Steinberg components of local deformation rings

Question

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)

Answer 1

If I understand correctly, I believe that you want $r:\Gamma\longrightarrow\operatorname{GL}_2(R)$ to factor through $R^{\operatorname{St}}$ if $r$ is a non-trivial extension of $\beta$ by $\alpha$ with $\alpha/\beta=\kappa$ (only these correspond to Steinberg representation by the Local Langlands Correspondence). Whether I'm correct or not, the problem you want to solve has been solved by Jack Shotton in the paper

Local deformation rings for $\operatorname{GL}_2$ and a Breuil-Mézard conjecture when $\ell\neq p$. Algebra Number Theory (2016).

The best for you is probably to read it, but for completeness here follows the answer to your question (assuming that you indeed want the genuinely Steinberg component, otherwise the answer is more complicated, but still contained in Shotton's paper).

First you correctly noticed that the answer will be different when $\ell\equiv 1\!\!\!\!\mod p$, but you missed that the case where $\ell\equiv-1\!\!\!\!\mod p$ is also special. If $\ell\not\equiv \pm1\!\!\!\!\mod p$, then $R^{\operatorname{St}}$ is a formally smooth irreducible component of $R^{\square}$ (Prop 5.5 of Shotton's paper).

If $\ell\equiv-1\!\!\!\!\mod p$, then $R^{\operatorname{St}}$ is the union of two formally smooth irreducible components of $R^{\square}$ that do not intersect in the generic fiber (Prop 5.6 of Shotton's paper).

Finally, if $\ell\equiv1\!\!\!\!\mod p$, then $R^{\operatorname{St}}$ is a reduced Cohen-Macaulay but non-Gorenstein domain and $R^{\operatorname{St}}[1/p]$ is formally smooth (Prop 5.8 of Shotton's paper and its proof).