nLab
double coset

Contents

Context

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

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

This construction generalises to topological groups and Lie groups, where it is called a double coset space. The double coset space for GG a compact Lie group and HH and KK closed connected subgroups, with HH acting freely on the singe coset space G/KG/K, is also known as a biquotient space.

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

Properties

Mackey’s formula

the following needs to state the assumpotion 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 HH of a module induced from KK. Let WW be a linear representation of KK. Then

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

where K gHgKg 1K_g \coloneqq H \intersection g K g^{-1}, and W gW_g is the representation of K gK_g which is WW as a vector space, but with action xw=(g 1xg)wx \cdot w = (g^{-1} x g) w where now g 1xgKg^{-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).

References

  • Serge Bouc, Mackey functors, (pdf)

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

Last revised on April 27, 2019 at 11:56:55. See the history of this page for a list of all contributions to it.