Irregular loci in the Emerton–Gee stack for GL₂

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>K</m:mi> <m:mo>/</m:mo> <m:msub> <m:mi mathvariant="bold">Q</m:mi> <m:mi>p</m:mi> </m:msub> </m:mrow> </m:math> K/Qₚ be unramified. Inside the Emerton–Gee stack <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">X</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> X₂ , one can consider the locus of two-dimensional mod 𝑝 representations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Gal</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mover accent="true"> <m:mi>K</m:mi> <m:mo>̄</m:mo> </m:mover> <m:mo>/</m:mo> <m:mi>K</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Gal(K̄/K) having a crystalline lift with specified Hodge–Tate weights. We study the case where the Hodge–Tate weights are irregular, which is an analogue for Galois representations of the partial weight one condition for Hilbert modular forms. We prove that if the gap between each pair of weights is bounded by 𝑝 (the irregular analogue of a Serre weight), then this locus is irreducible. We also establish various inclusion relations between these loci.