Consider a group $\,G\,$, a vector space $\,{\mathbb{V}}\,$, and a space $\,{\cal{L}}^G\,$ of functions
$$
{\cal{L}}^G\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\longrightarrow{\mathbb{V}}\,\right\}~~.
$$
In the space $\,{\cal{L}}^G\,$, acts a natural representation of $\,G$
$$
U~:\quad G~\longrightarrow~GL({\cal{L}}^G)
$$
implemented with
$$
U_g\varphi(x)=\varphi({g^{-1}}x)~~,\qquad g,\,x\in G~~.
$$
Consider a subgroup $\,K<G\,$ and assume that it possesses a representation $$ D~:\quad K~\longrightarrow~GL({\cal{L}}^G)~~. $$ My understanding is that in the mapping $$ k\in K\;:\quad \varphi(x)~\longmapsto~D(k)\varphi(x) $$ the function $\,\varphi(x)\,$ is still defined on the entire $\,G\,$, with its domain NOT restricted to $\,K\,$. Please correct me if I am wrong.
I wish to induce $\,D(K)\,$ up to $\,G\,$, then restrict the result down to $\,K\,$, to see if the outcome is different from $\,D(K)\,$.
INDUCTION
The induced representation $\,\operatorname{Ind}_K^GD\,$ is implemented with the left translations $$ U_g\varphi(x)=\varphi({g^{-1}}x)~~,\qquad g,\,x\in G \qquad\qquad\qquad (1) $$ acting in the subspace $\,\Gamma\in{\cal{L}}^G$ of the Mackey functions: $$ \Gamma\;=\;\left\{~\varphi~\Big{|}~~~\varphi:\,~G\longrightarrow{\mathbb{V}}\;,\quad \varphi(xk)=D^{-1}(k)\varphi(x)\,\right\}~~.\qquad\qquad\qquad (2) $$
RESTRICTION, for finite-dimensional representations
A restriction maps a representation of $\,G\,$ to that of $\,K<G\,$ by ``forgetting'' how to act by elements outside of $\,K\,$.
Applied to an induced representation, a restriction, generally, fails to map $\,\operatorname{Ind}_K^G D\,$ back to $\,D(K)\,$. To illustrate this, consider the case of a finite group $\,G\,$ and finite-dimensional representations. When an $\,n$-dimensional representation $\,D(K)\,$ gets induced to the representation $\,\operatorname{Ind}_K^G D\,$ of dimension $\,|G/K|\cdot n\,$, the subsequent restriction of $\operatorname{Ind}_K^G D\,$ to $\,K\,$ still has the dimension $\,|G/K|\cdot n\,$. Hence it cannot be the original representation, unless $\,K=G\,$.
RESTRICTION, for infinite-dimensional representations
My understanding is that the restriction is achieved by substituting $\,g\in G\,$ with $\,k\in K\,$ in the induced representation (1): $$ U_k\varphi(x)~=~\varphi(k^{-1}x)~~,\quad x\in G\,,\quad k\in K\qquad\qquad (3) $$ where $\,\varphi(x)\,$ is still a function defined on the entire $\,G\,$, not restricted to $\,K\,$. (Is this key caveat correct?)
With aid of the equivariance condition, the above equation can be continued as $$ U_k\varphi(x)~=~\varphi(k^{-1}x)~=~\varphi(k^{-1}xkk^{-1})~=~D(k)\varphi(k^{-1}xk)~~.\qquad\qquad\qquad (4) $$ My understanding is that, for spherical functions, $\,\varphi(k^{-1}xk)=\varphi(x)\,$. Thence I conclude that, if the restriction of $\,\mbox{Ind}^{\,G}_{K}D\,$ to $\,K\,$ can be expanded into irreducible components each of which has spherical functions as its basis, then $$ \operatorname{Res}_D^K\,\operatorname{Ind}_K^G D\,=\,D\,~. $$
This conclusion leaves me in doubt. Indeed, above I explained that for finite-dimensional representations the restriction and induction cannot be reciprocal because of their different dimensionalities.
Is it really true that this counterargument is no longer valid in infinite dimensions, and the two operations can sometimes be reciprocal?
Could it be that I made a mistake by assuming that a restriction restricts only the operations but leaves the space of functions unchanged?