= Local Steinberg representation of GL2
{title2=$\operatorname{St}=C^\infty(\mathbb P^1(F))/\mathbb C$}
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 $0\to1\to V(1)\to\operatorname{St}\to0$. Its scalar endomorphism algebra obstructs a splitting. This local infinite-dimensional representation differs from the finite-field Steinberg representation.
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