A generalisation of induced representations

Question

Let $G$ be a finite group, and $H\subseteq G$ a subgroup. Let $F$ be a field. Let $W$ be a finite-dimensional $F[H]$-module. Let $T$ be a left transversal of $H$ in $G$. Then we can define:

$W^G=\sum_{t\in T}t\otimes W$

which is the $F[G]$-module induced from $W$. I'm following the notation used in I.M. Isaacs' "Character Theory of Finite Groups" (see Chapter 6).

This induced module "mixes" the $F$-representation afforded by $W$, and the permutation representation given by the action by $G$ on the left cosets of $H$ via left multiplication. I say this because $G$ acts on the subspaces $\{t\otimes W\}_{t\in T}$ by permuting them (one might call this the "coarse" action, since we are considering the action at the level of subspaces and not at the level of individual elements of the module).

Is there a more general operation we can perform on modules, which "mixes" any given pair of $F$-representations in some analogous way?

Clarification:

Is there an operation $f$ that can "mix" any pair $(W,V)$, where $W$ is a $F[H]$-module and $V$ is a $F[G]$-module, and yield an $F[G]$-module $f(W,V)$, such that $f(W,V)=W^G$ in the special case that $V$ is the permutation module given by the action by $G$ on the cosets of $H$?

Answer 1

I'm not sure this answers your question, but here is a general procedure:

Fix any representation $W$ of $G\times H$. Then for any representation $V$ of $H$, the hom space $\hom_H(W,V)$ carries a $G$-action, and hence produces a functor $H\mathrm{-Rep}\to G\mathrm{-Rep}$ (here, note that I do not require $H\subset G$). Some examples:

• Suppose $H\subset G$, and let $W=\mathbb C[G]$, which carries a $G\times H$-action, by making $G$ act by left multiplication and $H$ act by right multiplication. Then $\hom_H(\mathbb C[G],V)$ is exactly the induced representation $V^G$.
• Suppose you have a homomorphism $\varphi\colon G\to H$, and let $W=\mathbb C[H]$, which carries a $G\times H$-action, as above (explicitly, $(g,h)\cdot x:=\varphi(g)xh^{-1}$). Then the $G$-action on $\hom_H(\mathbb C[H],V)=V$ is the composition $G\xrightarrow{\varphi}H\xrightarrow\pi\mathrm{GL}(V)$, where $\pi$ is the representation of $H$. In particular: if $G\subseteq H$, then this is the restriction, and if $G\to H=G/N$ is a quotient, then this is the inflation.

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• A bit different in flavor, but Deligne-Lusztig theory provides a similar construction. Letting $X_n$ be the variety $(\det(t_i^{q^{j-1}})_{i,j})^{q-1}=(-1)^{n-1}$, the étale cohomology $H_c^i(X_n,\overline{\mathbb Q}_\ell)$ carries an action of $\mathrm{GL}_n(\mathbb F_q)\times\mathbb F_{q^n}^\times$, which allows the construction of a representation of $\mathrm{GL}_n(\mathbb F_q)$ from a character of $\mathbb F_{q^n}^\times$.