Prasad's Conjecture on Distinguished Representations

• Prasad’s Conjecture is a central prediction in distinguished representation theory, establishing explicit criteria using Langlands parameters and period integrals.
• It employs harmonic analysis on Bruhat–Tits buildings to demonstrate the uniqueness of invariant linear forms, notably for the Steinberg representation.
• The conjecture underpins multiplicity one results, epsilon factor identities, and applications to trace formulas, modularity, and automorphic classification.

Prasad's Conjecture is a central prediction in the theory of distinguished representations for p-adic and finite groups, symmetric spaces, and their automorphic forms. It formulates explicit criteria—primarily involving Langlands parameters and period integrals—that determine whether an irreducible admissible representation of a group G is “distinguished” by an action of a smaller subgroup H, often through the existence and uniqueness of invariant linear functionals. It is tightly linked to the structure of harmonic analysis on symmetric spaces, the local and global Langlands program, and the interplay between representation theory, arithmetic geometry, and number theory.

1. Formulation and Canonical Setting

Prasad's conjecture, in its most basic local form, asserts the existence and uniqueness of invariant functionals on representations realized on symmetric spaces of the form $G(E)/G(F)$, with $G$ a split reductive group over a non-archimedean local field $F$, and $E/F$ an unramified quadratic extension. The archetypal statement, originally proven for the Steinberg representation $\mathrm{St}_E$ (Broussous et al., 2012), claims:

• $\mathrm{St}_E$ admits a unique (up to scalar) $G(F)$-equivariant linear form which is equivariant with respect to a specific quadratic character $E_G_F$ on $G(F)$.
• No other character distinguishes $\mathrm{St}_E$; $H^1_c(\mathrm{Chambers}(X_F),\mathbb{C})$ evaluates periods via sums over symmetric sets within the Bruhat–Tits building of $G(E)$.

This pattern generalizes to a wide array of groups (quasi-split, inner forms, etc.), extensions (not necessarily unramified), and representations (regular supercuspidal, depth-zero, or generic), always relating the existence of a distinguished period to arithmetic data encoded in the Langlands parameter.

2. Harmonic Analysis via Bruhat–Tits Buildings

A notable methodological advance is the realization of certain representations (notably the Steinberg) via harmonic functions on affine Bruhat–Tits buildings. For G split over $F$ and $E/F$ unramified, the representation space for $\mathrm{St}_E$ is identified with the top compactly supported cohomology $H^d_c(X_E,\mathbb{C})$, with $d$ the $E$-rank of $G$. Harmonic functions $f:C\to\mathbb{C}$ on chambers $C$ satisfy the cancelation condition over codimension-1 faces:

$$\forall\, D\,\text{(codimension-1 simplex)},\quad \sum_{C\supset D} f(C)=0.$$

The $G(E)$-action factors via a sign $\epsilon(g)$ on chambers $C$, naturally producing a quadratic character upon restriction to $G(F)$. The construction crucially depends on the symmetric subcomplex $X_F$, a fixed-point set for the Galois action. The resulting period operator,

$$\lambda(f) = \sum_{C \in \mathrm{Chambers}(X_F)} f(C),$$

is shown to be absolutely convergent for Iwahori–spherical vectors and equivariant:

$\lambda(T_E(g)\cdot f) = E_G_F(g)\cdot \lambda(f),$

where $E_G_F$ is the explicit quadratic character determined by the chamber labelling.

3. Multiplicity One and Transitivity Properties

A strong multiplicity-one result is established: the space $$\operatorname{Hom}_{G_\mathrm{der}(F)}(\mathrm{St}_E,\mathbb{C})$$ is at most one-dimensional. This is demonstrated using transitivity properties of $G(F)$ on the set of chambers at fixed combinatorial distances from $X_F$ and induction by the harmonicity relation. Any harmonic function invariant under $G(F)$ is determined by its values on $X_F$, and the uniqueness of the equivariant functional follows. Such multiplicity-one phenomena are critical for applications to trace formulas and to the uniqueness of periods in global contexts.

4. Langlands Parameters, Quadratic Characters, and Symmetry Constraints

The analytic properties predicted by the conjecture translate into arithmetic or Galois-theoretic conditions on Langlands parameters. In several contexts, distinguished representations are precisely those whose parameters factor through a specific subgroup (often an “opposite” or twisted subgroup) of the dual group, reflecting symmetries under involution (e.g., Chevalley, Galois, or contragredient). For regular depth-zero supercuspidal L-packets (Zhang, 2016), distinction forces $\pi^\vee \simeq \pi^\sigma$; this, in turn, translates at the parameter level into:

$$(C\circ\phi, \mu) \sim (\delta\circ\phi, \mu\circ\delta^{-1}),$$

and often implies functorial lifting from $H^\mathrm{op}$.

Quadratic characters are central; Prasad's character $\omega_{G(F),E}$, Hakim–Murnaghan's correction $E^{\mathrm{HM}}$, and Kaletha's rectifying character $E^{\mathrm{Kal}}$ arise from distinct sources (Galois cohomology, special isomorphisms, $x$-data). Their product often equals a fourth quadratic character from $(-)$-data. These matches account for toral contributions and subtleties in matching arithmetic invariants on both sides of the conjectural identity (Wang, 2022).

5. Relations to Multiplicity Formulas and Epsilon Factors

Prasad's Conjecture interweaves multiplicity formulas relating automorphic/invariant-function side counts to sums over Langlands-lifts, with precise weights respecting enhancements via component groups and cohomological factors. A representative formula (Lu, 2019):

$$\dim \operatorname{Hom}_{H(F)}(\pi, \omega_H) = \sum_{\phi \in F(\phi_\pi)} m(\lambda, \phi)\frac{\deg \Phi(\phi)}{d_0(\phi)}$$

connects analytic and Galois-side quantities. In the context of linear periods and inner forms of $\mathrm{GL}_n$, the Prasad–Takloo–Bighash conjecture (often reformulated via $\epsilon$-factors (Suzuki, 2023, Chommaux, 2018)) postulates that distinction is equivalent to

1. The parameter factoring through a symplectic similitude group with prescribed similitude.
2. An explicit root number identity:

$$\epsilon\left(\tfrac{1}{2}, \pi \otimes \operatorname{Ind}_E^F(\mu^{-1})\right) = (-1)^n\,\omega_{E/F}(-1)^{nd/2}$$

The space of distinguished representations in an L-packet is characterized via characters of the finite component group and such $\epsilon$-factors.

6. Applications: Classification, Trace Formulas, and Modularity

The conjecture underpins classification of distinguished representations in various settings: Steinberg representations for symmetric spaces (Broussous et al., 2012), regular depth-zero supercuspidals (Zhang, 2016), U(2), SO(4), Sp(4) (Lu, 2019), and representations with symplectic periods (Sharma et al., 2023). It also informs the structure of degenerate Whittaker spaces for representations over local principal ideal rings (Parashar et al., 14 Aug 2025), yielding explicit multiplicity-free decompositions predicted by the conjecture.

Via Prasad's volume formula, even geometric problems—such as reflective obstructions in modular varieties (Maeda, 2022)—are governed by arithmetic invariants paralleling those in representation theory. This duality between analytic period integrals and arithmetic parameter symmetries is emblematic of the deep connections between automorphic forms, harmonic analysis, and algebraic geometry.

7. Broader Impact and Directions

The evidence amassed for Prasad's Conjecture extends to diverse contexts: local and global fields, inner and outer forms, classical and exotic groups, even to finite groups via dualizing involutions (Arote et al., 2022). It bridges period integrals, trace formulas, Langlands parameters, and arithmetic invariants. The explicit computations, local proofs, and analytic–arithmetic correspondences provide models for generalizations, as well as suggest approaches for unresolved cases (ramified extensions, non-split groups, or more complicated period problems).

The conjecture thus represents a central organizing principle in the paper of distinguished representations, elucidating both the harmonic analytic and arithmetic structure of periods, their uniqueness, and their role in automorphic theory and beyond.