Formal degree of principal series of quasi-split groups

Abstract: Let $\mathcal{G}$ be a quasi-split connected reductive group over a non-archimedean local field $F.$ In this paper, we prove the formal degree conjecture for discrete series representations contained in a principal series of $\mathcal{G}(F)$. We first construct a type for each Bernstein component attached to a principal series representation of $\mathcal{G}(F).$ We then use these types and the local Langlands correspondence for principal series representations defined in [Sol25] to verify the formal degree conjecture. Our approach follows a similar strategy to [Ric25], reducing the problem to the case of unipotent representations of some other quasi-split group.

• The paper establishes the formal degree conjecture for discrete series subquotients in the principal series of quasi-split groups using the HII formula.
• The methodology leverages explicit construction of types and reductions via affine and graded Hecke algebras to relate group representations to unipotent cases.
• The results bridge harmonic analysis with the local Langlands correspondence, paving the way for broader applications in p-adic representation theory.

Formal Degree Conjecture for Principal Series of Quasi-split Groups

Introduction and Background

The paper "Formal degree of principal series of quasi-split groups" (2604.14823) establishes the formal degree conjecture of Hiraga, Ichino, and Ikeda (HII) for discrete series representations that occur as subquotients of principal series representations of quasi-split connected reductive groups over non-archimedean local fields. The formal degree conjecture asserts a precise formula connecting the formal degree of a discrete series representation, the dimension of an enhancement of its associated $L$-parameter, the size of a component group, and a local adjoint $\gamma$-factor.

This work generalizes prior results known for split groups (Ricci, 24 Jun 2025) by constructing explicit types for all Bernstein components in the principal series of quasi-split groups and by pushing through the computation of the formal degree via affine Hecke algebra and graded Hecke algebra techniques. The paper rigorously ties the computation of the formal degree to recent advances in the local Langlands correspondence (LLC) for principal series of quasi-split groups (Aubert et al., 2017), and it reduces the problem to the known case of unipotent representations [UnipotentFormalDegree].

Principal Series Types for Quasi-split Groups

A significant technical obstacle for quasi-split groups has been the lack of general types for the principal series representations, a gap famously resolved for split groups by Roche [RochePrincipal] but never for general quasi-split groups due to issues in the Galois descent of root data and the behavior of conductors of characters. This paper constructs types in all principal series blocks under mild residue characteristic assumptions (excluding $p = 2, 3$ when necessary). The approach relies on a careful adaptation of the Bruhat-Tits building and the stratification of the group via concave functions, inspired by the work of Miyauchi–Stevens [MyauchiStevens] and Morris [morris1999level].

The constructed types $(J_f, \chi_0)$ are shown to satisfy the necessary cover properties for Bernstein blocks. The analysis exploits concavity of the conductor functions and an explicit study of Jacquet module injectivity, culminating in a proof that the idempotent corresponding to the type cuts out the desired block. Blondel's criteria for covers and injectivity of Jacquet functors play a critical role [Blondel1, Blondel2].

Reduction to Graded Hecke Algebras

To relate the formal degree on the group to manageable algebraic objects, the paper uses Lusztig's two-step reduction—first from extended affine Hecke algebras to those associated with residual tori (via idempotent reduction and Morita equivalence), then further to graded Hecke algebras appropriate for the central character of the given representation [AffineHeckeGradedVersionLusztig, OnClassificationHeckeAlgebrasUnequalPar]. The analytic localization techniques and category equivalences are developed to ensure all functorial constructions are compatible with the induction by diagram automorphisms, yielding commutative diagrams of equivalences between module categories.

The graded Hecke algebra modules are classified via geometric data: nilpotent and semisimple elements in the relevant Lie algebra, together with local system data [GradDisc, GeneralizationSpringer]. This enables a precise correspondence between the discrete series under consideration and certain unipotent representations for twisted endoscopic groups.

Hecke Algebra Parameters and LLC for Principal Series

The LLC for the principal series components of quasi-split groups passes through an explicit construction of equivalences between module categories for group blocks and affine Hecke algebras, extending the constructions in [quasisplitps, AffineHeckeForLparameters]. This paper shows that after fixing the type, the Hecke algebras on both automorphic and Galois sides match by a canonical identification of root data and finite group extensions, permitting the LLC parametrization to transport between representations unambiguously.

Key is the relation between the Hecke algebra for a Bernstein block associated to $[T, \chi]_G$ and the one attached to its corresponding inertial class of $L$-parameters. These are matched by the explicit isomorphisms, with the root datum reflecting ramification data of the field extension and centralizer structure in the dual group.

Computation of Formal Degrees and Epsilon Factors

The formal degree is computed by transferring it to a calculation over Hecke algebras, where explicit expressions are available [TypesPlancherel, OpdamSpectralCorresp]. The paper carefully tracks the normalization of Haar measures to coincide with those prescribed in the context of the Artin conductor [HaarMeasureArtinCond], ensuring that local constants and volume terms in the formal degree formula are canonical.

The computation of the local adjoint $\gamma$-factors, essential for the HII formula, is handled via a detailed analysis of the Galois representation on the Lie algebra and its decomposition into root spaces. The absolute value of the $\varepsilon$-factor is analyzed through Artin conductors and the wild ramification filtration, using additive character normalization and the explicit behavior of local constants as per Gross–Reeder [ArithmeticInv] and Feng–Opdam–Solleveld [FengOpdamSolleveld]. The crucial property that the $\varepsilon$-factor is inductive in degree zero ensures compatibility with the descent to the relevant unipotent representations.

Reduction to Unipotent Cases and Comparison

The core of the argument is the reduction of the given discrete series representation to a unipotent representation of another quasi-split group $\mathcal{J}$, constructed as the centralizer of the image of inertia in the dual group. All terms in the formal degree formula—dimension of enhancement, component group order, and $\gamma$0-factor—are compared between the original representation and its unipotent avatar, and the necessary correction factors (volumes, induction indices) are explicitly evaluated.

The paper proves that the only remaining differences are tracked by the known behavior of formal degrees under induction and the explicit normalization of measures, and that the volumes and $\gamma$1-factors are matched exactly as predicted by the conjecture. In particular, the extension datum and stabilizer in the diagram automorphism group are precisely related to the index appearing in the dimension and component group term.

Statement and Proof of the Main Theorem

The main result establishes that for any discrete series subquotient of a principal series of a quasi-split connected reductive group over a non-archimedean local field, the HII formula holds: $\gamma$2 where $\gamma$3 is the enhanced $\gamma$4-parameter attached to $\gamma$5 via the LLC for principal series, $\gamma$6 is a certain explicitly defined component group, and $\gamma$7 is the adjoint $\gamma$8-factor.

This is achieved by a reduction to the case of unipotent representations (where the conjecture is known [UnipotentFormalDegree]), explicit comparison of all group-theoretic, Galois-theoretic, and measure-theoretic terms, and the use of the stratification by types to control the passage from group representations to Hecke modules. All steps are supported by precise Morita equivalences, analytic localization, and calculation of Fourier transforms of matrix coefficients.

Implications and Future Directions

This result affirmatively settles the formal degree conjecture for a wide class of representations for quasi-split $\gamma$9-adic groups, expanding the scope from previously known split cases. Its methodology links harmonic analysis, affine/graded Hecke algebras, and the structure of the LLC in new depth. The compatibility with the intricate local Langlands parametrization, even in the absence of strong genericity or unramifiedness hypotheses, is especially striking.

Beyond confirming the HII conjecture in this context, the methods suggest further generalizations:

• Extension of the formal degree formula to more general Bernstein blocks (not just principal series).
• Potential applications to explicit Plancherel formula computations and types for classical and exceptional groups.
• Deeper study of the local Langlands correspondence, particularly its intricacies for wild ramification and disconnected center phenomena.
• Influence on the harmonic analysis of $p = 2, 3$0-adic groups, with likely implications for the explicit computation of Plancherel measures and multiplicity-one results in the non-split and wildly ramified settings.

Conclusion

The paper achieves a complete and rigorous proof of the formal degree conjecture for principal series discrete series of arbitrary quasi-split $p = 2, 3$1-adic groups. The approach via explicit types, affine and graded Hecke algebras, reduction to unipotent representations, and careful measure normalizations, sets a new methodological standard for the analysis of the harmonic and representation-theoretic structure of reductive $p = 2, 3$2-adic groups. The results will have lasting influence on both the LLC program and the analytic theory of automorphic forms.

References:

• (2604.14823) "Formal degree of principal series of quasi-split groups"
• (Aubert et al., 2017) "Affine Hecke algebras for Langlands parameters"
• [quasisplitps] M. Solleveld, "On principal series representations of quasi-split reductive $p = 2, 3$3-adic groups"
• [UnipotentFormalDegree] Y. Feng, E. Opdam, M. Solleveld, "On formal degrees of unipotent representations"
• [RochePrincipal] A. Roche, "Types and Hecke algebras for principal series representations of split reductive $p = 2, 3$4-adic groups"
• [MyauchiStevens] M. Miyauchi, S. Stevens, "Semisimple types for $p = 2, 3$5-adic classical groups"
• [Blondel1, Blondel2] C. Blondel, "Critère d’injectivité pour l’application de Jacquet", "Quelques propriétés des paires couvrantes"
• (Ricci, 24 Jun 2025) G. Ricci, "The Hiraga-Ichino-Ikeda Conjecture for Principal Series of Split $p = 2, 3$6-adic Groups"
• [UnipotentCorresp] M. Solleveld, "A local Langlands correspondence for unipotent representations"