Branching Formula for $q$-Toda Function of Type B
Ayumu Hoshino, Yusuke Ohkubo, Jun'ichi Shiraishi
TL;DR
This work solves the explicit branching problem for q-Toda functions by proving a formula that expresses the B_N q-Toda eigenfunction in terms of A_{N-1} q-Toda eigenfunctions: f^{B_N Toda}(x|s|q) = ∑_{θ∈Z_{≥0}^N} e^{B_N/A_{N-1}}_{θ}(s|q) · ∏_{i=1}^N x_i^{−θ_i} · f^{A_{N-1} Toda}(x|q^{−θ_1}s_1, ..., q^{−θ_N}s_N|q). The proof combines a contiguity relation for the A_{N-1} Toda eigenfunctions with a recursion for the branching coefficients e^{B_N/A_{N-1}}_{θ}(s|q), facilitated by a key rational identity, to verify that the RHS satisfies the B_N Toda eigen-equation and normalization (constant term equal to 1). This branching rule provides a rigorous link between B_N and A_{N-1} q-Toda theories and has potential implications for representation-theoretic interpretations and related geometric frameworks such as Demazure modules and equivariant K-theory.
Abstract
We present a proof of the explicit formula for the asymptotically free eigenfunctions of the $B_N$ $q$-Toda operator which was conjectured by the first and third authors. This formula can be regarded as a branching formula from the $B_N$ $q$-Toda eigenfunction restricted to the $A_{N-1}$ $q$-Toda eigenfunctions. The proof is given by a contigulation relation of the $A_{N-1}$ Toda eigenfunctions and a recursion relation of the branching coefficients.