Alladi Ramakrishnan Hall

Holomorphic Vector Bundles and Intertwining Operators over Symmetric Domains

Harald Upmeier

University of Marburg, Germany

A symmetric space $D=G/K$ of non-compact type, equipped with a hermitian structure, can be realized as a symmetric domain in $\Cl^n.$ These domains generalize the unit disc (bounded model) and the upper half-plane (unbounded model). Higher dimensional examples include the matrix unit ball and more general unit balls in the so-called Jordan algebras.

Every (finite-dimensional) representation of $K$ induces a holomorphic vector bundle over $D$ equipped with a $G$-action. Of particular importance are the so-called spin representations indexed by positive integers $n,$ which include the tautological bundles associated with projective space and the Grassmannian. The associated homogeneous vector bundles over $D$ are not irreducible under the $G$-action, and it is a fundamental problem to find an explicit decomposition, indexed by all partitions with largest part $\le n,$ into irreducible $G$-submodules.

In the talk we describe this decomposition, obtained in joint work with Gadadhar Misra, via explicit intertwining differential operators. As a second main result it is shown that the $C^*$-algebra generated by holomorphic multiplication operators (called the Toeplitz $C^*$-algebra), acts irreducibly although the group action is reducible. These results are new even for rank $1$ domains (unit ball).