On local integrability results for $p$-adic reductive groups

Abstract: We present a short proof, based on local character expansions, of the celebrated theorem of Harish-Chandra about local integrability of complex characters of $p$-adic reductive groups. The proof gives an algebraic incarnation of the local integrability that works for some coefficients different from $\mathbb{C}$, verifies local integrability in cases that appear not covered in the literature, and shows that a character is locally-$L<sup>α$ for some specified $α&gt;1$ as in [GGH23].

• The paper presents a new algebraic proof of local integrability for characters of p-adic reductive groups using local character expansions and nilpotent orbital integrals.
• It establishes explicit L^alpha estimates for locally constant functions representing distribution characters, extending the result to various coefficient fields.
• The approach leverages harmonic analysis techniques and carefully uses Weyl integration and scaling arguments to generalize traditional methods in both zero and positive characteristic.

Local Integrability for $p$-adic Reductive Groups: Character Expansions and New Algebraic Proofs

Introduction and Problem Statement

The paper "On local integrability results for $p$-adic reductive groups" (2604.15079) revisits Harish-Chandra’s local integrability theorem for complex characters of admissible smooth representations of $p$-adic reductive groups, offering a short and fundamentally algebraic proof. The result ensures that the distribution character $\Theta_\pi$ of a finitely generated admissible representation $\pi$ of a $p$-adic reductive group $G$ is represented by a locally constant, locally $L^1$ function, and is ubiquitous in the harmonic analysis and representation theory of $p$-adic groups.

A central contribution is a concise proof leveraging local character expansions and nilpotent orbital integrals, which clarifies the underlying mechanisms of local integrability. The argument accommodates coefficient fields beyond $\mathbb{C}$, extending to algebraically closed fields of positive characteristic and provides local $p$0-integrability for explicit exponents $p$1, addressing cases not previously covered in the literature.

Framework and Main Results

Let $p$2 be a non-archimedean local field (characteristic zero or $p$3), and $p$4 a $p$5-adic reductive group, possibly disconnected. For an admissible finitely generated representation $p$6 with character distribution $p$7, the main assertions are:

• $p$8 is represented by a locally constant, $p$9-invariant function $p$0.
• $p$1 is locally integrable ($p$2) on $p$3.
• The distribution coincides with integration against $p$4.

The argument generalizes to Fourier transforms of nilpotent orbital integrals, showing these too are locally represented by locally constant, locally integrable functions. These assertions hold under mild restrictions on $p$5 and the residual characteristic.

A novel and crucial technical element is the use of local character expansions near arbitrary semisimple elements, which describe $p$6 locally as linear combinations of nilpotent orbital integrals. The proof systematically reduces local integrability to statements about $p$7-integrability of explicit functions on the Lie algebra, leveraging invariance, scaling, and structure theory of maximal tori and the Weyl integration formula.

Proof Strategy and Technical Innovations

A central observation is that the local character of $p$8 can, in an open neighborhood of any semisimple $p$9, be written via the local character expansion as a linear combination of distributions induced from nilpotent orbits on the centralizer $\Theta_\pi$0. Thus, integrability is reduced to the integrability of corresponding Fourier transforms of nilpotent orbital integrals on the Lie algebra.

The core of the proof involves:

• Localization near semisimple points, using Jordan decomposition and descent to centralizers;
• Application of the Weyl integration formula to reduce integrals to maximal tori;
• Shalika germ expansion and scaling arguments, exploiting invariance properties and explicit homogeneity for terms in the local expansion;
• An elementary lemma showing $\Theta_\pi$1-integrability for functions with prescribed scaling (homogeneity) behavior on free $\Theta_\pi$2-modules under dilations by powers of the uniformizer.

The precise quantitative $\Theta_\pi$3-integrability is controlled by computing the scaling exponents associated with nilpotent orbits and their duals, leading to explicit bounds for $\Theta_\pi$4 in terms of dimensions of orbits and centralizers.

Extensions, Algebraic Context, and Coefficient Fields

The method is algebraic, realizing the analysis in terms of sums of geometric progressions, and replaces complex-analytic arguments by formal manipulations valid over general coefficient fields, provided certain simple conditions are met (e.g., the characteristic does not divide $\Theta_\pi$5 and some additional constraints on the residue field and representation). This allows extending the integrability results to fields such as $\Theta_\pi$6 and, for $\Theta_\pi$7 sufficiently large, $\Theta_\pi$8.

For classical groups, the Cayley transform is used in place of the exponential map, with the results carrying over to unitary, symplectic, and special orthogonal groups over local fields, with restrictions on the residual characteristic.

Coverage is provided for cases in positive residual characteristic under the assumption that the characteristic is very good for the group under consideration. The approach also clarifies subtleties in wild and bad characteristic cases and connects to recent advances in harmonic analysis on $\Theta_\pi$9-adic groups.

Implications and Relation to Existing Literature

The paper not only recovers Harish-Chandra’s result through local algebraic and geometrical arguments but also sharpens prior analyses by providing explicit $\pi$0 estimates for the characters and the Fourier transforms of nilpotent orbital integrals. For generic representations and regular nilpotent orbits, the bounds align with those obtained by Gurevich, Gordon, and Howe [GGH23], and the method covers non-overlapping cases not treatable by previous techniques.

Additionally, the result delivers the sharpness for the $\pi$1-condition in cases where the minimal exponent is attained, particularly for the group $\pi$2 and its natural Levi subgroups. The work also demonstrates that character and orbital integral distributions are determined by their regular semisimple parts for large positive characteristic coefficients, elucidating the limitations in wild characteristic.

These results have direct applications in representation theory, automorphic forms, the local Langlands correspondence, and the harmonic analysis of $\pi$3-adic groups. The extension to general coefficients contributes to the theory of mod-$\pi$4 representations and their harmonic analysis.

Conclusion

The paper provides a transparent, algebraically grounded proof of local integrability for characters of $\pi$5-adic reductive groups and their nilpotent orbital integral Fourier transforms, valid for a wide class of coefficient fields and groups. The approach renders the integrability fundamentally algebraic, clarifies the role of local character expansions, and establishes explicit and optimal local $\pi$6 conditions under general hypotheses. These advances both solidify the theoretical foundations and broaden the practical applicability of harmonic analysis on $\pi$7-adic groups.