 Mackey restriction formula

ID: mackey-restriction-formula

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Mackey restriction formula by

Codex  0 Created 2026-09-24 Updated 2026-10-03

For

H, K \leq G

and a

K

-representation

V

,

Res_{H}^{G} Ind_{K}^{G} V ≅ ⨁_{x \in H \ G / K} Ind_{H \cap x K x^{- 1}}^{H} Res_{H \cap x K x^{- 1}}^{x K x^{- 1}} (^{x} V),

where

^{x} V (x k x^{- 1}) = V (k)

.

To prove the formula, write induction as

C [G] \otimes_{C [K]} V

and decompose the

(H, K)

C [G] = ⨁_{x \in H \ G / K} C [H x K] .

For

L_{x} = H \cap x K x^{- 1}

, the map

C [H] \otimes_{C [L_{x}]}^{x} V ⟶ C [H x K] \otimes_{C [K]} V, h \otimes v ⟼ h x \otimes v,

is a well-defined

H

-equivariant isomorphism. Taking the direct sum over the double cosets proves the result.

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