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On Connected Components of Shimura Varieties

Published online by Cambridge University Press:  20 November 2018

Thomas J. Haines
Thomas J. Haines*
Affiliation:
University of Toronto Department of Mathematics 100 St. George Street Toronto, ON M5S 3G3, email: haines@math.toronto.edu
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Abstract

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We study the cohomology of connected components of Shimura varieties ${{S}_{Kp}}$ coming from the group $\text{GS}{{\text{p}}_{2g}}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\bar{\omega }$ on the group of connected components of ${{S}_{Kp}}$ we define an operator $L(\omega )$ on the cohomology groups with compact supports $H_{c}^{i}\left( {{S}_{Kp,}}{{\overset{-}{\mathop{\mathbb{Q}}}\,}_{\ell }} \right)$, and then we prove that the virtual trace of the composition of $L(\omega )$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi _{p}^{r}$, can be expressed as a sum of $\omega$-weighted (twisted) orbital integrals (where $\omega$-weighted means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\bar{\omega }$). As the crucial step, we define and study a new invariant ${{\alpha }_{1}}\left( {{\gamma }_{0}};\gamma ,\delta \right)$ which is a refinement of the invariant $\alpha \left( {{\gamma }_{0}},\,\gamma ,\,\delta \right)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

Keywords

14G3511F70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 54 , Issue 2 , 01 April 2002 , pp. 352 - 395
Copyright
Copyright © Canadian Mathematical Society 2002

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