# Spaltenstein varieties of pure dimension

## TL;DR

This paper proves that Spaltenstein varieties for classical groups are pure dimensional under certain conditions and are Lagrangian in related geometric structures, extending results to $\sigma$-quiver-varieties.

## Contribution

It establishes pure dimensionality and Lagrangian properties of Spaltenstein varieties for classical groups, extending to $\sigma$-quiver-varieties.

## Key findings

- Spaltenstein varieties are pure dimensional for even or odd partitions.
- They are Lagrangian in partial resolutions of nilpotent Slodowy slices.
- Results extend to the $\sigma$-quiver-variety setting.

## Abstract

We show that Spaltenstein varieties of classical groups are pure dimensional when the Jordan type of the nilpotent element involved is an even or odd partition. We further show that they are Lagrangian in the partial resolutions of the associated nilpotent Slodowy slices, from which their dimensions are known to be one half of the dimension of the partial resolution minus the dimension of the nilpotent orbit. The results are then extended to the $\sigma$-quiver-variety setting.