Non‑negativity of twisted local traces needed for self‑dual density theorems

Establish non‑negativity of the twisted local trace tr(π_v(θ)|_{V_{π_v}(p_v^m)}) for θ‑stable unitary representations π_v of GL_n(F_v) in the self‑dual cases of Theorem 1.1 (ii) SO(2m+1)-type, (iii) Sp(2m)-type, and (iv) SO(2m)-type, so that automorphic density theorems analogous to Theorem 1.4 can be derived for these self‑dual settings.

Background

The paper proves an automorphic density theorem for conjugate self‑dual representations (Case (i), E≠F) leveraging a non‑negativity result (Lemma 6.1) for certain twisted traces at split non‑dyadic places. For the self‑dual cases (ii)–(iv) in Theorem 1.1, the same approach would yield density theorems if an analogous non‑negativity of twisted traces were available.

The authors explicitly note that without such non‑negativity the proof strategy fails, leaving the extension of the density theorem to self‑dual cases contingent on establishing this property. The twisted traces in question are those arising from the θ‑action on fixed‑vector spaces at principal congruence levels.