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## Theory of local representations for *GL*(*n*)

**14.1 Generalities on representations of**

Fix an integer *n* ≥ 1 and a prime *p*. A representation of is a pair (*π, V*) where *V* is a complex vector space and *π* : → *GL*(*V*) is a homomorphism. Such a representation (*π, V*) is smooth if every vector has an open stabilizer (see Definition 6.1.1). For every integer *r* ≥ 1, define the open subgroup

where *I*_{n} is the *n* × *n* identity matrix and Mat(*n*, ) is the set of all ...