Exercise 22, § 2

Let $G$ be a group and $B$ a subgroup of $G$.

a) Show that the following conditions are equivalent: (i) $B \cap gBg^{-1}$ is of finite index in $B$ for all $g \in G$; (ii) every double coset $BgB$ with respect to $B$ is a finite union of left cosets with respect to $B$.

More precisely, show that, for all $g \in G$, the index $q_g$ of $B \cap gBg^{-1}$ in $B$ is equal to the number of left cosets with respect to $B$ contained in the double coset $BgB$. Show that $q_{gh} \leq q_g q_h$ for all $g, h \in G$.

Assume from now on that conditions (i) and (ii) are satisfied and denote by $k$ a commutative ring. For $t \in G/B$ (resp. $t \in B\backslash G/B$), denote by $a_t$ the map from $G$ to $k$ defined by $a_t(g) = 0$ if $g \notin t$ and $a_t(g) = 1$ if $g \in t$. Let $L$ (resp. $H$) be the $k$-module generated by the $a_t$ for $t \in G/B$ (resp. $t \in B\backslash G/B$).

b) Show that there exists a unique linear form $\mu$ on $L$ such that $\mu(a_t) = 1$ for all $t \in G/B$.

c) Let $\varphi \in L$ and $\psi \in H$. Show that, for all $x \in G$, the map

belongs to $L$ and that the map $\varphi * \psi : x \mapsto \mu(\theta_x)$ belongs to $L$. Moreover, if $\varphi \in H$ then $\varphi * \psi \in H$. Show that the map $(\varphi, \psi) \mapsto \varphi * \psi$ makes $H$ into an algebra over $k$, having $a_B$ as its unit element, and makes $L$ into a right $H$-module.

The algebra $H$ is called the Hecke algebra of $G$ with respect to $H$ and is denoted by $H_k(G, B)$.

d) Show that, for $t, t' \in B \backslash G / B$,

where $m(t, t'; t'')$ is the number of cosets with respect to $B$ contained in $t \cap g {t'}^{-1}$ for all $g \in t''$.

e) Let $G$ act on $L$ by left translations. Show that the action of $H$ on $L$ defines an isomorphism from $H$ to the commutant of the linear representation of $G$ on $L$ thus obtained.

f) Assume that $G$ is finite and that the characteristic of $k$ does not divide the order of $G$. Show that $H_k(G, B)$ is absolutely semi-simple over $k$ (Algebra, Chap VIII, § 7, no. 5) (use Maschke’s Theorem (Chap. V, Appendix) and Prop. 3 of Algebra, Chap. VIII, § 5, no. 1).

g) Assume that $G$ is a topological group and that $B$ is a compact open subgroup of $G$. Show that conditions (i) and (ii) are satisfied and that, when $k = \mathbf{R}$ or $\mathbf{C}$, the product $\varphi * \psi$ is simply the convolution product relative to the right Haar measure on $G$, normalised by the condition $\mu(B) = 1$ (cf. Integration, Chap. VIII, § 4, no. 5).