Dirac index and associated cycles of Harish-Chandra modules

Salah Mehdi (1) , Pavle Pandžić (2) , David A Vogan (3) , Roger Zierau (4)

1 IECL - Institut Élie Cartan de Lorraine
2 University of Zagreb
3 Department of Mathematics [MIT]
4 OSU - Oklahoma State University [Stillwater]

Let G_R be a simple real linear Lie group with maximal compact subgroup K_R and assume that rank(G_R) = rank(K_R). For any representation X of Gelfand-Kirillov dimension 1/2 dim(G_R /K_R), we consider 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. Under a technical condition involving the Springer correspondence, we establish an explicit relationship between this polynomial and the multiplicities of the irreducible components occurring in the associated cycle of X. This relationship was conjectured in [12].

Mots clés: Dirac index; Equivariant K-theory; Nilpotent orbit; Associated cycle; Springer correspondence

Domaines: Mathématiques [math]

Salah Mehdi, Pavle Pandžić, David A Vogan, Roger Zierau. Dirac index and associated cycles of Harish-Chandra modules. Advances in Mathematics, 2020, 361, pp.106917. ⟨10.1016/j.aim.2019.106917⟩. ⟨hal-03336278⟩