# Hecke-Clifford superalgebras and crystals of type $D^{(2)}_{l}$

## TL;DR

This paper extends the understanding of the representation theory of affine Hecke-Clifford superalgebras at roots of unity, connecting it to the Lie theory of type D^{(2)}_{l} instead of A^{(2)}_{2l}.

## Contribution

It demonstrates that similar structural theorems hold for primitive 4l-th roots of unity, replacing the Lie type from A^{(2)}_{2l} to D^{(2)}_{l}.

## Key findings

- Representation theory controlled by type D^{(2)}_{l} Lie theory at specific roots of unity.
- Extension of Brundan and Kleshchev's results to new roots of unity.
- New connections between superalgebras and Lie types.

## Abstract

Brundan and Kleshchev showed that some parts of the representation theory of the affine Hecke-Clifford superalgebras and its finite-dimensional "cyclotomic" quotients are controlled by the Lie theory of type $A^{(2)}_{2l}$ when the quantum parameter $q$ is a primitive $(2l+1)$-th root of unity. We show in this paper that similar theorems hold when $q$ is a primitive $4l$-th root of unity by replacing the Lie theory of type $A^{(2)}_{2l}$ with that of type $D^{(2)}_{l}$.