Let G be a connected reductive group over a nonarchimedean field k and let \Gamma<\Gamma_1 be congruences subgroup of G(k). Let \pi be an irreducible smooth admissible representation of G. Suppose \pi is supercuspidal and \pi^{\Gamma_1}\neq 0.

Then \pi|_\Gamma contains a cuspidal representation \rho of \Gamma/\Gamma_1.

Therefore \pi contains an extension of \rho to Z\Gamma, \tilde{\rho}. So {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma})\neq 0. By Frobenius reciprocity,

{\rm Hom}_G({\rm ind}_{Z\Gamma}^G \tilde{\rho},\pi)\cong {\rm Hom}_{Z\Gamma}(\tilde{\rho},\pi|_{Z\Gamma}).

Since {\rm ind}_{Z\Gamma}^G\tilde{\rho} is irreducible, it must be isomorphic to \pi.

Now consider the decomposition of the induced representation into irreducible components

{\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}=\oplus_{x\in Z\Gamma\backslash G/\Gamma_1} {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}).

where Z\Gamma^x=x^{-1}Z\Gamma x and \tilde{\rho}^x is the obvious representation of Z\Gamma^x (transport \tilde{\rho} through conjugation by x, i.e., \tilde{\rho}^x(x^{-1}hx) = \tilde{\rho}(h) for h\in Z\Gamma.)

When does {\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1} (\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1}) contain the trivial representation of \Gamma_1? Equivalent questions are

When is {\rm Hom}_{\Gamma_1}({\rm ind}_{Z\Gamma^x\cap\Gamma_1}^{\Gamma_1}(\tilde{\rho}^x), \textbf{1}_{\Gamma_1})\neq 0?

When is {\rm Hom}_{Z\Gamma^x\cap\Gamma_1}(\tilde{\rho}^x|_{Z\Gamma^x\cap\Gamma_1},\textbf{1}_{Z\Gamma^x\cap\Gamma_1})\neq 0?

One can show x\in Z\Gamma by considering {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_{\Gamma_1}. If this contains the trivial representation, then {\rm ind}_{Z\Gamma}^G\tilde{\rho}|_\Gamma contains an irreducible representation, say \tau, such that \tau_{\Gamma_1}\supset\textbf{1}_{\Gamma_1}.

This implies that \tau is trivial on \Gamma_1. So \pi contains \rho and \tau. (A weaker approach is to use explicit representatives for Z\Gamma\backslash G/\Gamma_1.)

General theory implies that \rho and \tau intertwine, i.e., there exist x\in G such that

{\rm Hom}_{\Gamma^x\cap\Gamma}(\rho^x,\tau)\neq 0.

This implies x\in Z\Gamma. Using cuspidality of \rho plus reps for \Gamma\backslash G/\Gamma, we have that \rho\cong\tau.

It follows that \pi^{\Gamma_1}=\tilde{\rho}|_{\Gamma_1}. In particular,

{\rm dim}\, \pi^{\Gamma_1}={\rm dim}\, \tilde{\rho}={\rm dim}\, \rho.

L. Morris, Tamely ramified supercuspidal representations, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 5, 639–667.
A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994), no. 1-3, 393–408.