Thursday, January 1 2015
15:30 - 16:30

#### 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).

Download as iCalendar

Done