Let be a connected reductive group over a nonarchimedean field and let be congruences subgroup of . Let be an irreducible smooth admissible representation of . Suppose is supercuspidal and .

Then contains a cuspidal representation of .

Therefore contains an extension of to , . So . By Frobenius reciprocity,

Since is irreducible, it must be isomorphic to .

Now consider the decomposition of the induced representation into irreducible components

where and is the obvious representation of (transport through conjugation by , *i.e.*, for .)

When does contain the trivial representation of ? Equivalent questions are

When is ?

When is ?

One can show by considering . If this contains the trivial representation, then contains an irreducible representation, say , such that .

This implies that is trivial on . So contains and . (A weaker approach is to use explicit representatives for .)

General theory implies that and intertwine, i.e., there exist such that

This implies . Using cuspidality of plus reps for , we have that .

It follows that . In particular,

**References**

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.

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