A fixed point formula of Lefschetz type in Arakelov geometry. III: Representations of Chevalley schemes and heights of flag varieties (Q1608562)

It is well-known that the classical Weyl character formula for irreducible representations of a compact Lie group is a consequence of the classical Lefschetz fixed point formula applied to the corresponding generalized flag variety. In the context of Arakelov geometry, a fixed point formula of Lefschetz type has recently been formulated and proved by \textit{K. Köhler} and \textit{D. Roessler} [Invent. Math. 145, 333-396 (2001; Zbl 0999.14002)]. Again by applying that formula to generalized flag varieties (now over Spec(\(\mathbb Z\))), the authors present, in the paper under review, a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over \(\text{Spec}(\mathbb Z)\) [see \textit{J. C. Jantzen}, ``Representations of algebraic groups'' (1987; Zbl 0654.20039)], except for the three exceptional cases \(G_2\), \(F_4\) and \(E_8\). The proof involves the computation of the equivariant Ray Singer analytic torsion associated with certain vector bundles on the corresponding complex generalized flag variety. In the special case the flag variety is Hermitean symmetric, this computation has been carried out by \textit{K. Köhler} in a previous paper [J. Reine Angew. Math. 460, 93-116 (1995; Zbl 0811.53050)]. In the general case, the authors decompose the flag variety into Hermitean symmetric flag varieties by various fibrations and inductively apply a special case of a formula of \textit{X. Ma} [Ann. Inst. Fourier 50, 1539-1588 (2000; Zbl 0964.58025)], which relates the equivariant analytic torsion of the total space of a fibration to the equivariant analytic torsion of its base and its fibre. The authors in fact give a proof of this special case based on the arithmetic Lefschetz formula again. In the final chapter of the paper under review, the authors use the Jantzen sum formula to derive explicit formulae for the global height of ample line bundles on an arbitrary generalized flag variety. This way they recover formulas for projective spaces proved by \textit{H. Gillet} and \textit{C. Soulé} [Ann. Math. (2) 131, 163-203 (1990; Zbl 0715.14018) and 205-238 (1990; Zbl 0715.14006)], and for quadrics proved by \textit{J. Cassaigne} and \textit{V. Maillot} [J. Number Theory 83, 226-255 (2000; Zbl 1001.11027)].