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Basic smooth representations of unitriangular groups over p-adic fields
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Resumo(s)
Let F be a field. If A is a finite-dimensional nilpotent F-algebra, then the set G = 1+A of all formal expressions1+ 𝑥 with x ∈ A is a group under the binary operation (1+𝑥)(1+γ)=1+(𝑥 + γ + 𝑥γ). Such groups are called algebra groups over F. The prototypical example of an algebra group over F is the group U𝑛(F) of upper unitriangular matrices of order 𝑛 with entries in F, where A = u𝑛(F) is the algebra of strictly upper-triangular matrices of order 𝑛 with entriesi n F. When F is finite, we have the family of basic characters of U𝑛(F); these span the irreducible (complex) characters, in the sense that each irreducible character of U𝑛 (F) is the constituent of a unique basic character. If F is a non-Archimedean local field, then its topology turns every algebra group over F into a locally compact and totally disconnected group.In this case, it is natural to consider the class of smooth (complex) representations. Although their treatment can be done in an algebraic setting, smooth representations fail to be completely reducible in general; a technical caveat when adapting and extending results about characters of finite groups to the smooth case. In this work, we have extended the construction of basic characters of U𝑛(F) to the case where F is a p-adic field(i.e. F is a non-Archimedean local field of characteristic zero), by introducing the family of basic smooth representations of U𝑛 (F). As expected, these span (up to isomorphism) the irreducible smooth representations of U𝑛(F), in the sense that each irreducible smooth representation is now a quotient of a unique basic smooth representation. The positive characteristic case is more delicate, since the orbit method fails. However, we still showed that, in this case, each irreducible smooth representation of U𝑛(F) is the quotient of a smooth representation which is “almost” a basic smooth representation.
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Palavras-chave
corpo local e não-Arquimedeano grupo-álgebra representação suave método das órbitas de Kirillov non-Archimedean localfield l-group algebra group smooth representation Kirillov’s orbit method