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On the spectral side of Arthur’s trace formula – combinatorial setup. (English) Zbl 1241.52006

Let \(P\) be a \(d\)-dimensional convex polytope in the vector space \(V\), \(L_P(\lambda)=\int_P\exp \langle x,\lambda\rangle dx\) the Laplace transform of its characteristic function and \(H_P(\lambda)=\max_{x\in P}\langle x,\lambda\rangle\) its support function, both defined on the dual space \(V^*\). Considering the normal fan \(\Sigma_P\) of the polytope \(P\), M. Brion [Tohoku Math. J., II. Ser. 49, No. 1, 1–32 (1997; Zbl 0881.52008)] derives the functional relation \(\pi_{\Sigma_P} (\exp H_P)=L_P\): the natural map \(\pi_{\Sigma}\) is defined on the algebra of piecewise polynomial functions with respect to the complete simplicial fan \(\Sigma\) in \(V^*\) and with values in the algebra of polynomial functions on \(V^*\), this map \(\pi_{\Sigma}\) is of degree \(-d\) and has an extension to convenient piecewise formal power series. Hence, the volume \(\mathrm{vol}\,P = L_P(0)\) appears as a canonical \(d\)-th order derivative denoted by \(\mathcal D_{\Sigma_P}\exp H_P\). One geometric-combinatorial identity for this derivative will be given by each formula for the volume of \(P\): the paper recalls thes Lawrence-Varchenko algebraic cones alternating sum for the characteristic function of \(P\) (which gives a formula similar to the Brion’s one), the formula by inductive reduction to faces volumes and the extension of McMullen’s formula for zonotopes with mixed volumes.
Here, by purely algebraic methods, the authors generalize the last two identities to a non-commutative framework defined by a family \((\mathcal A_\sigma)_{\sigma\in\Sigma_0}\) of formal power series on \(V^*\) and with values in a non-abelian unital algebra \(\mathcal E\). The family is indexed by the chambers \(\sigma\) of a polyhedral fan \(\Sigma\) in \(V^*\) and satisfies some compatibility rules: \(\mathcal A_\sigma(0)=1\) and \(\mathcal A_\sigma\mathcal A_\tau^{-1}\in\mathcal E[[(\sigma\cap\tau)^\bot]]\) for adjacent chambers \(\sigma,\tau\). In the abelian context, an example of such compatible power series is given by exponentials of continuous piecewise linear functions on the normal fan of a polytope. The authors introduce the \(d\)-th order derivative \(\mathcal D_\Sigma \mathcal A\) and prove formulæ similar to those obtained for a polytope \(P\) and its normal fan \(\Sigma_P\).
Such non abelian combinatorial setups arise ubiquitously in intertwining operator terms for representations induced from parabolic groups for reductive groups over local fields: particular cases have been proved by J. Arthur [“An introduction to the trace formula”, American Mathematical Society (AMS). Clay Mathematics Proceedings 4, 1–263 (2005; Zbl 1152.11021)]. In a companion paper, T. Finis, E. Lapid and W. Müller [Ann. Math. (2) 174, No. 1, 173–195 (2011; Zbl 1242.11036)] use these results to study the spectral side of the trace formula.

MSC:

52A38 Length, area, volume and convex sets (aspects of convex geometry)
52A39 Mixed volumes and related topics in convex geometry
52B45 Dissections and valuations (Hilbert’s third problem, etc.)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)
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References:

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