Orbital integral bounds the character for cuspidal representations of $GL_n(mathbb{F}_{ell}((t)))$

We will denote by C -∞ (G) the space of generalized functions on G, i.e. functionals on the space of smooth compactly supported measures.

1.1. Orbital integrals. Our main result involves the notion of the orbital integral of a function on G. Let us first define this notion:

Let G rss be the collection of regular semisimple elements in G.

• Denote by µ G the Haar measure on G normalized such that the measure of a maximal compact subgroup in G is 1. • For x ∈ G rss denote by µ Gx the Haar measure on the torus G x normalized such that the measure of the maximal compact subgroup of G x is 1. • For x ∈ G rss denote by µ G•x the Ad(G)-invariant measure on the conjugacy class G • x :=Ad(G) • x that corresponds to the measures µ G and µ Gx under the identification Ad(G)

1.2. Main results. For an irreducible representation ρ of G we denote by χ ρ its character, which is a generalized function on G. Our main result consists of a bound on this character in terms of the orbital integral of a function on G. In order to formulate the bound we need some notation:

• For x ∈ G rss denote by ∆(x) the discriminant of the characteristic polynomial of x. • For x ∈ G let ov G (x) := max(max i,j (-val(x ij )), val(det(x))) where

x ij are the entries of x.

• For x ∈ G rss let ov G rss (x) = max(ov G (x), val(∆(x))).

Theorem A ( §1.4). Let ρ be a cuspidal irreducible representation of G. Let m be a matrix coefficient of ρ m(1) ̸ = 0. Then there exists a polynomial α ρ,m ∈ N[t] such that for every η ∈ C ∞ c (G) we have

where f ∈ C ∞ (G rss ) is defined by f (g) = α ρ,m (ov G rss (g)).

Remark. A priori, the right hand side of the above inequality can be infinity. We interpret the statement in that case as void.

1.3. Background and motivation. When the characteristic of F is zero, Theorem A is proven in [HC70, page 102]1 . This is an important step in the proof of Harish-Chandra’s integrability theorem: “The character of an irreducible cuspidal representation of a p-adic reductive group is given by a locally integrable function”, [HC70]. The proof in [HC70, page 102], as well as our proof of Theorem A, is based on the fact that averaging of cuspidal functions on G is bounded (up to a logarithmic factor) by their orbital integral. See Theorem B below.

This fact (in characteristic 0) is also an important step in the proof of Harish-Chandra’s integrability theorem for general (not necessarily cuspidal) irreducible representations.

In a sequel work [AGKS] we use Theorem A in order to prove an analog of Harish-Chandra’s integrability theorem for cuspidal representations of GL n (F ℓ ((t))) under some additional assumptions.

1.4. Idea of the proof. In our argument we will use the following language.

Several statements in this paper will concern the existence of certain polynomials in N[t] that satisfy some conditions. In the formulation of each such statement we assign a name for the corresponding polynomial. It is implied that after each such statement we fix such a polynomial and we will refer to it later by this name. Nothing significant will depend on the choices of these polynomials.

We note that in many of the statements one can actually choose this polynomial to be a linear function, but this is not essential to our argument.

Following [HC70] our proof can be divided into 2 steps:

(1) The character of ρ is, up to a scalar, the (weak) limit of the sequence of functions A i (m), where m is a matrix coefficient of ρ, and A i (m) is the averaging of m w.r.t. a ball G i in G. See Theorem 1.4.2 below. (2) Given x ∈ G rss one can bound all A i (m)(x) in terms of ov G rss (x) and Ω(m)(x), uniformly in i. We now provide a formal description of these ingredients. In order to formulate the first ingredient, let us define the notion of averaging: Definition 1.4.1. Denote

• Z(G) to be the center of G.

• G ad := G/Z(G).

• G i := {x ∈ G|ov G (x) ≤ i}. • (G ad ) i to be the image of G i under the map G → G ad .

• µ Z(G) to be the Haar measure on Z(G) normalized such that the measure of the maximal compact subgroup of Z(G) is 1. • µ G ad to be the Haar measure on G ad corresponding to µ G and µ Z(G) .

• For a function f ∈ C ∞ (G), denote its averaging A i (f ) ∈ C ∞ (G) by

where dg is the Haar measure µ G ad .

Let us recall the notion of matrix coefficient of a representation (ρ, V ρ ). For a pair v ∈ V ρ , φ ∈ V ρ , the corresponding matrix coefficient is a smooth function on G defined by m v,φ (g) = φ(ρ(g)v).

We can now formulate the formula for the character of a cuspidal representation:

Theorem 1.4.2 ( §3, cf. [HC70, Theorem 9]). Let (ρ, V ρ ) be an irreducible cuspidal representation of G. Then there exists a positi