ON THE (g,K)-COHOMOLOGY OF CERTAIN THETA LIFTS
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چکیده
It is well-known that one of the most important ways of constructing cohomology for an arithmetic quotient Γ\X, where X is the symmetric space associated with a semi-simple Lie group G with finite center and Γ is a discrete subgroup of G, is through the use of Matsushima’s formula, namely by exhibiting irreducible unitary representations with non-zero continuous cohomology which occur in L2(Γ\G). (See [BW].) In this connection, the machinery of theta correspondence plays a major role, and the technique of lifting a discrete series representation of G′, where (G′, G) forms a dual pair, proves to be an especially powerful tool. The basis of such applications lies in the fact that when G′ is relatively small compared to G, a sufficiently regular discrete series representation of G′ will be lifted to a unitary representation of G with non-zero cohomology ([RS], [Ad], [Li2]). Examples of such applications were given by Kazhdan [Ka] (from U(1) to SU(n, 1)), Borel-Wallach [BW] (from U(1) to SU(p, q)), Anderson [An] (with G′ compact), and in general by Li for reductive dual pairs of type 1 [Li3]. We note that all the representations of G′ × G involved in these applications are in the discrete spectrum of the restriction of the oscillator representation, and are automatically unitary.
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تاریخ انتشار 2001