I don’t know if this is an adequate question for MO. But I cannot understand many aspects of the said paper

https://link.springer.com/content/pdf/10.1023%2FB%3AMATH.0000027748.64886.23.pdf

by Krähmer.

He writes $ $ S:=\{\psi \in \mathbb{C}_q[G] \otimes \Sigma_{2m}| X \rhd \psi = \sigma(S(X)) \psi \; \forall X \in U_q(\frak{l})\}$ $ at the beginning of Section 6.

How is $ \sigma$ defined?

He explains $ \sigma$ in Section 5. But for me it is not clear.

How does the multiplication $ \sigma(S(X)) \psi$ look like?

Also:

Having the isomorphism $ $ S \cong \bigoplus_{\lambda \in P^+} V_\lambda \otimes \mbox{Hom}_{U_q(\frak{l})}(V_\lambda,\Sigma_{2m})$ $ explicitely written down would be useful. Isomorphism in which category?

It perhaps uses $ \mbox{Hom}(V_\lambda,V_\lambda) \cong V_\lambda \otimes V_\lambda^*$ linearly and the Peter-Weyl theorem.