Homological multiplicities in representation theory of p-adic groups

  • Avraham Aizenbud
  • Eitan SayagEmail author


For a spherical variety X of a reductive group G over a non-archimedean local field F, and for a smooth representation \(\pi \) of G we study homological multiplicities \(\dim {\text {Ext}}_{G}^{*}(\mathcal {S}(X),\pi )\). Based on Bernstein’s decomposition of the category of smooth representations of G, we introduce a sheaf that measures these multiplicities. We show that these multiplicities are finite whenever the usual mutliplicities
$$\begin{aligned} \dim {\text {Hom}}_{G}(\mathcal {S}(X),\pi ) \end{aligned}$$
are finite. The latter are known to be finite for symmetric varieties and for many other spherical varieties and conjectured to be finite for all spherical varieties. Furthermore, we show that the Euler–Poincaré characteristic is constant in families parabolically induced from finite length representations of a Levi subgroup \(M<G.\) In the case when \(M=G\) we compute these multiplicities more explicitly.


Branching laws Homological multiplicities Spherical spaces 

Mathematics Subject Classification

22E45 20G25 



We thank Dipendra Prasad for a number of inspiring lectures on Ext braching laws. We also thank Joseph Bernstein and Dmitry Gourevitch for many useful discussions and Yotam Hendel for his careful proof reading. Finally, we thank the referee for his remarks and questions, and for his many useful suggestions that improved the readability of the paper. This project was conceived while both authors were in Bonn as part of the program “Multiplicity problems in harmonic analysis” in the Hausdorff Research Institute for Mathematics. A.A. was partially supported by ISF Grant 687/13, and a Minerva foundation Grant. E.S. was partially supported by ISF 1138/10 and ERC 291612.


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Copyright information

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Authors and Affiliations

  1. 1.Faculty of Mathematical SciencesWeizmann Institute of ScienceRehovotIsrael
  2. 2.Department of MathematicsBen-Gurion University of the NegevBe’er ShevaIsrael

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