double coset

Given a group $G$ and two subgroups $H$ and $K$, then the set of **double cosets**, $H \backslash G/K$, is composed of equivalence classes of elements of $G$, where two elements are regarded as equivalent if they differ by left multiplication with an element in $H$ and right multiplication with an element of $K$.

This construction generalises to topological groups and Lie groups, where it is called a **double coset space**. The double coset space for $G$ a compact Lie group and $H$ and $K$ closed connected subgroups, with $H$ acting freely on $G/K$, is also known as a **biquotient** (but note that this term is also used, e.g., here, in a more general situation for the orbit space where $H$ is a closed subgroup of $G \times G$, and acts via $(h_1, h_2) g = h_1 g h_2^{-1}$).

From the nPOV, a weak form of the double coset construction is often more natural, $H \backslash \backslash G//K$, and can be defined as the homotopy pullback of maps of delooped groups,$\mathbf{B} H \to \mathbf{B} G$ and $\mathbf{B} K \to \mathbf{B} G$. This holds more generally for any three ∞-groups, with any homomorphisms $H \to G$ and $K \to G$.

Double coset decompositions are useful in representation theory, for example in George Mackey‘s formula for the restriction back to $H$ of a module induced from $K$. Let $W$ be a representation of $K$. Then

$Res^G_H Ind^G_K W
\cong
\underset{[g] \in H \backslash G/K} \bigoplus Ind^H_{K_g} W_g
\,,$

where $K_g \coloneqq H \intersection g K g^{-1}$, and $W_g$ is the representation of $K_g$ which is $W$ as a vector space, but with action $x \cdot w = (g^{-1} x g) w$ where now $g^{-1} x g \in K$. This formula, sometimes called *Mackey’s decomposition theorem*, amounts to the Beck-Chevalley condition for the homotopy pullback square. It appears, as the *Mackey axiom*, in one of the original definitions of Mackey functor (Bouc 02).

- Serge Bouc,
*Mackey functors*, (pdf)

Last revised on August 7, 2017 at 04:19:18. See the history of this page for a list of all contributions to it.