Q:

Question on a paper by U. Krähmer ("Dirac operators on quantum flag manifolds")

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.

A:

The map $\sigma$ is a representation of $U_q(\mathfrak{l})$ on the vector space $\Sigma_{2m}$, let's say for the moment just any such representation.
To answer the second question, $\sigma(S(X))\psi$ should really be thought as $(1\otimes\sigma(S(X)))\psi$. Let $\psi$ be decomposable of the form $\psi_1\otimes\psi_2$ with $\psi_1\in \mathbb C_q[G]$ and $\psi_2\in \Sigma_{2m}$, then: 
$$\sigma(S(X))\psi=\psi_1\otimes \sigma(S(X))\psi_2$$ 
is the action of $U_q(\mathfrak l)$ on the second component.
Similarly, on the left hand side of the equality you have $X$ acting only on the first component.
The equality defining ${\cal S}$ should be thought as translating with Hopf algebra maps (and at the infinitesimal level) the equivariant condition defining homogeneous vector bundles on homogeneous space:
$$ f(l\cdot g)=\sigma(l)f(g)$$
when $l\in L$, $L$ subgroup of $G$ wih a representation on a vector space $V$ and $f:G\to V$. This condition characterizes sections of a homogeneous vector bundle on $G/L$.
The last isomorphism is a vector space isomorphism: exactly what you have thought: first decompose $\mathbb C_q[G]$ via Peter-Weyl theorem, then do not forget to use the equivariance condition mentioned above to turn morphism of vector spaces into morphism of $U_q(\mathfrak l)$--modules.
Lastly: $\sigma$. Start from the spin representation of $\mathfrak{so}_{2m}(\mathbb C)$ on $\Sigma_{2m}$ (this you can find in books). Then call with the same name the $q$-spin representation of $U_q(\mathfrak{so}_{2m}(\mathbb C))$ (category of irreps are the same so you have a spin representation also for the quantized universal enveloping algebra), then restrict to the subalgebra $U_q(\mathfrak l)$. There you have a representation of the algebra $U_q(\mathfrak l)$ on the vector space $\Sigma_{2m}$.
Hope this helps. 

