A multiplicative analogue of complex symplectic implosion

TL;DR

This paper introduces a multiplicative version of complex-symplectic implosion for SL(n,C), constructing an affine variety that generalizes symplectic reduction and relates to Steinberg fibers and real symplectic group-valued implosion.

Contribution

It develops a novel multiplicative analogue of complex symplectic implosion as an affine variety with torus action, connecting to Steinberg fibers and real symplectic implosion.

Findings

Constructed the universal multiplicative implosion as an affine variety.

Showed reductions yield Steinberg fibers of SL(n,C).

Connected the space to real symplectic group-valued implosion.

Abstract

We introduce a multiplicative version of complex-symplectic implosion in the case of $SL(n, \C)$. The universal multiplicative implosion for $SL(n, \C)$ is an affine variety and can be viewed as a nonreductive geometric invariant theory quotient. It carries a torus action. and reductions by this action give the Steinberg fibres of $SL(n, \C)$. We also explain how the real symplectic group-valued universal implosion introduced by Hurtubise, Jeffrey and Sjamaar may be identified inside this space.