Geometry of second adjointness for p-adic groups

2011/12/29 by Bezrukavnikov, Roman, Kazhdan, David · 1 citation
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1112.6340

Abstract

We present a geometric proof of Bernstein's second adjointness for a reductive p-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a "co-specialization" map between spaces of functions on various varieties with G× G action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of functors lead to the second adjointness. We also get a formula for the "co-specialization" map expressing it as a composition of the orishperic transform and inverse intertwining operator; a parallel result for D-modules was obtained in arXiv:0902.1493. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra, generalizing a result by Opdam.

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