Glasnik Matematicki, Vol. 53, No. 2 (2018), 275-330.

COMPUTING THE ASSOCIATED CYCLES OF CERTAIN HARISH-CHANDRA MODULES

Salah Mehdi, Pavle Pandžić, David Vogan and Roger Zierau

Institut Elie Cartan de Lorraine, CNRS - UMR 7502, Université de Lorraine, Metz, F-57045, France
e-mail: salah.mehdi@univ-lorraine.fr

Department of Mathematics, Faculty of Science, University of Zagreb, 10000 Zagreb, Croatia
e-mail: pandzic@math.hr

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
e-mail: dav@math.mit.edu

Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
e-mail: roger.zierau@okstate.edu

Abstract.   Let Gℝ be a simple real linear Lie group with maximal compact subgroup Kℝ and assume that rank(Gℝ)=rank(Kℝ). In [17] we proved that for any representation X of Gelfand-Kirillov dimension 1/2dim(Gℝ/Kℝ), the polynomial on the dual of a compact Cartan subalgebra given by the dimension of the Dirac index of members of the coherent family containing X is a linear combination, with integer coefficients, of the multiplicities of the irreducible components occurring in the associated cycle. In this paper we compute these coefficients explicitly.

2010 Mathematics Subject Classification.   22E47, 22E46

Key words and phrases.   (𝔤,K)-module, Dirac cohomology, Dirac index, nilpotent orbit, associated variety, associated cycle, Springer correspondence

Full text (PDF) (free access)

DOI: 10.3336/gm.53.2.05

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