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Paul Broussous
Paul Broussous
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The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are algebraicunitary. The reference is :

Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a non-achimedean local field, Russian Math. Surveys, 1976.

where smooth representations are called algebraic. In this paper you can also find the theory of Whittaker (and Kirillov) models for ${\rm GL}(n)$.

In the more general case of ${\rm GL}(n)$, irreducible supercuspidal representations are generic (they have a Whittaker model) and one has multiplicity $1$. This is not true in general (i.e. for certain non supercuspidal representations, or for certain supercuspidal representations of other reductive groups) : representations are not always generic and when they are generic one not necessarily has multiplicity $1$.

The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are algebraic. The reference is :

Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a non-achimedean local field, Russian Math. Surveys, 1976.

where smooth representations are called algebraic. In this paper you can also find the theory of Whittaker (and Kirillov) models for ${\rm GL}(n)$.

In the more general case of ${\rm GL}(n)$, irreducible supercuspidal representations are generic (they have a Whittaker model) and one has multiplicity $1$. This is not true in general (i.e. for certain non supercuspidal representations, or for certain supercuspidal representations of other reductive groups) : representations are not always generic and when they are generic one not necessarily has multiplicity $1$.

The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are unitary. The reference is :

Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a non-achimedean local field, Russian Math. Surveys, 1976.

where smooth representations are called algebraic. In this paper you can also find the theory of Whittaker (and Kirillov) models for ${\rm GL}(n)$.

In the more general case of ${\rm GL}(n)$, irreducible supercuspidal representations are generic (they have a Whittaker model) and one has multiplicity $1$. This is not true in general (i.e. for certain non supercuspidal representations, or for certain supercuspidal representations of other reductive groups) : representations are not always generic and when they are generic one not necessarily has multiplicity $1$.

Source Link
Paul Broussous
Paul Broussous
  • 6.3k
  • 1
  • 19
  • 32

The answer is yes. If $H$ is a closed subgroup of a locally profinite group $G$ then the functor of restriction to $H$ has a right adjoint : the functor of smooth induction (we work in categories of smooth representations). You do not need to assume that your representations are algebraic. The reference is :

Bernstein and Zelevinsky : Representations of the group ${\rm GL}(n,F)$, where $F$ is a non-achimedean local field, Russian Math. Surveys, 1976.

where smooth representations are called algebraic. In this paper you can also find the theory of Whittaker (and Kirillov) models for ${\rm GL}(n)$.

In the more general case of ${\rm GL}(n)$, irreducible supercuspidal representations are generic (they have a Whittaker model) and one has multiplicity $1$. This is not true in general (i.e. for certain non supercuspidal representations, or for certain supercuspidal representations of other reductive groups) : representations are not always generic and when they are generic one not necessarily has multiplicity $1$.