Lethbridge Number Theory and Combinatorics Seminar: Sean Fitzpatrick

Let $G$ be a compact semisimple Lie group, and let $H$ be a closed subgroup of $G$. Given a linear representation $\tau:H\to\mathrm{End}(V)$ of $H$, one can form an associated vector bundle $\mathcal{V}_\tau \to G/H$ over the homogeneous space $G/H$, and define an induced representation of $G$ on the space of $L^2$ sections of $\mathcal{V}_\tau$, after the method of Frobenius. Despite the resulting representation of $G$ being infinite-dimensional, Berline and Vergne showed that it is possible to give a formula for its character.

If we assume that $H$ is a maximal torus in $G$, then $G/H$ is a complex manifold, and the vector bundle $\mathcal{V}_\tau$ can be equipped with a holomorphic structure. In this case one can define the holomorphic induced representation of $G$ by restricting to the space of holomorphic sections of $\mathcal{V}_\tau$, which is a finite-dimensional vector space. We'll show that both extremes can be viewed as special cases of a family of induced representations, whose characters can be computed as the index of a transversally elliptic operator on the homogeneous space $G/H$.