The celebrated Harish-Chandra's integrability theorem states that the distributional character of an irreducible smooth representation of a p-adic group $G(F)$ is integrable, that is represented by an $L^1_{loc}(G(F))$ function. Here $F$ is a non-Archimedean local field of characteristic $0$ and $G$ is a reductive algebraic group defined over $F$. In this paper we focus on cuspidal representations of $GL_n(F)$ for a field $F$ of positive characteristic. We show that in this case the integrability holds under the hypothesis of existence of desingularization of (certain) algebraic varieties in positive characteristics. Furthermore, in the case $char(F)>n/2$ we establish the regularity of such characters unconditionally.
Throughout the paper we fix a non-Archimedian local field F of arbitrary characteristic. Denote by ℓ the size of the residue field of F . All the algebraic varieties and algebraic groups that we will consider are defined over F . We will also fix a natural number n and set G = GL n , considered as an algebraic group defined over F . Denote G = G(F ).
We will denote by C -∞ (G) the space of generalized functions on G, i.e. functionals on the space of smooth compactly supported measures. We also denote by L 1 loc (G) the space of locally L 1 -functions on G and consider it as a subspace of the space of generalized functions C -∞ (G) in the usual way.
1.1. Main results. We study the following conjecture:
Conjecture A. Let ρ be an irreducible cuspidal smooth representation of G and let χ ρ ∈ C -∞ (G) be its character. Then χ ρ ∈ L 1 loc (G). When the characteristic of F is zero, this is a special case of a well known result of Harish-Chandra [HC70]. In this paper we show that this conjecture follows from the conjectural existence of resolution of singularities in positive characteristic.
More precisely, consider the following:
Conjecture B. Let Z be an algebraic variety defined over the finite field F ℓ .
Then there exists a proper birational map γ : Z → Z s.t.
• Z is smooth.
• γ is an isomorphism outside the singular locus of Z.
• The preimage of the singular locus of Z (considered as a subvariety of Z) is a strict normal crossings divisor.
In this paper we prove:
Theorem C ( §12). Conjecture B implies Conjecture A.
We also prove the following unconditional result:
Proposition D ( §12). If char(F ) > n 2 then Conjecture A holds. Remark 1.1.1. In fact, for given F and n it is enough to assume Conjecture B for a specific variety defined over F ℓ . We also give some other alternatives that replace the role of Conjecture B in Theorem C, see §13.
Remark 1.1.2. We also prove analogues of Theorem C and Proposition D for orbital integrals. See §1.4 below.
1.2.1. Previous results. In [CGH14, Theorem 2.2] it was established that local integrability of characters of irreducible representations of reductive groups over F ℓ ((t)) holds true for large enough characteristics (depending on the group G). However, no explicit bound was given.
The case of GL 2 (F ) was already proven in [JL70,Chapter 9].
In [Rod85] it was established that local integrability of characters of irreducible representations of GL n (F ℓ ((t))) holds true in neighborhoods of elements with separable characteristic polynomials. In particular the local integrability holds whenever char(F ℓ ) > n.
In a series of papers ( [Lem96], [Lem04], [Lem05]) it was claimed that local integrability holds true in arbitrary characteristics for the family of groups GL n (F ), GL n (D), SL N (D) where F = F ℓ ((t)) is a local non-Archimedean field and D a division algebra over F . However the arguments in these papers have a flaw. See more detailed explanation in Appendix B.
On the other hand, it seems that the argument in [Lem96] can give a proof for Proposition D of the present paper. 1.2.2. The original argument of Harish-Chandra. Let us shortly present the main parts of the original Harish-Chandra’s proof of the local integrability of cuspidal characters from [HC70]. This presentation differs slightly from the original, as it is adapted to better suit our purposes. One can roughly divide Harish-Chandra’s proof into two parts:
(1) Bound the character (up to a logarithmic factor) by the inverse square root of the discriminant -|∆| -1 2 .
(2) Prove the integrability of |∆| -1 2 . In more details, let p : G → C := (G//Ad(G))(F ) be the Chevalley map. one can divide the first step into the following sub-steps:
(a) Locally bound the character by the orbital integral Ω(f ) of a smooth function f ∈ C ∞ c (G) (up to a logarithmic factor). See Notation 3.0.1 for the definition of Ω(f ). We did this in [AGKSc]. (b) Bound the orbital integral Ω(f ) by a product |∆| -1 2 • p * (p * (f )) where, the push forward is taken w.r.t. some fixed, smooth, nowhere vanishing measures on G and C. (c) Bound p * (f ). 1.2.3. Difficulties with Harish-Chandra’s argument in positive characteristic.
Step (2) does not hold in positive characteristic (even for the case of GL 2 ((t))). So, one should replace |∆| -1 2 with a better bound (like the function κ described in §4 below).
Both substep (1)(a) and step (2) are done for each torus in G separately. This is enough in characteristic zero, as there are only finitely many conjugacy classes of tori. However, the latter is no longer true in positive characteristic. See more details in [AGKSb,§1.5] Substep (1)(c) uses the assumption on characteristic in many places. See more details in [AGKSb, §1.5.1].
1.2.4. The approach of [JL70, Chapter 9]. The proof of [JL70, Chapter 9] in the GL 2 case goes essentially along the same lines as the proof of [HC70]. All the bounds are much more explicit, and the bound |∆| -1 2 is replaced by
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