Global normalization of intertwining operators making all scalar intertwiners trivial while keeping the cocycle constant

Determine whether, for an arbitrary reductive p-adic group G, a Levi subgroup M with discrete-series representation σ, the torus X of unramified unitary characters of M, and W=(X ⋊ W_M)_σ, there exists a choice of normalizing isomorphisms T_w (for w in W) defining the normalized intertwining operators I_{w,χ} such that (i) for every χ in X and every w in the scalar-intertwiner subgroup W'_χ={w in W_χ | I_{w,χ} is scalar}, the operator I_{w,χ} equals the identity (equivalently, the scalar ε_{w,χ}=1), and simultaneously (ii) the associated 2-cocycle γ: W×W→T remains independent of χ (i.e., constant across X).

Background

In the construction of normalized intertwining operators for the parabolically induced representations Ind_PG(σ⊗χ), the authors introduce for w in W=(X⋊W_M)σ and χ in X the unitary operators I{w,χ}. For χ where I_{w,χ} is scalar, they write I_{w,χ}=ε{w,χ} id and define the scalar-intertwiner subgroup W'χ={w in W_χ | I_{w,χ} scalar).

It is known one can choose the normalizing operators T_w so that ε{w,χ0}=1 for a fixed χ0, and in many cases the 2-cocycle γ associated to the product T{w2}T_{w1}=overline{γ(w2,w1)}T_{w2w1} can be made trivial. The open question asks for a global normalization achieving ε{w,χ}=1 simultaneously for all χ and all w in W'χ while keeping γ independent of χ, which would simplify later K-theoretic computations based on equivariant spectral sequences.

References

It would simplify the computations in Section \ref{sec:KCXEW} if we could normalise the intertwining operators so that ${w,\chi}=1$ for all $w\in {W}'\chi$, for all $\chi$ at once, while keeping the cocycle $\gamma$ constant in $\chi$. We do not know whether this is possible in general, though it certainly is possible in some cases: e.g., when $G$ is a split Chevalley group and $M$ is a minimal Levi subgroup: see the proof of Theorem 1 in Section 3 of .

— Reduced $C^*$-algebras and $K$-theory for reductive $p$-adic groups  (2603.29965 - Clare et al., 31 Mar 2026) in Remark (Scalar intertwiners), Section on Reduced C*-algebras of p-adic reductive groups, subsection "Scalar intertwiners"