Local Steinberg representation of GL2 ($\mathrm{St}=C^{\infty }({\mathbb{P}}^1(F))/\mathbb{C}$)

The local Steinberg representation is the quotient of locally constant functions on the projective line by constant functions, with the action induced from GL2. In the unnormalized convention, $V(1)$ fits into $0\to 1\to V(1)\to \mathrm{St}\to 0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.