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A proof of Vogan's conjecture about Dirac operators
by
Pavle Pandžić
Department of Mathematics, University of Zagreb
Coauthors: Jing-Song Huang (Hong Kong University of Science and Technology)
Let G be a connected real reductive group with maximal compact subgroup K. Let g0=k0+p0 be the Cartan decomposition of the Lie algebra of G. Let C(p0) be the Clifford algebra of p0 (with respect to the Killing form), and let S be a complex simple module for C(p0) (the space of spinors). The double cover [K\tilde] corresponding to Spin(p0) acts on S.
The Dirac operator can be defined as D=\sumi Zi\otimesZi in \Cal U(g)\otimesC(p0), where {Zi} is an orthonormal basis of p0. If X is a (g, K)-module, and \tau a (genuine) [K\tilde]-type, then D induces an operator D\tau on the space \operatornameHom[K\tilde](V\tau, X\otimesS).
D.Vogan conjectured that if X is irreducible unitary, and if the kernel of D\tau is not zero, then the infinitesimal character of X is determined by \tau; modulo certain identifications, it is given by the highest weight of \tau plus a \rho-shift. This was proved for the discrete series by W.Schmid.
Vogan also made an algebraic conjecture from which the one just described can be deduced. Namely, consider the diagonal embedding of k into the algebra \Cal U(g)\otimesC(p); let k\Delta be its image. The conjecture states that for any element Z in the center of \Cal U(g), Z\otimes1 can be written as a sum of an element of the center of \Cal U(k\Delta) with an element of the form aD+Db, for some (K-invariant) a and b.
I will present a proof of this last conjecture (this then also proves the first conjecture). The idea is to look at a differential on the K-invariants of the above algebra, defined by d(a)=Da-sgn(a)aD, where sgn(a) is 1 (resp. -1) if the Clifford part of a is even (resp. odd). The result follows once we identify the cohomology of this differential.
Date received: April 1, 2000
Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-54.