductive integral model of G over O F [1/N ] for a sufficiently large integer N .
Let v be a place of F . Write µ can v for the canonical measure on G(F v ) (denoted by L(M ∨ (1)) · |ω G | in [Gro97]), and if G(F v ) has compact center, denote by µ EP v the Euler- When G is unramified over F v it is known that µ can v assigns volume 1 to hyperspecial subgroups. From §5.3 on we will fix a finite set of places S and consider the (possibly negative) measure (This is different from the convention of [ST16]; there we used µ can v at all finite places. Also note that µ can,EP depends on the set S.) Define the volume of the adelic quotient τ (G, S) := µ can,EP (G(F )\G(A F )), relative to the counting measure on the discrete subgroup G(F ). This volume is finite if G(F ∞ ) has compact center, which will always be the case by part (iii) of the assumptions made in the next subsection. 6 The Tamagawa volume of G(F )\G(A F ) is denoted by τ (G). 5.2. The simple trace formula. As the trace formula is going to play a central role in the proof, we recall some basic facts. Let T ell , T disc : H(G(A F )) → C designate the invariant distributions consisting of contributions from the elliptic conjugacy classes and the discrete automorphic spectrum, respectively. Arthur's trace formula is the equality of the two invariant distributions I geom and I spec on the geometric and spectral sides. In general I geom (resp. I spec ) is the sum of T ell (resp. T disc ) and other very complicated terms, but we will always be in the situation where the simple trace formula applies, i.e.
where the sum runs over the set of F -elliptic conjugacy classes in G(F ), and µ Iγ (I γ ) is the volume of I γ (F )\I γ (A F ) for the quotient measure of µ Iγ (A F ) . (Note that I γ (F ∞ ) has compact center by ellipticity of γ and assumption (iii) above.) The discrete part of the trace formula is where m disc (π) denotes the multiplicity of π in the discrete automorphic spectrum. Under the above hypotheses Arthur [Art88, Cor 7.3, Cor 7.4] provides us with the simple trace formula Indeed the assumption at v 1 (resp. at v 1 and v 2 ) in Corollary 7.3 (resp. 7.4) of that paper is satisfied by any v 1 ∈ S (resp. any v 1 ∈ S and any v 2 ∈ S ∞ ) by (ii) and (iii) above. Here we use the property of Euler-Poincaré functions [Art89,p.270,p.281] that their orbital integrals vanish outside elliptic conjugacy classes.

5.3.
Counting measures for automorphic representations. Let G be a connected reductive group over a totally real field F as in the preceding subsection. Let S 0 , S, S, ξ, Π ∞ (ξ), K S 0 , and K S be as in the introduction. (We allow S 0 to be empty.) Throughout this section G is assumed to be unramified away from the finite set of places S. This is always ensured by increasing the set S 0 if necessary. We make the following additional hypothesis, which is technically helpful as it was in [ST16].
• The highest weight of ξ is regular.
to be the multi-set of π ∈ A disc (G) whose multiplicity is zero unless π S is unramified, π S σ S , and π ∞ ∈ Π ∞ (ξ), in which case the multiplicity of π is By Harish-Chandra's finiteness theorem, a F (π) = 0 only for finitely many π. We may replace m disc (π) by the multiplicity in the cuspidal spectrum since every automorphic representation with a supercuspidal component (or with π ∞ in discrete series) is cuspidal. For each σ ∞ ∈ Π ∞ (ξ) let f σ∞ be a pseudo-coefficient for σ ∞ . Set Then tr π ∞ (f ξ ) = 0 if and only if π ∞ ∈ Π ∞ (ξ), in which case the trace equals 1. (The only if part follows from the results of Vogan-Zuckerman on Lie algebra cohomology.) Moreover f ξ is an Euler-Poincaré function up to a nonzero constant, cf. [Kot92, Lem 3.2]. Write f σ S ∈ C ∞ c (G(F S )) for the product of explicit supercuspidal coefficients f σv (see §3.2).
Proof. It follows from the definition that the left hand side equals which is none other than µ can S 0 (K S 0 ) −1 T disc (φ). We conclude by (5.3).

Bounds on the geometric side.
Here we recollect various bounds on the terms appearing on the geometric side, mostly from [ST16]. Given each semisimple element γ ∈ G(F ), fix a maximal torus T γ in G over F containing γ and write Φ γ for the set of roots of T γ in G outside I γ , namely the set of roots α in G such that α(γ) = 1. Thus Φ γ is nonempty if and only if γ / ∈ Z(F ). Define S γ for the following set of finite places of F : Evidently S γ is independent of the choice of T γ . In the same way we defined Ram(G), we have the set Ram(I γ ). For each v / ∈ Ram(G), we have a maximal split torus where v runs over places of F outside Σ. For simplicity we will write Q a+bκ to mean Here || · || is an Ω G -invariant norm on X * (A v ), which depends on the choice of an R-basis of X * (A v ) R .
Write d(G ∞ ) for the cardinality of Π ∞ (ξ), which is independent of ξ. Let q(G ∞ ) denote the real dimension of G(F ∞ ) modulo (any) maximal compact subgroup. Given a constant C ≥ 1 and ∈ {alg, reg}, we define a set where max(ξ) and min(ξ) are given as follows. Let T be a maximal torus in G over C and choose a Borel subgroup B containing T . Let λ ξ ∈ X * (T ) denote the B-dominant weight of ξ. Write Φ + for the set of B-positive roots of T in G, and ρ for the half sum of roots in Φ + . Then max(ξ) (resp. min(ξ)) is the maximum (resp. minimum) value of the natural pairing α, λ ξ + ρ as α runs over Φ + . The value is independent of the choice of T and B. (We introduce Irr C (G(F ∞ )) so that we can vary ξ in a controlled manner in that set.) Proposition 5.2.
(i) There exists A 1 > 0 depending only on G such that the following holds for any Σ and κ = (κ v ) as above: Choose U Σ to be a compact subset of For every C ≥ 1, ξ ∈ Irr alg C (G(F ∞ )) and γ ∈ G(F ∞ ) ss with γ ∈ Z(F ∞ ), we have where ν ∞ ∈ R >0 depends only on G(F ∞ ) and the implicit constant is independent of γ and ξ. One can choose ν ∞ to be the minimum of 1 − (dim R I γ − rk R I γ )/(dim R G ∞ − rk R G ∞ ) as γ runs over noncentral elements in G(F ∞ ) ss . (iii) Let v / ∈ S γ ∪ Ram(G). Suppose that γ ∈ G(F v ) ss is conjugate to an element of K hs v . Then I γ is unramified over F v , and O γ (1 K hs v , µ can v ) = 1.
(iv) There exists a lower bound p 0 > 0 and A 3 , B 3 > 0 depending only on G such that for every finite place v whose residue characteristic is greater than p 0 , for every γ ∈ G(F v ) ss , and for every λ ∈ X * (A v ) with λ ≤ κ v , Proof. Part (i) follows from [ST16,Prop 8.7]. (Take S 0 and S 1 there to be our Σ\S ∞ and {v / ∈ Σ : κ v = 0}, respectively. The proposition there assumes that the nonzero values of κ v are all equal, but the same proof works when κ v are different. Finally observe that A 3 can be absorbed into B 3 in that proposition.) Let us prove (ii). It is a standard fact ([Kot92, p.659]) for a discrete series representation π ∞ that O γ (f π∞ ) vanishes unless γ is elliptic semisimple, in which case if the Euler-Poincaré measures are used on G(F ∞ ) and I γ (F ∞ ). This implies everything but the last bound in (ii) as f ξ is the sum of f σ∞ over σ ∞ ∈ Π ∞ (ξ). It remains to bound D G ∞ (γ) 1/2 |tr ξ(γ)| = O(dim(ξ) 1−ν∞ ) with ν ∞ described as in the proposition. We get the bound from [ST16, Lem 6.10.(ii)], observing that m(ξ) there is equal to min(ξ) and that dim(ξ)/ min(ξ) |Φ + | is bounded both above and below (in terms of C).
We get ( Write Mot Iγ for the Artin-Tate motive associated to I γ by Gross [Gro97]. We have the decomposition Mot Iγ = ⊕ d∈Z ≥1 Mot Iγ ,d (1 − d), where Mot Iγ ,d is an Artin motive, and (1 − d) denotes the Tate twist. For any Artin-Tate motive M over F , denote by L(Mot Iγ ) (resp. L v (Mot Iγ )) the global (resp. local) L-function evaluated at s = 0. Write Ω Iγ for the absolute Weyl group of I γ . There is a certain local cohomological invariant c v (I γ ) ∈ Q >0 defined in [Gro97, (8.1)]. We do not need the definition but only the property that Let γ ∈ G(F ) ell so that I γ has F -anisotropic center. (This ensures that µ can,EP Iγ (I γ ) is finite, cf. [Gro97,Prop 9.4].) We have the identity [Gro97, Thm 9.9] (5.5) |µ can,EP . Lemma 5.3. Let γ vary over the set G(F ) ell and retain the above notation. Let S iso denote the set of finite places v of F such that Z contains a nontrivial F v -split torus. (So S ∩ S iso = ∅.) (i) There exist constants 0 < c 1 < c 2 such that for all γ ∈ G(F ) ss , (ii) There exist constants c 0 , c 3 , A 2 > 0 depending only on G such that the following holds: for all S such that G is unramified at all places in S, we have |µ can,EP

Proof. (i) Observe that
where I d is a finite index set, and a d,i ∈ C has absolute value q d−1 for all i. (Since v / ∈ S iso we always have a d,i = 1.) For each d and i we have the obvious bounds q d−1 − 1 ≤ |1 − a d,i | ≤ q d−1 + 1. Part (i) is now easily deduced from the following facts [Gro97,§1], cf. [ST16, Prop 6.3]: • |I d | ≤ dim Mot Iγ ,d with equality when I γ is unramified at v, (ii) From (5.5) (applied one more time with γ = 1) we have the bound |µ can,EP Using (i) we bound Note that the implicit constants in both O(·) depend only on G, not on γ or S. (This is obvious in the latter. For the former, it is enough to observe that |Ω Iγ | and τ (I γ ) are uniformly bounded as γ varies. The Weyl group is bounded by |Ω Iγ | ≤ |Ω G |. The Tamagawa measure formula [Kot88] tells us that τ (I γ ) ≤ |Z( I γ ) Gal(F /F ) |, and the latter is uniformly bounded by [ST16,Cor 8.12].) From (5.4) and [Kot86, Thm 1.2], we have Since Gal(F /F ) acts on Z( G) through a finite quotient, say Gal(F /F ), if we take c 2 ∈ Z ≥1 be the maximum of |π 0 (Z( G) H )| as H runs over all subgroups of Gal(F /F ), then clearly c v (G) ≤ c 2 . Note that c 2 is independent of v. Finally we have for a uniform constant A 2 > 0, with an implicit constant of O(·) which is uniform for all γ, by [ST16, Cor 6.16]. Now the bound of (ii) follows by putting c 3 := c 2 c 2 .
5.5. Equidistribution results. Fix S, S 0 , and K S 0 . We keep the notation from the previous subsection with S = S ∞ ∪ S 0 ∪ S. Throughout this subsection we suppose that • the residue characteristic of every v ∈ S is sufficiently large such that Theorem 3.11 applies to G(F v ) at each v ∈ S, • S ∪ S 0 contains the places of F with small residue characteristics such that the result by Cluckers-Gordon-Halupczok on uniform bound on orbital integrals [ST16, Thm 14.1] applies to places outside S ∪ S 0 . Note that the lower bound on the residue characteristic for Theorem 3.11 can be made effective, whereas the lower bound for [ST16,Thm 14.1] to hold is ineffective by the nature of its proof.
Our interest lies in statistics of the family F = F(ξ, σ S , K S 0 ) as ξ and σ S vary. Write Since we are assuming Z(F S ) and Z(F ∞ ) are compact, the intersection Z(F ) ∩ K S 0 K S (taken in G(A F )) is finite. (The same is true with any compact K Sbi-invariant subset of G(A S F ) in place of K S .) As in [ST16, §2.3], define the truncated Hecke algebra H ur,≤κ (G(A S F )) as the space of locally constant bi-K S -invariant functions on G(A S F ) whose support is inside the compact subset v ∈S U ≤κv v .
Theorem 5.4. There exist constants ν S , ν ∞ > 0 and A > 0 such that for every ξ ∈ Irr reg C (G(F ∞ )), for every σ S ∈ Irr Yu (G(F S )), for every κ = (κ v ) v / ∈S , and for every φ S ∈ H ur,≤κ (G(A S F )) which is the characteristic function of a bi-K S -invariant compact subset, The implicit constant in O(·) depends on G, S, S 0 , K S 0 , and C (but is independent of ξ, σ S , κ, and φ S ).
Remark 5.5. The proof shows that ν ∞ can be chosen to be as in Proposition 5.2. We have restricted to the set ξ ∈ Irr reg C (G(F ∞ )) to underline the analogy between the finite places S and the infinite places S ∞ . Without the restriction the error bound could be stated in terms of min(ξ) as in [ST16,Thm 9.19]. The same remark applies to Theorem 6.4 below.
Remark 5.6. If we fix κ and φ S (while allowing ξ and σ S to vary) then the same proof shows the asymptotic formula with error bound O(dim(ξ) 1−ν∞ deg(σ S ) 1−ν S ). This holds under a weaker assumption on S, namely that the residue characteristic of each v ∈ S has to be large enough so that only Theorem 3.11, but not [ST16, Thm 14.1], applies. Hence the lower bound for the residue characteristic can be explicitly given. Thus an explicit lower bound is possible for Corollaries 5.8, 5.11, and 6.5.
Proof. By Lemma 5.1 the left hand side is equal to For each v ∈ S, write U v for the (finite) union of a set of representatives for G(F v )conjugacy classes of elliptic maximal tori in G(F v ). Take U v to be an elliptic maximal torus in G(F v ) for infinite places v ∈ S ∞ . Thus U v is compact in both cases. Clearly the summand in the preceding formula vanishes unless γ ∈ Y(κ), where Y(κ) is as in Proposition 5.2 taking Σ = S and The contribution from central elements z ∈ Z(F ) is computed as in the first line on the right hand side of (5.6). For this it is enough to observe that µ can,EP Harish-Chandra's Plancherel theorem, and f ξ (z) = (−1) q(G∞) ω ξ (z) dim ξ by Proposition 5.2. This implies that the contribution of γ ∈ Z(F ) in the above sum equals the main term in the right hand side of (5.6). Hence it suffices to show that for some uniform constants ν S , ν ∞ , A > 0, We will bound the summand for each γ ∈ Y(κ). Without loss of generality we assume that γ belongs to v / ∈S U ≤κv v × K S 0 × v∈S∪S∞ U v . Define Φ γ and S γ as in the last subsection (with S = S 0 ∪ S ∪ S ∞ ). Write S γ for the set of v ∈ S γ with v / ∈ Σ. The subset of v ∈ S γ with κ v = 0 is written by S γ,0 .
According to Lemma 5.3 (iii), the group I γ is unramified at v if v / ∈ Ram(G), v / ∈ S γ , and κ v = 0. (The last condition ensures that the v-component of φ S is supported on K hs v so that γ is conjugate to K hs v .) Hence Now Lemma 5.3, Proposition 5.2, Theorem 3.11 and Proposition 4.3 tell us that there exist positive constants c 0 , c S , c S 0 , c ∞ , A 2 , A 3 , B 3 , ν S , ν ∞ such that |µ can,EP At this point, the main remaining task is to bound q S γ and q S γ,0 independently of γ.
We introduce some invariants of the group G over F . Write d G for the dimension of G, w G for the order of the absolute Weyl group, and s G for the minimal degree over F of an extension field over which G splits.
Define δ ∞ to be the supremum of v∈S∞ |1 − α(γ ∞ )| v as γ ∞ runs over v∈S∞ U v , and as α runs over the set of absolute roots for the compact maximal torus v∈S∞ U v in G(F ∞ ).
(It makes no difference if we impose α(γ ∞ ) = 1.) Then δ ∞ < ∞ as the supremum of a continuous function on a compact set is finite. Likewise define (5.10) as γ v runs over the v-component of K S 0 if v ∈ S 0 , and as γ v runs over elliptic semisimple elements of G(F v ) if v ∈ S, and α runs over the set of absolute roots of a maximal torus containing γ v in G. By definition δ v with v ∈ S ∪ S 0 depends only on G, S, S 0 , and K S 0 . We see that δ v < ∞ for v ∈ S ∪ S 0 by continuity and compactness as for δ ∞ . (For instance when v ∈ S 0 , there are only finitely many G(F v )-conjugacy classes of elliptic maxial tori, which are compact as G(F v ) has compact center. For each compact maximal torus, the value |1 − α(γ)| v is bounded as α varies over the set of absolute roots and γ on the maximal torus.) For our purpose below, we may and will arrange that δ v , δ ∞ ≥ 1 by redefining each of them to be 1 if smaller than 1. Noting that |1 − α(γ)| v = 1 at v / ∈ S γ ∪ S ∞ with α ∈ Φ γ , we deduce from the product formula that By [ST16, Lem 2.18], for some B 1 > 0 which is independent of γ and κ, For each v ∈ S γ,0 , we have |1 − α(γ)| v ≤ 1 for every α ∈ Φ γ . By the definition of S γ,0 , there exists α ∈ Φ γ such that |1 − α(γ)| v < 1. The argument as in the proof of [ST16,Prop 8.7] (also see the proof of Theorem 9.19 there) shows that a fortiori |1 − α(γ) Taking the product of (5.11) over α ∈ Φ γ and applying the estimates (5.10) at v ∈ S γ \S γ (which is contained in S ∪ S 0 ), (5.12) at v ∈ S γ \S γ,0 , and (5.13) at v ∈ S γ,0 , we see that To summarize so far, the absolute value of each summand in (5.7) is bounded by, if we set Applying the above bounds on q S γ and q S γ ,0 , the summand admits a bound of the form O(Q A 4 +B 4 κ deg(σ S ) 1−ν S dim(ξ) 1−ν∞ ). The number of nonzero summands is bounded as O(Q A 1 κ ) by Proposition 5.2. All in all, the absolute value of the left hand side of (5.7) is

The proof is complete by taking
Remark 5.7. An affirmative answer to the question in Remark 3.12 would immediately improve Theorem 5.4 with the hypotheses relaxed accordingly, by exactly the same argument.
Consider the set of pairs (ξ, σ S ) ∈ Irr reg C (G(F ∞ )) × Irr Yu (G(F S )). We partition the set into P = and P = according as whether ω ξ = ω σ S or not on Z(F ) ∩ K S 0 K S . Recall that Z(F ) ∩ K S 0 K S is finite.
Corollary 5.8. We have the limit multiplicity formulas Remark 5.9. It is clear from the definition that m(ξ, σ S , K S 0 ) ≥ 0. This is consistent with the signs in the second formula above. Indeed the sign of the measure µ EP ∞ µ EP S is the same as that of τ (G, S) since the canonical measure µ can v is a positive measure for v / ∈ S ∪ S ∞ . The signs of µ EP ∞ and µ EP S are the same as those of (−1) q(G∞) and deg(σ S ), respectively. Proof. We start by claiming that dim(ξ) deg(σ S ) → ∞ if and only if dim(ξ) ν∞ deg(σ S ) ν S tends to infinity. To see this, we partition the set of all σ S into two sets where | deg(σ S )| ≥ 1 and | deg(σ S )| < 1. The claim is obvious in the first set. (Recall that ν ∞ , ν S > 0 and note that dim(ξ) ≥ 1.) In the second set, the claim follows from the fact [HC99, Thm 7] that deg(σ S ) is bounded below by a positive constant. (More precisely deg(σ S ) is an integral multiple of a constant depending only on a Haar measure on G(F S ).) Now the corollary readily follows from the preceding theorem by plugging in κ v = 0 for all v / ∈ S and φ S = 1 K S .
We can restate Theorem 5.4 in terms of m(ξ, σ S , K S 0 ), assuming G is split and semisimple for simplicity. The L 1 -norm of φ S ∈ H ur (G(A S F )) is given by φ S 1 := G(A S F ) φ S (g) dµ can,S . Corollary 5.10. Suppose that G is a split semisimple group over F . There exist constants , A > 0 such that for every (ξ, σ S ) ∈ P = and for every φ The implicit constant in O(·) depends on G, S, S 0 , K S 0 , and C (but is independent of ξ, σ S , and φ S ).
Proof. Since G has finite center, the center is contained in every maximal compact subgroup. So Z(F ) ∩ K S 0 = Z(F ) ∩ K S 0 K S and ω σ S = ω ξ on Z(F ) ∩ K S 0 for (ξ, σ S ) ∈ P = . Our task is to turn the right hand side of (5.6) to the right hand side as in the corollary. Let κ = (κ v ) v / ∈S be chosen such that φ S ∈ H ur,≤κ (G(A S F )). We know that from Theorem 5.4, cf. the proof of Corollary 5.8. So it is enough to show that for some constants A , B > 0 (whose independence is as in the corollary), . This immediately follows from [ST16,Prop 8.7].
It remains to check (ii). We adopt the notation about truncated Hecke algebras from [ST16, §2.3]. By Cartan decomposition we may write g = ( for a suitable R-basis B = {e 1 , ..., e r } of X * (T ) ⊗ Z R, where X * (T ) is the cocharacter group of a maximal torus T over F . (Thus r = dim T = rkG.) Of course κ v = 0 for all but finitely many v. Choose a Borel subgroup containing T so that we have a set of positive coroots Φ ∨,+ for T . We take B to consist of simple coroots in Φ ∨,+ . Similarly we choose a Borel subgroup B v ⊃ T v and a set of positive coroots Φ ∨,+ v for T v . Without loss of generality we assume that We claim that there exists a constant c > 1 such that for every v / ∈ S we have Indeed [Gro98,Prop 7.4] tells us that 1 KvgvKv 1 /q µv,ρ ∨ is equal to |(G/P µv ) for a suitable parabolic subgroup P µv of G. Of course there are finitely many parabolic subgroups (up to conjugation). We see that the quotient (which is a quotient of two polynomials in q v ) tends to one as q v → ∞. The claim follows.
Let us record a sample application to the existence of certain automorphic representations. For simplicity we assume that G is split over F . We fix a reductive model over the ring of integers O F , giving rise to hyperspecial subgroups K hs v at each finite place v. Corollary 5.11. Suppose that G is a split reductive group over F . Fix ξ ∈ Irr reg (G(F ∞ )) and S a nonempty finite set of finite places. Suppose that the residue characteristic of each v ∈ S is sufficiently large in the sense at the start of §5.5. Then there exists d 0 > 0 with the following property: For every π 0 S ∈ Irr Yu (G(F S )) with | deg(π 0 S )| ≥ d 0 and ω π 0 S = ω ξ on Z(F ) ∩ K S 0 K S , there exists a cuspidal automorphic representation π of G(A F ) such that • π S π 0 S , • π ∞ ∈ Π ∞ (ξ), • π S,∞ is unramified.
Proof. In the preceding corollary, it is enough to take S 0 to be sufficiently large and contain all places of small residue characteristics, and set K S 0 := v∈S 0 K hs v . Then fix ξ and let deg(σ S ) go to infinity.
According to Corollary 5.8, it is reasonable to restrict our attention to (ξ, σ S ) ∈ P = when studying equidistribution problems on the following counting measure for the multi-set F = F(ξ, σ S , K S 0 ), where δ π S denotes the Dirac delta measure supported at π S on the unramified unitary dual G(A S F ) ∧,ur := v / ∈S G(F v ) ∧,ur : Of course this makes sense if |F| = 0. To obtain a clean formula we will further assume that Z(F ) ∩ K S 0 = Z(F ) ∩ K S 0 K S . (Alternatively we may instead restrict to the pairs (ξ, σ S ) ∈ P = such that ω ξ = ω σ S on Z(F ) ∩ K S 0 .) Given z ∈ Z(F ) let ω z denote the function on G(A S F ) ∧,ur whose value on each representation is its central character evaluated at z. Define a measure µ pl,ur,S z on G(A S F ) ∧,ur to be ω z · µ pl,ur,S , where µ pl,ur,S := v / ∈S µ pl,ur v is the product of the Plancherel measure µ pl,ur v on the unramified unitary dual of G(F v ). 7 Recall that φ S defines a function φ S : π → tr π(φ S ) on v / ∈S G(F v ) ∧,ur . Note that we can integrate φ S against the (possibly infinite) sum measure µ pl,ur,S Z(F )∩K S 0 := z∈Z(F )∩K S 0 µ pl,ur,S z . Indeed µ pl,ur,S z ( φ S ) is nonzero for only finitely many z ∈ Z(F ) ∩ K S 0 since Z(F ) intersects an open compact subset of Z(A F ) at only finitely many points.
This confirms Conjecture 1 in [SST16], or more precisely its analogue as explained in the remark below it. In our case the limiting measure is the product of the unramified Plancherel measures so (i) and (ii) of the conjecture are true. Part (iii) is essentially [ST16,Prop 5.3], from which (iv) follows immediately.
Remark 5.14. The results above should carry over to the case where neither G(F S ) nor G(F ∞ ) has compact center, at least if G is a cuspidal group in the sense that the center of Res F/Q G has the same split Q-rank and split R-rank. This requires some modification in the statements (e.g. pseudo-coefficients of a supercuspidal representation have compact support only modulo center) but would not lead to any significant change in the proof. Alternatively one could work with the trace formula with fixed central character (one could use [Dal19, §6] for instance), in which case representations and test functions also have fixed central characters which are inverses of each other.
5.6. Possible generalizations. It is sensible to ask whether the method of this paper applies to non-supercuspidal discrete series representations π but there are difficulties. In that case we still have a somewhat explicit construction of pseudo-coefficients for π, cf.
[SS97, §3.4] but they are not as simple as in §3.2 to be useful. In the trace formula, if we impose a pseudo-coefficient of π at a local place v then one still has the simple trace 7 Since we consider only those φ S coming from H ur (G(A S F )), the formulas remain valid if we use the Plancherel measure on the whole unitary dual.
formula, but the spectral side picks up automorphic representations whose v-components are not only π but possibly nontempered representations in a finite list. This means that one has to control these spectral error terms. Alternatively one could allow more general test functions at v but then the trace formula will have more terms to be dealt with (unless the global reductive group is anisotropic modulo center).

Steinberg representations and horizontal families
The results in this section do not rely on Sections 2 through 4. However the reader will see a strong analogy both in the statements and proofs between the vertical families in the last section and the horizontal families in this one.
Let ξ, S, S 0 , S ∞ , and S be as in §5.3, cf. §1.2. In particular the finite sets S, S 0 , and S ∞ are mutually disjoint, and G is unramified outside S. We will fix S 0 and assume that • the residue characteristic of every v ∈ S is sufficiently large. Throughout this section we keep a large residue characteristic assumption as in §5.5: • S ∪ S 0 contains the places of F with small residue characteristics such that the result by Cluckers-Gordon-Halupczok on uniform bound on orbital integrals [ST16, Thm 14.1] applies to places outside S ∪ S 0 . We also assume that (to use Lemma 6.2 relying on strong approximation) • the simply connected cover of G der has no F -simple factor that is F v -anisotropic for every v ∈ S. Let F(ξ, St S , K S 0 ) denote the multi-set of π ∈ A disc (G), with m disc (π) dim(π S 0 ) K S 0 as the multiplicity of π, which satisfies the following conditions: first, π ∞ ∈ Π ∞ (ξ); second, π v is isomorphic to the Steinberg representation for all v ∈ S; third, π v is unramified for all finite places v ∈ S ∪ S 0 . In this section we study F(ξ, St S , K S 0 ) as we vary the set S. The situation is somewhat complementary to that in the previous section. We refer to F(ξ, St S , K S 0 ) as a horizontal family. 8 Kottwitz [Kot88] constructed Euler-Poincaré functions for p-adic groups. For any place v ∈ S, we denote it by φ EP v ∈ C ∞ c (G(F v )). We have that [Kot88, Thm 2] We will assume that • G is a simple 9 algebraic group, i.e. every proper normal subgroup of G over an algebraic closure of F is finite. In that case Casselman's theorem, cf. [Cas, Thm 2] and [Kot88, Thm 2'], tells us that an irreducible unitary representation π v with tr π v (φ EP v ) = 0 can occur only in the following two cases (which may not be mutually exclusive): (i) π v is the trivial representation and tr π v (φ EP v ) = 1, or Set f St S := v∈S f Stv , q S := v∈S q v , and q(G S ) := v∈S q(G v ).
Remark 6.1. The formal degree of St v is equal to (−1) q(Gv) for the Euler-Poincaré measure by the Plancherel theorem, (ii) above, and φ EP v (1) = 1 recalled above. See also [Bor76]. We will state the analogue of Lemma 5.1 for F(ξ, St S , K S 0 ) after recalling the following well-known result on one-dimensional automorphic representations.
Lemma 6.2. Let H be a connected reductive group over a number field E. Let v be a place of E, and assume that the simply connected cover of H der has no E-simple factor that is E v -anisotropic (in particular, H(E v ) is not compact modulo center). If π is an automorphic representation of H(A E ) with dim π v = 1 then π is one-dimensional.
Proof. Using a z-extension of H ([Kot82, §1]) we can reduce to the case when the derived subgroup H der of H is simply connected. Now the groups H der (E) and H der (E v ) are subgroups of H(A E ) so they act on π viewed as a subspace of the regular representation of H(A E ) on automorphic forms of H(A E ) (with a fixed central character). The onedimensionality of π v implies that H der (E v ) acts trivially on π. Then any smooth vector of π, as a function φ : H(A E ) → C, has the property that φ(h 0 hh v ) = φ(h) for h 0 ∈ H der (E), h ∈ H(A E ), and h v ∈ H der (E v ). Hence φ(hh ) = φ(h) for h ∈ h −1 H der (E)hH der (E v ) = h −1 H der (E)H der (E v )h. By the strong approximation theorem, h runs over a dense subset of H der (A E ). Therefore φ is constant on left H der (A E )-cosets, i.e. H(A E ) acts through an abelian quotient. Hence π is one-dimensional. tr π S φ S , µ can,S = µ can S 0 (K S 0 ) −1 T ell (φ, µ can,EP ).
Suppose that the summand for π is nonzero. Then π ∞ cannot be one-dimensional by regularity of ξ. Let v ∈ S so that tr π v (f Stv ) = 0. If G(F v ) is compact modulo center then St v is the trivial representation, so π v St v by Casselman's theorem above. If G(F v ) is not compact modulo center then Lemma 6.2 tells us that π v cannot be one-dimensional so π v St v by the same theorem. In either case tr π v (f Stv ) = 1. We see that (6.2) equals the left hand side of the lemma.
Theorem 6.4. There exist real constants ν ∞ , A > 0 and an integer B ∈ Z ≥1 such that for every ξ ∈ Irr reg C (G(F ∞ )), for every κ = (κ v ) v / ∈S , and for every φ S ∈ H ur,≤κ (G(A S F )) which is the characteristic function of a bi-K S -invariant compact subset, The implicit constant in O(·) depends on G, S 0 , K S 0 , and C (but is independent of ξ, S, κ, and φ S ).
Proof. The proof proceeds exactly as for Theorem 5.4. The main term coming from γ ∈ Z(F ) is computed similarly. (Note that f St S (z) = (−1) q(G S ) = deg(St S ) for z ∈ Z(F S ). So deg(σ S ) in (5.6) is replaced by (−1) q(G S ) here.) The issue is to bound the contribution from noncentral elements. To explain the mild modifications in the argument we freely use the notation from the proof there. It suffices to show the following analogue of (5.7) for uniform constants ν ∞ , A > 0 and B ∈ Z ≥1 : The only nontrivial change is to replace (5.8) and (5.9) with the following inequalities for noncentral elements γ ∈ Y(κ) for suitable uniform constants c 0 , A 5 > 0: The bounds for orbital integrals away from S as well as the rest of the proof are exactly the same as in the proof of Theorem 5.4. So we content ourselves with justifying the two inequalities, beginning with (6.5). (Along the way we obtain a bound on q Sγ . Evidently the same bound works for q S γ,0 . Together with (5.9), this is used to bound the orbital integral away from S.) As we recalled in (6.1), |O γ (f St S )| is bounded by 1 and nonzero only on γ that is elliptic in G(F v ) for all v ∈ S. Then γ is contained in an F v -anisotropic maximal torus T v of G(F v ) so |α(γ)| v = 1 for all absolute roots α of T v in G by compactness, implying that |1 − α(γ)| v ≤ 1. Therefore D S (γ) ≤ 1 and (6.5) holds.
It remains to check the uniform bound (6.4) for noncentral elements γ. If γ is not in the center of G, we have that 1 2 (dim I γ − rkI γ ) − 1 2 (dim G − rkG) is a negative integer. Take B to be the maximum of such integers as γ varies. Our initial assumption tells us that Ram(G) ⊂ S 0 . At each v ∈ S 0 we define δ v < ∞ as in (5.10) or δ v = 1, whichever is bigger. We define d G , s G , w G ∈ Z >0 as in the proof of Theorem 5.4. From Lemma 5.3 and the fact that it follows that |µ can,EP Sγ Q A 2 ). (6.6) We will bound q Sγ independently of γ, by considering the partition S γ = (S γ ∩ S) (S γ ∩ S 0 ) (S γ \S γ,0 ) S γ,0 .
Corollary 6.5. We have the limit multiplicity formulas More precisely, for each > 0, there exists δ > 0 with the following property: for every finite set of finite places S such that S ∩ Ram(G) = ∅ and for every ξ ∈ Ξ =1 such that q S dim(ξ) > δ (while S 0 is fixed), we have |m(ξ, St S , K S 0 )| < · d(G ∞ ) dim(ξ)τ (G, S). The second limit formula is interpreted in a similar way.
Proof. This follows from the preceding theorem exactly as Corollary 5.8 does from Theorem 5.4. Example 6.6. Consider the case when G = PGL(2) over a totally real field F and S 0 = ∅. Then d(G ∞ ) = 1, q(G ∞ ) = (−1) [F :Q] , and the right hand side of (6.7) is (−1) [F :Q] . (When F = Q, corresponding to classical holomorphic modular forms of even weight k ∈ Z ≥2 10 with trivial Nebentypus character is the representation ξ k = Sym k−2 of PGL(2) so that 10 The regularity condition on ξ excludes k = 2 but we can easily work out the case k = 2. The simple trace formula is still valid for the same test function φ. The only extra work is to bound the extra spectral terms from one-dimensional automorphic representations, cf. Lemma 6.2, which do not show up when k > 2. dim(ξ k ) = k − 1. Similarly dim(ξ) is computed for general F .) Gross's motive for PGL(2) is Q(−1). We can easily compute τ (G) = 2 and τ (G, S) = ζ F (−1)2 1−[F :Q] where ζ F is the Dedekind zeta function (in particular ζ Q (−1) = −1/12). Since dim G − rkG = 2, we can take B = 1 in Theorem 6.4 by its proof and ν ∞ = 1 by Remark 5.5. So we have the asymptotic formula Here the bound O(1) comes from q −1 S v∈S (q v − 1) = O(1). In the special case when F = Q and ξ k = Sym k−2 is fixed, we have where φ(·) is Euler's phi-function. The difference from [ILS00, Cor 2.14] by the factor 2 |S| is explained as follows. At each v ∈ S there is another square-integrable representation St v such that St v and St v differ as representations of GL 2 (Q v ) by the unique nontrivial unramified quadratic character of Q × v . The result in loc. cit. can be interpreted as σ S m(ξ k , σ S , ∅), where the sum runs over σ S such that σ v ∈ {St v , St v }. Thus their count is 2 |S| times ours. In this special case, observe that our O(1) in (6.8) improves on the error bound O((kq S ) 2/3 ) obtained in loc. cit.
Corollary 6.7. Suppose that G is a split simple reductive group over F . There exist constants , A, B > 0 such that for every ξ ∈ Irr reg C (G(F ∞ )) and for every φ S = 1 K S gK S ∈ H ur,≤κ (G(A S F )) with some g ∈ G(A S F ), The implicit constant in O(·) depends on G, S, S 0 , K S 0 , and C (but is independent of ξ and φ S ).
Proof. This is proved in the same way as Corollary 5.10.
As an immediate consequence of Theorem 6.4, we deduce the existence of representations with very mild ramification (e.g. one can take S to be a singleton) and fixed weight.
Corollary 6.8. Let G be a split simple reductive group over F . Fix ξ ∈ Irr reg (G(F ∞ )). There exists a constant q 0 > 0 with the following property: for every finite set of places S such that G is unramified away from S, if q S > q 0 then there exists a cuspidal automorphic representation π such that • π ∞ ∈ Π ∞ (ξ) (in particular it is a discrete series representation), • π v is the Steinberg representation at each v ∈ S, • π is unramified at every finite place v / ∈ S.
Proof. We take S 0 = ∅. Theorem 6.4 implies that F(ξ, St S , ∅) is nonempty if q S is sufficiently large since the main term in the right hand side of the theorem is nonzero. Any π ∈ F(ξ, St S , ∅) is (not only discrete but) cuspidal since it has a Steinberg component, cf. [Lab99,Prop 4.5.4].