nLab double coset

Idea

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 the singe coset space $G/K$, is also known as a biquotient space.

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 $K \to G$.

Properties

Mackey’s formula

the following needs to state the assumption on the kind of group, e.g. finite group, compact Lie groups, …

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 linear 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).

Examples

Gromoll-Meyer sphere

The Gromoll-Meyer sphere – an exotic 7-sphere – arises as the biquotient of Sp(2) by two copies of Sp(1).

Related concepts

• biquotient space

• Clifford-Klein form

• locally Klein geometry

• Hecke algebra

• Bruhat decomposition

References

• Serge Bouc, Mackey functors, (pdf)

• Burt Totaro, Cheeger manifolds and the classification of biquotients (arXiv:math/0210247)