Proposition 2

Let H be a subgroup of G, and let $(M, \sigma )$ be a linear representation of H. The canonical homomorphism $\iota :$ Ind$^G_H(\sigma )\rightarrow$ Coind$^G_H(\sigma )$ is injective. If the subgroup H has finite index in G, then $\iota$ is an isomorphism of K[G]-modules.

Let $S\subset G$ be a system of representatives of $G/H$. The family $(s)_{s\in S}$ is a basis of the right K[H]-module K[G]. It follows that the mapping $M^{(S)}\rightarrow$ Ind$^G_H(\sigma )$ defined by $(m_s)_{s\in S}\mapsto \sum_{s\in S}s\otimes m_s$ is an isomorphism of K-modules. For all $s, s'\in S$ and every $m\in M$, we have the relations

It follows that $\iota '$ is injective.

Suppose that H has finite index in G. The set S is then finite; let $\rho$ be the mapping from Coind$^G_H(\sigma )$ to Ind$^G_H(\sigma )$ given by $u\mapsto \sum_{s\in S}s\otimes u(s^{-1})$. It satisfies $(\iota \circ \rho (u))(s^{-1}) =u(s^{-1})$ for $u\in$ Coind$^G_H(\sigma )$ and $s\in S$. Since the family $(s^{-1})_{s\in S}$ is a basis of the left K[H]-module K[G], the mapping $\iota \circ \rho$ is the identity mapping. Consequently, $\iota$ is bijective with inverse bijection $\rho$.

Let H be a subgroup of G of finite index. Let $u$ be a central function on H; denote by $u^0$ the function on G that extends $u$ and that is zero at every point of G - H. Let S be a system of representatives of $G/H$. For $g\in G$, set

(4) Ind$^G_H(u)(g) =\sum_{s\in S}u^0(s^{-1}gs)$.

Note that for all $x, g\in G$ and every $h\in H$, we have $u^0((xh)^{-1}gxh) =$ $u^0(x^{-1}gx)$. It follows that Ind$^G_H(u)$ is a central function on G that does not depend on the choice of S. We thus define a K-linear mapping Ind$^G_H$ from $\mathscr{Z}_K(H)$ to $\mathscr{Z}_K(G)$.