Dual Pairs and Disconnected Reductive Groups

Abstract

In R. Howe’s seminal paper, “Remarks on classical invariant theory,” he introduces the notion of a Lie algebra dual pair (a pair (g₁, g₂) of reductive Lie subalgebras of a Lie algebra g such that g₁ and g₂ equal each other’s centralizers in g) and the notion of a Lie group dual pair (a pair (G₁, G₂) of reductive subgroups of a reductive Lie group G such that G₁ and G₂ are each other’s centralizers in G). Both notions have since been widely used and studied. This thesis extends what is known about the classifications of complex reductive Lie group and Lie algebra dual pairs, and establishes a step towards a more general framework for understanding complex reductive Lie group dual pairs. In the first part of this thesis, we classify the reductive dual pairs in the complex classical Lie groups: GL(n, C), SL(n, C), O(n, C), SO(n, C), and Sp(2n, C). We also establish some general relationships between Lie group dual pairs and dual pairs in corresponding Lie algebras and quotient groups. These relationships lead to complete classifications of the reductive dual pairs in the complex classical Lie algebras (gl(n, C), sl(n, C), so(n, C), and sp(2n, C)) and preliminary progress towards classifying dual pairs in the projective classical groups (P GL(n, C), P Sp(2n, C), P O(n, C), and P SO(n, C)). In the second part of this thesis, we complete an explicit classification of the semisimple Lie algebra dual pairs in the complex exceptional Lie algebras, initially outlined by H. Rubenthaler in a 1994 paper. This explicit classification makes Rubenthaler’s 1994 result more complete, usable, and understandable. A major obstacle to understanding reductive Lie group dual pairs is their potential disconnectedness. Inspired in part by this obstacle, in the third part of this thesis we describe the possible disconnected complex reductive algebraic groups E with component group Γ = E/E₀. We show that there is a natural bijection between such groups E and algebraic extensions of Γ by Z(E₀). Finally, in the last part of this thesis we classify the reductive dual pairs in P GL(n, C). While the connected dual pairs in P GL(n, C) can be easily understood using tools from the first part of this thesis, the classification of the disconnected dual pairs in P GL(n, C) is much more difficult and requires tools from the third part of this thesis. This serves as the first complete classification of dual pairs in a non-classical group and as a step towards understanding how disconnectedness factors into the classification of dual pairs more generally.