Harish-Chandra j-Function

• The Harish-Chandra j-Function is a central object in harmonic analysis on reductive groups, normalizing parabolic induction via rational intertwining operators.
• It is constructed through functorial methods using the Bernstein center and is compatible with base changes in both complex and modular settings.
• Its universal nature bridges representation theory with the local Langlands correspondence by equating normalized gamma factors to arithmetic invariants.

The Harish-Chandra $j$-function is a fundamental object in harmonic analysis on reductive groups over local fields, representation theory, and the theory of automorphic forms. Its construction governs the normalization of intertwining operators between parabolically induced representations and encapsulates deep structural properties, including functoriality, compatibility with base change, and the interpolation of Plancherel measures. Recent advancements, notably the construction of a "universal" Harish-Chandra $j$-function as a rational function in the Bernstein center, have extended the theory to broad categorical and arithmetic frameworks, serving as a bridge to the local Langlands correspondence in families (Moss et al., 24 Sep 2025).

1. Algebraic Construction of the Harish-Chandra $j$-Function

Given a nonarchimedean local field $F$ with residue field of cardinality $q$, let $G$ be the $F$-points of a connected reductive group and $M$ a Levi subgroup, viewed as $A[M]$-module objects for a commutative Noetherian algebra $A$ over $\mathbb{Z}' = \mathbb{Z}[{\sqrt{q}}^{-1}]$. For any module $\sigma_0$, form its "universal unramified twist" $\sigma = \sigma_0 \otimes_A R$ with $R = A[M/M^0]$ and $M^0$ generated by compact subgroups of $M$.

Parabolic induction is defined as $i_P^G(\sigma)$, and comparison between realizations for different parabolics is achieved via rational intertwining operators $$J_{Q|P}(\sigma_0) \in \operatorname{Hom}_{R[G]}(i_P^G(\sigma), i_Q^G(\sigma))$$. The geometric lemma isolates the unique subquotient corresponding to $\sigma$ inside $r_Q^Gi_P^G(\sigma)$ and an injective signature map

$$\operatorname{sig}_{PQ}: \operatorname{Hom}_{R[M]}(r_Q^Gi_P^G(\sigma), \sigma) \to \operatorname{End}_{R[M]}(\sigma)$$

is constructed. If $\operatorname{End}_{A[M]}(\sigma_0) = A$, the identification $\operatorname{End}_{R[M]}(\sigma) \cong R$ applies. The rational intertwining operator $J_{Q|P}(\sigma_0)$ is then the unique element mapping to $1 \in R$ under $\operatorname{sig}_{PQ}$.

The $j$-function arises specifically for the pair of opposite parabolics $P$, $\overline{P}$: $$j^G(\sigma_0) = J_{P|\overline{P}}(\sigma_0) \circ J_{\overline{P}|P}(\sigma_0)$$ and is an element of the fraction ring $S^{-1}R$, with $S$ being suitable localizing elements. This is a rational, functorial object determined entirely by the geometry of parabolic induction and the module-theoretic properties of $\sigma_0$.

2. Functoriality and Scalar Extension

The construction is fully functorial in the category of smooth, finitely generated $A[M]$-modules and is compatible with extension of scalars: for any ring homomorphism $A \rightarrow A'$, the intertwining operators and hence the $j$-function descend canonically. This compatibility with base change allows for the framework to work uniformly in characteristic zero and positive characteristic, subject only to $p$-adic constraints. The structure generalizes earlier work in the $A = \mathbb{C}$ setting (e.g., works by Waldspurger and Dat) and removes the requirement of finite length or irreducibility, which previously limited the generality of the $j$-function’s definition (Moss et al., 24 Sep 2025).

3. Generic Schur's Lemma and Endomorphism Rings

The paper proves a generic version of Schur's lemma for parabolic induction: If $\sigma_0$ is a finitely generated $A[M]$-module with $\operatorname{End}_{A[M]}(\sigma_0) = A$, then $i_P^G(\sigma)$ also satisfies Schur's lemma, i.e.,

$$\operatorname{End}_{R[G]}(i_P^G(\sigma)) \cong \operatorname{End}_{R[M]}(\sigma) \cong R$$

This is achieved via the signature map and the uniqueness of the isotypic subquotient in the geometric lemma’s filtration, bypassing generic irreducibility requirements.

4. Universal $j$-Function and Specialization via Bernstein Center

By applying the construction to a finitely generated projective generator $P_{0,r}$ of the appropriate block in the category of $A[M]$-modules, the resulting $j$-function,

$$j^G(P_{0,r}) \in S^{-1} R, \qquad R = Z_{Z',M}[M/M^0]$$

is "universal" in the sense that, for any smooth finitely generated $\pi_0$ of the same block, one has

$j^G(\pi_0) = \lambda(j^G(P_{0,r}))$

for the central character homomorphism $\lambda: Z_{Z',M} \rightarrow B$. Thus, the $j$-function interpolates rationally over points of the Bernstein scheme, serving as a universal normalization factor for intertwining operators and parabolic induction in families.

5. Application to Local Langlands in Families

A key application is the characterization of local Langlands "in families" morphisms via equality of $j$-functions. Given a candidate morphism

$$\mathcal{L}: R_{\widehat{L G}, Z'_\ell}^{\widehat{G}} \longrightarrow Z_{G, Z'_\ell}$$

one demands that the canonical universal $\gamma$-factor (or its reciprocal, the $j$-function) constructed from Langlands-Deligne parameters matches the universal Harish-Chandra $j$-function from the representation-theoretic side. Explicitly,

$$(\mathcal{L} \otimes \mathcal{L}_{GL_m})(j_\psi(\phi_{univ}^{e(r)} \otimes \phi_{univ}^{e(r')}, X, R, \psi)) = j_\psi^{G'}(P_{0,r} \otimes P_{0,r'})$$

This criterion (Theorem 8.16) uniquely characterizes such morphisms when they exist for quasisplit classical groups, thereby bridging representation theory, harmonic analysis, and the arithmetic of Galois parameters.

6. Comparison to Classical Constructions and Generalizations

Earlier works constructed the Harish-Chandra $j$-function analytically or for complex representations, typically requiring irreducibility or finite length. The universal construction at the level of the Bernstein center allows for interpolation and compatibility with congruences, crucial in modular representation theory and $\ell$-adic families. The rationality and functoriality extend the normalization factors to categorical settings beyond classical harmonic analysis.

7. Summary and Significance

The universal Harish-Chandra $j$-function is an explicitly computable rational function whose coefficients reside in the Bernstein center. It determines the normalization of all parabolic induction and intertwining operators and specializes faithfully to each point of the Bernstein scheme, encompassing both complex and modular settings. Its role in identifying local Langlands morphisms via equality of $j$-functions is a powerful new tool for bootstrapping and compatibility arguments in the theory of automorphic forms and their associated Galois representations (Moss et al., 24 Sep 2025). This framework generalizes analytic and categorical perspectives on intertwining operators, providing a unified algebraic solution to normalization and parametrization questions in nonarchimedean harmonic analysis and representation theory.