Characters and growth of admissible representations of reductive p-adic groups

We use coefficient systems on the affine Bruhat-Tits building to study admissible representations of reductive p-adic groups in characteristic not equal to p. We show that the character function is locally constant and provide explicit neighbourhoods of constancy. We estimate the growth of the subspaces of invariants for compact open subgroups.

Let F be a non-Archimedean local field, possibly of nonzero characteristic, and let G be a reductive algebraic group over F, briefly called a reductive p-adic group.Let π be an admissible representation of G on a complex vector space V .Since V K has finite dimension for every compact open subgroup K ⊆ G, the operator π(f ) has finite rank for all test functions f .The resulting distribution θ π (f ) := tr(π(f ), V ) is called the character of π.Since V usually has infinite dimension, the operators π(g) need not be trace-class for g ∈ G. Nevertheless, Harish-Chandra could show that the character is described by a locally integrable function: Theorem 1.1 (Harish-Chandra).Let π : G → Aut(V ) be an admissible representation of a reductive p-adic group.
(a) The operator π(g) has a well-defined trace tr π (g) when g belongs to the set G rss of regular semisimple elements.(b) The function tr π : G rss → C is locally constant.(c) The function tr π , extended by 0 on G\ G rss , is locally integrable with respect to the Haar measure µ on G, and for any test function f , g) tr π (g) dµ(g).
Supported by the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.
(d) Let D(g) for g ∈ G rss be the determinant of Ad(g) − 1 acting on Lie(G) / Lie(T ) for a maximal torus T in G containing g.The function G ∋ g → |D(g)| 1/2 tr π (g) is locally bounded.
The original proof of this deep theorem is distributed over various papers of Harish-Chandra collected in [7].A complete account of it can be found in [8].The proofs of (c) and (d) use the exponential mapping for G, which only works well if the characteristic of F is zero.It is reasonable to expect that (c) and (d) are valid in non-zero characteristic as well, but the authors are not aware of a proof.According to [24, paragraph E.4.4] Harish-Chandra's proof of (a) and (b) remains valid if one replaces C by an algebraically closed field of characteristic unequal to p.
In this article we generalise part of Theorem 1.1 to representations on modules over unital rings in which p is invertible.In this purely algebraic setting, we can only define the character as a function where it is locally constant.To prove (a) and (b), we describe explicit neighbourhoods on which tr π is constant.In characteristic 0, similar results are due to Adler and Korman [1].
Parts (c) and (d) seem specific to real or complex representations because they involve analysis.Unfortunately, our methods are insufficient to (re)prove them, as we discuss in the last section.
As a substitute we estimate the dimension of invariant subspaces V K for certain compact open subgroups K in G.The authors have not found growth estimates for these dimensions in the literature.Since V K is the range of an idempotent K in the Hecke algebra associated to K, we get But the estimate in (d) is not strong enough to control these integrals.
Our methods are of a geometric nature and involve the affine building of G. Thus we will make extensive use of Bruhat-Tits theory, including some hard parts.At the same time, we use only little representation theory.Both of our main results use the resolutions constructed by Schneider and Stuhler [18].These resolutions are based on a family of compact open subgroups U (e) x for e ∈ N, indexed by vertices of the affine Bruhat-Tits building.These generate subgroups U (e) σ indexed by polysimplices in the building.The invariant subspaces V U (e) σ in an admissible representation V form a locally finite-dimensional coefficient system on the building.It is shown in [11] that this coefficient system is acyclic on any convex subcomplex of the building.In particular, it provides a resolution of V of finite type.
Here we need acyclicity also for finite subcomplexes of the building because this provides chain complexes of finite-dimensional vector spaces, which are used in [11] to express the character of V as a sum over contributions of polysimplices in the building.We use this formula to find for each regular semisimple element γ and each vertex x in the building a number r such that the character is constant on U (r) x γ; the constant r depends the distance between x and a subset of the building corresponding to the maximal torus containing γ, on the (ir)regularity of γ, and on the level of the representation V , that is, on the smallest e ∈ N such that V is generated by the U (e) y -invariants for all vertices y.Along the way, we also prove some auxiliary results that may be useful in other contexts.We prove that the parabolic subgroup contracted by an element of a reductive p-adic group is indeed parabolic and, in particular, algebraic (Proposition 2.3).We describe which points in the building are fixed by a semisimple element in Section 4. We establish that the level of representations is preserved by Jacquet induction and restriction (Proposition 5.8).The relationship between character function and distribution is made precise in an algebraic setting in Section 6.

The structure of reductive algebraic groups
We fix our notation and recall some general facts from the theory of linear algebraic groups.Nothing in this section is new and most of it can be found in several textbooks, for example [20].
Let G be a linear algebraic group defined over a field F. The collections of characters and cocharacters of G are denoted by X * (G) and X * (G), respectively.Let G := G(F) be its group of F-rational points.By definition, an algebraic (co)character of G is a (co)character of G that is defined over F. The corresponding sets are denoted by X * (G) and X * (G).Let Z(G) be the centre of G and let Z c (G) be the maximal connected algebraic subgroup of Z(G).We denote the centraliser in G of an element g ∈ G by Z G (g).
We will assume throughout that G is connected and reductive.An algebraic subgroup P of G is parabolic if G/P is a complete algebraic variety.We denote the unipotent radical of P by R u (P).A Levi factor of P is a reductive subgroup M such that P = M ⋉ R u (P).
We write Z(G), Z c (G), P , R u (P ), and M for the groups of F-points of Z(G), Z c (G), P, R u (P), and M, respectively.We denote the space of F-points of the Lie algebra of G by Lie F (G).
We say that an algebraic torus T splits over F if T (F) ∼ = (F × ) dim T as F-groups.We say that G splits (over F) if there is a maximal torus T of G that splits over F.

Proposition 2.1.
There is a finite Galois extension of F over which G splits.
Proof.For tori this was first proven by Ono [14,Proposition 1.2.1].This implies the result for general reductive groups.
Let S be maximal among the tori in G that split over F and let S := S(F).We call S a maximal split torus in G. Notice that every algebraic (co)character of S is defined over F, as S is split.Let Φ = Φ(G, S) ⊂ X * (S) be the root system of G with respect to S, and let Φ ∨ ⊂ X * (S) be the dual root system.Let Z G (S) and N G (S) denote the centraliser and the normaliser of S in G and let Z G (S) and N G (S) be their groups of F-points.The Weyl group of Φ is The root system Φ need not be reduced if G is not split.The corresponding reduced root system is For every root α ∈ Φ(G, S) there is a unipotent algebraic subgroup U α ⊂ G with group of F-points U α , characterised by the following two conditions: ) is the sum of the S-weight spaces for α and 2α, with respect to the adjoint action of S on Lie F (G).
If α, 2α ∈ Φ then U 2α U α , and it is convenient to write The groups U α /U 2α and U 2α are naturally endowed with the structure of an F-vector space and are isomorphic to their respective Lie algebras.The subset α∈Φ red U α ∪ Z G (S) generates the group G. Let Φ + be a system of positive roots in Φ and let ∆ ⊆ Φ red be the corresponding basis.Any subset D ⊆ ∆ is a basis of a root system Φ D := ZD ∩ Φ.The algebraic subgroup P D of G generated by Z G (S) and the U α with α ∈ Φ D ∪ Φ + is parabolic.Its unipotent radical is generated by the U α with α ∈ Φ + \ Φ + D .The group M D that is generated by α∈ΦD U α ∪ Z G (S) is a Levi subgroup of P D .Moreover, We note that P ∅ is a Borel subgroup of G, that P ∆ = M ∆ = G, and that S ∆ (F) is the unique maximal split torus of Z(G).Definition 2.2.Groups of the form P D are called standard parabolic (with respect to S and Φ + ).
Every parabolic subgroup of G is conjugate to exactly one standard parabolic subgroup.Let Φ − := −Φ + be the set of negative roots and let PD be subgroup of G generated by Z G (S) and the U α with α ∈ Φ D ∪ Φ − .The parabolic subgroup PD is opposite to P D in the sense that We shall also need the pseudo-parabolic subgroup ( 2) for an algebraic cocharacter χ : F × → G.This limit is meant purely algebraically, by definition it exists if and only if the corresponding map F × → G extends to an algebraic morphism F → G.In a reductive group, any pseudo-parabolic subgroup is the group of F-points of a parabolic subgroup by [20,Lemma 15.1.2].
From now on we assume that the field F is endowed with a non-trival discrete valuation v : F → Q ∪ {∞}.We fix a real number q > 1 and we define a metric on F by d(λ, µ) = q −v(λ−µ) .Via an embedding G → GL n , the metric d yields a metric on G = G(F) as well.Even though there is no unique way to do this, the resulting collection of bounded subsets of G is canonical.This bornology on G is compatible with the group structure, in the sense that B −1 1 B 2 is bounded for all bounded subsets B 1 and B 2 of G.It follows directly from the properties of a valuation that every finitely generated subgroup of (F, +) is bounded, and this implies that every unipotent element of G generates a bounded subgroup.
Following Deligne [6], we assign to any g ∈ G the parabolic subgroup contracted by g, (3) The following result, which will be needed in Section 7.2, was proved in [15, Lemma 2] under the additional assumptions that G is semisimple and almost F-simple.Although it is apparently well-known that it holds for general reductive groups, the authors have not found a good reference for this.
Proposition 2.3.The subgroups P g and M g for g ∈ G have the following properties: (a) P g is a parabolic subgroup of G.
Proof.We first establish (a).Clearly, P g is a subgroup of G that contains g.The difficulty is to show that P g is an algebraic subgroup of G, although it is defined in topological terms.Choose a finite extension field F g of F which contains the roots of the characteristic polynomial of g.Then we have a Jordan decomposition g = g s g u = g u g s in G(F g ), see [20,Section 2.4].Let T be a maximal torus in G defined over F g that contains g s , and let F be a finite extension field of F g over which T splits (Proposition 2.1).We may and will assume that F is normal over F. According to [19,Section I.4] the valuation v extends to a valuation ṽ on F. We abbreviate G( F) = G, and similarly for its algebraic subgroups.Let Φ be the root system of G with respect to T .
Since g u is unipotent, K := {g n u : n ∈ Z} is a bounded subgroup of G, and it centralises g s .For α ∈ Φ and p ∈ Ũα \ {1}, the following are equivalent: We may choose a system of positive roots Φ+ with ṽ α(g s ) ≥ 0 for all α ∈ Φ+ .Let D ⊆ ∆ be the set of simple roots with ṽ α(g s ) = 0.The group Pg is generated by T := T ( F) and all Ũα with α ∈ Φ+ ∪ Φ D .Thus Pg is the group of F-points of the parabolic subgroup P D of G, and the collection of non-zero weights of T in Lie F(P D ) equals (5) α ∈ Φ : ṽ α(g s ) ≥ 0 =: Φ(P g , T ).
Thus Pg = P ( χΓ ).The cocharacter χΓ is defined over FΓ .The field extension F ⊆ FΓ is finite and purely inseparable, see for example [10,Section 7.7].Hence some positive multiple χ of χΓ is defined over F and still satisfies (6).This yields Pg = P (χ) and finishes the proof of (a).Now we prove (b).Lie F(P g ) is spanned by the vectors X ∈ Lie F(G) with Ad(g s )X = λX with ṽ(λ) ≥ 0. Similarly, Lie F R u (P g ) is spanned by the root subspaces Lie F(U α ) with α ∈ Φ(P g , T ) but −α / ∈ Φ(P g , T ).These are precisely the α ∈ Φ with ṽ α(g s ) > 0. Therefore Since all powers of g u are contained in the bounded subgroup K, these statements are also equivalent to lim n→∞ g n hg −n = 1.Now (b) follows because R u (P g ) = R u ( Pg ) ∩ P g .
Next we establish (c).Let χ be a cocharacter of G defined over F with P g = P (χ).The same reasoning as in the proof of (a) shows that P g −1 = P (−χ).The assertion (c) now follows by applying [20,Theorem 13.4.2] to P g and P g −1 .
Finally, we turn to (d).The eigenvalues of Ad(g s ) acting on Lie F(M g ) all have valuation 0. Hence Ad(g) lies in a bounded subgroup of the adjoint group of Mg .Equivalently, the image of g in Mg / Z( Mg ) generates a bounded subgroup.Finally, we note that M g / Z(M g ) can be identified with a subgroup of Mg / Z( Mg ).

Some Bruhat-Tits theory
We keep the notation from Section 2. Let F be a non-Archimedean local field with a discrete valuation v.We normalise v by v(F × ) = Z.Let O ⊂ F be the ring of integers and P ⊂ O its maximal ideal.The cardinality q of the residue field O/P is a power of a prime number p.We briefly call F a p-adic field.
Bruhat and Tits [3,4,21] constructed an affine building for any reductive p-adic group G = G(F).More precisely, they constructed two buildings, one corresponding to G and one corresponding to the maximal semisimple quotient of G.We call the latter the Bruhat-Tits building of G and denote it by B(G, F).Relying on [18, § 1.1] and [23, Section 1], we now recall its construction.The main ingredients are certain subgroups U α,r and H r of G.
3.1.The prolonged valuated root datum.Let •, • : X * (S) × X * (S) → Z be the canonical pairing.There is a unique group homomorphism be the maximal compact subgroup of Z G (S).
Bruhat and Tits [4] defined discrete decreasing filtrations of H and U α by compact open subgroups H r and U α,r , respectively.These groups satisfy the properties of a "prolonged valuated root datum" [3, § 6.2].We first describe these subgroups in the special case where G splits over F. Then each U α is a one-dimensional vector space over F, and a Chevalley basis of Lie F (G) gives rise to an isomorphism U α ∼ = F. Chevalley bases are known to exist but they are not unique.We fix one, and we use suitable subsets as bases of Lie F (P D ) and Lie F (M D ), for any standard parabolic subgroup P D with Levi factor M D .Thus U α is endowed with a discrete valuation v α and one defines (7) U α,r By assumption, the maximal split torus is a maximal torus, that is, S = Z G (S).For r < 0 we may put H r = H, but H 0 is more difficult to define.According to [4, 5.2.1] there is a canonical smooth affine O-group scheme Z such that Z(F) = Z G (S).Let Z c be the neutral component of Z and put H 0 := Z c (O).The inclusions are all of finite index.We define for r > 0 as in [18,Proposition I.2.6].Now we extend the above construction to a non-split group G. Proposition 2.1 provides a finite Galois extension F of F over which G splits.The strategy of descent is explained in [3,Chapitre 9]; the basic idea is to construct the required groups first in G( F) and then to intersect them with G(F).This does not work as such because the root system of G( F) is usually larger than that of G(F), so that must be taken into account as well.Bruhat and Tits descend in two steps: first from split to quasi-split, then from there to the general case.This is, in all probability, necessary for the proof, but the conclusions can be written down in one step.Of course it is by no means obvious that the groups we will define below form a (prolonged) valuated root datum: proving this is precisely what most of the work in [4] is dedicated to.
If X is any object constructed over F, then we will denote the corresponding object over F by X.According to [19,Proposition I.2.3]F is also a local field, and there is a unique discrete valuation ṽ : where e F/F ∈ N is the ramification index of F over F. The constructions above still work for this non-normalised valuation ṽ.
For α ∈ Φ red and r ∈ R the descent [4, 4.2.2 and 5.1.16]boils down to (10) Ũβ,2r , These groups do not depend on the chosen ordering of the factors.For a standard Levi subgroup M D ⊆ G and α ∈ Φ D , our consistent choice of Chevalley bases ensures that it does not matter whether we consider the groups U α,r in G or M D .We can use (10) to define a valuation on U α by (11) v α (u α ) := sup {r ∈ R : u α ∈ U α,r }.

The affine Bruhat-Tits building. The image of any cocharacter
The affine Bruhat-Tits building B(G, F) will be defined as G × A S / ∼ for a suitable equivalence relation ∼.
Let •, • AS be a W (Φ)-invariant inner product on A S .Then the different irreducible components Φ ∨ i of Φ ∨ are orthogonal and on RΦ ∨ i the inner product is unique up to scaling.Thus we may assume that α ∨ , α ∨ AS = 1 for all short coroots α ∨ ∈ Φ ∨ .
The centraliser Z G (S) acts on A S by This extends to an action of N G (S) on A S by affine automorphisms, such that the linear part of x → g • x is given by the image of g ∈ N G (S) in W (Φ). In particular, the action of g on A S is a translation if and only if g ∈ Z G (S).The affine hyperplanes ( 14) • F cannot be extended to a larger set with the first property.
Thus the closure of a facet is a polysimplex, and a facet is closed if and only if it is a single point.Moreover, a facet is open in A S if and only if it is of maximal dimension, in which case we call it a chamber.
The affine action of N G (S) on A S respects the polysimplicial structure.In fact, N G (S) is generated by the translations coming from Z G (S) and the reflections in the hyperplanes A S,α,k : where α ∨ ∈ Φ ∨ is the coroot corresponding to α.For a non-empty subset Ω ⊆ A S we define This gives rise to the following subgroups of G: U Ω := subgroup generated by The latter is a group because nU Ω n −1 = U nΩ for all n ∈ N G (S).For Ω = {x} we abbreviate U Ω = U x , which should not be confused with the root subgroups U α .
Given a partition Φ = Φ + ∪ Φ − of Φ(G, S) in positive and negative roots, we let U ± be the subgroup of G generated by α∈Φ ± U α .We write 3, 6.4.9]).These subgroups have the following properties: is an isomorphism of algebraic varieties, for any ordering of the factors.
As announced, the Bruhat-Tits building of G is The group G acts naturally on B(G, F) from the left, and the map . Since all maximal split tori of G are conjugate by [2, Théorème 4.21], there is a bijection between apartments in B(G, F) and maximal split tori in G.
A facet of B(G, F) is a subset of the form g • F , where g ∈ G and F is a facet of A S .For a polysimplicial complex Σ, we denote the set of vertices by Σ • and the set of n-dimensional polysimplices in Σ by Σ n for n ∈ N.
For any subset Ω ⊆ B(G, F), we denote the pointwise stabiliser of Ω by P Ω .This is consistent with (16) when Ω ⊆ A S .

Fixed points in the building
An element g of G is called compact if its image in G/Z(G) belongs to a compact subgroup of G/Z(G).According to the Bruhat-Tits Fixed Point Theorem (see [3, § 3.2]), the compact elements of G are precisely those that fix a point in the building B(G, F).In this section, we study how the fixed point subset B(G, F) γ depends on γ.
Let H be a group of polysimplicial automorphisms of B(G, F).If x, y ∈ B(G, F) H , then H fixes the geodesic segment [x, y] pointwise by [3, 2.5.4].Consequently, B(G, F) H is a convex subset of B(G, F).Recall that a chamber complex is a polysimplicial complex Σ such that: • all maximal polysimplices of Σ (the chambers) have the same dimension; • given any two chambers C 1 and C 2 of Σ, there exists a gallery of chambers connecting C 1 and C 2 .If g ∈ G is compact and belongs to a maximal split torus S of G, then there is a chamber in the corresponding apartment A S that is fixed pointwise by g.There exist, however, regular semisimple elements γ ∈ G that fix no chamber in the building pointwise.For such elements the fixed point subcomplex is not necessarily a chamber complex.But once g fixes a chamber, say, because it belongs to a maximal split torus, the fixed point subset is automatically a chamber complex: Proof.This is well-known, but we include a proof anyway.Let x ∈ B(G, F) H and let A x be an apartment that contains C and x.Since dim C = dim A x and B(G, F) H is convex, it contains an open subset of some chamber C x ⊆ A x with x ∈ C x .Thus H fixes C x pointwise and B(G, F) H is the union of all its closed chambers.
Suppose that C is any collection of chambers of an apartment A S of B(G, F).Then C∈C C is convex if and only if all minimal galleries between elements of C are contained in C. Hence B(G, F) H ∩ A S contains all minimal galleries between its chambers.
4.1.The split case.Let γ be a compact element of the maximal split torus S. Then v χ(γ) = 0 for all χ ∈ X * (S), so that γ fixes the apartment A S pointwise.The subcomplex B(G, F) γ ⊆ B(G, F) is convex and S-invariant.Its core is formed by the apartment A S and from there "hairs" extend in all directions.This terminology applies quite well to one-dimensional buildings, but in general such a hair is a (not necessarily bounded) chamber complex.Since S acts by translations on A S , it shifts all these hairs.If γ ∈ S is regular, then B(G, F) γ /S is compact by [9, Section 9.1]: the length of the hairs is finite.Now we study when an arbitrary point x ∈ B(G, F) is fixed by γ ∈ S. Choose a chamber C 0 ⊆ A S and let ρ be the retraction of B(G, F) to A S centred at C 0 .Let Φ + be a system of positive roots in Φ such that f ρ(x) (α) ≥ f C0 (α) for all α ∈ Φ + ; equivalently, Φ + contains all roots with f ρ(x) (α) > f C0 (α).Let ∆ be the basis of Φ corresponding to Φ + . Then , which together with Proposition 3.1.(c)shows that P C0 ⊆ U + C0 P ρ(D) .Since P C0 acts transitively on the set of apartments containing C 0 by [3, 7.4.9],there is u ∈ U + C0 with x = uρ(x).Thus we want to know which part of the apartment uA S is fixed by γ.
By definition, u ∈ U + C0 fixes all y ∈ A S satisfying −α(y) ≤ f C0 (α) for all α ∈ Φ + .These points constitute a cone in A S ∩ uA S , which is fixed by γ.We are interested in the larger subset (uA S ) γ , which is a convex subcomplex of B(G, F) γ .Hence the complex Y := u −1 (uA S ) γ is convex as well.Concretely, this means that Y ⊆ A S is determined by a system of equations −α(y) ≤ r α for certain r α ∈ R, α ∈ Φ + .We need some notation to make this more explicit.The singular depth of γ in the direction α ∈ Φ is sd α (γ) := v(α(γ) − 1).
Recall that the height of a positive root is defined as follows: Proposition 4.2.Let S, γ, u, and the u α be as above and let x .(b) The decomposition ( 17) is unique once we fix an ordering on Φ red , but the terms u α may depend on this ordering.Let Φ * := Φ + \ ∆ be the set of non-simple positive roots.Then α∈Φ * (U α ∩ U C0 ) generates a normal subgroup U * C0 of U + C0 .The quotient U + C0 /U * C0 is abelian and can be identified with a lattice in the F-vector 17) are independent of the ordering.
Suppose now that γ fixes uy ∈ uA S .By part (a), we have [γ, u −1 ] ∈ U + x and since γ normalises the groups U α,r for α ∈ Φ + , r ∈ R, this implies But on the vector space U α /U 2α the map a → [γ, a] can be identified with multiplication by α(γ) − 1. Hence ( 18) is equivalent to for all α ∈ ∆, which implies the statement (b).
So by construction [γ, u −1 ] α ∈ U α,fC 0 (α) and γ fixes uy if and only if moreover Assuming that this is the case, we will show by induction on ht(α) that For roots α of height 1 this is (19).For other roots α a closer look at (20) shows that [γ, u −1 ] α is a product of terms of the form (23) [ where all the β n ∈ Φ+ are different and ) is at least the minimum of −α(y) and the v α -values of the other terms (23) that occur in [γ, u −1 ] α .By the induction hypothesis and (13), these other terms have valuation at least (24) which completes our induction step.Now consider a nonreduced root α = 2β with ht(β) minimal.The same calculation shows that all terms (23) with This yields a more precise version of ( 22): Then another, similar, induction argument shows that (25) is also valid when ht(β) is not minimal.Finally, (25) implies that We remark that, given an arbitrary point y ∈ A S , the condition of Proposition 4.2.(c) on the u α is in general insufficient to ensure that γ fixes uy.Counterexamples can be found whenever Φ contains an irreducible root system of rank greater than one.
Moreover, Proposition 4.2 only says something about the fixed points of semisimple elements that lie in a maximal split torus.(We will not consider the fixed points of non-semisimple elements of G in this article.)Other F-split tori can be dealt with in the same way, since they are subconjugate to S. For elements of non-split maximal tori we need yet another aspect of Bruhat-Tits theory.

4.2.
The non-split case.The construction of the Bruhat-Tits building over p-adic fields is functorial with respect to finite field extensions by [3, 9.1.17].For any such extension F/F, the group In particular, for every g ∈ G(F) we have an inclusion where g ⊆ G( F) denotes the subgroup generated by g.
In general, B(G, F) is strictly smaller than B(G, F) Γ , even if F/F is a Galois extension (in which case Γ is its Galois group).Rousseau [17] proved that B(G, F) = B(G, F) Γ if F/F is a tamely ramified Galois extension, see also [16].Consequently, (26) is an equality for such extensions.
Let T = T (F) be a maximal torus and F/F a finite Galois extension over which T splits, as in Proposition 2.1.Since T is defined over F, it is Γ-stable, and hence the corresponding apartment ÃT ( F) of B(G, F) is Γ-stable.The action of Γ on ÃT ( F) is linear, so that the origin of ÃT ( F) is fixed.Thus Rousseau's above result implies that Any g ∈ G acts on Lie F (G) Lie F Z G (g) by the adjoint representation.The collection E(g) of eigenvalues (in some algebraic closure of F) is finite and does not contain 1. Assume that G is not a torus and that g is regular, that is, Z G (g) has the smallest possible dimension.The number sd(g) := max is well-defined because every eigenvalue lies in a finite field extension of F. For irregular g ∈ G we put sd(g) = ∞, because in that case the multiplicity of the eigenvalue 1 of Ad(g) ∈ End F Lie F (G) is too high.Finally, if G is a torus, then we define sd(g) = 0 for all g ∈ G.This definition stems from [1, Section 4], where sd(g) is called the singular depth of γ.We note that (28) sd(gz) = sd(g) = sd(hgh −1 ) for z ∈ Z(G) and h ∈ G.
Now we specialise to a compact regular semisimple element γ ∈ T .Then B(G, F) γ is non-empty by the Bruhat-Tits Fixed Point Theorem.If T /Z c (G) is anisotropic, then B(G, F) γ is a finite polysimplicial complex (see [18, p. 53]) and there is an open neighbourhood If T /Z c (G) is not anisotropic, we have a weaker substitute.Since B(G, F) γ /T is compact, there exists an open neighborhood V of γ in T such that B(G, F) g = B(G, F) γ for all g ∈ V .Let Hr be as in (12), but with respect to G( F), T ( F) .First the authors believed that one could take V = γ Hr ∩ T for any r > sd(γ), but this turns out to be incorrect in general.We thank the referee for pointing out the weakness in our former argument.Proof.In view of (26) it suffices to prove the corresponding statement for fixed points in the building B(G, F).We use the notation from the proof of Proposition 4.2, but with some additional tildes.We want to know when γ fixes uy, for some point y ∈ ÃS .According to (21), this is equivalent to and [γ, u −1 ] α is a product of terms of the form (23).
Recall from Section 3.1 that we have a Chevalley basis of Lie F(G) and corresponding isomorphisms of algebraic groups Ũα ∼ = F.Under these isomorphisms [γ, u −1 ] α becomes a sum of terms of the form It is the leading term, in the sense that λ α does not occur in the equations for U β with ht(β) < ht(α), whereas the factors λ β of the other terms do.Supposing that u β has already been fixed for all such roots β, (29) determines which u α ∈ Ũα can give rise to fixed points uy.
So, if the λ β with ht(β) < ht(α) have already been fixed, then the leading terms (1 − α(γ))λ α and (1 − α(γ)α(h))λ α both lead to certain sets of solutions for λ α , and these sets differ only in the parts of valuation at least But these parts do not influence the point uy.Hence γh fixes such a point uy if and only if γ does.Since this holds for all y ∈ ÃS we conclude that

The groups U (e) Ω
Schneider and Stuhler introduced an important system of compact subgroups of G, which they used to derive several interesting results on complex smooth G-representations in [18].These subgroups were also studied by Moy and Prasad in [12,13] for their theory of unrefined minimal types, and by Vignéras in [23] in the context of G-representations on vector spaces over general fields.
Let R be the set R ∪ {r+ : r ∈ R} ∪ {∞} endowed with the ordering We define addition and multiplication with positive numbers on R in the obvious way, so that they respect the ordering.For example r + (s+) = (r + s)+ and 2 • r+ = (2r)+.
Starting with the filtrations ( 10) and ( 12) we define for α ∈ Φ and r ∈ R: Since the filtrations are discrete, we have U α,r+ = U α,r+ǫ for sufficiently small ǫ > 0, and similarly for H r+ .For a function f : Φ ∪ {0} → R, let U f be the subgroup of G generated by α∈Φ U α,f (α) ∪ H f (0) .For non-empty Ω ⊆ A S we vary on (15) by For e ∈ R ≥0 , we define Notice that the closure Ω of Ω yields Example 5.1.Let G = GL n (F).We identify the standard apartment A S of B(GL n , F) with R n / R(1, 1, . . ., 1), such that the set of vertices is the image of Z n .Denote the smallest integer larger than r+ ∈ R by ⌈r+⌉.Recall the fractional ideals P m in F for m ∈ Z.For a point x = (x 1 , . . ., x n ) ∈ A S and e ∈ R ≥0 we have and Ω ⊂ A S is the standard chamber, defined by 1 + P e+1 P e P e P e+1 1 + P e+1 P e Ω is contained in the standard Iwahori subgroup of GL n (F), and that they are not equal because the diagonal entries differ.
The groups U (e) Ω satisfy the following unique decomposition property.Proposition 5.2 ([3, 6.4.48]).For any ordering of Φ red the product map By a diffeomorphism between p-adic algebraic varieties we mean a homeomorphism f , such that f and f −1 are given locally by convergent power series.The above product map is obviously algebraic, but its inverse need not be.
There is a version of the unique decomposition property with Φ red ∪ {0} instead of Φ red .It follows easily from Proposition 5.2, since H e+ normalises U α,r .
The above decomposition implies that the subgroups U Ω behave well with respect to field extensions and Levi subgroups.Proof.Let S and ρ S be as on page 7 and let Ã S ⊇ A S be the corresponding apartment of B(G, F).Then f * Ω (α) = f * Ω (ρ S (α)) for all α ∈ Φ.Now apply Proposition 5.2 and Equations ( 10) and (12).
Let M D = M D (F) be a standard Levi subgroup of G. Then a maximal split torus S of G is a maximal split torus of M D as well, and the standard apartment of B(M D , F) is Since S ∆ ⊆ S D , there is a quotient map between the apartments (35) in the buildings for G and M D .Proof.For Ω ⊆ A S and α ∈ Φ D we clearly have f * ΩD (α) = f * Ω (α).As the groups U α,r and H r are the same in M D and in G, the statement follows from Proposition 5.2.
We are mainly interested in the cases where Ω is a point, a facet or a polysimplex.Theorem 5.5.For a point x, a polysimplex σ, and a general subset Ω of an apartment A S , the following hold: g −1 Ω g −1 for any non-empty subset Ω of an apartment gA S .Now Theorem 5.5 holds in the entire building B(G, F).
We need one more important property.We define the hull H(σ, τ ) of two polysimplices σ and τ as the intersection of all apartments containing σ ∪ τ .This finite polysimplicial complex is a combinatorial approximation to the closed convex hull of σ ∪τ .Similarly, we can define the hull H(x, z) of two arbitrary points x, z ∈ B(G, F).The proof of [ The fixed points of the groups U α,k in the standard apartment are described by [3, 7.44 for all α ∈ Φ and k ∈ Γ α .Let ⌊r⌋ Γα for r ∈ R denote the largest element of Γ α that is strictly smaller than r.For x ∈ A S , (37), Proposition 5. x ) x∈B(G,F) • for fixed e ∈ Z ≥0 is a "consistent equivariant system of subgroups" in the terminology of [11, § 2.2] because of properties (b), (e), and (a) in Theorem 5.5 and (36).The main result of [11], which was inspired by [9, Section 7.1], uses these subgroups to construct resolutions of G-representations and suitable subsets thereof.We now describe this in greater detail.
Let π be a representation of G on a Z[1/p]-module V , where p is the characteristic of the residue field of F. For any polysimplicial subcomplex Σ ⊆ B(G, F) we define σ by Theorem 5.5.(e)above, so that V U (e) τ ⊇ V U (e) σ .Fix any orientation of B(G, F) and declare σ endowed with the opposite orientation to be equal to −σ ∈ Z{σ}.We define a boundary map (39) Here ∂(σ) is the usual boundary of σ, a weighted sum of codimension-one faces of σ.This yields a chain complex C * (Σ; V ), ∂ * , that is, ∂ 2 = 0. We augment it by (40) where g • σ is endowed with the orientation coming from σ.
This level is similar to the depth of a representation defined by Vignéras in [22, II.5.7], generalising [12].More precisely, if V is irreducible and e is the smallest integer such that V has level e, then the depth of V lies in (e − 1, e].The category of G-representations of level e is studied in [11,Section 3].If V is a complex G-representation of level e and Σ = B(G, F), then Theorem 5.6 recovers a result of Schneider and Stuhler [18, II.3.1].As we will see later, Theorem 5.6 for finite subcomplexes has independent significance.
Let P be a parabolic subgroup of G with unipotent radical R u (P ).We let The representation (π Ru(P ) , V Ru(P ) ) of P or P/R u (P is called the (unnormalised) parabolic restriction of V .Let (ρ, W ) be a smooth representation of P/R u (P ).Inflate it to a representation of P and construct the smoothly induced G-representation Ind G P (W ).This is known as the (unnormalised) parabolic induction of W . Proof.We first establish (a).We may assume that P = P D is a standard parabolic subgroup.Then U + ⊆ P D .[3, Proposition 7.3.1]yields G = P D N G (S)U C for any chamber C ⊆ A S .Since C is a fundamental domain for the action of G on B(G, F), The definition of the level and Lemma 5.4 yield x D .
This implies that V Ru(PD ) has level e as well: x D x D Ru(PD ) .
Now we establish (b).For notational convenience, we assume that P = P D is standard parabolic, so that we may identify P/R u (P ) with xD on C ∞ c (G).This coinvariant space for an increasing union of compact subgroups is the inductive limit x ∩ P D )γ −n .Here x is a pre-image of x D in the building for G for the map in (35).Thus U (e) x ∩ R u ( PD )).Any smooth compactly supported function on G/γ n (U (e) x ∩ P D )γ −n is invariant under right translation by Km for sufficiently large m because Km = 1.Hence we may rewrite x γ −n .

Since the regular representations on C
x ) have level e, so has their inductive limit.Hence xD ) has level e as asserted.

Characters of admissible representations
We define the character of an admissible representation first as a distribution and then describe how to interpret it as a locally constant function on suitable open subsets.Our discussion is purely algebraic and also works for representations over arbitrary fields whose characteristic is different from the characteristic p of the residue field of F.
There is a Haar measure µ on G such that µ(K We call H(G, Z[1/p]) endowed with this multiplication the Hecke algebra.It is an associative idempotented, non-unital Z[1/p]-algebra.Every element of G naturally defines a multiplier of be the corresponding idempotent.
A smooth representation π of G on a Z[1/p]-module V becomes a H(G, Z[1/p])module in a natural way, and we have , where the limit runs over all pro-p compact open subgroups K of G.There is a natural equivalence between the following categories: (see [11,Proposition 1.3]).We say that a representation G on a K-vector space V has good characteristic if the characteristic of the field K does not equal p.
In good characteristic, we may define the algebra H(G, K), whose smooth modules are in bijection with smooth representations of G on K-vector spaces.Such a representation (π, V ) is called admissible if V K has finite dimension for all compact open subgroups K ⊆ G.An admissible representation in good characteristic gives rise to a distribution If K = C, then Harish-Chandra's Theorem 1.1 shows that this distribution is associated to a locally integrable function, that is, θ π (f ) = f (g) • tr π (g) dµ(g) for all f ∈ H(G, C) and a locally integrable function tr π .Furthermore, tr π is locally constant on the subset of regular semisimple elements.Since this subset has full measure, the distribution θ π is determined by the values of tr π on regular semisimple elements.If V has infinite dimension, then tr π is not locally constant near a unipotent element u because the closure of the conjugacy class of u contains 1 and tr π (1) = dim V = ∞.
Since integration requires analysis, the notion of a locally integrable function is unclear for a general field K.The following definition of a character function makes sense for any field K: Definition 6.1.Let (π, V ) be an admissible K-linear representation of G and let g ∈ G.We write tr π (g By definition, the domain of definition dom tr π of tr π is open in G, and tr π is locally constant on dom tr π .Moreover, the trace property of θ π forces the function tr π to be a class function, that is, dom tr π is invariant under conjugation and tr π (gxg −1 ) = tr π (x) for all g ∈ G and x ∈ dom tr π .
In the following sections, we will show that dom tr π contains all regular semisimple elements, and given such an element g, we will describe a subgroup K for which tr π is locally constant on KgK.We begin with some preparatory results.First we describe the trace distribution as a limit of locally constant functions and relate the latter to the trace function.
Let K be a compact open pro-p subgroup of G (these exist by [11,Lemma 1.1]).Since the space V K of K-invariants in V is finite-dimensional, the linear operator π( K g K ) has finite rank for all g ∈ G. Hence defines a K-biinvariant function on G; here we used that π(g K ), π( K g K ), and π( K g) have the same trace.By construction, for all K-biinvariant compactly supported functions f on G. Let (K n ) n∈N be a decreasing sequence of compact open pro-p subgroups with K n = {1}.Then any locally constant, compactly supported function is K n -biinvariant for some n ∈ N, so that (41) holds for K = K n for all sufficiently large n.In this sense, the trace distribution is the limit of the locally constant functions χ K in a distributional sense.The following lemma is trivial: Lemma 6.2.The trace function exists at γ ∈ G and has value τ if and only if there is n 0 ∈ N with χ Kn (g) = τ for all g ∈ K n0 γK n0 and all n ≥ n 0 .Furthermore, then tr π is defined and constant on K n0 γK n0 .
Let γ ∈ G be a regular semisimple element.Then γ is contained in some maximal torus T .Let T rss ⊆ T be the subset of regular elements.It is well-known that the map is open.We are going to quantify this statement by providing compact open subgroups K, K G ⊆ G, and K T ⊆ T such that ψ(K G T × K T γ) contains KγK for a given regular element γ of T .We first consider the split case.
Lemma 6.3.Suppose that T contains the maximal split torus S of G. Then the map where we interpret U 0 as Z G (T ).Let U (n) be the group generated by the U α with α ∈ Φ + of height at least n.Then ⊇ {1} is a filtration of U + by normal subgroups.Moreover, as algebraic groups where Φ (n) denotes the set of roots of height n.The group U α /U 2α carries a canonical F-vector space structure, so we can speak of λu α for λ ∈ F and u α ∈ U α /U 2α .( 10) and ( 12) are discrete).Now Theorem 5.5.(i)yields x .In other words, ψ(u − u + , y 0 ) = y in P x /U (r ′ ) x .Next we try to find a solution of the form ψ(u − u + g, ty 0 ) = y.By (43) this is equivalent to , the right hand side lies in U (r ′ ) x y 0 .Thus we transformed the original problem x γ, repetition of this process yields a solution ψ −1 (y).Now we consider a regular element γ of a non-split maximal torus T = T (F).Furthermore, we want to generalise the statement by allowing the choice of an arbitrary x ∈ A S .Let F be a splitting field of T , let G = G( F), and let T := T ( F).This is a split maximal torus in G, which therefore corresponds to an apartment Ã T in the building B(G, F).Recall the subgroups Hr ⊆ Z G( F) T ( F) .
For x ∈ B(G, F), let π T (x) be the point of Ã T that is nearest to x.Let Ψ be the root system corresponding to an apartment of B(G, F) that contains x and π T (x).We define (44) d T (x) := max If F/F is tamely ramified, then (27) shows that Ã T ∩ B(G, F) is non-empty, that is, there is x with d T (x) = 0.Alternatively, let C ⊆ Ã T be a chamber containing π T (x), let ρ Ã T , C : B(G, F) → Ã T be the associated retraction.Then Lemma 5.3 and (34) yield ( 46) Proof.Equation (46) and Proposition 6.4 show that every element of Ũ (r+dT (x)) x γ is conjugate in G( F) to an element of Hr+ γ ∩ T ( F).Since the maps ψ : are injective and open, respectively on Ũ (0) πT (x) / H0+ × Hr+ γ ∩ T ( F ) and on the intersection of this set with G, Moreover, by Proposition 6.4 the right hand side contains There is a decreasing sequence (K n ) n∈N of normal compact open subgroups in K x with K n = {1}.Since K x is open in G, we may use this sequence to approximate the trace distribution as in (41).Since K n is normal in K x , then the space of K n -biinvariant functions is invariant under conjugation by elements of K x .This implies that the function χ Kn is invariant under conjugation by elements of K x .Therefore, Lemma 6.5 shows that χ Kn is constant on U (r+dT (x)) x γ once it is constant on Hr+ γ ∩ T .In the following, we may therefore restrict attention to elements of a torus in G.

The local constancy of characters
Let (π, V ) be an admissible representation of G in good characteristic, of level e ∈ Z ≥0 .Let γ be a regular semisimple element of a maximal torus T ⊆ G and let x ∈ B(G, F) • be a vertex in the building of G.We are going to find r(γ) ∈ N depending only on γ and the level e of the representation, such that tr π is defined and constant on U (r(γ)+dT (x)) x with d T (x) as in (44).

7.1.
Local constancy for compact elements.First we assume, in addition, that γ is a compact element, so that γ fixes some point in the affine building.The assertions for general elements are reduced to the compact case in Section 7.2.
Our definition of r(γ) is somewhat complicated and probably not optimal.It is likely that r(γ) = max{sd(γ), e} works, but we can only prove this if T has a subtorus S that is a maximal F-split torus of G.
Let T = T (F) ⊆ G be a maximal torus containing γ and let F be a splitting field of T .Recall the subgroups Ũ + ⊂ G( F) and Hr ⊆ Z G( F) T ( F) .Let B be a Borel subgroup of G( F) containing T ( F). Definition 7.1.For x ∈ B(G, F) define d T (x) as in (44) and let d(γ) ∈ R be the smallest number such that We have d(γ) < ∞ because B(G, F) γ /T is compact.If F/F is tamely ramified, then (27) shows that there is a point x ∈ B(G, F) with d T (x) = 0, so that tr π is constant on U (r(γ)) x γ.The number r(γ) will reappear frequently in the following.We will not need the definition of r(γ) but only Theorem 7.2.(a).That is, the following results remain true for a smaller value of r(γ) provided Theorem 7.2.(a) can be established for it.
Proof.(a) Theorem 5.6 implies a formula for tr(π(f ), V ), which is worked out in [11,Proposition 4.1].We need some notation to state this trace formula.For g ∈ G, let Σ g be the set of all polysimplices σ with gσ = σ and let ǫ σ (g) = ±1, depending on whether the automorphism of σ induced by g preserves or reverses orientation.For a locally constant function f supported in P x , [11,Proposition 4.1] asserts (49) tr(π(f ), V ) = lim (−1) dim σ ǫ σ (g) tr π(g), V U (e) σ dµ(g), where the limit means that there is a finite convex subcomplex Σ 0 such that the right hand side is the same for all P x -invariant finite convex subcomplexes Σ of B(G, F) with Σ ⊇ Σ 0 .Thus we want to show that the function γ for all sufficiently large P x -invariant finite convex subcomplexes Σ.The function τ Σ is invariant under conjugation by elements of P x because Σ is P x -invariant.
Lemma 4.3 yields B(G, F) g = B(G, F) γ for all g ∈ Hr(γ)+ γ ∩ T , because r(γ) ≥ ht(Φ)sd(γ).Since ( 51) the operator π(g −1 γ) restricts to the identity on V U (e) x , for all x with d T (x) ≤ d(γ).Let D be a set of simplices in B(G, F) γ , such that D is a fundamental domain for the action of B on B • B(G, F) γ and every σ ∈ D contains an interior point x with d T (x) ≤ d(γ).Equation (50) becomes We want to show that τ Σ (γ) = τ Σ (g).Write b 1 = t 1 u 1 ∈ T ( F) Ũ + , where Ũ + is the unipotent radical of B. By Lemma 6.3 the map Ũ Hence we can find This implies that γ and g fix u 2 σ, so u 2 σ occurs in the sum τ Σ (γ), although it not necessarily equals b 1 σ.Now Since Σ g = Σ γ , g −1 γ fixes σ pointwise, while in view of (51) and the definition of D, π(g −1 γ) acts as the identity on V U (e) σ .Therefore which shows that every term of the sum (52) also occurs in τ Σ (γ).The converse also holds and both sums have the same number of terms, so we can conclude that τ Σ (γ) = τ Σ (g).(b) Lemma 6.5 shows that any element of U (r(γ)+dT (x)) x γ is P x -conjugate to one of γ Hr(γ)+ ∩ T .Hence (b) follows from (a).
(c) To a large extent we will copy the proof of part (a), but we take advantage of U + • A S = B(G, F).This clearly implies d(γ) = 0, so that D is a collection of simplices of A S that form a fundamental domain for the action of Z G (S) on A S .This D works for both γ and g = γh.
With these choices the proof of (a) mostly goes through, even though we do not know whether B(G, F) g equals B(G, F) γ or not.The only problem arises in the last line, where we still have to justify that the sums τ Σ (γ) and τ Σ (g) involve the same number of terms.It suffices to show this for the number of terms n(σ, γ) (respectively n(σ, g)) corresponding to a particular simplex σ ∈ D. For sufficiently large Σ these numbers equal the number of simplices of B(G, F) γ (respectively B(G, F) g ) of the form uσ with u ∈ U + .Guided by Proposition 4.2 we have a closer look at the maps , and similarly for n(σ, g).Like in the proof of Lemma 6.3, the generalised eigenvalues of the differentials Dφ γ , Dφ g : Lie F (U + ) → Lie F (U + ) are {1 − α(γ) : α ∈ Φ + } and {1 − α(g) : α ∈ Φ + }, and they occur with multiplicity for all α ∈ Φ.Let µ U + be a Haar measure on the locally compact group U + .For any compact open subset But φ γ and φ g are diffeomorphisms, so φ −1 γ and φ −1 g also multiply volumes by the same factor.Together with (53) this shows that n(σ, γ) = n(σ, g), as required.

7.2.
Local constancy for non-compact elements.We would like to generalise Theorem 7.2 to all regular semisimple elements.This is possible using Jacquet modules and parabolic restriction as in [5].Although the methods in [5] are algebraic and not restricted to complex coefficients, Casselman refers to earlier work which was written with complex representations in mind.This makes it hard to judge whether Casselman's proofs work for representations in good characteristic.Fortunately, Vignéras [22] proved the required results in this generality.
Let γ ∈ T be a semisimple element and let P γ ⊆ G be the parabolic subgroup contracted by γ, which is defined in (3).Since F is complete with respect to the valuation v, Proposition 2.3.(d)shows that γ is compact in M γ .It follows from Proposition 2.3.(b) that Lie F R u (P γ ) ⊆ Lie F (G) is the sum of all eigenspaces of Ad(γ) corresponding to eigenvalues with strictly positive valuation.(Although the eigenvalues may lie in a field extension of F, this subspace is defined over F.) Similarly, R u (P γ −1 ) corresponds to the γ-eigenvalues with strictly negative valuation.
The description of (standard) parabolic subgroups in Definition 2.2 shows that M γ contains a maximal split torus of G, say S γ .It may happen that γ / ∈ S γ .Let x be a point of the apartment A γ of B(G, F) corresponding to S γ .Proposition 5.2 implies (55) or, in other words, U x is well-placed with respect to (P γ , M γ ).The collection X = {gx ∈ B(G, F) : g lies in the maximal compact subgroup of T } is finite and γ-invariant.Since T ⊂ M γ , the subgroup U (e) x ′ is well-placed with respect to (P γ , M γ ) for every x ′ ∈ X.The group K (e) := x ′ ∈X U (e) x ′ is also wellplaced: + , so that the sequence K (e) for e ∈ N has all the properties claimed in [6].Theorem 7.3 ([22,II.3.7]).Let (π, V ) be an admissible smooth G-representation in good characteristic and let g ∈ G be such that P g = P γ .There exist increasing sequences of finite-dimensional vector spaces V (e) ⊆ V K (e) and V (e) for all g ∈ G with P g = P γ .Notice that the set of such g is contained in M γ , so it is not open in G unless γ is compact in G.
Proof.Since γ is compact in M γ , Theorem 7.2 tells us that tr π R u (Pγ ) is well-defined and constant near γ.Pick an e ∈ N such that it is constant on γK (e) 0 .Now (56) yields As the subsets K (e) γK (e) form a neighbourhood basis of γ in G, taking the limit e → ∞ and invoking Lemma 6.2 shows that tr π (γ) is well-defined and equals tr π R u (Pγ ) (γ).This theorem, which Casselman [5] proved for complex representations, enables us to reduce the computation of traces from general semisimple elements to compact semisimple elements.Theorem 7.2 tells us on which neighbourhood of γ the function tr π Ru(Pγ ) is constant.But this is only a neighbourhood in M γ .We also want to know on which neighbourhood in G the function tr π is constant.Let r(γ) be such that Theorem 7.2.(a)holds when we view γ as a compact element in M γ .Theorem 7.5.Let γ be a regular element of a (not necessarily split ) maximal torus T of G. Let (π, V ) be an admissible representation of G of level e in good characteristic.Proof.For every root α ∈ Φ G( F), T ( F) and every g ∈ Hr(γ)+ γ ∩ T we have ṽ α(g) = ṽ α(γ) because gγ −1 is compact.Together with (5), this implies P g = P γ , so that Theorem 7.4 applies to all g ∈ Hr(γ)+ γ ∩ T and tells us that tr π (g) = tr π R u (Pγ ) (g).Theorem 7.2 and Proposition 5.8 show that tr π R u (Pγ ) is constant on Hr(γ)+ γ ∩ T , so the same goes for tr π .This proves (a), from which (b) follows upon applying Lemma 6.5.This theorem is similar to [1,Corollary 12.11], which was proved only for complex representations and "tame" elements γ.Our neighbourhoods of constancy are usually smaller than those in [1], because Theorem 7.2.(a) is not optimal.The results of Adler and Korman suggest that Theorem 7.2.(c) could be valid whenever the maximal torus T splits over a tamely ramified extension of F. Possibly this has something to do with Rousseau's result (26).

A bound for the dimension of V K
In this section, we will use the resolutions of [11] to estimate the dimension of V U (e) x for an admissible representation (π, V ) of G in good characteristic.We abbreviate K e := U (e) x .First we estimate the growth of some related double coset spaces in order to show that our later estimates are optimal, at least for GL n .
Since every irreducible smooth representation is a subquotient of a parabolically induced one, the essential case is V = Ind G P (W ), where P is a parabolic subgroup of G and (ρ, W ) is a supercuspidal representation of P/R u (P ).There is a natural isomorphism (57) V Ke ∼ = P gKe where the sum runs over all double (P, K e )-cosets.The space P \G/K e is finite because P \G is a complete algebraic variety (and hence compact in the p-adic topology) and K e is open.We will discuss how |P \G/K e | grows as e increases, under some simplifications.If P is a Borel subgroup and ρ is a character, then |P \G/K e | and dim V Ke have equivalent growth rates.Suppose that G is split.Let S be a split maximal torus of G and let P D be a standard parabolic subgroup of G.The dimension of Let x ∈ A S .By construction, the groups K e decrease equally fast in every direction; if K e corresponds to a lattice L (e) in Lie F (G), then K e+1 corresponds to PL (e) , where P is the maximal ideal in the maximal compact subring of F. Hence a double coset P D gK e contains approximately q dim(PD \G) double P D , K e+1 -cosets.Therefore, |P D \G/K e | grows, in first approximation, like q e dim(PD \G) .Now we focus on the easier example G = GL n and let P and S be the standard Borel subgroup and the standard maximal torus in GL n (F).The irreducible representations of S = P/R u (P ) are characters.Let (ρ, C) be such a character and let V be the parabolically induced representation of G. Since any character is trivial on K e ∩ S for large enough e, C P ∩gKeg −1 ∼ = C for large enough e, so that and for any map ǫ : Φ → {+, 0, −} we write A S .We may pick a non-zero vector δ ǫ ∈ A S such that (1) δ ǫ is orthogonal to A ǫ S,⊥ , (2) A ǫ S,r−e + R ≥0 δ ǫ ⊆ A ǫ S,r−e , (3) δ ǫ lies in the span of an irreducible root subsystem Ψ ∨ of Φ ∨ (here we decompose Φ ∨ as a direct sum of irreducible root systems).
This number is an upper bound for the number of U o -orbits in U o • F .Since it does not depend on F , we only need to multiply it with the number of facets of A b S,r−e .While this number is not easily expressible in a formula, it clearly grows like r dim AS .Theorem 8.5.Let (π, V ) be an admissible G-representation of level e ∈ Z ≥0 in good characteristic.Let r ∈ R ≥e and define Q and m V as in Lemma 8.4 and (63).
with constants independent of V and r. for all s ∈ Z ≥0 .A calculation like the one in (64) and (65) shows that this index is at least q rd0 α∈Φ red q rnαdα q rn2αd2α/2 .
(We cannot be exact because we do not know at which points the filtration of H jumps.)This yields the second estimate.
These estimates are sharp in some examples: (58) shows that (a) and (c) cannot be improved for GL n .Here all n α and d α are 1, Φ is reduced, and there are n(n − 1)/2 positive roots, so that Q = q n(n−1)/2 .

Conclusion
Let G be a reductive p-adic group and let (ρ, V ) be an admissible representation of G on a vector space V of characteristic not equal to p.We have seen that the character of (ρ, V ) is a locally constant function on the set of regular semi-simple elements, and we have described explicit open subsets on which it is constant.Furthermore, we have estimated the growth of the dimensions of the fixed-point subspaces V U (e) x for e → ∞.Both results are based on the main result of [11] about the acyclicity of certain coefficient systems on the affine Bruhat-Tits building.
It is still unclear whether Harish-Chandra's theorem about the local integrability of the character function for complex representations can be established using these resolutions.This may depend on a better understanding of the character formulas.While the resolution in [11] does provide an explicit formula for the character, more work is required to understand and simplify this formula.

Lemma 5 . 3 .
Let F/F be a finite field extension and let Ũ (e) Ω ⊆ G( F) be defined like U (e) Ω ⊆ G(F).Then U

Lemma 5 . 4 .
Let Ω D be the image of Ω in the standard apartment A D of the building for M D .Then U (e) ΩD = U (e) Ω ∩ M D and U (e) Ω = U (e) Ω ∩ R u (P D ) U (e) Ω ∩ M D U (e) Ω ∩ R u ( PD ) .
fixes the star of x pointwise.(e) U (e) σ = x vertex of σ U (e) x if e ∈ Z ≥0 .(f) If x is an interior point of σ and e ∈ Z ≥0 , then U ) Ω whenever e ≤ e ′ .(h) The groups U (e) σ for e ∈ N form a neighbourhood basis of 1 in G. (i) The group generated by the commutators U (e) Ω , U (e ′ ) Ω is contained in U (e+e ′ ) Ω .Since U α,r = {1} if and only if r = ∞, (a) follows from Proposition 5.2.Statements (c) and (d) show that the order of the product in (e) does not matter.The proofs of (b)-(e) and (g)-(h) may be found in [18, Section I.2].Property (f) is [23, Proposition 1.1], whereas (i) follows from [3, 6.4.41].Notice that so far these properties hold only for subsets of the standard apartment A S .However, (c 23, Lemma 1.28] yields (a) If x, z ∈ B(G, F) and y ∈ H(x, z), then U

Proposition 5 . 8 .
Let P ⊆ G be a parabolic subgroup.(a) If V is a G-representation of level e, then V Ru(P ) is a representation of P/R u (P ) of level e.(b) If W is a representation of P/R u (P ) of level e, then Ind G P (W ) has level e.
and only if it is a quotient of a direct sum of copies of the regular representation on C ∞ c (M D /U (e) xD ) for points x D in the building of M D ; here C ∞ c denotes the space of locally constant functions with compact support.Since Jacquet induction preserves direct sums and quotients, it suffices to prove that the Jacquet induction of C ∞ c (M D /U (e) xD ) has level e. Inspection shows that this Jacquet induction is isomorphic to the regular representation on C ∞ c (G/R u (P D )U (e) xD ).The subgroup R u (P D )U (e) xD of G is an inductive limit of compact subgroups because U (e)xD is compact and R u (P D ) is unipotent.It is useful to choose a special sequence of compact subgroups exhausting R u (P D ), namely,K n := γ n (U (e)xD ∩ R u (P D ))γ −n , where γ is a central element of M D that is strictly positive, that is, K n = R u (P D ).We also consider the subgroups Kn := γ n (U (e) xD ∩R u ( PD ))γ −n in the opposite unipotent group; then Kn = {1}.The space C ∞ c (G/R u (P D )U (e) xD ) is the coinvariant space for the right action of R u (P D )U (e)

( a )
The function tr π is defined and constant on Hr(γ)+ γ∩T , and on all G-conjugacy classes intersecting this set.(b) The function tr π is constant on U (r(γ)+dT (x)) x γ, for any x ∈ B(G, F).

.Lemma 8 . 2 .(− 1 )F
Most of the sets A ǫ S,r are empty, some are compact, and the others are unbounded.The non-empty A ǫ S,r partition A S .Let A b S,r be the union of the bounded A ǫ S,r ; this is a polysimplicial subcomplex of A S which is star-shaped around o.The subcomplex B r := P o • A b S,r of B(G, F) is obviously stable under the action of all the groups U (s) o for s ∈ R ≥0 .We may think of B r as a combinatorial approximation to a ball of radius r around o. Let r ∈ Z ≥e and let Σ ⊆ B(G, F) be any finite convex subcomplex that contains B r−e .ThenU (r) deg σ U (r) o U (e) σ .Proof.Fix ǫ : Φ → {+, 0, −} such that A ǫ S,r−e is unbounded.First we establish U ′ for certain facets F, F ′ ⊆ A ǫ S,r−e .The coroots α ∨ ∈ Φ ∨ with ǫ(α) = 0 span a proper subspace A ǫ S,⊥ for F ⊆ A ǫ S,r−e .In view of the unique decomposition property (Proposition 5.2) this is equivalent toU (r) o ∪ U (e) F ∩ U α = U (r) o ∪ U (e) M(F ) ∩ U α for all α ∈ Φ red .By definition, U(r) o∩ U α = U α,r+ and U (e)F ∩ U α = U α,−α(x)+e+ for x ∈ F .If ǫ(α) = −, then −α + e > r on F ∪ M (F ), so that U α,r+ ⊇ U α ∩ U (r) o ∪ U (e) M(F ) .If ǫ(α) = −, then sup x∈F −α(x) ≤ sup x∈M(F ) −α(x), which combined with U