Absence of S-torsion subquotients for admissible bi-Whittaker modules

The paper studies admissible modules over the bi-Whittaker quantum Hamiltonian reduction \mathbb{W} (quantum Toda lattice) of D(G) for a complex connected reductive group G. A key geometric input is the big Bruhat cell Bw_0B ⊂ G, whose complement is cut out by a bi-invariant function d ∈ [G]^{N_\ell × N_r}. The authors define the multiplicative set \mathcal{S} = {d^k : k ≥ 0} in \mathbb{W} and analyze torsion properties of \mathbb{W}-modules relative to \mathcal{S}.

They prove that for admissible \mathbb{W}-modules, subquotients of \mathcal{S}-torsion vanish after passing to the completion with respect to the Kazhdan filtration (Theorem 6.9 and Corollary 6.10). This yields, via the monodromic equivalence, minimal-extension properties for the corresponding D(G)-modules over the big cell in certain regular cases. However, the authors cannot show the torsion-freeness directly at the \mathbb{W}-module level (without completion), which would strengthen these results and imply minimal extension statements in full generality.