# Verlinde formulas for nonsimply connected groups

## TL;DR

This paper proves a symplectic version of Verlinde formulas for moduli spaces of surface group representations in compact nonsimply connected Lie groups, extending previous conjectures to surfaces with at most one boundary component.

## Contribution

It provides a proof of a symplectic Verlinde formula for nonsimply connected groups, using Kostant's maximal torus in apposition, for surfaces with limited boundary components.

## Key findings

- Established a symplectic Verlinde formula for nonsimply connected groups.
- Extended Verlinde formula applicability to surfaces with at most one boundary.
- Utilized Kostant's maximal torus in apposition as a key computational tool.

## Abstract

In 1999, Fuchs and Schweigert proposed formulas of Verlinde type for moduli spaces of surface group representations in compact nonsimply connected Lie groups. In this paper, we will prove a symplectic version of their conjecture for surfaces with at most one boundary component. A key tool in our computations is Kostant's notion of a maximal torus in apposition.