Small representations and minuscule Richardson orbits (Q2778023)

Let \(G\) be a simple adjoint algebraic group over \(\mathbb{C}\) with maximal torus \(T\) and Weyl group \(W\) and let \(G^{sc}\) be the simply connected cover of \(G\). The central theme of the paper is the attachment of a representation of \(G\) to a unipotent class \(C\) in \(\widehat G\) and this is realised by proving two important propositions and four theorems stated in the paper as theorems 1.1, 1.3, 1.4 and 5.1. If \(V_\mu\) is an irreducible representation of the simply connected cover \(G^{sc}\) of \(G\) with highest weight \(\mu\), then the author discovers that not all representations occur in some \(\text{End} (V_\mu)\). The intention of the author in proving Theorem 1.1 was to use the theory of miniscule representations to give more uniform proofs of certain facts about small representations for simply laced groups observed in earlier papers of the author and to extend these results to the non simply laced class. This attempt of the author was only partially successful. The main result in the remaining part of the paper is the reduction of the problem of classification of small representations for non simply laced groups to the case of simply laced groups, using graph automorphisms and Langlands functoriality and the description of their zero weight spaces in terms of the Springer representations.