The Bethe Equation at q=0, The Mobius Inversion Formula, and Weight Multiplicities: III. The X^{(r)}_N case

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It is shown that the numbers of off-diagonal solutions to the Uq(X(r)N ) Bethe equation at q = 0 coincide with the coefficients in the recently introduced canonical power series solution of the Q-system. Conjecturally the canonical solutions are characters of the KR (Kirillov-Reshetikhin) modules. This implies that the numbers of off-diagonal solutions agree with the weight multiplicities, which is interpreted as a formal completeness of the Uq(X(r)N ) Bethe ansatz at q = 0.

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