Let $F=\mathbb Q_l$ and $E$ be a finite extension of $\mathbb Q_p$ with residue field $k \cong O_E/m_E$ ($p \neq l$). Let $\bar r: \Gamma_F=Gal(\bar F/F) \to GL_2(k)$ be a continuous representation. Let $\mathrm{CLN}_{O_E}$ be the category of complete local notherian $O_E$-algebras and let $\mathrm{Lift}_{\bar r}: \mathrm{CLN}_{O_E} \to Sets$ be the functor defined by $\mathrm{Lift}_{\bar r}(R)$={continuous homomorphisms $r: \Gamma_F \to GL_2(R)$ such that $r$ mod $m_R$ equals $\bar r$}. $\mathrm{Lift}_{\bar r}$ is representable, and we denote the representing ring (i.e. the universal lifting ring) by $R^{\Box}$. It is known that generic fiber $R^{\Box}[1/p]$ is nonzero. An $E$-point $x$ of $R^{\Box}[1/p]$ induces a homomorphism $r_x: \Gamma_F \to GL_2(E)$ that lifts $\bar r$.

Suppose $\bar r \cong \bar\kappa \oplus 1$ where $\bar\kappa: \Gamma_F \to \mathbb F_p^{\times}$ is the $p$-th cyclotomic character. There is an reduced, p-torsion free quotient $R^{St}$ of $R^{\Box}$ such that a map $R^{\Box} \to R$ induced by a lifting $r: \Gamma_F \to GL_2(R)$ of $\bar r$ factors through $R^{St}$ iff $r$ is conjugate to $\begin{pmatrix} \alpha & *\\ 0&\beta \end{pmatrix}$ with $\alpha/\beta=\kappa$, where $\kappa: \Gamma_F \to \mathbb Z_p^{\times}$ is the p-adic cyclotomic character. (Such representations correspond to the Steinberg representations of $GL_2(F)$ under local Langlands.) Let $\mathcal C^{St}$ be the Zariski-closure of the image of $Spec R^{St}[1/p] \to Spec R^{\Box}[1/p]$. Is it irreducible? smooth? Note that the following two cases should be different: the case $l \not\equiv 1 (p)$ (equivalently, $\bar\kappa$ is nontrivial) and the case $l \equiv 1 (p)$ (equivalently, $\bar\kappa$ is trivial)