The full asymptotic expansion of analytic torsion on homogeneous spaces

The full asymptotic expansion of the equivariant complex Ray-Singer torsion for high powers of line bundles on symmetric spaces is given in an explicit form. In the case of isolated fixed points this expansion is given for general complex homogeneous spaces. Furthermore the full asymptotic expansion is given for the complex analytic torsion form associated to fibrations by projective curves. The expansions are compared with results by Bismut-Vasserot, Finski and Puchol. The results are applied to lattice representations of Chevalley groups.

💡 Research Summary

The paper presents a complete asymptotic expansion of the equivariant complex Ray‑Singer analytic torsion for high tensor powers of positive line bundles on complex homogeneous spaces, with a focus on both symmetric spaces and general homogeneous spaces with isolated fixed points. The author builds on the classical Ray‑Singer torsion, which is defined via regularised determinants of Laplace operators on Hermitian holomorphic vector bundles, and connects its asymptotic behaviour to arithmetic Hilbert‑Samuel theorems in Arakelov geometry.

The technical heart of the work lies in two intertwined ingredients: (1) an explicit asymptotic expansion of the derivative of the Lerch zeta function Φ(z,s,v) for integer negative s, and (2) a systematic treatment of sums of the form Σ P(k) log k where P(k) is a polynomial multiplied by exponential phases e^{ikφ}. The author refines earlier results of Katsurada, Elizalde and others, providing precise coefficient formulas involving Bernoulli numbers, Hurwitz zeta values, and explicit remainder bounds of the form (N−1)! (2π)^{−(N+1)} C(z,−N,a). This analytic machinery is then applied to the representation‑theoretic data of a compact Lie group G acting on a complex homogeneous space M=G/K.

For a G‑invariant positive line bundle L_{ρ_K+λ} on M, with λ a dominant weight, the equivariant torsion T_t(M,L^ℓ) for a group element t∈G is expressed in Theorem 1.2 as a combination of three types of terms:

• a leading logarithmic term −log ℓ ∑{α∈Ψ} χ{ρ+ℓλ+kα}(t);
• a sum of derivatives of Lerch‑zeta combinations ζ′_{ρ+ℓλ+kα}(t);
• a correction involving log⟨α∨,λ⟩ multiplied by characters χ_{ρ+ℓλ+kα}(t).

All coefficients are written explicitly in terms of the root system Ψ, the Weyl vector ρ, and the weight λ. The error term R₁ is bounded by (N−1)! (2πℓc₂)^{−N} c₁ with constants c₁,c₂ independent of ℓ and N. When M is a Hermitian symmetric space, Theorem 1.3 gives a more uniform formula valid for any t∈G, where the characters appear both linearly and inside a formal logarithmic series log^{#}(⟨α∨,ρ⟩+k⟨α∨,ℓλ⟩+1). Again the remainder R₂ enjoys an explicit factorial‑type bound.

The paper then verifies that the first four terms of the expansion coincide with the classical results of Bismut‑Vasserot and Finski (Theorem 6.5). It also shows that, for isolated fixed points, the expansion matches the Janzen sum formula, thereby providing asymptotic formulas for the arithmetic characters of Chevalley groups (Theorem 7.1).

Further extensions include:

• The Lie‑algebra‑equivariant torsion on the projective line ℙ¹_ℂ (Theorem 9.1), where the author recovers the Bismut‑Goette construction and supplies the full series.
• The full asymptotic expansion of torsion forms associated with fibrations by rational curves (Theorem 10.3). Lemma 10.6 confirms that the leading terms agree with Puchol’s formulas, while Proposition 10.1 supplies a previously unpublished closed form for a combinatorial sum needed for higher‑order terms.
• Detailed comparisons with results of Bismut‑Ma‑Zhang, Liu, and Q. Ma, illustrating that the new expansion refines and unifies these earlier works.

Throughout, the author makes extensive use of representation‑theoretic characters χ_{ρ+ℓλ+kα}(t), Weyl denominator identities, and explicit evaluations of Lerch‑zeta at roots of unity. The paper also provides a self‑contained appendix on the analytic continuation of Φ, its relation to Hurwitz zeta and Bernoulli polynomials, and the derivation of the remainder estimates.

In summary, the work delivers a comprehensive, explicit, and rigorously error‑controlled asymptotic expansion of complex analytic torsion on homogeneous spaces. It bridges analytic number theory (via Lerch and Hurwitz zeta functions), Lie theory (roots, Weyl groups, characters), and arithmetic geometry (Arakelov Hilbert‑Samuel theorems, lattice representations). The results not only generalise known leading‑term formulas but also give the full series, opening the way for precise arithmetic applications and for further investigations into equivariant analytic torsion in both mathematical physics and number theory.