A proof of Harish-Chandra's integrability theorem for cuspidal representations of $\mathrm{GL}_n(\mathbb F_\ell((t)))$

Abstract: Consider the Chevalley map $$p:\mathfrak{gl} <em>n(F)\to (\mathfrak{gl}_n//\mathrm{GL}_n)(F),$$ where $F=\mathbb{F}</em>\ell((t))$. We show that the push forward via $p$ of every smooth compactly supported measure on $\mathfrak{gl}_n(F)$ is a measure whose density belongs to $L<sup>q$ for every finite $q$. As a consequence, using the main result of [AGKSc], we obtain local integrability for Harish--Chandra's characters of irreducible cuspidal representations of $\mathrm{GL}_n(F)$.

• The paper establishes a proof of local integrability for Harish-Chandra characters of cuspidal representations of GLₙ over Fₗ((t)).
• The methodology innovatively combines an explicit analysis of the Chevalley map with a concrete resolution of singularities to control the Jacobian determinant.
• The results eliminate previous restrictions, bridging the gap in local character theory for p-adic groups in positive characteristic.

Proof of Harish-Chandra's Integrability Theorem for Cuspidal Representations of $\mathrm{GL}_n(\mathbb F_\ell((t)))$

Introduction and Context

The work establishes an unconditional proof of local integrability for Harish-Chandra's characters associated to irreducible cuspidal representations of $\mathrm{GL}_n(F)$, where $F = \mathbb{F}_\ell((t))$ is a local field of positive characteristic. This result addresses a major open case in the representation theory of reductive $p$-adic groups in positive characteristic, where previous approaches either relied on resolution of singularities or imposed explicit bounds or restrictions on the characteristic or the group structure.

The methodology synthesizes tools from algebraic geometry, invariant theory, and harmonic analysis, with a particular emphasis on an explicit analysis of the Chevalley map and its singularity structure in the context of positive characteristic.

Chevalley Map and Pushforward Integrability

The principal technical object is the Chevalley map $$p : \mathfrak{gl}_n(F) \to (\mathfrak{gl}_n // \mathrm{GL}_n)(F)$$, concretely realized as the passage from a matrix to the coefficients of its characteristic polynomial. The key is to study, for a smooth compactly supported measure $\mu$ on $\mathfrak{gl}_n(F)$, the pushforward $p_*\mu$. Establishing absolute continuity of $p_*\mu$ relative to Haar measure $\mu_\mathfrak{c}$ on $\mathrm{GL}_n(F)$0 leads to investigation of the Radon-Nikodym derivative $\mathrm{GL}_n(F)$1.

The central assertion is that for any such $\mathrm{GL}_n(F)$2, the induced density belongs to $\mathrm{GL}_n(F)$3 for all finite $\mathrm{GL}_n(F)$4, i.e.,

$\mathrm{GL}_n(F)$5

which, by translation, yields the local integrability of Harish-Chandra's character for cuspidal smooth irreducible representations.

Structural and Technical Innovations

Reduction via Affine Partial Structure

A critical observation is that the Chevalley map is affine linear in the last column of the input matrix, conditioned on fixing the remaining columns. This reduction frames $\mathrm{GL}_n(F)$6 as a family of affine linear maps parametrized by the spaces of $\mathrm{GL}_n(F)$7-matrices. This structure enables application of a pushforward criterion—if the determinant of the linear part of the family, as a function of the parameters, has controlled local singularities, then the pushforward density admits $\mathrm{GL}_n(F)$8-integrability for all $\mathrm{GL}_n(F)$9.

Explicit Jacobian Formula

The Jacobian determinant of the partial map (with respect to the last column) is computed explicitly. Up to sign, it can be expressed as

$F = \mathbb{F}_\ell((t))$0

for $F = \mathbb{F}_\ell((t))$1 and standard basis vector $F = \mathbb{F}_\ell((t))$2. The absolute value of $F = \mathbb{F}_\ell((t))$3 encodes the non-degeneracy locus of the Chevalley map and its singular behavior.

Log-Canonicity via Embedded Resolution

To analyze the integrability of $F = \mathbb{F}_\ell((t))$4 for $F = \mathbb{F}_\ell((t))$5, the authors establish an explicit embedded resolution of singularities for the divisor defined by $F = \mathbb{F}_\ell((t))$6, exploiting the geometry of full flag varieties and associated incidence correspondences. The crucial outcome is a local normal crossings structure, permitting local calculation of singularity exponents and allowing a direct check of $F = \mathbb{F}_\ell((t))$7-integrability up to the threshold $F = \mathbb{F}_\ell((t))$8.

Consequences for Character Theory

Applying the main geometric and analytic result to the pushforward criterion for affine-linear partial maps, the proof concludes integrability of the character function for irreducible cuspidal representations:

• The character of any such representation is given by a locally integrable function.
• No restriction on the residue characteristic or field is required, nor on the existence of resolution of singularities—contradicting prior necessary assumptions in the literature.

Implications and Future Directions

The unconditional proof for $F = \mathbb{F}_\ell((t))$9 has several substantive implications:

• Demonstrates that logarithmic canonicity of key divisors for the Chevalley map is tractable in positive characteristic via explicit geometric construction, sidestepping reliance on resolution of singularities in general algebraic geometry.
• The method provides a template to extend the result to more general classes of representations, notably relaxing the condition of cuspidality, and to encompass broader classes of groups beyond type $p$0 (general linear).
• The result aligns positive and zero characteristic representation theory with respect to local character integrability, solidifying the analytic theory of characters for $p$1-adic groups in full generality.
• The explicit pushforward criteria are relevant for harmonic analysis, transfer problems, and the study of orbital integrals in the Langlands program, especially where singularity analysis of invariant-theoretic quotients arises.
• Ongoing work targets extensions to classical groups in odd characteristic and to all irreducible representations, suggesting eventual comprehensive resolution of Harish-Chandra's integrability conjecture in full generality over local fields of positive characteristic.

Conclusion

This paper establishes that for all $p$2 and all cuspidal smooth irreducible representations of $p$3, the Harish-Chandra character is locally integrable. The proof is achieved by an explicit analysis of the Chevalley map's Jacobian and its singularities, realized via a concrete resolution adapted to the positive characteristic setting. The approach circumvents previous barriers related to resolution of singularities and provides a pathway toward complete resolution of local character theory for reductive groups in positive characteristic (2606.14279).