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What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...

Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...