TY - JOUR

T1 - Cauchy-szegö maps, invariant differential operators and some representations of su(n + 1, 1)

AU - Meaney, Christopher

PY - 1989

Y1 - 1989

N2 - Fix an integer n < 1. Let G be the semisimple Lie group SU(n+l, l) and K be the subgroup S(U(n+l)xU(l)). For each finite dimensional representation (τ, Kτ) of K there is the space of smooth r-covariant functions on G, denoted by C蜴(G, τ) and equipped with the action of G by right translation. Now take the representation of K on the space of harmonic polynomials on C+l which are bihomogeneous of degree (p, p). For a real number v there is the corresponding spherical principal series representation of G, denoted by. In this paper we show that, as a (g, K)-module, the irreducible quotient of I1, 1-n-2p can be realized as the space of the AT-finitee lements of the kernel of a certain invariant first order differential operator acting on C蜴(G, τP, P). Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of G are realized in a manner similar to Schmid’s realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of G with an end of complementary series representation.

AB - Fix an integer n < 1. Let G be the semisimple Lie group SU(n+l, l) and K be the subgroup S(U(n+l)xU(l)). For each finite dimensional representation (τ, Kτ) of K there is the space of smooth r-covariant functions on G, denoted by C蜴(G, τ) and equipped with the action of G by right translation. Now take the representation of K on the space of harmonic polynomials on C+l which are bihomogeneous of degree (p, p). For a real number v there is the corresponding spherical principal series representation of G, denoted by. In this paper we show that, as a (g, K)-module, the irreducible quotient of I1, 1-n-2p can be realized as the space of the AT-finitee lements of the kernel of a certain invariant first order differential operator acting on C蜴(G, τP, P). Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of G are realized in a manner similar to Schmid’s realization of the discrete series. The kernels of the differential operators which we use here are the intersection of kernels of some Schmid operators and quotient maps, which we call Cauchy-Szegö maps, a generalization the Szegö maps used by Knapp and Wallach. We also identify this representation of G with an end of complementary series representation.

UR - http://www.scopus.com/inward/record.url?scp=84966233754&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1989-0930080-6

DO - 10.1090/S0002-9947-1989-0930080-6

M3 - Article

AN - SCOPUS:84966233754

VL - 313

SP - 161

EP - 186

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -