Journal of the Institute of Mathematics of Jussieu

Research Article

Boundary behaviour of special cohomology classes arising from the Weil representation

Jens Funkea1 and John Millsona2

a1 Department of Mathematical Sciences, University of Durham, Science Laboratories, South Rd, Durham DH1 3LE, UK (jens.funke@durham.ac.uk)

a2 Department of Mathematics, University of Maryland, College Park, MD 20742, USA (jjm@math.umd.edu)

Abstract

In our previous paper [J. Funke and J. Millson, Cycles with local coefficients for orthogonal groups and vector-valued Siegel modular forms, American J. Math. 128 (2006), 899–948], we established a correspondence between vector-valued holomorphic Siegel modular forms and cohomology with local coefficients for local symmetric spaces attached to real orthogonal groups of type . This correspondence is realized using theta functions associated with explicitly constructed ‘special’ Schwartz forms. Furthermore, the theta functions give rise to generating series of certain ‘special cycles’ in with coefficients.

In this paper, we study the boundary behaviour of these theta functions in the non-compact case and show that the theta functions extend to the Borel–Sere compactification of . However, for the -split case for signature , we have to construct and consider a slightly larger compactification, the ‘big’ Borel–Serre compactification. The restriction to each face of is again a theta series as in [J. Funke and J. Millson, loc. cit.], now for a smaller orthogonal group and a larger coefficient system.

As an application we establish in certain cases the cohomological non-vanishing of the special (co)cycles when passing to an appropriate finite cover of . In particular, the (co)homology groups in question do not vanish. We deduce as a consequence a sharp non-vanishing theorem for -cohomology.

(Received August 26 2011)

(Accepted March 01 2012)

(Online publication July 03 2012)

Keywords

  • Weil representation;
  • cohomology of locally symmetric spaces

AMS 2010 Mathematics subject classification

  • 11F27;
  • 22E40