Let G be a finite group and H any subgroup of G. Furthermore let (π, V) be a representation of H. Let n = [G : H] be the index of H in G and let g1, ..., gn be a full set of representatives in G of the left cosets in G/H.
(This is just another way of saying that g1, ..., gn is a full set of representatives.)
In the case of finite groups and finite-dimensional representations, the Frobenius reciprocity theorem states that, given representations σ of H and ρ of G, the space of H-equivariant linear maps from σ to Res(ρ) has the same dimension over K as that of G-equivariant linear maps from Ind(σ) to ρ.
[2] The universal property of the induced representation, which is also valid for infinite groups, is equivalent to the adjunction asserted in the reciprocity theorem.
, then there exists a H-equivariant linear map
with the following property: given any representation (ρ,W) of G and H-equivariant linear map
is the unique map making the following diagram commute:[3]
The Frobenius formula states that if χ is the character of the representation σ, given by χ(h) = Tr σ(h), then the character ψ of the induced representation is given by where the sum is taken over a system of representatives of the left cosets of H in G and If G is a locally compact topological group (possibly infinite) and H is a closed subgroup then there is a common analytic construction of the induced representation.
Let (π, V) be a continuous unitary representation of H into a Hilbert space V. We can then let: Here φ∈L2(G/H) means: the space G/H carries a suitable invariant measure, and since the norm of φ(g) is constant on each left coset of H, we can integrate the square of these norms over G/H and obtain a finite result.
The group G acts on the induced representation space by translation, that is, (g.φ)(x)=φ(g−1x) for g,x∈G and φ∈IndGH π.
This construction is often modified in various ways to fit the applications needed.
A common version is called normalized induction and usually uses the same notation.
The definition of the representation space is as follows: Here ΔG, ΔH are the modular functions of G and H respectively.
With the addition of the normalizing factors this induction functor takes unitary representations to unitary representations.
This is just standard induction restricted to functions with compact support.
Suppose G is a topological group and H is a closed subgroup of G. Also, suppose π is a representation of H over the vector space V. Then G acts on the product G × V as follows: where g and g′ are elements of G and x is an element of V. Define on G × V the equivalence relation Denote the equivalence class of
Note that this equivalence relation is invariant under the action of G; consequently, G acts on (G × V)/~ .
The latter is a vector bundle over the quotient space G/H with H as the structure group and V as the fiber.
This is the vector space underlying the induced representation
as follows: In the case of unitary representations of locally compact groups, the induction construction can be formulated in terms of systems of imprimitivity.
In Lie theory, an extremely important example is parabolic induction: inducing representations of a reductive group from representations of its parabolic subgroups.
This leads, via the philosophy of cusp forms, to the Langlands program.