Cauchy-Szegő maps, invariant differential operators and some representations of 𝑆𝑈(𝑛+1,1)

Abstract

Fix an integer n > 1 n > 1 . Let G G be the semisimple Lie group SU ( n + 1 , 1 ) {\text {SU}}(n + 1,1) and K K be the subgroup S(U ( n + 1 ) × U ( 1 ) ) {\text {S(U}}(n + 1) \times {\text {U}}(1)) . For each finite dimensional representation ( τ , H τ ) (\tau ,{\mathcal {H}_\tau }) of K K there is the space of smooth τ \tau -covariant functions on G G , denoted by C ∞ ( G , τ ) {C^\infty }(G,\tau ) and equipped with the action of G G by right translation. Now take ( τ , H τ ) (\tau ,{\mathcal {H}_\tau }) to be ( τ p , p , H p , p ) ({\tau _{p,p}},{\mathcal {H}_{p,p}}) , the representation of K K on the space of harmonic polynomials on C n + 1 {{\mathbf {C}}^{n + 1}} which are bihomogeneous of degree ( p , p ) (p,p) . For a real number ν \nu there is the corresponding spherical principal series representation of G G , denoted by ( π ν , I 1 , ν ) ({\pi _\nu },{{\mathbf {I}}_{1,\nu }}) . In this paper we show that, as a ( g , K ) (\mathfrak {g},K) -module, the irreducible quotient of I 1 , 1 − n − 2 p {{\mathbf {I}}_{1,1 - n - 2p}} can be realized as the space of the K K -finite elements of the kernel of a certain invariant first order differential operator acting on C ∞ ( G , τ p , p ) {C^\infty }(G,{\tau _{p,p}}) . Johnson and Wallach had shown that these representations are not square-integrable. Thus, some exceptional representations of G 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 G with an end of complementary series representation.