Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Cauchy-Szegő maps, invariant differential operators and some representations of $\textrm {SU}(n+1,1)$
HTML articles powered by AMS MathViewer

by Christopher Meaney PDF
Trans. Amer. Math. Soc. 313 (1989), 161-186 Request permission

Abstract:

Fix an integer $n > 1$. Let $G$ be the semisimple Lie group ${\text {SU}}(n + 1,1)$ and $K$ be the subgroup ${\text {S(U}}(n + 1) \times {\text {U}}(1))$. For each finite dimensional representation $(\tau ,{\mathcal {H}_\tau })$ of $K$ there is the space of smooth $\tau$-covariant functions on $G$, denoted by ${C^\infty }(G,\tau )$ and equipped with the action of $G$ by right translation. Now take $(\tau ,{\mathcal {H}_\tau })$ to be $({\tau _{p,p}},{\mathcal {H}_{p,p}})$, the representation of $K$ on the space of harmonic polynomials on ${{\mathbf {C}}^{n + 1}}$ which are bihomogeneous of degree $(p,p)$. For a real number $\nu$ there is the corresponding spherical principal series representation of $G$, denoted by $({\pi _\nu },{{\mathbf {I}}_{1,\nu }})$. In this paper we show that, as a $(\mathfrak {g},K)$-module, the irreducible quotient of ${{\mathbf {I}}_{1,1 - n - 2p}}$ can be realized as the space of the $K$-finite elements of the kernel of a certain invariant first order differential operator acting on ${C^\infty }(G,{\tau _{p,p}})$. 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.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E46
  • Retrieve articles in all journals with MSC: 22E46
Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 313 (1989), 161-186
  • MSC: Primary 22E46
  • DOI: https://doi.org/10.1090/S0002-9947-1989-0930080-6
  • MathSciNet review: 930080