The holomorphic discrete series of an affine symmetric space and representations with reproducing kernels

Abstract

Abstract. Consider a semisimple connected Lie group G with an affine symmetric space X . We study abstractly the intertwining operators from the discrete series of X into representations with reproducing kernel and, in particular, into the discrete series of G ; each such is given by a convolution with an analytic function. For X of Hermitian type, we consider the holomorphic discrete series of X and here derive very explicit formulas for the intertwining operators. As a corollary we get a multiplicity one result for the series in question. IntroductionThe purpose of this work is twofold. The first is to study abstractly the intertwining operators between discrete series representations of an affine symmetric space X and discrete series of the corresponding semisimple group. The second is to obtain very explicit formulas for certain discrete representations on X, in effect giving the analogue of Harish-Chandra's original construction of the holomorphic discrete series in the group case. Not surprisingly, these two problems are highly connected. Thus Theorem 5.4 below gives the formula for the lowest /C-type (the Flensted-Jensen function) on X, whereas in Theorem 7.3 essentially the same function gives the intertwining operator. Much of our analysis builds on the framework in [14], but otherwise we have tried to make the paper reasonably self-contained. This is, in particular, the case in §1, §2, and §4, where we abstract the properties of our intertwining operators with an eye towards later applications.We shall always let G be a connected, noncompact semisimple Lie group such that the associated Riemannian symmetric space is a bounded symmetric domain D c C" . The best known and most interesting discrete series representations of G are the holomorphic discrete series representations. As explained above, one of our aims in this paper is to generalize this construction to affine symmetric spaces of Hermitian type and relate those with other well-known representations.

Spherical unitary highest weight representations

“…The fact that z(k) ⊆ q implies that we can choose ∆ C is injective and via this inclusion mapping we identify in the sequel a * C with a subspace of t * C . For a H ∩ K-spherical highest weight λ ∈ a * ⊆ it * the condition for L(λ) to belong to the relative holomorphic discrete series of G/H is given by [29], [19]; see also [24] …”

“…Discrete series on G/H were constructed in [12] (see also [30]) and in [3] it was shown that (H −∞ ) H is one-dimensional for all discrete series on G/H except for four types of exceptional symmetric spaces. Holomorphic discrete series, i.e., unitary highest weight representation of G which can be realized in L 2 (G/H), were constructed in [28], [29] (this construction is essentially different from the one in [12]) and it was shown that (H −∞ ) H is always one-dimensional (this also covers half of the exceptional spaces in [3]). …”

Unitary representations and Osterwalder-Schrader duality

“…The first deep results in this direction are due to Ol'shanskii [80] and Stanton [98], who realized the holomorphic discrete series of the group G in a Hardy space of a local tube domain containing G in the boundary. The generalization to noncompactly causal symmetric spaces was carried out in [33,34,75,76].…”

“…What it does not do is give a natural analytic construction of the inner product on H(Ω C , W λ ). This is known only for some special representations, e.g., the holomorphic discrete series of the group G c [23,7,35] or symmetric spaces of Hermitian type [75,76]. At this point we will only discuss the holomorphic discrete series, which was constructed by Harish-Chandra in [23], in particular Theorem 4 and Lemma 29.…”

Causal compactification and Hardy spaces

“…The image of β is given by the direct sum of the holomorphic discrete series constructed in [17,18]. Let π be an irreducible unitary representation of K on the finite dimensional Hilbert space V π .…”